Anomalies in Quantum
Field Theory
REINHOLD A. BERTLMANN
Institute for Theoretical Physics
University of Vienna
CLARENDON PRESS • OXFORD
1996

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FOR RENATA

Preface
When I was a fellow at CERN A978-1980) I had the great fortune to meet
John Bell. In the course of my stay I had the chance to become acquainted
with his deep understanding of (nonrelativistic) quantum mechanics and of
(relativistic) quantum field theory (see [Bell 1987]). This led to an intense
and wonderful collaboration in the field [Bell, Bertlmann 1980-1984].
In the mid-eighties the anomalies of quantum field theory were a very
exciting subject of research. During this time I visited John regularly. As
one of the fathers of the anomaly he had a great interest in these current
developments. The main subject of our discussions was the work of Stora
[Stora 1984] and Zumino [Zumino 1984], who found within an elegant ma-
mathematical formalism the so-called chain of descent equations. Subsequently
we discussed the differential-geometric and topological aspects of quantum
field theory, especially the important Atiyah-Singer index theorems. Retur-
Returning to Vienna I gave a series of lectures on this topic; these lectures form
the basis of this book.
Anomalies are the key to a deeper understanding of quantum field theory.
The aim of this book is, therefore, to present all the different aspects of an-
anomalies in quantum field theory. Much emphasis is placed on a modern
mathematical formulation using differential geometry and topology, a for-
formulation which will prove to have great importance in the future. The dif-
different aspects of anomalies are introduced step by step to make the whole
Held accessible to a broad audience. The derivations and calculations are
«iven explicitly to make the subject more attractive to students entering
this field.
All the mathematics needed is not presupposed but explained at the very
I x'ginning. I also give an introduction to the path integral formalism which
will be needed for studying anomalies. The reader should be familiar with
i ho fundamentals of quantum field theory. Finally I consider gravitation and
study gravitational anomalies because I believe that gravitation has to be
included in a (unified) quantum field theory.
Although this field is still evolving and many gaps remain to be filled,
нч-ent years have nevertheless been a period of consolidation, so that I have
1хч!п able to present a rather coherent and self-contained description of the
anomalous nature of quantum field theory. It is my sincere hope that, at the

viii
end, the reader will have a deeper understanding of the subject as a whole.
There are many people whose assistance and criticism have been of great
value to me while I was writing and revising the manuscript of this book.
I would like to recognize in a special way the contribution of Raymond
Stora, who has taken the time and trouble to read parts of the book, pointing
out errors and making critical comments and helpful suggestions on several
subjects. I am very greatful to Gerard Emch for his valuable and encouraging
advice on early drafts of some chapters and for his fruitful suggestions for
improvements. I have benefited very much from stimulating discussions with
Helmuth Urbantke and I would like to thank him for his numerous detailed
critical comments.
I would also like to thank Jifi Hofejsi for his careful reading of some
sections. I would like to thank Herbert Pietschmann for his interest and
Gerhard Ecker, Walter Grimus, Helmut Kiihnelt, Hanns Stremnitzer, Hel-
Helmut Neufeld and Franz Schoberl for fruitful discussions.
My great and especial thanks go to Pranziska Wagner for her masterly
work transforming a long and complex manuscript into WFf$. and for her
great patience and endurance, which made it possible to complete this book
on time. I would also like to single out for special thanks Christoph Adam
for his great help and assistance over a long time until the book reached
its final form, for reading the manuscript, for making the index and for the
final ЖЕеХ corrections.
I also wish to thank Khosrow Chadan and Hellmuth Satke for private
communications and their kind support.
Finally, and most of all, I would like to thank my wife very warmly for
her unfailing encouragement and support over a long period.
Vienna
August 1995 R. A. B.

Contents
1 Introduction 1
2 Differential geometry, topology and fibre
bundles 9
2.1 Topology 10
2.2 Homotopy 13
2.2.1 Homotopy of maps 14
2.2.2 Homotopy groups 18
2.3 Differentiable manifolds 29
2.4 Differential forms 40
2.5 Homology and de Rham cohomology 58
2.5.1 Homology 58
2.5.2 de Rham cohomology 67
2.6 Flow, Lie derivative and Lie group 73
2.6.1 Differential map 73
2.6.2 Pullback 76
2.6.3 Flow 78
2.6.4 Lie derivative of a vector field 81
2.6.5 Lie derivative of a differential form 84
2.6.6 Lie group and Lie algebra 88
2.7 Fibre bundles 95
2.7.1 Bundle set-up 95
2.7.2 Connection 105
2.7.3 Curvature 113
Л Path integrals, FP method and BRS
transformation 118
3.1 Quantum mechanics 118
3.1.1 Propagator 118
3.1.2 Typical examples for the propagator 120
3.1.3 Feynman's path integral 124
3.1.4 Connection between Lagrange- and
Hamilton formalism 126
3.1.5 Field theory analogies 129

Contents
3.2 Scalar field theory 135
3.2.1 Free scalar fields 135
3.2.2 Free Green functions 139
3.2.3 Interacting fields 142
3.2.4 Green functions for interacting fields 144
3.2.5 Connected Green functions 147
3.3 Fermion field theory 149
3.3.1 Grassmann algebra 149
3.3.2 Dirac fields 154
3.4 Abelian gauge fields 156
3.5 Faddeev-Popov method for non-Abelian gauge
fields 159
3.5.1 Yang-Mills theory 159
3.5.2 Faddeev-Popov determinant and ghosts 165
3.6 BRS transformation 172
Anomalies in QFT 177
4.1 Classical conservation laws and symmetries 177
4.1.1 Abelian fields 177
4.1.2 Non-Abelian fields 180
4.2 Ward identities and anomaly 185
4.2.1 Green functions 185
4.2.2 Generating functional 191
4.3 ABJ anomaly calculations 197
4.3.1 Triangle graph regularization 197
4.3.2 Pauli-Villars regularization 205
4.3.3 n-dimensional 't Hooft-Veltman
regularization 207
4.3.4 Singular current operator 210
4.4 2-dimensional anomaly and dispersion relations 214
4.4.1 Ward identities 215
4.4.2 n-dimensional regularization procedure 220
4.5 The anomaly and the Dirac sea 227
4.6 Decay тг° -+ 77 and PCAC 233
4.7 Singlet anomaly 238
4.8 Non-Abelian anomaly—Bardeen's result 241
4.9 Importance of anomalies 244
Path integral and anomaly 249
5.1 Fermionic measure and chiral transformation 249

Contents xi
5.2 Fujikawa's method and singlet anomaly 255
5.3 2-dimensional anomaly 260
5.4 Regularization independence of the anomaly 261
5.5 Fujikawa's uncertainty principle 263
5.6 Non-Abelian anomaly 265
5.7 Heat kernel and zeta function regularization 272
5.7.1 Heat kernel regularization 273
5.7.2 Zeta function regularization 277
6 Physics in terms of differential forms 287
6.1 Abelian fields, electrodynamics 287
6.2 Non-Abelian fields, Yang-Mills theory 293
6.3 Anomalies 296
6.4 Dirac monopole 297
6.5 Aharonov-Bohm effect 306
6.6 Instantons 311
7 Chern-Simons form, homotopy operator and
anomaly 321
7.1 Invariant polynomials 321
7.2 Transgression formula and Chern-Simons form , 325
7.3 Poincare lemma and homotopy operator 328
7.4 Cartan homotopy formula 333
7.5 Chern-Simons form, gauge transformations and
anomaly 335
7.6 Chern-Simons form, variations and anomaly 339
8 Consistent anomaly 342
8.1 Infinitesimal gauge operator, BRS and geometry 342
8.2 Wess-Zumino consistency condition 350
8.3 Algebra, cocycles and cohomology 356
8.3.1 Faddeev-Popov ghosts and gauge elements 356
8.3.2 Algebra 358
8.3.3 Cocycle 360
8.3.4 Cohomology 363
0 Stora-Zumino chain of descent equations 366
9.1 Stora's approach to the chain 366
9.2 Chain for nontrivial gauge bundles 370

xii Contents
9.3 Zumino's approach to the chain 372
9.4 Explicit solutions for the chain terms 381
9.5 Zumino's formulae for chain terms 384
10 Covariant anomaly 390
10.1 Bardeen-Zumino polynomial and covariant
anomaly 390
10.2 Covariant anomaly and differential forms 395
10.3 Geometry in the space of gauge potentials 400
10.4 Nonlocal extensions of anomalies in Sp A 405
11 Index and anomaly 408
11.1 Singlet anomaly and index 408
11.2 Fredholm and elliptic operators and index theory 412
11.3 Heat kernel and index 422
11.4 Atiyah-Singer index theorem 425
11.5 Non-Abelian anomaly and generalized index
theorem _ 427
11.5.1 Differential operator D 427
11.5.2 Anomalous Jacobian 432
11.5.3 Alvarez-Gaume and Ginsparg's index
procedure 436
12 Gravitation 451
12.1 Riemannian geometry 452
12.1.1 Metric 452
12.1.2 Equivalence principle 458
12.1.3 Christoffel connection and curvature 460
12.1.4 Variations and derivatives 465
12.1.5 General coordinate transformations 468
12.2 Tangent frame 476
12.2.1 Vielbein 476
12.2.2 Spin connection and curvature 479
12.2.3 Examples in 2 dimensions 482
12.2.4 Relation between Christoffel- and spin
connection 487
12.2.5 Local Lorentz transformations 490
12.3 Action principle 491
12.3.1 Einstein-Hilbert action 491
12.3.2 Energy-momentum tensor 493

Contents xiii
12.4 Fermionic action 496
12.5 Invariances 502
12.5.1 Lorentz invariance 503
12.5.2 Einstein invariance 504
12.5.3 Weyl invariance 506
12.6 Gravitational anomalies 508
12.6.1 Einstein-, Lorentz-, Weyl anomaly 508
12.6.2 Consistency conditions 511
12.6.3 Equivalence of Einstein- and Lorentz
anomaly 515
12.6.4 Covariant gravitational anomaly 518
12.7 BRS algebra and descent equations 520
12.7.1 BRS algebra 520
12.7.2 Descent equations 525
12.7.3 BRS for nontrivial fibre bundles 528
12.8 Index theorem for gravitation 531
12.8.1 Alvarez-Gaume and Ginsparg's index
procedure 532
12.8.2 Examples in two dimensions 537
12.8.3 Mixed anomalies 541
12.8.4 Axial gravitational anomaly 542
Bibliography 545
Index 557

1
Introduction
Symmetries and their corresponding conservation laws play an important
role in describing the fundamental forces of nature. However, it might turn
out that a certain conservation law, valid in the classical theory, is violated
in the quantized version. Then we speak of an anomaly. Actually, this is
not so surprising; we know by now that 'naive' classical conceptions are
often demolished by quantum effects. Familiar examples are the uncertainty
relation of space and momentum, or the nonvanishing ground state energy of
the oscillator, etc. Why then are the anomalies so extraordinary in quantum
field theories?
The basis of modern quantum field theory (QFT)—gauge theory—is the
principle of gauge symmetry. There an anomaly—the violation of a classi-
classically conserved current—signals the breakdown of the gauge symmetry and,
in consequence, the ruin of the consistency of the theory. Avoiding, on one
hand, the anomaly—which may be possible—leads to severe constraints on
the physical content of the theory; for example, to the prediction of the
top quark. But, on the other hand, anomalies are also needed to describe
certain experimental facts. It is this double-feature which makes anomalies
so important for physics.
The anomalies which we consider in our book are the axial- or the chiral
anomaly corresponding to an axial- or a chiral fermion current. The disco-
discovery of the axial anomaly has quite a long history. It began in 1949 with
.lack Steinberger [Steinberger 1949], who calculated in his Ph.D. a Feynman
diagram in an at that time fashionable pion-nucleon (тг-N) model (which,
n.s we know now, represents the anomaly) in order to describe the decay
""" —> 77- Independently H. Fukuda and Y. Miyamoto [Fukuda, Miyamoto
1949] performed similar calculations. Steinberger's тг-N model containing a
7r>-vertex was in excellent agreement with experiment. However, his compar
rison with a тг-N amplitude containing a 7^75-vertex (which, as we know
now, corresponds to the axial Ward identity), of course, failed. Noting this
puzzle, Steinberger left theory and became a Nobel Laureate in experimen-
tal physics.

2 Introduction
Two years later, in 1951, Julian Schwinger [Schwinger 1951] pointed out
that the conservation of the axial current in QED—an immediate conse-
consequence of the axial symmetry—is violated when the current operator is
appropriately regularized. After a pause of 12 years it was Ken Johnson
[Johnson 1963] who remarked that in a massless 2-dimensional QED one
cannot have both the conservation of the gauge current and the conservation
of the axial current. However, little attention was paid to the importance of
these results in the subsequent years.
In the sixties Gell-Mann's current algebra became rather popular, and
within this framework, relying on PCAC (partial conservation of the axial
current), Sutherland and Veltman [Sutherland 1967], [Veltman 1967] proved
the theorem that the neutral pion cannot decay into two photons, in con-
contradiction to experiment! Impressed by the analysis of his friend Martinus
Veltman, John S. Bell from CERN stressed that 'the subject of current al-
algebra must not be closed until this puzzle is resolved' (citation from Roman
Jackiw's anomaly recollections [Jackiw 1991]) and urged the study of it. In
1969 Bell and Jackiw [Bell, Jackiw 1969] (Jackiw was a CERN fellow in
1967) solved the 7г° —> 77 puzzle by using the cr-model: the anomaly—the
quantum mechanical breaking of the axial symmetry—corrects the decay
rate resulting from the Sutherland-Veltman theorem by a definite amount
which is in excellent agreement with experiment. Independently, in the same
year, Stephan L. Adler [Adler 1969] from the Institute of Advanced Study,
Princeton, arrived at similar conclusions by working in spinor electrodyna-
electrodynamics.
By calculating the now famous triangle Feynman diagram made up of
one axial- and two vector currents, containing an UV divergency, the authors
found the following result: while conservation of the vector current can be
maintained, the conservation law of the axial current is broken (we consider
the massless case, m = 0, for simplicity)
l = A
where Л represents the celebrated ABJ anomaly
°~f>Fi°'Faft (L2)
(ABJ in honour of the three discoverers).
Now the ice was broken, the ABJ anomaly opened the door to a deeper
understanding of QFT, a new era for field theory research began.
It is a remarkable fact that the anomaly is already totally given by
the 1-loop calculation, the anomaly receives no contributions from radiative
corrections. This is the content of the important Adler-Bardeen theorem
[Adler, Bardeen 1969].

Introduction
The extension of the theory to non-Abelian fields A^ = A^
F?uTa, leads on one hand to the singlet anomaly
^Ta,
л = dXj"= Yb?^a0tlFfiuFa^ (L3)
and on the other to Bardeen's non-Abelian anomaly result [Bardeen 1969]
-Ga[A»\ = (Z^JT = ±^etiVf"TtiTadtl(AudpA(r + \AvApAa), A.4)
where ± corresponds to positive or negative chiral fields.
Soon after these perturbation calculations, in the year 1971, Dolgov and
Zakharov [Dolgov, Zakharov 1971] proposed a different method, with certain
merits, to calculate the anomaly: the technique of dispersion relations. There
the source of the anomaly is the existence of a superconvergence sum rule
for an infrared divergent amplitude.
However, the anomaly is not just a perturbation effect, resulting from
the regularization of some divergent diagrams, but reflects the deep laws of
quantum physics. So the anomaly must be seen from a different point of
view as well.
In the years 1976-77 there occurred the first topological investigations
of the anomaly. Several authors like Jackiw, Rebbi, Nielsen, Romer, Schroer
[Jackiw, Rebbi 1976,1977], [Nielsen, Schroer 1977], [Nielsen, Romer, Schroer
1977] discovered that the singlet anomaly is determined by an index theo-
theorem. The reason is that the anomaly can be expressed by a sum of eigenfunc-
tions of the Dirac operator, where only the zero-modes of a given chirality
(n+,n-) survive
— / dxA{x) = / dx^tplix^btpnix) =n+-n- = index D+. A.5)
The difference in the chirality zero-modes represents the index of the Weyl
operator D+ =0P+. Using now the distinguished Atiyah-Singer (AS) index
theorem [Atiyah, Singer 1968a,b,c, 1971a,b] one can express the index in
terms of characteristic classes, here in terms of the Chern character in 4
dimensions
index D+ = -—^ / trFF, A.6)
that establishes the anomaly on pure topological grounds.
Another important step towards understanding anomalies happened in
1979. Kazuo Fujikawa worked with the path integral of quantized fermions
in an external gauge field (in Euclidean space)

Introduction
] = e~w^ = I d-4)d$es{A»] A.7)
with the classical action
S = fdx$(W -т)ф , p = 7"(a/1 + Лм). A.8)
Naively, no anomaly would appear since under a chiral transformation
the classical action clearly remains invariant. However, Fujikawa [Fujikawa
1979, 1980] discovered that the path integral measure does transform too
ехр[-2г / dx/3(x) J^ ip^x^ip^x)] A-10)
3 n
(tpn are the eigenfunctions of the Dirac operator ^), and the Jacobian
contains—after appropriate regularization—precisely the anomaly
ЛF^. A.11)
So this procedure correponds to a local evaluation of the index (described
before). Nowadays we know that Fujikawa's procedure—also called the non-
perturbative approach—is closely related to the heat kernel method and to
the zeta function regularization procedure.
In the eighties the field of anomalies began to flourish again. Modern ma-
mathematical techniques—differential geometry, cohomology and topology—
became very popular and helped to shed more light onto the anomalous
phenomena of QFT. The anomalies are now rewritten in terms of differen-
differential forms
singlet anomaly
A = d*j5 = ^ tr FF = ^dti {AdA + | A3) A.12)
non-Abelian anomaly
Ga[A] = -(D*j)a = ±^trTad{AdA+\A% A.13)
The equation which defines the anomaly is the Wess-Zumino (WZ) consi-
consistency condition; on a BRS level it can be formulated in a very compact
way

Introduction 5
sG{v, A) = 8 j vaGa[A] = 0 A.14)
(v = vaTa denotes the Faddeev-Popov ghost, s the BRS operator with
s2 = 0). Generally, any nontrivial solution represents a consistent anomaly;
of course, the trivial solution is
Gtriv = sG[A]. A.15)
Mathematically, the WZ consistency condition corresponds to cocycles and
cohomologies.
The modern, differential geometric treatment of anomalies started with
the lectures of Stora [Stora 1984] and Zumino [Zumino 1984]. They found a
chain of descent equations where several polynomials in v,A, F—so-called
chain terms—are linked together in different dimensions
sQ°2n-l + dQL-2 = 0
sQL-2 + dQln-3 = 0
sQl"'1 = 0. A.16)
P(Fn) is a symmetric, gauge invariant polynomial in F. The chain terms
@2п-1-к> where the lower index denotes the form degree, the upper index
the power in v, can be solved explicitly in a rather easy way. (Actually, the
above chain can be traced back to Stora's work in 1977 [Stora 1977].)
The amazing feature of this pure mathematical chain is that it has coun-
counterparts in physics. For example, the third equation of the Stora-Zumino
(SZ) chain represents a local version of the WZ consistency condition, so
that we can identify the chain term <92n-2 (mathematics) with the anomaly
G(v, A) (physics) in Bn - 2) dimensions
G(v,A) = n[ Q\n-2{v,A) A.17)
(N is some normalization not determined by the chain).
Choosing as the invariant polynomial the symmetrized trace str F, we
recover the singlet anomaly in In dimensions. So the two anomalies, the
singlet- and the non-Abelian anomaly, although different in their nature,
are linked together in different dimensions.
A second boom of topological activities started with the work of Atiyah
and Singer [Atiyah, Singer 1984] and Alvarez, Singer and Zumino [Alva-
[Alvarez, Singer, Zumino 1984] who discovered that the non-Abelian anomaly is
related to a more refined index theorem—the family index theorem. Equi-

6 Introduction
valently, Alvarez-Gaume and Ginsparg [Alvarez-Gaume, Ginsparg 1984] re-
related the anomaly in In dimensions to an AS index theorem in Bn 4- 2)
dimensions. The extra 2 dimensions arise from considering two-parameter
families of gauge potentials. The result is
—G(v,A) = 2ni Index i
The anomaly is given by the SZ chain term Q\n with the correct normali-
normalization.
Parallel to the activities in gauge theories there was rapid development in
gravitation. Gravitation is an important part of field theory which cannot be
left aside. It began with the pioneering work of Alvarez-Gaume and Witten
[Alvarez-Gaume, Witten 1983] on gravitational anomalies, and the enthu-
enthusiasm culminated in the discovery of Green and Schwarz [Green, Schwarz
1984] that gauge and gravitational anomalies may cancel each other, howe-
however, in a supersymmetric theory in 10 dimensions.
Here gravitation is regarded as a gauge theory where the gauges are the
general coordinate transformations—the diffeomorphisms—or the rotations
in the tangent frame—the Lorentz transformations. Then the classical con-
conservation law of the energy-momentum tensor can be broken in the quantum
case—an Einstein anomaly occurs
^Т^) =: GE@ A.19)
(? is an infinitesimal coordinate shift). On the other hand, a Lorentz anom-
anomaly may also occur
= J dxeaab(Tab) =: Gb(a) A.20)
(a is the parameter of the infinitesimal Lorentz transformation), which
is equivalent to the existence of an antisymmetric part of the energy-
momentum tensor. Bardeen and Zumino [Bardeen, Zumino 1984] discovered
that both types of gravitational anomaly are not independent, but can be
shifted into each other with help of a counterterm.
The organization of the whole material is as follows:
In Chapter 2 we introduce all necessary mathematical concepts like ho-
motopy, manifolds and differential forms, homology and cohomology, Lie
derivative and Lie groups, and finally the fibre bundle set-up. In Chapter 3

Introduction 7
we explain the path integral formalism, specifically the Faddeev-Popov me-
method for quantizing gauge theories and the BRS transformations.
We spend the whole of Chapter 4 on a thorough investigation of the
anomaly within perturbation theory. The anomaly is calculated within all
commonly used regularization techniques (e.g. Pauli-Villars and 't Hooft-
Veltman regularization, point splitting method), and we also work out the
method of dispersion relations. The famous 7г° —> 77 decay is described in
detail. We establish the anomalies for non-Abelian fields—the singlet- and
the non-Abelian anomalies, which are the topic throughout the book—and
we close this chapter with a discussion about their importance for physics.
In the subsequent Chapter 5 we present the nonperturbative view. We
work in path integral formalism and explain Fujikawa's method for calcula-
calculating the anomaly. We demonstrate the case of the singlet anomaly as well
as the covariant non-Abelian anomaly. (The consistent anomaly calculation
is preferably shifted into Chapter 11.5.2.) The connection to the heat kernel
method and to the zeta function regularization is carried out in detail.
A little practice in differential forms and fibre-bundle concepts for our
geometric point of view of field theories like electrodynamics, Yang-Mills
theory, Dirac monopole, the Aharonov-Bohm effect and instantons is given
in Chapter 6; while Chapter 7 is committed to invariant polynomials, to
the Chern-Simons form and 'transgression', and to the use of the homotopy
operator.
The modern treatment of anomalies begins in Chapter 8, where we in-
introduce the gauge operator and the BRS machinery. The WZ consistency
condition is established and illuminated from an algebraic, cocycle and co-
homological point of view. In Chapter 9 we derive the SZ chain of descent
equations, for both, for trivial and nontrivial gauge bundles. We find explicit
solutions and formulae for the chain terms and we discuss the identifications
with physics. The covariant type of anomaly is studied separately in Chap-
Chapter 10; its relation to the consistent type is analysed in terms of differential
forms on the space-time manifold M as well as in terms of differential forms
on Sp Л, the space of all gauge connections.
The topological investigations of anomalies are performed in Chapter 11.
We begin with the relation of the singlet anomaly to the AS index theorem,
and we present a bit of index theory afterwards, emphasizing the relation
of the index to the heat kernel. Introducing a modified Dirac operator we
calculate the consistent non-Abelian anomaly first a la Fujikawa and then
a la Alvarez-Gaume and Ginsparg, with help of an AS index theorem in
Bn + 2) dimensions.
The final Chapter, 12, is devoted to gravitation. We first review some
basic concepts of Riemannian geometry and establish our notation. Then we
discuss the gravitational action, specifically the fermionic action in detail.
We introduce the Einstein-, Lorentz- and Weyl anomalies by violating the

8
Introduction
corresponding Einstein-, Lorentz- and Weyl symmetries and we establish
the consistency conditions. The equivalence of the Einstein- and Lorentz
anomaly is demonstrated, the covariant gravitational anomaly type is also
discussed. Finally we treat gravitation on a BRS level, deriving the SZ chain
of descent equations, and we use the index theorems to carry out explicit
anomaly examples.
Summarizing, we would like to show these many different aspects of the
anomalies, which make them so fascinating, in the following picture:
Dispersion Relations
Perturbation Theory Nonperturbative
Approach
Feynman Diagrams ж Path Integral
ANOMALY
Differential Geometry Cohomology
Descent Equations Gauge Cocycles
Index Density
Topology
Index Theorems

Differential geometry, topology and
fibre bundles
In the last decade theoretical physicists have made extensive use of diffe-
differential geometry, topology and fibre bundles. Particularly in quantum field
theory—which is our main interest—these modern mathematical techniques
prove to be fruitful. In fact, it is our aim to formulate physics in a differen-
differential geometric and topological way, which is of increasing importance in the
future. We do not assume the knowledge of these techniques, but introduce
the basic ideas, especially those concepts we use continuously in the book.
In Section 2.1 we briefly summarize some topological definitions; in Sec-
Section 2.2 we explain the homotopy of maps and the homotopy of groups. In
Section 2.3 we introduce the concept of differentiable manifolds and in Sec-
Section 2.4 we present the differential forms together with their Hodge duals,
along with the differentiation and integration. This is one of the basic secti-
sections since we will formulate all laws of physics in terms of differential forms.
In Section 2.5 we speak about homology and de Rham cohomology and in
Section 2.6 we explain important concepts like the pullback of a differential
form, the Lie derivative, the Lie group and the Lie algebra. Finally, Section
2.7 is devoted to the construction of fibre bundles including connection and
curvature. They turn out to be a suitable mathematical concept to describe
the physics of gauge theories.
The mathematics is sometimes presented in an intuitive way avoiding
such a rigorous derivation. For more rigor and a broader presentation the
reader should consult the classical mathematical literature: [Flanders 1963],
[Kobayashi, Nomizu 1963, 1969], [Choquet-Bruhat, DeWitt-Morette 1982],
[Singer, Thorpe 1967], [Greub, Halperin, Vanstone 1972], [Bott, Tu 1982],
[Warner 1983], or the books which have recently appeared and are close to
our main interest: [Nash, Sen 1983], [Gockeler, Schucker 1987], [Isham 1989],
[Nakahara 1990]. We refer to the appropriate literature in each section.

10 Differential geometry, topology and fibre bundles
2.1 Topology
A topological space is a set with a structure such that we can define a
neighbourhood and continuous functions in a general way.
Topological space: Let U be a system of subsets of a set X. U defines a
topology on X if
i) 0 € U and X 6 U,
ii) for any finite or infinite subsets {Щк} the union satisfies |Jfc Щк € U,
iii) for any finite subsets {Щг,... ,Щ } the intersection satisfies
Then X or the pair (X, U) is called a topological space and the Uik are
open sets.
For example, let X = R, then all open intervals (a, b) and their unions
define the usual topology.
Closed set: A set А с X is closed if Ac = X \ A is open. Ac = {x €
X\x $ A} is the complement of A.
Neighbourhood: Let X be a topological space. N{x) is a neighbour-
neighbourhood of a point x € X if N(x) contains an open set U(x) which contains
x : N{x) D U{x).
For example, the closed interval [a, b] is a neighbourhood for any point
x € (a, b) of the open interval.
Theorem: A subset Л С X is open iff it is a neighbourhood for each of
its points.
Limit point or accumulation point: A point x € X is a limit point
of the set А С X if each neighbourhood N(x) contains at least one point
о G A (different from x): {N{x) - {x}) П А ф 0, V N(x).
Theorem: A is closed iff it contains all its limit points.
Closure: The closure A of A in X is the union of A with all its limit
points. It is the smallest closed set containing A.
Interior: The interior A0 of A is the largest open set contained in A.
Dense: The set A is dense in X if A = X.

Topology
11
D
Fig. 2.1. Open disk D with open covering {Ya}
Boundary: A boundary dA of a set Л is the complement of the interior
in the closure of A : dA = A\A°. A closed set contains its boundary while
an open set is disjoint.
For example, all sets @,6), (a,b], [a,b), [a,b] have the same boundary
ЭА = А\А° = {a,b} since A = [a,b] and A0 = (a,b).
For example, the disk A = {(x, y)\x2 +y2 < a2} has as its boundary the
circle dA = {(x,y)\x2 + y2 = a2}.
Remark: Let A be a set, then
А П dA = 0
дА с А
A is open
A is closed.
Metric: A metric is a map d : X x X —> R such that
i) d{x,y) = d{y,x), symmetric,
ii) d(x, y) > 0, equality iff x = y,
iii) d(x, y) + d(y, z) > d(x, z), triangle inequality, Vi,t/,zgX.
X becomes a topological space via the metric d whose open sets are
open balls Ue(x) = {y € X\d(x,у) < ё) and all their unions. In this way
we get a metric topology and X is called a metric space.

12
Differential geometry, topology and fibre bundles
/
/ / /
I I I
I
\\ \
\ \ \
\
D V \
1 ' 1
J' 1
Vе
Уп=\
x\
Fig. 2.2. Closed disk D with a finite open subcovering V^=1
Compactness: Given a topological space X and a family of sets {Ya} —
Y. Then У is a covering of X if Ц, Ya D X.
If all Ya are open sets, the covering Y is called an open covering.
The set X is compact if for every open covering {Ya} there exists a
finite subcovering {Yi,... ,Yn} of X.
Example: The open disk D = {(xi,X2)|xf + x\ < 1} is not compact.
Choose an open covering of D, namely concentric disks
Ya = Uxi,X2)\xl
< 1-
with a = 1,2,....
B.1)
Obviously Ua^a = D but {Ya} has no finite subcovering (see Figure 2.1),
hence D is not compact.
The closed disk D = {(xi, X2)|xf + x\ < 1} is compact. All open cover-
coverings
Ya ~ { {xi,x2)\xi + x\ < 1 + ? \ with a = 1,2,... and e > 0
B.2)
now have a finite subcovering Y? with n > 1/e — 1, the condition that the
radius is larger than 1 or ? > l/(n+l) (see Figure 2.2). Hence D is compact.

Homotopy 13
Theorem: The subset X С R" is compact iff it is closed and bounded.
Thus the n-dimensional spheres Sn are compact since they are closed
and bounded in Rn+1.
Compactification: The compactification of a topological space X is
the pair {ip, Y) where У is a compact space and ip is a homeomorphism:
X -+ dense subspace of Y.
The compactness is a very important notion in topology. The mathema-
mathematical analysis and therefore also the physics is much simpler on a compact
space. In fact, we only work on compact sets—compact manifolds—like
spheres.
Continuous: Let X, Y be two topological spaces. The map ip : X —> Y
is continuous iff for every open set V с Y the set U = ip~l(V) С X is
open. Alternatively, ip is continuous at x € X iff ip(xi) —> ip{x) whenever
Xi —> x (convergence of sequences).
Homeomorphism—diffeomorphism: Let X, У be two topological
spaces. The map ip is a homeomorphism if ip : X —* Y is a bisection
A — 1 and onto) and ip and ip'1 are continuous. Then X and Y are homeo-
morphic to each other.
If ip is a bisection and ip and ip~l are continuously differentiable ip, ip~l €
C°° then ip is called a diffeomorphism.
2.2 Homotopy
The main aim in topology is to classify spaces with certain properties. For
instance, two spaces are equivalent if they can be continuously deformed
into each other. More precisely, two topological spaces belong to the same
equivalence class if they are homeomorphic. Such equivalence classes are
characterized by topological invariants. They are numbers like the num-
number of connected components of a space, or they typify algebraic structures
like groups constructed on a space (winding numbers), etc. A complete cha-
characterization in terms of invariants is rather difficult; instead what we know
is:
Proposition:
• If two topological spaces have different topological invariants they are
not homeomorphic!

14
Differential geometry, topology and fibre bundles
/o
Fig. 2.3. The homotopy Ft: a continuous deformation of /o to Д
In search of invariants another notion—a bit coarser but still powerful—
is homotopy. There the continuous function mapping one space into ano-
another need not have an inverse. Homotopy creates equivalence classes of con-
continuous maps. As we shall see it is also very helpful to visualize the objects
we investigate—the topological spaces, the homotopic maps, the loops—by
an ideal rubber which we can squeeze and deform smoothly in any direction.
2.2.1 Homotopy of maps
Homotopy: Let X, Y be two topological spaces. Consider two continuous
maps /o, /1 : X -+ У, then /o is homotopic to Д (/o can be continuously
deformed into Д):
/о ~ Д B.3)
(the symbol ~ means homotopic throughout the ensuing discussion) if there
exists a continuous family of functions Ft parametrized over the product
space X x I:
B.4)
XxI-^UY with I = [0,1],
) and Ft=1|Xx{1} =Д.
B.5)
such that
^*=о1хх{о} =
When t € [0,1] varies from 0 —¦ 1 then the map /b is deformed continuously
into /i like a rubber band (see Figure 2.3).
Ft is called a homotopy. It defines homotopy classes for functions. Two
functions lie in the same class if they are homotopic.

Homotopy
15
R»\{0}
8
n-\
Fig. 2.4. The space Rn \ {0} is squeezed to a sphere
r»-l
Transitivity: Let /, g, ft be continuous maps in topological spaces. Then
transitivity is satisfied:
if / ~ g and g ~ ft, then / ~ ft.
B.6)
Homotopic equivalent spaces: Two topological spaces X and Y are
homotopic equivalent if there exist continuous maps f,g:
X Л Y Л X,
such that
f • g
g ¦ f
identity on Y
identity on X.
B.7)
B.8)
Example of homotopic spaces: Choose the Rn space with origin ex-
extracted X = Rn \ {0} and the (n - l)-dimensional sphere Y = S"'1, then
R"\{0}
B.9)
A homotopic space is like a rubber block which can be smoothly squeezed
everywhere (see Figure 2.4).
Proof. It is illustrative to prove the above example. We construct the two
maps /, g:
:X = Rn\{0} -f Sn~1=Y
x
via x i-+ —-
\x\
B.10)

16 Differential geometry, topology and fibre bundles
and
g . i = о —> xv \ \Uj = л
via x (-» a; with |x| = 1. B-11)
There exists a continuous map Ft:
X x [0,1] -^ X, B.12)
such that
F' -(rr\ — 1 (тЛ — т япН F. , (<тЛ — n ¦ f(r\ — @ 1 4"\
Explicitly it is
therefore g ¦ f ~ ix. This construction is typical for homotopic families and
we shall use it frequently in this book. The second part / ¦ g ~ ly is trivial.
Q.E.D.
Example of homotopic maps: Let X = {9} be the unit circle described
by the angle в where в and в + 2тг are identified and let Y = {ei<p} be the
set of unimodular complex numbers, the group f/(l). Consider the map
/ : X —f Y; then the continuous functions
f{6) = е«пв+а\ B.15)
which are such mappings, generate a homotopy class for different a and
fixed integer n; explicitly
/o@) = ei(ne+Qo) and Л@) = е^пв+а1\ B.16)
Then /o is homotopic to /i, /о ~ Л for fixed n, since there exists a conti-
continuous family of functions FtF):
Xxl -^ У with / € [0,1], B.17)
such that
*Uo@) =/o@) and Ft=1@) = Д@). B.18)
It is the homotopy
Ft@) = exp{i[n0 + A - t)ao + tan}}. B.19)
Next we consider the covering of this map. The function f(9): Sl —>
U(l) ~ S1 maps a circle—a l-dimensional sphere—onto a circle since ?7A)

Homotopy
17
Fig. 2.5. While S1F) is covered once the group U(l) or the sphere
is wound around n times
is isomorphic to 51. When в varies from 0 -+ 2тг the sphere X = S1F) is
covered once. But then C — пв varies from 0 —> 2-кп and the other sphere
Y = 51(/3), the group U(l), will be covered n times or wound around n
times (see Figure 2.5). Therefore n is named the winding number.
Proposition:
• Each homotopy class is characterized by its winding number n!
For a given mapping fF) the winding number can be expressed by
an integral
B.20)
If we have a map of winding number 1, say fF) = егв, then a mapping
of a higher winding number k is obtained by taking k powers of this map:
№ = {№)}"¦
We can also express the map in terms of Cartesian coordinates
f(x,y) = x + iy with x2 + y2 = 1. B.21)
We may expand the domain of the map from the sphere S1 to the whole
R1:-oo<x<ooby identifying the endpoints x = — oo and x = oo to
be the same point; thus the map satisfies f(x = oo) = f(x = — oo). In this
way we achieve a 1-point compactification of R1 which is topologically
equivalent to the S1.
For example, the map

18
Differential geometry, topology and fibre bundles
Fig. 2.6. Space X with a hole. Loop a surrounds the hole, loop /3 does not.
Loop a cannot be shrunk to a point, whereas loop /3 can be shrunk
f(x) = exp \in
B.22)
is of this type and has winding number 1. Generally the winding number
is given by
A~. t~^(<r\ t(<r\ О 01\
The important applications in physics are the Dirac monopole and the
instantons which we shall discuss in Sections 6.4 and 6.6.
2.2.2 Homotopy groups
Introducing loops in a topological space it turns out that the equivalence
classes of loops reveal a group structure.
Loop: Let X be a topological space and / = [0,1] an interval. The conti-
continuous map a(t) : I —> X with a@) = a(l) = x0 e X and t € I is called a
loop with base point xo.
There exist two kinds of loop (see Figure 2.6):
i) a loop containing a hole—it cannot be shrunk to a point,
ii) a loop not containing a hole—it can be shrunk.
Connectedness: A topological space X is disconnected if there exist
two disjoint open subsets X\, X?. Xx П Х2 = 0, such that Xy U X2 = X;
otherwise the space X is connected.

Homotopy
19
(a)
(b)
Fig. 2.7. a) The homotopy H(s,t) between a(s) and /3(s). b) The loop a
is continuously deformed into the loop /3 by the homotopy Я
If any loop in X can be continuously shrunk to a point then X is simply
connected. X is arcwise connected (or path connected) if there exists a
path between any pair of points xo,xi e X.
Note: arcwise connected
connected.
Again we imagine the loop to be an ideal rubber ring which can be
smoothly stretched or shrunk. Then we consider two loops as equivalent
or homotopic if they can be continuously deformed into each other. They
form an equivalence or homotopy class which is characterized by an
integer n, the number of times a loop surrounds a hole (n > 0 for clockwise).
For example, in Figure 2.6 the loop a is not homotopic to the loop /3.
Homotopic loops: Two loops a and /3 based at xo are homotopic to
each other
a ~ /3 B.24)
if there exists a continuous map H(s,t):
I xl "-^P X, with s,t€l=[0,1]
B.25)
such that

20
Differential geometry, topology and fibre bundles
Fig. 2.8. For the product а о /3 we first pass along a then along /3
a
Fig. 2.9. The loop 7 is homotopic to the product ao/3:7~ao/3
H{s, 0) = a(s) and H{s, 1) = C{s), V s e I,
B.26)
H@,t) = H{l,t) = xo, VteJ.
H(s,t) is called a homotopy between the loops a and /3 (see Figure 2.7).
Fundamental group or first homotopy group Hy(X,xo): To equiva-
equivalence classes of loops one can assign a group structure. For that we first
have to define a product of loops, the inverse loop and the unit.
Product of loops: The product of two loops a(t) and /3(t) based at
x0 e X, a@) = a(l) = 0@) = /3A) = i0 6 X, is denned by

Homotopy
21
a-1
Fig. 2.10. Loop a and its inverse a
-1
Fig. 2.11. A loop a times its inverse a x is homotopic to the identity:
~l ~ 1
а о a
0*00 =
aBt)
for 0 < t < 1/2
P{2t - 1) for 1/2 < t < 1.
B.27)
So starting at xo we first pass along the loop a and then along the second
loop /3 (see Figure 2.8).
Remark: Loops homotopic to а о /3 start and end at the base point xo
but need not pass through xo in the midjourney; for instance, 7 ~ а о /3 in
Figure 2.9.
Inverse loop: The inverse loop a~1(t) of a based at xo € X is denned
by
a-l{t) = a{l-t), VteJ. B.28)

22 Differential geometry, topology and fibre bundles
So for the inverse we pass along the loop in the opposite direction (see
Figure 2.10).
Identity or unit element:
1@ = so, VteJ. B.29)
The identity is just the 'constant' loop at xo € X.
However, the product of a loop with its inverse is not the identity: а о
а Ф l! But it is homotopic to the identity: а о a~x ~ 1 as we demonstrate
in Figure 2.11 where the loops surround a hole.
Thus the loops themselves cannot form a group but it is the equivalence
class of loops which admits a group structure. Therefore we consider the
homotopy class of loops denoted by [a] and we finally define the product
and inverse of these equivalence classes
[a]o\0\:=[ao0\ B.30)
[a] := [a]. B.31)
Then the group properties are satisfied and the group is named the funda-
fundamental group or first homotopy group IIi(X,xo):
i) [а] о [0\ = [7] e IIi(X,xo) closed
ii) ([а] о [Ь]) о [с] = [а] о ([Ь] о [с]) associative
iii) 1 о [а] — [a] 3 unit element
iv) [а] о [a] = [1] 3 inverse element
v) if [а] о [0\ = [0\ о [a] Abelian.
Examples:
i) Consider the maps into the circle; there the homotopy classes are
typified by the winding number n. A product of classes has winding
number n + m, therefore the group is Abelian and it is isomorphic to
the group of integers Z
n^QO^II^S^-Z B.33)
(the symbol ^ denotes isomorphic). In physics the Dirac monopole is
characterized by the fundamental group IIi(J7(l)) whose integers—
the winding numbers—represent the magnetic monopole charge, see
Section 6.4.

Homotopy 23
Fig. 2.12. Plane with two holes; a / /3 but with the help of 7 we convert
a into /3, see Figure 2.13
ii) The mapping of a loop into the sphere S2 can be shrunk to a point.
Therefore the fundamental group of the S2 contains only the identity
and we have (in an additive group notation)
= 0 B.34)
and furthermore
riiOS) =0 for n > 2. B.35)
iii) Similarly the fundamental group of the n-dimensional Euclidean space
is trivial
IIi(Rn)=0. B.36)
iv) The first homotopy group П1 need not necessarily be Abelian. A non-
Abelian homotopy group is, for example,
П1B?2-{х1}-{х2}), B.37)
where the space is a disk (or plane) with two holes or with two points
extracted. This can be readily understood from Figures 2.12 and 2.13
or 2.14. The loop a is not homotopic to the loop /3: а ф /3; a cannot
be continuously deformed into /3 because of the lower hole. But with
the help of loop 7 we can convert a into /3 (see Figure 2.13)

24 Differential geometry, topology and fibre bundles
о
Fig. 2.13, The product of loops 7 о а о 7 Ms homotopic to /3
a°Y
Fig. 2.14. The loop producte 7 о a Ф а о ¦у are not homotopic

Homotopy 25
70a 07 ~ /3 ф а
=>¦ 7oa^ao7
=>¦ [7] о [«] T6 [«] о [7]. B-38)
Alternatively we could directly consider the loop products 70a and
а о 7. Figure 2.14 illustrates that there exists no homotopy between
them: 7 о a 9^ а о 7.
Considering finally the fundamental groups based at different points xo,
Xi, one finds the following theorems.
Theorem: Let X be an arcwise connected topological space and xo, x\ e
X, then
IIxOX.xoJ^IIxCX.xi). B.39)
Theorem: Let X and Y be arcwise connected topological spaces, then
П1(ХхУ,AО,2/о))-П1(Л:,хо)©П1(УJ/о). B.40)
Both theorems can be generalized to the higher homotopy groups Пп.
Examples: Choose the torus T2 = S1 x Sl, then
Пу{т2) = Uiis1 x s1) ~ Uiis1) en^s1) ~ z e z, B.41)
whereas for the cylinder С = S1 x R we have
n1(s1 xR)~ni(s1)en1(R)~ze{i}~z. B.42)
Higher homotopy groups: Our homotopy discussion with 1-dimen-
sional loops can now be generalized. We introduce loops in n dimensions.
n-loop: Let X be a topological space and In = I x .. .x I the unit n-cube
where I = [0,1] and let dln be its geometric boundary. The continuous map
an : In —> X, B.43)
such that
an :дГ —>xoeX B.44)
is also called an n-loop in X based at the point xo.
So an n-loop an is a continuous map of an n-dimensional cube In into
и topological space X such that the boundary dln is mapped into a single
point xq of that space X. Effectively we deform the cube In—by identifying

26
Differential geometry, topology and fibre bundles
Fig. 2.15. A 2-loop: the square is 'blown up like chewing gum'
its boundary dln with a single point x0—into a sphere 5" analogous to
the previous 1-dimensional case where we have bent the interval / to a
1-dimensional loop a.
Illustration in two dimensions: The square I2 is the source of a 2-loop
«2 when the boundary 9/г is identified with the point xo € S2 of the sphere
S2. Then the square is homotopic (even homeomorphic) to the sphere which
represents the 2-loop. This we have illustrated in Figure 2.15 by 'blowing
up chewing gum'!
Product of n-loops: Be an and /3n two n-loops based at xo G X then
the product is denned by
,t2,.-.,tn) forO<fi<l/2
— l,t2 «r») for 1/2 < ti < 1.
B.45)

Homotopy
27
a2
t
h
x0
a2
h
*•*
4
fk
«2
0 x0 1
0 »0 1
О х„ 1 Xr. 1 <i
Fig. 2.16. The product of two 2-loops a2 о fa
Again in two dimensions, we glue together the two blades a.4 and /З2 at
an edge x0 (see Figure 2.16).
Inverse n-loop:
a~1(ti,t2,...,«„) =an{l-t1,t2,...,tn), V(tut2,...,tn) e Г. B.46)
Identity or unit element:
i(tb...,«„) = so, V(ti,...,tn)er*.
B.47)
Homotopic n-loops: Two n-loops an and 0n based at Xo are homotopic
l о each other
an ~ /3n B.48)
if there exists a continuous map—a homotopy—Hn(s\,..., sn; t):
Г xl ^*X with Si,tel = [0,1] B.49)

28
Differential geometry, topology and fibre bundles
Fig. 2.17. The two loops with a thin and thickened boundary are homotopic
to each other
such that
Hn(si,...,sn;0) = an(su...,sn)
#„(«!,..., en;l) = /3n(su...,sn)
#„(«!,..., en>t) = xQ?X ii(su
B.50)
This homotopy generates equivalence classes of n-loops denoted by
[an] and, defining again a product and inverse,
Ы ° \Pn] = К
B.51)
B.52)
then these equivalence classes of n-loops admit a group structure—the n-th
homotopy group Пп(Х,Яо) of the topological space X with base point
x0.
Examples: Recall the isomorphy between SUB) ~ S3, then the mapping
of a S3 into a S3 gives
U3(SUB)) ~ П3E3) ~ Z.
B.53)
This is the homotopic formulation of the physical example of instantons,
see Section 6.6, where the integers Z represent the instanton numbers.
Generally we have
ПпE") * Z. B.54)
The higher homotopy groups are trivial if the sphere dimension—the
space where the loop 'lives'—is larger than that of the loop
Tln{Sk) = 0 for к > п.
B.55)

Differentiable manifolds
29
a
P
a
Fig. 2.18. Higher homotopy groups are Abelian. The product of n-loops is
commutative an о /3n ~ /3n о ап
But this is not necessarily so for the reversed к < п; for example,
П3E2) = Z. B.56)
Finally we show that in contrast to Щ the Пп is Abelian.
Proposition:
• The n-th homotopy group Пп (n > 1) is Abelian!
This is easily understood by viewing the n-loop as a rubber which can
be squeezed around. The n-loop an based at xo is homotopic to the loop
we obtain by thickening the boundary dln and mapping this thickened
boundary into x0, see Figure 2.17 in the case of n = 2. Then we can
immediately deform the loop product an o/3n into /3n °otn as in Figure 2.18.
2.3 Differentiable manifolds
Now we turn to differential geometry where the basic concepts are the dif-
differentiable manifolds and the differential forms.
What is a manifold?
For example, a so-called 'Schnuller' is such an object (see Figure 2.19).
(lenerally, a differentiable manifold M is some smooth surface—a topolo-

30
Differential geometry, topology and fibre bundles
M
Fig. 2.19. A so-called 'Schnuller' represents a manifold
gical space—which locally looks Euclidean but not necessarily in its global
extent. Via a homeomorphism to RTO we attach to each point of M a set of
coordinates and require the transition from one set to another to be smooth.
In addition to the literature quoted at the beginning the reader will
find the following literature helpful: [Thirring 1992], [Sexl, Urbantke 1983],
[Visconti 1992], [Curtis, Miller 1985].

Differentiable manifolds 31
Chart: Let M be a topological space. A chart (Va,<pa) is a homeomor-
phism tpa from an open set Va С M into an open set Ra С Rm (see Figure
2.19)
Va^Ra. B.57)
Two charts are compatible if the overlap maps are continuously differen-
differentiable (diffeomorphisms)
or if Vi П V2 = 0 (see Figure 2.19).
Atlas: The set of compatible charts {(Va, ipa)} covering M is named, for
obvious reasons, an atlas. Two atlases are compatible if all their charts
are compatible.
Differentiable manifold: We speak of a differentiable manifold if the
following four conditions are satisfied:
i) M is a topological space.
ii) M is equipped with a family of pairs {(Va, </>«)} (charts),
iii) {Va} is a family of open sets which cover M : \Ja Va — M and ipa is a
homeomorphism from Va onto an open subset Ra С Rm: Va -^> Ra
(see Figure 2.19).
iv) Given two open sets Va, Vp with Va П Vp Ф 0, the overlap maps (see
Figure 2.19) <P0-<p~l from subset (ра(УаГ\Ур) to <pp(VanVp) or fa-'Pp'1
from subset y>p{Va П Vp) to fa(Va П V^) are infinitely differentiable
(C°° functions, diffeomorphisms).
Hence a differentiable manifold is given by a topological space M and the
equivalence class of atlases (which is determined by the compatibility). The
features ii) and iii) imply that the topological space—the manifold M—is lo-
locally Euclidean. M is covered with patches Va and via the homeomorphisms
ipa we attach coordinates in Rm to these patches. So within one patch the
manifold looks like the Euclidean Rm but not necessarily globally—we have
to know how the several patches fit together.
Feature iv) means that in the overlap region of two patches Va, Vp we
establish two coordinates in RTO, <pa(Va П Vp) and <pp(Va П Vp) and we can
change these coordinates smoothly via 4>p-4>al and ц>а-ц>~2х. More explicitly:
x := <pa(p) € Rro with у € Va B.59)
and
У ¦= ЫР) 6 Rm with PtVp B.60)

32
Differential geometry, topology and fibre bundles
Fig. 2.20. A 'Schnuller' manifold with boundary
then the coordinate transformation is determined by the map ц>$ ¦ <p~l
<p0-4>Z4x) = y(x)- B-61)
The dimension of the manifold is given by the dimension of the
Euclidean vector space
dimM = dimRTO = m.
B.62)
If we replace the real space Rm —* Cm by a complex space we obtain a
complex analytic manifold.
Manifold with boundary: The manifold M is denned as a space which
locally looks like the Euclidean Rm. If we now replace Rm -»• R!p by the
positive half space
R™ = {(z\... ,xro) € Rro|zm > 0}
with the boundary
= {(i1,... ,iro) G Rro|iro = 0}
B.63)
B.64)
and xm denotes the m-th coordinate, then we achieve the notion of a ma-
manifold with boundary. In this case an atlas consists of charts {(VQ,tpa)}

Differentiable manifolds 33
where <pa : Va —> i?+ represents a homeomorphism onto the open subset
Rt С R+ (see Figure 2.20). The overlap maps are again infinitely differen-
differentiable <pp ¦ <p~l, tpa ¦ ip^1 ? C°°. Then &M, the boundary of a manifold,
is the set of all points which are mapped into dRip (see Figure 2.20)
-\<pa(Va)ndR™). B.65)
The dimension of dM is clearly less by 1
dimdM = m-l. B.66)
For a manifold without boundary we have dM = 0. The boundary dM itself
represents a (m — 1)-dimensional manifold without boundary: ддМ = 0.
Remarks:
i) If a manifold M is connected then its boundary dM is not necessarily
connected. For example, M = [a, b] is connected but 8M = {a} U {b}
is not.
Vice versa:
If dM is connected then M is not necessarily connected. For example,
the point a is the (connected) boundary dM = {a} of the disconnected
manifold M = [a, b] U (b,c).
ii) A manifold with boundary is not necessarily compact. For example,
the manifold M = @,1] has a boundary dM = {1} but M is not
compact.
Vice versa:
A compact manifold has not necessarily a boundary. For example, the
circle M = S1 is compact but boundaryless dM = dS1 = 0.
Examples:
i) The Rm is clearly a manifold where just one chart is needed and the
homeomorphism ц> may be the identity map.
ii) The closed interval M = [a, b] is a manifold with boundary dM =
{a} U {b}. This is intuitively obvious but let us apply our definitions.
We find two charts:
(Vi,(pi) with
Vi = [o,b) B.67)
<pi : x A x - a = tpi(x) e Rf С R+
and

34 Differential geometry, topology and fibre bundles
Fig. 2.21. Spaces which are not manifolds
(V2.V2) with
V2 = (a,b] B.68)
tp2 ¦ x н^+ b - x = <p2(x) 6i^C R+.
The boundary is then defined by
dM = <pT1(<Pl(V1)ndR+)U<P2\MV2)ndR+)
= y?r1@)U^2-1@) = {a}U{b}. B.69)
iii) The closed ball M = Bm = {(x\...,xm) € Rm|ES=i(^J < 1}
is a manifold with m dimensions whose boundary is the (m — 1)-
dimensional sphere dBm = S™'1.
iv) Note!
The closed cube M = Cm = {(ж1,... ,жто) е В.т\\х*\ < l,Vi =
1,... ,m} does not represent a manifold with boundary since there
are edges and corners which violate the differentiability of the overlap
maps. But the interior is again a manifold.
Similarly, spaces containing junctions (see Figure 2.21) do not repre-
represent manifolds since at the junctions the space does not look Eucli-
Euclidean.
v) The n-dimensional sphere is a compact manifold without boundary.
To get used to our manifold notations it is worthwhile demonstrating
the proof in one dimension:
S1 = {(xbx2) ? R2|x^ + x\ = 1} is the 1-dimensional sphere.
Searching for an atlas we find the two charts (at least two are neces-
necessary, see Figure 2.22) (Vby>i) with Vi = S1 \ {A,0)}, and defining
the inverse map for convenience
tpil:9 >->-* (cos0,sin0)
@,2*) ^ Sl B.70)

Differentiable manifolds
35
0 2n -n 0 n R
Fig. 2.22. An atlas for S1 containing at least two charts
and (V2,(P2) with V2 = Sl \ {(-1,0)} and
(-7Г,7Г)
(cos 0, sin в)
B.71)
The maps <pl1, <р2г are invertible. All maps ip\, ц>2, ipx l, ip2l are
continuous, thus tpi, </?2 represent homeomorphisms and the overlap
maps y>\ ¦ ц>2~Х,Ч>1 • 4>\X 6 C°° are smooth. So all conditions for a
manifold are fulfilled.
Product of manifolds: The Cartesian product M\ x M% of two manifolds
Mi and Mi with atlases {(Va, </>«)} and {(^s> V/з)} is defined by the product
of the charts
(Va,<pa)x(V
with the homeomorphic map
Va x Vp —$
/ \ 4>a X V
(Pl,P2) ' >'
= (Va X Vf),<pa X
B.72)
B.73)

36
Differential geometry, topology and fibre bundles
Fig. 2.23. The torus T2 = S1 x S1 as a product manifold
Sphere
Mobius strip
Outside
Outside
Outside
Fig. 2.24. A sphere is orientable, the Mobius strip is not
and pi e Mi, p2 € Mi- So the atlas {(Va x Vp,(pa x (fp)} determines a
diflferentiable structure on the (mi + m2)-dimensional manifold Mi x M2.
For instance, the 2-dimensional torus T2 — S1 x S1 is such a product
manifold with the coordinates (#i,#2) and #i,#2 6 [0,2тг) (see Figure 2.23).
Orientability of a manifold: Let us consider a disk, it has a top- and
bottom side; a sphere, it has an in- and outside. These 2-sided surfaces
are orientable since we can use this 2-sidedness to define an orientation in
R3. However, if we study the Mobius strip we do not find a definite in-
and outside. Moving around the normal, it changes its direction (see Figure
2.24). So the Mobius strip is not orientable.

Differentiable manifolds 37
H_
Fig. 2.25. The orientation of the boundaries dH+ and dH_ is reversed
Mathematically, the key object for the orientability is the Jacobian de-
determinant. Recall a vector space and choose two different bases {e»} and
{е'г}. A basis change is given by a matrix A : е'г — Ajej. Then the determi-
determinant controls the orientation of the two bases. If det A > 0 (det A < 0) the
bases {a} and {e?} have the same (opposite) orientation.
Analogously we have to consider a manifold which is covered by charts
{(Va,tpa)}. There the coordinates are given by the homeomorphisms ipa
and we have to study the coordinate change ipp ¦ ip^1. Then the manifold
is orientable if there exists an atlas such that
det((pfi ¦ <P~l) > 0, VVa,V0 with VanVfi? 0. B.74)
In the following we always work with compact and orientable manifolds
unless we specify it differently.
The orientation for the boundary dM of a manifold M is induced in a
quite natural way which we illustrate with upper and lower hemispheres in
Figure 2.25.
Tangent space to a manifold: Vectors on a manifold M always describe
tangent vectors to a curve in M. Let p(t), t ? I be some curve passing
through a chart. The coordinates of this curve are xl(p(t)), г = 1,... ,m
and the tangent vector to the curve is given by
\{t)) B.75)

38 Differential geometry, topology and fibre bundles
Next we consider a function f(p) : M —* Rm denned on M. The change of
the function along the curve is
B.76)
and in terms of local coordinates
djdx\p{t))
C/X Qit
Note that -^-rf actually means -r-r/ • <p-1(x).
дх1 дхг
Defining the differential operator
X = Хг—г with Xi = - ^ >' B.78)
we obtain
f(p(t)) = Xf. B.79)
We consider the differential operator X as the tangent vector to the
manifold M at the point p = p(to)> with t0 fixed, in the direction of the
curve p(t).
Example: Applying, for example, the operator X to the position coordi-
coordinates we obtain the velocity
For every differentiable curve through a point p € M of the manifold M
there exists a tangent vector. So we can define the tangent space TP(M)
to the manifold M at p to be the space of all possible tangents at the point
p. Tp(M) describes a vector space and in terms of local coordinates the set
of differential operators
{fix*}'
i = l,...,m B.81)
forms a basis in TP(M). Clearly the dimension is
dim TP{M) = dim M = m. B.82)
This notation of tangent vectors—they represent contravariant
vectors—is a coordinate independent description. If we choose, for instance,

Differentiable manifolds 39
another chart with local coordinates yl we obtain for the tangent vector
= yi_d_ = dy<_ dx
dy* dxi dt dt dxk
д , д
X
dt
Cotangent space to a manifold: To the contravariant vectors, which
we have considered up to now, there also exist their duals—the covariant
vectors. The dual space to TP{M) is the cotangent space T*(M) where
duality is denned via the inner product
So the set of differentials
{dx*}, i = l,...,m, B.85)
which are dual to equation B.81) forms a basis in T*(M).
For example, the differential
df = Jj/OOd*' € Т;{М) B.86)
is such an element of T*(M). It is a cotangent vector or already a (special)
1-form. Then the action of df e T*{M) on X € TP(M) is denned by the
inner product
{df,x) = (j-jdx^-^^JLj
= Xi-?-f = Xf€R. B.87)
Generally, an element of T* (M) is given by the so-called 1-form
и = Uidx1 e T?{M) B.88)
where w, represents some covariant vector. Again, the action of a 1-
form on a vector is determined by
{w,X) = Udz'.X's-r =шаЧ^=ШгХг B.89)
or in a slightly different notation we say: a 1-form evaluated at a vector
field

40 Differential geometry, topology and fibre bundles
w(X) = bJidx{Xi —. = WiX'S^j = шгХг е R. B.90)
The notion of a 1-form B.88) is also a coordinate independent descrip-
description; in the overlap region of two charts we have
Q = u>i{y)dyl = u}j(x)—-^dxk = Wj(x)dxJ = w. B.91)
Tensors: We can now construct tensors of type (a, b) by mapping a
elements of T*(M) and b elements of TP(M) into R. So the space of these
tensors is denned by
Tab = TP(M) 9... 9 Tp{M)®T;(M) ® ... g> Г;(М). B.92)
a factors b factors
An element of Таь, a mixed tensor with contravariant rank a and covaxiant
rank b, is given in terms of local coordinates
гоо = 2&:? («) ет • • • ^dxh ¦ ¦ ¦dxJb eTa"- Bl93)
The action of T on 1-forms w\,...,wa and vectors X\,..., Хь gives the
number
Allowing the point p to vary smoothly over the whole manifold M the
vectors and tensors also vary smoothly over M and we achieve so-called
vector fields and tensor fields on M.
2.4 Differential forms
Differential forms are very practical objects. First, their notation is inde-
independent of the choice of a coordinate system and, second, the whole mathe-
mathematical formalism becomes very simple.
Wedge product: Let us begin with the antisymmetric tensor product
dz" Л dxv := dx» <8> dx" - dx" ® dx" B.94)
which is the simplest wedge product.
Notation: For convenience from now on we drop the symbol wedge Л
which is habitually used in the literature
dx'dx1' ^dx^Adxr B.95)

Differential forms 41
Then we have
dx»dxv = -dxvdx»
dx»dx» = 0 B.96)
by definition.
Analogously we define the higher wedge products as totally antisymme-
antisymmetric tensor products
dx»dxvdxa = dx» <8> dxv <g> dx" + dxu <8> dx" <g> dx»
+ dx" ® dx» <g) dx" - dx» <g> dx" ® dxv
-dx° ® dx" ® dx» - dxu <g> dx» ® dx"
dxtidxudx"dxp = dx^ ® dx" ® dx" ® dxp
„ / +1 even \ ...
+ all I _ ,, 1 permutations,
etc. B.97)
In this way we obtain totally antisymmetric tensors of rank: 0,1,2,..., m =
dim M. If the rank p > m = dim M exceeds the dimension of the manifold
the wedge product vanishes.
Differential form: With the help of the wedge products we can define
now the several differential forms
0-form ш = u){x)
1-form ш = LJli(x)dx11
2-form ш = —iJlil/(x)dxtldxv
p-form и = —w/il.../ip(x)(ix'il... dx»", B.98)
where we understand summation over all indices. Шщ...цр{х) expresses a
totally antisymmetric covariant tensor field of rank p < т. ш vanishes for
p > m, ш = 0.
We denote the set of all p-forms by Лр. This is a vector space of dimension
B.99)
p!(m-p)!
For example, for a manifold of dimension m = 3 we get the following
notations

42 Differential geometry, topology and fibre bundles
vector space p-form basis dimAp
Л0 w0 {1} 1
Л1 wi {dx1,dx2,dx3} 3 B.100)
Л2 w2 {dx1dx2,dx2dx3,dx3dx1} 3
Л3 w3 {dx4x2dx3} 1
Note that the vector spaces Лр and Лт~р have the same dimensions.
We can construct (p + g)-forms out of p-forms and g-forms in a straight-
straightforward manner with help of the wedge product ap/3q € Лр+9.
All forms belong to the space
л* = л0фЛ1ел2е...елт, B.ioi)
which is closed under the wedge product operation (or exterior product). Л*
is a graded algebra, also named Cartan's exterior algebra (or Grass-
mann algebra).
Commuting the forms ap € Лр and @q € Л9 we obtain
B.102)
So odd forms always anticommute.
Exterior derivative: We differentiate the forms by introducing the ex-
exterior derivative
d = ~dx>i B.103)
acting on a p-form in the following way
= -~u}lil...lip{x)dxvdx^ ... dx»». B.104)
The exterior derivative is a map d : Лр —* Ap+1 which transforms p-forms
into (p + l)-forms. It satisfies the important property
d2 = 0 B.105)
—named nilpotency by the mathematicians—which can be quickly verified
by direct application to a p-form
ltlm..ltp() ... dx»* = 0. B.106)
This expression has to vanish since the derivatives —— —— are symmetric
ox" oxv
while the wedge product dx"dxv is antisymmetric.

Differential forms 43
Furthermore d obeys the antiderivation rule
d(ap0q) = (dap)/3q + (-)papd0q, B.107)
where again we have indicated the degrees of the forms.
There is an important property for the exterior derivative which actually
represents the basis for the anomaly equations in physics to be studied later
on.
Lemma:
dw(X, Y) = Xu(Y) - Yu{X) - w([X, Y}), B.108)
with w = w^dx» e Л1 and X = X"^, Y = Y»— e T{M).
The commutator, being again a vector field,
[Х,У] = [Х,У]^ B.109)
represents the Lie bracket (see Section 2.6.4)
[X,Yf = Xv-^-Y» - Y"-^X». B.110)
We evaluate the 2-form dw at the vector fields X, Y,
dxvdx'i(X, Y) = Х"У - УХ", B.111)
which we verify by applying our definitions.
Proof. We prove lemma B.108)
Xu(Y) - Yw{X) - w([X, Y\)
WM(xy уx) = ^WMdxd^(X, Y)
= du(X,Y). Q.E.D. B.112)
Lemma: Generalization to p-forms w € Лр
P+i
dw(Xlt..., Xp+1) = ^(-)i+1XMXu ...Ли---, Xp+1)
i=l
(-)*+M№,^i].Xb... ,Xif... .Xj,...,Xp+1), B.113)

44 Differential geometry, topology and fibre bundles
where Xi e T(M) is a vector field and the symbol ~ means that the corre-
corresponding vector field has been omitted.
Interior product: We construct a contracted multiplication—interior
product—of a p-form ибЛр with a vector field X 6 T(M). It is defined
by
i) ix antiderivative
ii) ixf = 0 B.114)
iii) ixdx» = X»,
where / denotes some function. Viewed as a mapping the operation ix :
Лр —> Лр~ * maps opposite to the exterior derivative d. Clearly we have
ix = 0. B.115)
For example, for a 1-form we obtain
гхш = ШцХ» = ш(Х) B.116)
for a 2-form
i B.117)
w
(note the minus sign stems from the antiderivative property) and finally for
a p-form
B.118)
(p-l)l- -•""¦¦•'
If we apply the interior product twice to a 2-form we get a scalar function
1
u
= u^XttYv=u>(X,Y). B.119)
Thus we find
iyixdx^dx" = X»Y" -УТ = dxltdxv{X,Y), B.120)
which we had before (equation B.111)). Analogously we can contract a
p-form

Differential forms 45
гХр...гХ1ш = ш(Х1,...,Хр) B.121)
to arrive at a 0-form.
Integration of differential forms: We can also integrate differential
forms. The integration is the inverse operation to the exterior derivative.
The integration on a manifold M requires an integration measure or a vo-
volume element. The differential forms automatically provide such an object
together with the correct transformation properties.
We always assume compact, orientable manifolds. An orienting Tri-
Triform is
= dxldx2 ... dxm ~ dVx B.122)
and provides the volume element dVx (for a right-handed coordinate sy-
system). The totally antisymmetric e-tensor or Levi-Civita tensor is defi-
defined by
Г
I
1 for any even permutations of 1,..., m
0 if two indices are equal B.123)
—1 for any odd permutations of 1,..., m.
Choosing a different coordinate system the m-form with its antisymmetry
generates the Jacobian determinant. For example in two dimensions we have
dxdx = (wdy + Wdy) Wdy + W
дх1дх2
1 ду2 dy2dy1
= det (^j dy'dy2. B.124)
Generally we obtain
и = dxldx2 ...dxm = det (S^) dy1 dy2 ...dym, B.125)
which is the correct transformation property for a volume element.
Now we take a smooth function f on M then any m-form can be written
like /w. We define the integral of such an m-form in the domain of a chart
(Va, <pa) with coordinates (ж1, x2,..., im) by

46 Differential geometry, topology and fibre bundles
f(x)dVx. B.126)
This is an ordinary multiple integration of a function with m variables.
For the integral of the m-form over the whole manifold we have to con-
consider an atlas of charts (a finite number)
f fij = E / *<*>> BЛ27)
Jm V Jv~
where / = ^2a fa and fa has its support on the chart (Va,<pa).
This is usually called a partition of unity and means that we have
functions
&a '• Ja — J&a
satisfying
i) 0< eQ(p) <1 forpe Va,
ii) ea(p)=0 ifp^4,,
iii) Yla ea(p) = 1 in the overlap region.
Stokes' theorem: Let M be an orientable compact manifold with di-
dimension dim M = m and with a nonempty boundary dM; lo € Am-1 be an
(m — l)-form, then
[ du)= I u. B.128)
Jm JdM
In fact, it expresses an interesting connection between a local information
of a quantity in the manifold M and a global information on the boundary
dM of the manifold.
Examples: Relation B.128) generalizes in a compact way theorems we
are already familiar with in 1, 2, 3 dimensions.
i) 1-dimensional case, dimM = 1:
M = [a,b] , dM = {a,b} B.129)
and
w = u(x) , du=—w(x)dx B.130)
ax
Then Stokes' theorem corresponds to the ordinary integration in R
I dw = I ^w(i)dx = ы(Ъ) - w(e) = f ш. B.131)
Jm Ja a* JdM

Differential forms 47
ii) 2-dimensionaJ case, dimM = 2:
M = {{xl,x2)\x\ +x\< 1}, ЭМ = {{xux2)\x{ + x\ = 1} B.132)
and
ш = u)x{x)dxx + w2(x)dx2 = A(x)dx, B.133)
dw = j—Wj(x)dxldx:' = -(diAj ~ djAi)dxldx:>
= \ефВ^хЧх^ =Bkdfk = Bdf, B.134)
with the curl Bk = ЕщдгА^ or В = V x A multiplied by an area
element
1 ¦ • - ( \
dfk = -ekijdxldx3 or df = 0 . B.135)
V 4fa /
Then we recover Stokes' theorem in the familiar notation
f dw= f (VxA)df= f Adx= f ш. B.136)
JM J disk ^circle JdM
In terms of physics, A represents the vector potential and В the ma-
magnetic field. The integral of the magnetic flux through the disk equals
the integral of the vector potential along the boundary, the circle,
iii) 3-dimensional case, dimM = 3:
M = {(xi,x2,x3)\xl +x\ +x\ < 1} ball
dM = {(xi,x2,x3)\xl+xl+x23 = l} sphere B.137)
and
ш — -u)ij{x)dx%dx:i
2
= leijkEk(x)dxidxj = Ekdfk = Edf B.138)
2
= \dtujijdxtdxidxi = ^eijkdeEkdxedxidxj
EdV.
B.139)
= ^deEkeijkeeijdx1dx2dx3 = dkEkdV = VEdV.

48 Differential geometry, topology and fibre bundles
In 3 dimensions Stokes' theorem B.128) represents what we usually
call the Gauss law
f dw= [ VEdV= f Edf= [ u. B.140)
JM ./ball Jsphere JdM
Again in terms of physics, E describes the electric field.
Hodge star operation: Let us recall the space of all p-forms Лр and
the space Лт~р. Both vector spaces have the same dimension dim Am~p =
dimAp. There is a duality between these 2 spaces, an isomorphism given by
the Hodge * operation:
Лр -JL> Am~P. B.141)
The star transforms p-forms into (m — p)-forms and its action is defined by
'Х1%Ь'х+1 • • • dx»™, B.142)
(т-р)Г ^"+
where
д^и is the metric tensor and g — det <7М„. In a curved space with metric
we have
?i...m = \/Ы —* 1 in flat space
and
t l
Riemannian —* 1 Euclidean
B.144)
1 - • - -1
Л...т _.
Note that we raise and lower the indices by
= 9~1eVl...um- B.145)
In the following we only work in flat space throughout the book, except in
Chapter 12, which is specially devoted to gravitation. So our metric is

Differential forms 49
1
ци = V/ли = I _¦• I Minkowskian
-1
1
9y.v = «V = I i I Euclidean. B.146)
1
Given a p-form
u>p = -:a;Ml...Mpdx'xl ... dx1^ B.147)
then the dual p-form denotes
1
p\(m-p)l"
*t,i — и
p ~ l()!
p){m-p)\
Lemma: When applied to a p-form we have
{(_)p(m-p) ( (_у{т-р)^ Euclidean
or *-1 = <
(_)P(m-P)+i у (_)р(т-р)+1„, Minkowskian.
B.149)
Proof. We need the useful identity
Мр+1...дт щ..мр ^^
which can be quickly verified. Then we calculate
B.151)
Reshuffling the indices in the second e-tensor gives a factor (—)р(ш~р) and
using identity B.150) yields

50 Differential geometry, topology and fibre bundles
= (-)p(m-p)Wp. B.152)
For Minkowski space identity B.150) and hence relation B.152) gets an
extra minus sign. Q.E.D.
Note that in Euclidean space we have
, _ .... = dx1...dxm = *l
dx1*1 ... dx*1™ = e^-^dx1 ...dxm B.153)
and in Minkowski space
—?m...^(fa^" • • • dx^m ~ dx^dx1... dx = *1
dx>il...dx>im = -e^-^dx0^1...^-1 B.154)
since
Mi-Mm _ / m' ^or Euclidean space , .
mi-Mm — ^ _mi for Minkowski space. ^ '
Examples:
i) 3-dimensionaJ space, dim M = 3:
*dxidxjdxk = eijfe
*dxidxj = eijkdxh B.156)
* dx* = —eljkdx:idxk
* 1 = —eijkdx%dxPdxh = dx1dx2dx3
o.
vector space dual p-form basis dim Лр
Л3 *ш0 {йхЧх2(&3} 1
Л2 *wi {da;2^3,^3^1,^1^2} 3
Л1 *w2 {dx3,^1^!2} 3
Л° *ш3 {1} 1
B.157)
ii) 4-dimensionai Miakowski space, dim M = 4 (eom = 1):
*1 = dx°dxldx2dx3

Differential forms 51
*dx° = ^g^
= dx1dx2dx3
tdx1 = j^e
= -dx2dx°dx3 = dx°dx2dx3
* dx2 = -dx°dx1dx3
* dx3 = -dx°dx2dxl = dx°dx1dx2
* dx1dx2 = dx°dx3
*dx4x3 = ^g^g^^^
= dx2dx° = -dx°dx2
*dx°dx1dx3 = g^g^g^e^^
= -dx2
*dx°dx'idx3 = dx1
*dx4x2dx3 = dx°
* dx°dx1dx2dx3 = * * 1 = -1. B.158)
Coderivative or adjoint exterior derivative: For the exterior deriva-
derivative we can also construct an adjoint via an inner product of p-forms.
The inner product of two p-forms ap,/3p € Ap is denned by
(ар, /Зр) = / ap * Cp € R. B.159)
JM
(Note that for a product of forms we always mean a wedge product unless
we specify it differently.) How does it look in explicit coordinates?
(ap,/3p) = Jm plp]^_p)l /m
ein-v"Vp+l...Vmelil-^v^-Vmdx1 ...dxm B.160)
using the identity
^...^e1^-"'^,...^^1-^^1-"" = P'("» ~ p)!/3M1-Mp B.161)
we obtain
(ap,0p)= I «P*/3P = -5i f dVam...ltp(xH^-^(x). B.162)
JM V- J
If the forms contain matrices

52 Differential geometry, topology and fibre bundles
ap = ol%T\ pp = C$Ta, B.163)
with Ta some matrices, the generators of a group, then we have to take the
(normalized) trace
(ap,/3p) = f tr ap*Cp = i IdVal^^^-^ix). B.164)
Jm P- J
The inner product is certainly symmetric
(ap,/3p) = (/?p,aP) B.165)
since ap * fip — CP * ap and it is also positive definite if the manifold M is
Riemannian
(ap,ap)>0, B.166)
with equality only for ap — 0.
Now we consider the following inner product
= /
J
m
/ / _!*/3p). B.167)
/
Jm
The second term vanishes via Stokes' theorem B.128). It corresponds to an
integral over the boundary of the manifold and we assume a boundaryless
manifold dM — 0. Inserting the operation **-1 we have
(dap_i,/3p)= / ap_1*{-)P*-1d*pp. B.168)
Jm
Definition: The 6-operator, coderivative, or adjoint derivative ap-
applied on p-forms is defined by
S := (-)P*~4*
= (_)mP+m+! * d*, B.169)
where the second line follows by respecting the inverse star *-1 (recall equa-
equation B.149)). For a compact manifold without boundary this leads to the
following theorem.
Theorem:
(dap_i,/3p) = / ap_! *60P = (ap_b6/3p). B.170)
The coderivative lowers the degree of a form by one unit and 6 : Ap —> Лр-1
operates opposite to the exterior derivative d : Лр —¦ Лр+1.

Differential forms 53
Properties: We have
S = (-y*d* form odd B.171)
S = - * d * for m even B.172)
<52 = 0. B.173)
Instead of 6 it is also customary to use the symbol dt.
Laplacian: Finally we have to introduce the Laplacian, which is denned
by
Д = dS + 6d = (d + 6J. B.174)
It is a map A : Ap —> Ap from p-forms to p-forms. Indicating the degrees of
the forms we write
ApUJp = dp_i<5p_iwp + SpdpUJp. B.175)
So altogether we have the following operators acting on the vector spaces
Example: Let us consider the Laplacian when acting on a l-form in 3-
dimensional Euclidean space. We need
8ш = (-K1+3+1 * d * uiidx*
where we have used etjk?e-'k = 2дге; then
<*"=-?*шш'**- B-178)
On the other hand, we calculate
-)*d*^T
oxk
d d
??* B-179)

54 Differential geometry, topology and fibre bundles
where we have used eklj?i:>я = {—)ek%jelj = (~)(дке9гз — 9ks9%t)- Alto-
Altogether we find
Aw = (d6 + Sd)u = - fu, B.180)
which is the familiar Laplacian with a negative sign.
Property: The Laplacian commutes with
Hodge star *A = A *
derivative dA = Ad B.181)
coderivative 8 A = AS.
Proposition:
• A is a positive operator!
Proof. We take the inner product
(wp, Au)p) = (bJp,d8u)p) + (wp, 6du>p)
= Fup,6up) + (dLJp,dujp)>Q. Q.E.D. B.182)
This equation immediately implies the following theorem.
Theorem:
= 0 «=*> dup = 0
8шр = О. B.183)
Definitions: We call a p-form шр
harmonic if Дшр = О
closed ]fdujp = 0 B.184)
coclosed if 6wp = 0.
Furthermore: if there exists globally a form a such that
Шр = dap-1 then wp is exact
Up = 6ap+i then o»p is coexact. B.185)

Differential forms 55
Remark: Note that
(co-)exact if (co-)closed. B.186)
Poincare's lemma:
• Any closed form, dw — 0, can be expressed locally as an exact form,
w = da!
This is a very important statement which we use frequently in this book.
We devote a separate section to its discussion and proof (Section 7.3).
Example: An example of a closed form which is not exact is the following.
Let us consider the manifold M = R2 \ {0}, then the 1-form
и = uJidx\ B.187)
with wi = —j, иъ = ~2 and x1 = x, x2 = y, r2 = x2 + y2
is well denned on M. It is closed since
dw = -^%dxldx2 = 0. B.188)
дхг г2
But w is not exact. Exactness would mean that there is a function / such
that
u = df = d(arctan -). B.189)
However, f(x, y) is not continuous (hence not differentiable) on the whole
manifold M = R2 \ {0}. Therefore и) Ф df globally, u) is not exact.
It is instructive to study this example also in a complex notation (see
e.g. [Thirring 1992])
, , ., dz xdx + ydy + i(-ydx + xdy)
z = x + iy, dz = dx + idy, — = g '-.
B.190)
Then
-ydx + xdy dz
= ^2 = T
^ Rfid? BЛ91)
r z
and
w = WRe + io;im. B.192)
We have du> = 0, ui is closed, since du>Re = dw\m = 0. On the other hand,

56 Differential geometry, topology and fibre bundles
ш = u>Re + iujim = Re \-ilm — = — = dlnz. B.193)
z z z
But the logarithm In 2 is not denned on the whole manifold M = R2 \ {0},
hence и) ф d In z globally and u) is not exact.
On the other hand, if we cut the plane along the negative i-axis, so if
we consider the manifold M = R2 \ {A,y)\x < 0,у = 0}, there w = dlnz
is well defined, w is exact.
Trace, (anti-)commutator: Finally we define the trace, the commutator
and anticommutator of differential forms.
Let otv € Lie G <g> Лр, /3q € Lie G <g> A4 be forms
otp = —alil...IMpdx'*1 ...dx»* p-form B.194)
/3, = -iPvl...Vqdar*...dx?'' q-ioim, B.195)
which are matrix valued, we say Lie algebra valued aM1...Mp, CVl...vq € Lie G
<W..Mp = <...»T B-196)
P*...vq = ^...VT\ B-197)
with Ta being the generators (matrices) of a Lie group G. Equations B.194),
B.195) and the subsequent equations actually mean the tensor product, like
а = Та <8> аа G Lie G ® Ap, between the basis {Ta} of the Lie algebra and
the wedge products of the differential form. Then we define the trace of
forms by
tr apCq = -L tr (altl...rPPv1...u,)dx*1 ... dx»» dxVl ...dxv\ B.198)
where the trace is to be taken over the matrix product Т"ТЬ.
We define the commutator and anticommutator of forms similarly
by separating the (anti-)commutator of the matrices and the wedge product
of the forms
x^...dx^ B.199)
B.200)
Since the matrices Ta satisfy the Lie algebra

Homology and de Rham cohomology
57
Fig. 2.26. Space Xi is homeomorphic to a triangle with interior; space X2,
containing a hole, is deformable to a triangle without interior
[Га,Гь] = fabcTc,
where fabc are the antisymmetric structure constants, we have
1
\a я ] = fabcaa В j
B.201)
and
Rules:
B.202)
B.203)
ii)
iii)
iv)
v)
vi)
[ap,/3q] = ap/3q - (-
{ap,/3J = apflq + (-
ap/3a = -(
[aPl3g,uJr\ = ар[13д,шг} + (-)qr[ap,uJr}l3q
tr ap/3q = (-)w tr /3qap
tr[ap,/3g]=0.
, 0q] + {ap, /
B.204)

58 Differential geometry, topology and fibre bundles
For instance, the commutator of a l-form a € Л1 (any odd-form) with itself
does not vanish but it is
[a,a] = 2aa=:2a2. B.205)
This is an important result which we use permanently later on when for-
formulating quantum field theory with differential forms.
2.5 Homology and de Rham cohomology
In the homotopy section (Section 2.2) we classified the topological spaces
with the help of loops; loops containing a hole or loops not containing a
hole. Now we characterize the spaces with their topological invariants in a
different way by studying the boundaries of the spaces. This is the topic of
homology and cohomology; for literature we refer to [Singer, Thorpe 1967],
[Nash, Sen 1983], [Nakahara 1990], [Bott, Tu 1982].
2.5.1 Homology
Let us consider some region Xi of a topological space, say R2. It is homeo-
morphic to a triangle with interior (see Figure 2.26). On the other hand,
a region X2, which contains a hole, can be continuously deformed into a
triangle without interior (see Figure 2.26). In the first case the edges of the
triangle are the boundary of a connected space, whereas in the second case
the closed edges do not represent a boundary of any space. So in classi-
classifying spaces we can detect a hole in a space by searching for some closed
area which is not itself a boundary. Mathematically, we formulate this in a
general way by introducing so-called simplexes.
Simplex: Let vo,...,vp be (p + 1) distinct points in R*™. The set of
points—vectors—{t>o, • • •, vp} is termed geometrically independent if
the p vectors: v\ — vq, v-i — vq, ..., vp — vo are linearly independent.
For example, in R2 the set of vectors {vo,vi,vi} is geometrically inde-
independent iff v0, vi, V2 are not collinear.
A p-simplex crp is defined by
p p
ap = {x e Km\x = YsWuCi > O.^Tci = 1}, B.206)
t=0 t=0
where the vectors vq,...,vp are geometrically independent. The coeffi-
coefficients (со,... ,Cp) express the barycentric coordinates of x; for instance
A/3,1/3,1/3) for p = 2, the triangle. The index p represents the dimension
of the simplex. It is customary to use also the notation ap = [t>ot>it>2].
A p-simplex <rp is compact since it is closed and bounded in Rm.

Homology and de Rham cohomology
59
Fig. 2.27. 0-, 1-, 2-, 3-simplexes
Fig. 2.28. A 2-face
opposite to the 0-face [vo] of the simplex аг =
We have drawn examples of simplexes in Figure 2.27: cr°—a point, a1—a
line, a1—a triangle with interior, a3—a tetrahedron with interior, etc.
Face: Recall the barycentric coordinates c» > 0. If aii с» > О then the
corresponding set of points represents the interior of a simplex ap—an
open simplex denoted by ap = (v0viV2). If some Q = 0 then x describes
11 face of the simplex ap opposite to the vertex Vi.
Consider, for instance, the tetrahedron (see Figure 2.28). [vq] is a 0-face;
11 0-simplex has no faces, [fi 1*21*3] is a 2-face opposite to vq of the simplex
\voV\V2Vz]. So we have one 3-face, four 2-faces, six 1-faces and four 0-faces.

60
Differential geometry, topology and fibre bundles
Fig. 2.29. Oriented simplexes
Simplicial complex: A simplicial complex К is a finite set of simplexes
in some Rm satisfying
i) if crp e К then all faces of av are elements of K,
ii) if arp, aq € К then either av П aq = 0 or av П cr9 is a common face of
crp and a9.
The dimension of the complex К is given by the maximum dimension of
the simplexes.
Orientation: We can assign an orientation to a p-simplex by directing
the line segments. We denote an oriented p-simplex by av = (vo ... vp). In
Figure 2.29 we illustrate some examples:
(vovi) = -
= (v2v0vi)
B.207)
p-chain: Let К be a. fc-dimensional simplicial complex containing I orien-
oriented p-simplexes (so p = 0,1,..., k). The p-chain group of K, denoted by

Homology and de Rham cohomology
61
{VOV1V2) 92<U0UjU2>
Fig. 2.30. Oriented boundary of the oriented 2-simplex
CP(K), is a free Abelian group generated by the oriented p-simplexes of K.
An element, the p-chain, is expressed by
= Y] ciior? € CP(K) with Oj € Z.
B.208)
Boundary: The boundary of an oriented simplex can be defined in a very
obvious manner. Let us begin with a0, which has no boundary (a point is
boundaryless)
doar° = d0 (v0) = 0. B.209)
Next we consider a 1-simplex a1 =
the boundary is
, an oriented line segment. Then
B.210)
The minus sign is determined by the orientation.
Considering a 2-simplex a1 — (v0viV2), which is an oriented triangle
with interior, then its boundary represents the triangle without interior,
clearly it is boundaryless, and its orientation has been induced (see Figure
2.30). So we have
B.211)
(v2v0) + (vovi)
and

62 Differential geometry, topology and fibre bundles
д2(г>2г>о) + ^M^i)
+ (v0) - (v2) + (ui> - (uo> = 0.
B.212)
As a rule for the boundary we write a plus sign for the first omitted vertex,
a minus sign for the second omitted vertex, etc.
Now we generalize to a p-simplex. The boundary dpcrb of an oriented
p-simplex is a (p - l)-chain defined by
dv°V = ?(") Wi ¦¦¦vi.--vp), B.213)
i=0
where the symbolл means that Vi is omitted. Then the boundary of a
p-chain is
t
t=l
So the boundary operator dp represents a map dp : CP(K) —* CP-\{K)
from the p-chains into the (p — l)-chains of a complex. The sequence
0 Л Cm(K) % Cm-!{K) ^ ..A d(K) ^ C0(K) ^ 0 B.215)
is called a chain complex and the map i an inclusion map.
Lemma: The boundary operator is nilpotent
др-хдр = О. B.216)
This is intuitively clear since the boundary of a simplex—a point, line,
triangle, tetrahedron, ...—is boundaryless.
Proof. Generally, from our definition B.213) we can quickly prove
dp- idp(Tp = dp- idp(v0...Vp)

Homology and de Rham cohomology 63
---«i •••«*• ••«P>=0. Q.E.D.
B.217)
Definitions:
p-cycle: If cp € Cp (K) satisfies
дрср = О B.218)
cp is called a p-cycle. The set of p-cycles
ZP{K) = {cp e Cp{K)\dpcp = 0} = ker dp B.219)
represents the kernel of the boundary operator dp.
p-boundary: If there exists a (p + l)-chain bp+i € CP+\(K) such that
Cp = dp+1bp+1 B.220)
then Cp is a p-boundary. The set of p-boundaries
BP(K) = {cp € CpiK^Cp = dp+1bp+1} = image dp+1 B.221)
represents the image of the operator dp+i.
p-homology: Returning now to our task, mentioned at the beginning, to
detect a hole in space we have to eliminate the boundaries BP(K) contained
in ZP(K). So the elements of ZP(K) which are not boundaries are the desired
objects. Clearly we have BP(K) С ZP(K) since d2 = 0. The quotient group
HP(K)=ZP(K)/BP(K) B.222)
is called the p-homology group of the complex K.
Generalization: What we have discussed so far in Euclidean space Rm
and with integer chain coefficients can be generalized to arbitrary topological
spaces or manifolds and real chain coefficients. By choosing the vertices
v0 - @,0,...,0)
«i = A,0,...,0)
vp = @,0,..., 1) B.223)

64
Differential geometry, topology and fibre bundles
R2
6-
0 1 z2
Fig. 2.31. Mapping a standard p-simplex into a manifold
we define the standard p-simplex in Rp
B.224)
i=\
Next we map this (oriented) p-simplex into a manifold M, s : Др —> M (see
Figure 2.31), and we obtain the singular p-simplex in M
S? = s{A").
B.225)
The map s is assumed to be smooth but need not have an inverse (for this
reason the term singular is used).
A p-chain in M is a sum of these p-simplexes with R-coefficients
с =
with
R.
The boundary of the p-chain in M is
dc =
B.226)
B.227)
where dsp = s{dAp) clearly represents the map of the boundary of the stan-
standard simplex into the manifold. So dsp describes the geometrical boundary
of sp with an induced orientation. The p-chains in M generate the chain
group in M denoted by CP(M). The boundary operator д then defines a
map д : CP(M) —> Cp_i(M) and is obviously nilpotent
d2 = 0. B.228)

Homology and de Rham cohomology 65
As before we define the set of p-cycles
ZP(M,K) = {c € Cp(M)\dc = 0} = kerd B.229)
and the set of p-boundaries
BP(M, R) = {c € Cp(M)\c = db} = image д B.230)
and by taking the quotient (BP(M, R) С ZP(M, R) since d2 = 0) we obtain
the singular p-homology group of M
HP(M, R) = ZP(M, R)/ BP{M, R). B.231)
Remark:
с = 56 ^ 5c = 0. B.232)
In homology we are interested in those cycles which are not themselves
boundaries.
Triangulation of space: Take a simplicial complex K. We regard each
simplex as a subset of Rm, then the union of all simplexes is again a subset
of Rm and is called a polyhedron of K.
A topological space X which is homeomorphic to a polyhedron is said
to be triangulable. The polyhedron is called a triangulation of X which
is not at all unique.
Proposition:
• All compact manifolds are tn&ngulahlel
With the help of triangulations we can calculate the homology groups
of manifolds. Let M be a compact manifold (in general a topological space)
and К an arbitrary triangulation, then we have for the homology groups
HP(M)=HP(K), p = 0,1,2,.... B.233)
This is valid independently of the choice of the triangulation by virtue of
the following proposition.
Proposition:
• Homology groups are topological invariants!
This means the isomorphy
Яр(#0 ~ Hp{K2) B.234)

66 Differential geometry, topology and fibre bundles
if K\ and K2 are two different triangulations of M (in general K\ and
K2 may be triangulations of the homeomorphic manifolds M\ and M2,
respectively).
Example: To triangulate the circle S1, a 1-dimensional space, we have to
find a set of suitably joined 1-simplexes which is homeomorphic to S1. This
is the set
К = {(wo), («!>, (v2), (vovi), (ища), {v2v0)}. B.235)
Let us calculate the homology groups HP(K):
Since there is no 2-simplex in К the 1-simplexes are not boundaries and
we have
BX(K) = 0
= Z1(K)/B1(K) = Z1(K). B.236)
Considering a 1-cycle
z - a(vovi) +b(viv2) + c(v2v0) € Z^K) with a,b,c € Z B.237)
we demand
= о,
which is satisfied only for о = b =
Z,{K) = {a{(voVl)
and
TT ( T/-\
tl 1 \J\. J
Calculating next the 0-homology
ZQ(K) = C0(K) = {a(v0)
and
B0(K) = {adi (voVi) + bi
= {(c-a)K) +
: c. So we
1 + (^1^2)
= ZX[K)
we clearlj
+ b(vi) +
{a-b)(vi
find
+ (v2v0))\a € Z}
~Z.
r have
c(u2)|a,6,c€Z}
> + (Ь-с)(«2>|о,Ь,с
B.
B.
B.
.238)
.239)
.240)
B.241)
;} B
eZ}.
.242)
Next we define a surjective (onto) homomorphism (this is а тар
preserving the group structure, e.g. / : Gi —^ G% with f(x+y) = f(x) + f(y)
/ : Z0(K) - Z B.243)

Homology and de Rham cohomology 67
by
f(a(v0) + 6(ui) + c(v2)) = a + b + c B.244)
and we observe that
keTf = f-1{0)=B0(K) B.245)
since
/((c - a)(v0) + (a - b)(vi) + (b- c)(v2)) = (c - a) + (a - b) + {b - c) = 0.
Then we use the theorem of homomorphism:
we recall that
and we find
Z0(K)/ ker / ~ image /,
image / = Z
H0(K) = Z0(K)/B0(K) ~ Z.
B.246)
B.247)
B.248)
The homology groups of the circle S1 are, finally,
H0(Sl) = H0{K)~Z
tfiOS1) = Hi(K)~Z. B.249)
2.5.2 de Rham cohomology
In a certain sense, dual to the chains discussed before are the differential
forms that the physicists are more familiar with. So we find a dual group
to the homology group, this is the de Rham cohomology group. What is it
more precisely?
Definitions: Let us consider the space of all differential forms Л* B.101)
and we define the p-cocycles (closed forms)
Zp{M,R) = {u€ Ap\dw = 0} = kerd B.250)
and the p-coboundaries (exact forms)
BP(M,R) = W € Ap\lj = dp} = image d. B.251)
Clearly we have BP(M, R) С ZP(M, R) since d2 = 0, so the quotient of both
groups B.250) and B.251) makes sense. The p-de Rham cohomology
group of M is defined by
tfp(M,R)= ZP(M,R)/BP{M,K). B.252)

68 Differential geometry, topology and fibre bundles
Let be w e ZP(M, R) then [w] € HP(M, R) represents an equivalence class.
Two forms u/, a> € ZP(M, R) are equivalent or cohomologous if they differ
by an exact form:
a/
if a/ = w 4-
So in cohomology we are searching all closed forms which are not exact.
The sequence
О Л Л°(М) Н А\М) ^ ... dp-^ Ap(M) ^...d^1 Am(M) U 0 B.253)
is called the de Rham complex and г is again the inclusion map.
Duality: Duality between chains and differential forms is defined via an
inner product. Take a p-chain с € CP(M) and a p-form a> e ЛР(М) where
1 < p < m = dimM, an inner product of a p-chain and a p-form is
defined by the bilinear map
( , ) : CP(M) x ЛР(М) -> R B.254)
с,ии (c,w> =: Iw e R. B.255)
Jc
Then Stokes' theorem B.128) introduced in Section 2.4
fdu}=fu B.256)
Ус Jdc
can be written as an inner product
= (dc,w). B.257)
In this sense the boundary operator д and the exterior derivative d are
adjoint to each other.
Period: An inner product of a cycle (9c = 0) with a cocycle (dw — 0) is
called a period П
П(с,ш) = (с,ш) = LeR, B.258)
Jc
with
ceZp(M,R) and weZp(M,R). B.259)
The mapping П on
П : ZP(M, R) x Zp(M, R) -» R, B.260)
however, may be degenerate since

Homology and de Rham cohomology 69
П(с, ш) = 0, Уше Zp(M, R) B.261)
does not imply с = 0. Identity B.261) holds whenever с = 96
0 B.262)
since do» = 0. But the map П is nondegenerate when working on the
(co)homologies HP(M, R), HP(M, R). This leads us to de Rham's theorem.
de Rham's theorem: The mapping
П : HP(M, R) xHp(M, R)-R B.263)
с,ии П(с, w) = (с,w) = /we R, B.264)
Jc
with
ce#p(M,R) and w€#p(M,R) B.265)
is bilinear and nondegenerate. The spaces HP(M, R) and HP(M, R) are
vector spaces and finite dimensional and
dim#p(M,R) = dim#p(M,R). B.266)
Thus HP(M, R) is dual to HP(M, R) with respect to the inner product
П. Both spaces are isomorphic
HP(M, R) ~ HP(M,R) B.267)
and can be identified. Choosing a set of independent p-cycles which form a
basis for HP(M, R)
with г = 1,...,dimHP(M,R) B.268)
we find the dual basis—a set of independent p-cocycles
{ujj} with j = 1,..., dimtfp(M, R) B.269)
via the inner product U(ci, u>i), an invertible matrix. The dimension of the
(co)homology space is the p-th Betti number of the manifold M
bp(M) := dim#p(M,R) = dim#p(M,R) =: b"(M). B.270)
Euler characteristic: Let us introduce here the Euler characteristic
of a manifold M

70 Differential geometry, topology and fibre bundles
X(M) = (number of vertices of K)
— (number of edges of K) + (number of faces of K),
B.271)
where К is a polyhedron homeomorphic to M. x(M) represents a pure
topological quantity depending only on the kind of manifold.
On the other hand, the Euler characteristic can be expressed by the
alternating sum of the Betti numbers
X(M) = Y,{-)pV. B-272)
p=0
This is a remarkable relation. We can calculate the topological Euler cha-
characteristic by studying the analytical de Rham cohomology. We shall meet
such connections between topology and analysis again when we discuss the
index theorems (see Chapter 11).
Poincare duality: There exists another type of duality—Poincare
duality—when considering the inner product of p-forms introduced in Sec-
Section 2.4 (equation B.159))
(<*p,0p)= f ap*/3peH. B.273)
Jm
The dual form */3p € Am~p is clearly a (m — p)-form. So it makes sense
to define an inner product of the cohomologies Hp and Цт~р by
the bilinear map (note that we consider only compact orientable manifolds
without boundaries)
[, ] : HP(M,R) x Hm-p(M,R) — R B.274)
a,C >-> [a,0\ =: I aCe R, B.275)
Jm
where
aeHp(M,R) and /? e #m~p(M,R). B.276)
This determines the Poincare duality between HP(M, R) and Hm~p(M,R).
Both spaces are isomorphic
HP(M, R) ~ Hm-p(M, R) B.277)
and the Betti numbers are related by
b>> = bm~p. B.278)

Homology and de Rham cohomology 71
Consequently the Euler characteristic of an odd-dimensional manifold va-
vanishes, x(-Modd) = 0.
Kiinneth formula: If we consider a product manifold M = M\ x M2 then
the cohomology can be decomposed into a sum of products—the Kunneth
formula
Hk(M,R)= ф [HP(M1,R)®H"(M1,R)} B.279)
P+q=k
(<g> denotes the tensor product and © the direct sum). Hence the Betti
numbers are related by
bk(M)= ]T V(M1)bq(M2), B.280)
p+g=k
implying for the Euler characteristics the formula
X(M) = X(M1)x(M2). B.281)
Harmonic forms and de Rham cohomology: There is an interesting
connection between the de Rham cohomology and the harmonic forms. The
starting point is a theorem which decomposes any p-form into a closed-,
coclosed- and harmonic form.
Hodge decomposition theorem: Let M be a compact orientable ma-
manifold without boundary, then any p-form wp e Лр can be uniquely decom-
decomposed into
wp = dap-г + 6{3P+1 + 7P, B.282)
with ap-i e Л*, /3p+i € Лр+1 and Ajp = 0.
Remark: Let us choose dwp = 0, ujp closed or ujp e HP(M, R), we have
0 = (dup, /?p+i) = (сК/Зр+i, /3p+i) = (ё/Зр+i, 6Cp+i), B.283)
which implies 6/3p+i = 0 and we get
wp = do;p_i+7p. B.284)
If we choose 6u)p = 0, ljp coclosed, we analogously obtain dap_i = 0 and
up = 6Cp+i + 7P. B.285)
If the form is harmonic, Awp = 0, then clearly a>p = 7P.

72 Differential geometry, topology and fibre bundles
So it is the harmonic component -fp which determines the cohomology
class of the form up. In fact, there exists an isomorphism between the set
of harmonic p-forms
Harmp(M) = {7p e Лр|Д7р = 0}
and the p-cohomology
Harmp(M)~Hp(M,R).
In particular for the space dimensions we have
dimHarmp(M) = dimHp(M,R)
or
dim ker Ap = bp
B.286)
B.287)
B.288)
B.289)
so that the Euler characteristic is given by
m
= ]T(-)pdiinkerAp. B.290)
So the topological quantity x(M) is determined by solving an analytical pro-
problem, the eigenvalues of the Laplacian Др. In index theory this connection
corresponds to the index of the de Rham complex.
Examples:
i) de Rham cohomology of the Euclidean space, M = Rn.
In Rn all closed forms are also exact except the solutions to df = 0,
the real constants / = const, e R. Therefore we have
#p(Rn,R) = 0, dim#p(Rn,R) = 0, l<p<n
tf°(Rn,R) = R, dimF°(R",R) = 1. B.291)
The de Rham cohomology of the Euclidean space is trivial. This is the
cohomology version of Poincare's lemma.
ii) de Rham cohomology of the sphere, M — Sn.
The only nonvanishing cohomologies for Sn are H° and Hn. H° con-
contains the real constant functions and Hn the constants times the vo-
volume element, an n-form. The result is
) = R, dim#°(Sn,R) = 1,
ЯрEп, R) = 0, dim #p(Sn, R) = 0, 1 < p < n, B.292)
Hn(Sn,R) = R, d\mHn{Sn,R) = 1.

Flow, Lie derivative and Lie group 73
iii) de Rham cohomology of the torus, M = T2 = S1 x S1.
With the knowledge of the sphere-cohomology we can calculate the
torus-cohomology by using the Ktinneth formula B.279)
H°(T2 = S1 xS'.R) =
= R©R
= S1xS1,R) = R®R = R B.293)
and
dimH°(T2 = S1 xS\R) = 1
dimF1(T2 = S1 xS\R) = 2
dim#2(T2 = S1 x S\R) = 1. B.294)
2.p Flow, Lie derivative and Lie group
Vector fields on a manifold generate flows. The change of a vector field,
generally of a tensor field, along such a flow is described by the Lie derivative.
This important mathematical conception is widely used in physics and we
shall discuss it now. For further literature we refer to [Choquet-Bruhat,
DeWitt-Morette 1982], [Nakahara 1990], [Thirring 1992], [Warner 1983].
2.6.1 Differential map
In the following we deal with mappings from one manifold to another. How
do the quantities defined on the manifold, the vector fields, transform under
such a mapping?
Consider a smooth map / : M —> N. It induces the differential map
/* or tangent map
B.295)
in the tangent spaces of the manifolds (see Figure 2.32).
Let us find an explicit expression for /». We denote by .F(iV) the set of
smooth functions on the manifold iV and we choose some function д е .F(iV);
then gf € .F(M). Concentrating on the chart (V, ip) in N the coordinate
representation of g is determined by the map дф~г : Rn —> R (see Figure
2.33).
Definition: We choose a vector X e TP(M) and we define the differen-
differential map or tangent map f*X € TP(N) by
f.X[g) = X[gf). B.296)

74 Differential geometry, topology and fibre bundles
T*p(M)
• /*<»
r
• (o
Tp(M)
Fig. 2.32. The map / together with the induced differential map /* and
the pullback /*

Flow, Lie derivative and Lie group
75
Fig. 2.33. The function g : iV —» R and its coordinate representation дф х
In terms of the charts (U, <p) on M and (V, ф) on iV this means
^ 1}^), B.297)
where x = f(p) and у = ф(/(р)) are the coordinates (or homeomorphisms)
of the charts. Furthermore we have
L](x) B.298)
and the choice g = yu provides the components of the tangent map
B.299)
Thus the two components X^ and f*Xv are related by the Jacobian of the
map /.
The composite map ft/ with f : M —> N and ft : N —> P clearly gives
= ft»/». B.300)
Example: Let us take two manifolds M and iV with coordinates (x1, x2)
and (у1,у2,у3) and the tangent vector

76 Differential geometry, topology and fibre bundles
* = 31ГТ+4А B-301)
дх1 дх2 v '
on M. We consider the map f : M —> N with the coordinate representation
(у\у2,у3) = (х\х2,A-г2I/2) and r2 = (z1J + (z2J. B.302)
Then the differential map or tangent map is
fX - frA-^A+FMA+I,?l
J* J* dyv дх» ду1 дх» ду2 дх» ду3
_ 3А+4 » -2»1±^» B.303)
ду1 ду2 у3 ду3
according to equation B.299).
2.6.2 Pullback
Certainly we also may consider the dual space—the cotangent space. Then
the smooth map f : M —> N induces a pullback map
/*:Т;(р)(ЛГ)^Г;(М) B.304)
in the cotangent spaces of the manifolds (see Figure 2.32). Whereas /» pu-
pushes forward the vector fields (in the direction of /), /* pulls the differential
forms backward (in opposite direction of /).
Definition: Choosing a vector X = Xv-^ € Tp(M) and a 1-form и =
bj^dy» e T}(p)(N) we define the pullback /*w = f*u^dx» e T;(M) by
(Гш,Х) = {и>,/.Х) B.305)
or in the notation of equations B.90) and B.116)
f*u}{X)=u){J*X). B.306)
We obtain the components by evaluating the inner products
(Гч,,Х) = Гч,,Ах)Х»(х) B.307)
and (recall equation B.299))
^. B.308)
Comparing both results B.307) and B.308) implies for the components
of the pullback

Flow, Lie derivative and Lie group 77
^ B-309)
Again the two components ши and /*wM are related by the Jacobian of the
map /. In local coordinates / is defined by yv{x^) with ц = l,...,m =
dimM and v = 1,... , n = dimiV.
It is straightforward to generalize the pullback /* for p-forms
^... dy"» e Ap(N). B.310)
Definition: The map
B.311)
is denned by (in the notation of equation B.121))
f*u(Xu ...,XP)= w(/.JCi,..., UXP). B.312)
Explicitly we have
fu> = ^/•uw./.p'k'4 • • •dx>Mp e ЛР(М). B-313)
with
B.314)
Properties: The pullback has the following properties
i) d(/*w) - /«dw,
ii) (ft/)* = /*ft* with f :M ^N and h:N ^ P,
iii) /*(aP/3g) = /*ap/*/3, with ap e A"(^), /3, €
B.315)
Remark: Let us return to our discussion of the inner product of a p-chain
с = 5Z«a«sf e CP(M) with a p-form и e AP(M) on a manifold M (recall
equations B.226) and B.255) of the previous section)
[
Jc
= [ш =
Jc
Studying the integration further we consider the integral of a p-form ш over
a p-simplex sp in M. By virtue of the pullback it is

78 Differential geometry, topology and fibre bundles
o(t,x)
Fig. 2.34. The integral curve a(t, x) of a vector field X
[ uj= f s*w€R, B.317)
JgP JAP
where s : Др —> M describes a smooth map such that sp — s(Ap) (recall
equations B.224), B.225)). The pullback s*w then represents a p-form on
Rp and Др С Rp denotes the p-simplex in Rp—the domain of integration.
Thus the r.h.s. of equation B.317) expresses the usual p-fold integration in
Rp giving a real number.
Example: Choose a map f : U —+ I x U from some area U to a cylinder
manifold I x U given by the coordinates
(t,x) = (x2,x) with (t,x) e I x U, x e U. B.318)
Consider a 1-form on the cylinder
w = tdx + xdt e A1 (I x U) B.319)
then its pullback is the following 1-form on the area
fw = x2dx + x ¦ 2xdx = 3x2dx € A\U). B.320)
2.6.3 Flow
A vector field on a manifold M describes, quite naturally, a flow in M. We
consider the integral curve <r(t, x) of a vector field X = X^ д/дх^ €
TX(M)—a curve whose tangent vector is X—passing through a; at a time
t — 0 (see Figure 2.34). In a given chart (U, <p) we have
= X»{a{ty x)) with ff"@,x) = i". B.321)
at

Flow, Lie derivative and Lie group 79
o-v(t,x) denote the local coordinates which are actually the components of
the homeomorphism ip{x{t)) and for simplicity we use x to indicate both
the point in M and its coordinate in Rm. Such an integral curve a(t,x),
representing a map a : R x M —+ M, is called a flow generated by the
vector field X.
Theorem: Let X € T(M) be a vector field on the manifold M. For any
point x e M there exists an integral curve of X, a flow a : R x M —> M,
such that a(t,x) is a solution of the differential equation B.321).
Lemma: A flow satisfies the group property
a(t,a(s,x)) = a(t + s,x). B.322)
This follows from the differential equation B.321).
One-parameter group: When keeping the parameter t fixed we denote
the flow by
a(t,x)=at(x). B.323)
So at is a diffeomorphic map: M —+ M and it represents the commutative
one-parameter group:
i) Gt(aa(x)) = at+s(x) or at-as^ at+a,
ii) Go = 1 unit element,
iii) o^1 = o-t inverse element. B.324)
Choosing the parameter t infinitesimal then the differential equation
B.321) determines the infinitesimal flow
o?(x) = x» + tX^ix) B.325)
and X is called the infinitesimal generator of the flow group at.
Remark: For a given vector field X the flow a can be expressed by the
exponentiation of X
<7"(t, x) = exp(tX) x». B.326)
It fulfils the properties:

80
Differential geometry, topology and fibre bundles
X(z)
Y(x')
Fig. 2.35. Change of the vector Y along the flow a of X. The vector Y(x')
must be moved back to the point x by the inverse differential map j(a-t)*
to be compared with the vector Y(x)
i) a@,x)=x,
ii) — a(t,x) = XexpCtX) x,
at
iii) a(t,a(s,x)) = a(t + s,x).
We can verify the exponentiation as follows
t=0
- exp(t— 1<т"
= exp(tX) x»,
t=o
B.327)
B.328)
where for the last step we used the differential equation B.321).
Example: On the plane M = R2 the vector field
v(r\ - -r2— + x1— x = (r1 t2) B 3291
generates the flow
<j(t,x) = (a1,a2) = (x1coet - x2sint,a;1sint + x2cost) B.330)

Flow, Lie derivative and Lie group 81
which represents a circle through the point x:
(a1J + (a2J = (x1J + (x2J = r2. B.331)
2.6.4 Lie derivative of a vector fleld
Let X and Y be two vector fields on a manifold M. We are interested now
in the change of the vector field Y along the flow a(t, x) which is generated
by the vector field X. So we have to compare the vector Y at the point x
with У at a nearby point x' = at(x). However, the vectors Y(x) e TX(M)
and Y(x1) e TX>(M) belong to different tangent spaces! Therefore we first
have to map Y(x') to TX(M). This can be done by the inverse differential
map
(аГ1). - (<7_t). : Tx,=at{x)(M) - Ta-iat{x){M) = TX(M) B.332)
(recall definition 2.295). Then we can take the difference of the two vectors
(a_t)»y(fft(a;)) € TX(M) and Y(x) ? TX(M) which are both lying in the
tangent space TX(M) (see Figure 2.35).
Definition: The Lie derivative of a vector field Y along the flow
of X we define by
CXY= Urn ±[(a-t).Y(at(x)) - Y(x)]. B.333)
Alternatively we also could consider the limits
CXY = lim|[y(a:)-(<7t).l>-t(aO]
= bm\[Y{Gt{x))-{at)*Y{x)}. B.334)
Now we perform the explicit calculation. On a chart (U, ip) with coor-
coordinates x we have the two vector fields X = Х^(х) д/дх^ and Y =
У (ж) д/дх» and the flow a?(x) = x» + tX^x). The vector field Y at
the point x' = at(x) is
B.335)
Y(at(x)) = yfotoj
'дхх
Remembering the transformation property B.299) of a vector under a dif-
differential map we obtain

82 Differential geometry, topology and fibre bundles
(a-t).Y(at(x)) = {а-±).У"Ых))^р B.336)
Then we find for the Lie derivative B.333)
Lie bracket: Next we introduce the Lie bracket [X, Y] defined by
[X,Y}f = X[Y[f}}-Y[X[f]}, B.338)
with / € T(M) some smooth function. Explicitly in components we get
or д
[X,Y] = [X,YY — , B.340)
with
РМ^Х'^^*"' B-341)
which again expresses a vector field (note: neither XY nor YX alone is a
vector field).
So the Lie derivative of the vector field Y along X
CXY = [X>Y) B.342)
is equal to the Lie bracket of X and Y.
Properties of the Lie bracket: The following properties are easy tc
verify
i) [X, aYi + bY2] = a[X, Yi] + b[X, Y2]
[aXi + bX2,Y) = a[XuY) + b[X2,Y]
with a, b some constants,
ii) [X,Y] = -\Y,X] antisymmetry,
iii) \[XtY],Z} + [[У,Z},X] + \\Z,X],Y] = 0 Jacobi identity,

Flow, Lie derivative and Lie group
83
4*)
Fig. 2.36. Geometrically the Lie bracket [X, Y] describes the noncommu-
tativity of the two flows at and r.
at*(o(at(x))
X(x)
ffl(x')
3f=at(x)
Fig. 2.37. Change of a p-form ш along the flow a of X. w(x') must be
pulled back to the point x via <7t* to be compared with uj(x)
iv) Cx fY = [X, fY] = f[X,Y] + X[f]Y
CfX Y = [fX, Y] = f[X,Y] - Y[f]X
with / a smooth function, / e !F(M),
v) MX, Y) = [UX, /.У] with map / : M
N.
B.343)
Geometry: The Lie bracket describes geometrically the noncommutati-
vity of two flows. Let us consider the flow Ut{x) generated by the vector
field X and the flow тд(х) generated by Y. Moving first along at and then
along Ts and vice versa (see Figure 2.36) we find for the difference of the
coordinates the Lie bracket

84 Differential geometry, topology and fibre bundles
х)) - <т?(тв(х)) = st[X, Y\». B.344)
So the parallelogram in Figure 2.36 does not close—the two flows at and ts
do not commute.
2.6.5 Lie derivative of a differential form
Now we consider a differential form ш(х) е Л?(М) and study its change
along a flow a(t, x). When comparing the differential forms at two nearby
points x and x' = at(x) they belong to different spaces. Therefore we have
to pull back
B.345)
the flowed differential form a*uj(at(x)) in order to be able to take the diffe-
difference with w(x) (see Figure 2.37).
Definition: We define the Lie derivative of a p-form li along the
flow of a vector field X by
Cx и = Urn -t\a*Mat{x)) - u(x)]. B.346)
The explicit calculation gives
a*tuj{at{x)) =
r\
B.347)
so that we find for the Lie derivative of the p-form ш along the vector
field X the result
Cx ш = [Х"дии!^..ф
\ , B.348)
p!
with dv = d/dxv.

Flow, Lie derivative and Lie group 85
For a 1-form w e Л1 we have
Cx w = [Г^ 4- w^X^dx». B.349)
For a 0-form ш = f G Л0, a function, the Lie derivative of a function
along the vector field X is
Cxf = limQ-t[f(at(x))-f(x)}
B.350)
and corresponds to the usual directional derivative along X.
Tensors: Considering next the tensor product of a covariant and con-
travariant vector or generally the tensor product of tensor fields t\, *2 of
arbitrary type, then the Lie derivative satisfies the Leibniz rule
® t2) = {Cx h) ®t2 + tx®Cx t2. B.351)
From this rule and from the previous results for vector fields and p-forms
B.337), B.348) we deduce the following formula for the Lie derivative of
a tensor field T^;;;(x) along the flow of a vector field X
LXT%;:: = XvdvT$;;;{x)
- dvXaTp-(x) - all upper indices
+ T?~{x)df}Xv + all lower indices. B.352)
Introducing the compact notation
matrix components
T «- T$;;;(x)
ЭХ «-¦ dvXa
Xd = Xvdv
[ЭХ,Т] = dvXaT%-;(x) + all upper indices
- T°;- {x)df}Xv - all lower indices B.353)
formula B.352) is re-expressed by
СхТ = ХдТ- [дХ, Т]. B.354)

86 Differential geometry, topology and fibre bundles
Tensor-valued p-forms: If, on the other hand, the tensor field also re-
represents a differential form
Г - 7?;- = ±Т?:-^р(х)с1х^ ... dx^ e Л", B.355)
which we need, for instance, in gravitation (see Chapter 12) then we obtain
for the Lie derivative of a tensor-valued p-form Tj?\" along X the
formula
— dvXaTp;;^ltlp(x) — all upper indices
+ V.:;;til...tip(x)d[,Xv + all lower indices
+ ^?.",V.../»1,(aO0/ii*1' + a11 form indices]
dx...dx B.356)
pi
and rewriting the sum of the form index contractions we have
CX 7$;;; = [Х»д„Т?::^р(х) + р 7?:;„/4я...^(хHМ1Х"
- диХаТ?;-^ фр(х) - all upper indices
v + all lower indices]
— dx ...dx^. B.357)
p\
When studying the first two terms we find that they correspond to a combi-
combination of interior product—exterior derivative for a p-form. We shall prove
the result below
(ixd + dix)T = [Xvda^1..^{x
—dx^1 ...da;"" B.358)
p!
which is valid for a p-form T e Лр, Т being tensor-valued or not, so that
expression B.357) can be reformulated in a compact way
Cx Г = (ixd + dix)T- [dX,T]. B.359)
For a scalar valued p-form ш € Лр, the case we usually consider, the com-
commutator clearly vanishes and we get the well-known result
B.360)

Flow, Lie derivative and Lie group 87
Formula: The Lie derivative satisfies the formula:
a) Cx = ixd + dix B.361)
b) Cx = ixd + dix - [dX, ) B.362)
a) when applied to a scalar valued p-form,
b) when applied to a tensor valued p-form.
Proof. Now we prove equation B.358).
ix du = [Xvdvuilil...llp - pXvdIXlwvlla...IXp)-dx^x ... Ac"'. B.363)
Hence we get
1
lix)u — \Xvdvu>^ ^p +pWj/^j...црдщХ"]—dx^1 ...dxIJ'p, B.364)
which is also true for a tensor-valued p-form since ix and d affect only the
form indices; Q.E.D.
Properties: The Lie derivative satisfies the following properties
i) [?x,ix] = 0
ii) [Cx,d\=0
iii) [Cx,iy] =i[x,Y]
iv)
v) (,) ()
for ш € Л1 being a 1-form,
vi)
P+i
t=l
B.365)
for ш ? Ap being a p-form, and the symbolл means that the corre-
corresponding vector field has been omitted.
Proof. Properties i) and ii) are obvious in the light of formula B.361).
Property iii): It is sufficient to check on 0-forms (which is trivial) and on

88 Differential geometry, topology and fibre bundles
1-forms; let w = w^dx^ € Л1 be a 1-form then we have
[?x,iY}u> = Ffl^yw.) - Г
(
= wu[X,Y]v = i[XiY]W. Q.E.D. B.366)
Property iv) follows from property v) which we rewrite
i = ?xiYw ~ ?Yixu — Цх,У]ш- B.367)
We choose ш — df and use formulae B.361) together with ix f = 0, then
we have
= 0
(CXCY-CYCX-C[xx])f = 0. B.368)
Since the Lie derivative commutes with d equation B.368) is valid for 1-
forms and so for all forms. Q.E.D.
Property v):
CX w(Y) - CY ш(Х) - u([X,Y])
). Q.E.D. B.369)
Properties v) and vi) are equivalent to equations B.108) and B.113) since
Cx = X when applied to a 0-form, a scalar function.
2.6.6 Lie group and Lie algebra
Lie groups and algebras play an important role not only in mathematics—
we shall need them in the geometry of fibre bundles—but also in physics.
In fact, we mainly work with Lie groups in this book. Let us discuss here
some basic features.
Lie group: A Lie group G is a differentiable manifold with a group struc-
structure such that the following group operations are differentiable (smooth):
i) G xG -»G given by E1,52) >-+ 9i92 € G,
ii) G —> G given by g ^ g~x.
The dimension of a Lie group G is determined by the dimension of G as a
manifold. The unit element we denote by e.
An important property of Lie groups is stated in the following proposi-
proposition.

Flow, Lie derivative and Lie group 89
Proposition:
• A closed subgroup H of a Lie group G is again a Lie subgroup!
Examples: U(N), SU(N), GL(m, R), SO(m), ....
Definitions: We need the following maps (diffeomorphisms):
Rg : G -* G
Rgh = hg right-translation B.370)
Lg : G -> G
Lgh = gh left-translation B.371)
with g,h € G.
These maps induce differential maps in tangent space
Rg*:Th{G) -> Thg{G) B.372)
Lg.:Th(G) -> Tgh{G). B.373)
Let X be a vector field on a Lie group G, then X remains invariant if
Rgt,X\h = X\hg right-invariant vector field B.374)
Lgt,X\h = X\gh left-invariant vector field. B.375)
A vector V € Te(G) generates a unique left-invariant vector field on G
by
Xv\g = Lg.V. B.376)
Indeed we have
Xv\gh = LghtV = {LgLh)tV
= Lgt,Lht,V=Lg*Xv\h. B.377)
We denote the set of all left-invariant vector fields on G by Q. The map
Te(G) —> G given ЬуУн Xv is an isomorphism, Te(G) ~ Q, hence dim? =
dim Te(G) =dimG.
We also can define a Lie bracket [ , ] on G. Let X, Y € Q be two
left-invariant vector fields on G, then we find
Lg.[X,Y}\h = [Lg.X\h,Lg.Y\h] = [X,Y}\gh. B.378)
So the Lie bracket [X, Y] is again a left-invariant vector field [X, Y] € Q; we
say Q is closed under the Lie bracket.

90 Differential geometry, topology and fibre bundles
Lie algebra: The set of all left-invariant vector fields Q with a Lie bracket
[ ) ] : Q x G —> Q is called the Lie algebra of the Lie group G. Prom now
on we use the notation Lie G = Q.
One-parameter subgroup: Let us next consider a map фу : Te (G) —> G
given by
0v(*)=exp(W), B.379)
with V € Te(G) and t € R. <l>v(t) represents a commutative one-
parameter subgroup of G generated by Xy = Lg»V. This means (j>v{t)
satisfies the properties
ii) 0@) = e, 0-1(*) = 0(-t). B.380)
At t = 1 it is conventionally named an exponential map
0v(l)=expV. B.381)
For a matrix group, say G = GL(m, R), we have
d>v(t) = expftV) = l + tV + ... + —Vn + .... B.382)
n\
Flow: The curve
at(g)=gexp{tV) B.383)
represents a flow through g € G. Thus we have
d(Tt{g)
dt
t=o
dt
B.384)
t=o
In the case of a matrix group G we can use expansion B.382) and we get
Xv\g = Lg.V = gV. B.385)
The flow B.383) can be also interpreted as a right-translation of the
group G
°t = ReMtv) B.386)
which gives at{g) = gexp(tV).
Basis: Let us choose a basis for the Lie algebra Lie G
{Xa} o = l,...,fc; fc = dimLie G = dimG,
then the commutator of these vector fields satisfies the relation

Flow, Lie derivative and Lie group 91
fabcXc, B.387)
where fabc express the totally antisymmetric structure constants of the
Lie group/algebra. We say that the commutator closes. The structure con-
constants obey the familiar Jacobi identity (see e.g. equation C.289) in Section
3.5.1).
We also can introduce a dual basis {ва} of the dual Lie algebra Q* =
*Lie G via the inner product
ea-Xb = <5V B.388)
The set {0a} is the basis for the left-invariant 1-forms on G.
By defining next wedge products
еаеь = еа л еь = ea ® eb - eb <g> ea
еаеьес = еа л еь л ес = ea <g> eb <g> ec + -... B.389)
(etc.) one can construct p-forms on G.
Analogously to vectors a p-form ш € AP(G) on the Lie group G remains
invariant if
L*w| = Hs_ih left-invariant p-form B.390)
R*u)\ = u\h i right-invariant p-form. B.391)
The dual basis reveals a very peculiar property. It satisfies the following
theorem.
Theorem: Maurer—Cartan structure equation on the Lie group
dea = -i/abCebec. B.392)
Proof. Using relation B.108)
du(X, Y) = Хш(Х) - Yu{X) - U([X, Y}) B.393)
we calculate the l.h.s.
d&a(xb, xe) = xb ea ¦ xc - xc вахь-еа- [хь, xc]
= Xb6ac - Xc6ab - fbcd Qa ¦ Xd
= -Fbc, B.394)
using (recall equation B.111))
е69г(Хь, Xe) = 6bb6\ - 6bc6% B.395)

92 Differential geometry, topology and fibre bundles
we calculate the r.h.s.
-\fab-c еьес(хь, xe) = -\fa-bs F\6ec - бъсб%) = -Ac Q.e.d.
B.396)
Definition: Next we define the Maurer—Cartan form on the Lie
group G
e = va®ea, B.397)
where
{Va} is the basis of Te(G)
{Qa} is the dual basis of T*(G). B.398)
So the Maurer-Cartan form 9 is a Lie algebra valued 1-form where the
tangent vector X € Tg(G) is moved back to the unit
9:Xh Lg-uX = L~}X € Te(G). B.399)
Then we clearly have for the derivative
dB = Va ® йва B.400)
and for the commutator
[в, 9] = [Va, Vb] ® 9a9b = 299. B.401)
Multiplying equation B.392) by Va® and using
[Vb,Vc] = fabcVa B.402)
we obtain the Maurer—Cartan structure equation in terms of 9
d9 = -i[9,9] = -92. B.403)
Remark: The Maurer-Cartan form on the Lie group has a very import-
important correspondence in physics. It will be identified with the Faddeev-Popov
ghost in a non-Abelian gauge theory. We recover the Maurer-Cartan struc-
structure equation in the gauge transformation procedure of Becchi, Rouet and
Stora. This we will discuss extensively in Sections 3.6, 8.1 and 8.3.2.
Lie group action on manifold: The left-action of a Lie group G on a
manifold M is a differentiable map a : G x M —*¦ M such that

Flow, Lie derivative and Lie group
i) cr(e,p) = p Vp e M, e e G,
ii) <T(ffi,<7(ff2,P)) = <Kffiff2iP),
The usual notation will be
with
i) ep = p,
93
B.404)
B.405)
B.406)
Example: Take for example the group G = GL(m, R) and the manifold
M = Rm. Let be A € GL(m, R) and x e Rm, then the action of GL(m, R)
on Rm
a(A,x) = Ax B.407)
corresponds to the standard matrix action on a vector.
Definition: Analogously we define the right-action of G on M by
<r(9,P)=P9 B.408)
with
i) pe = p,
ii) {P9i)92 =
B.409)
Orbit: The group action shifts a given point p e M of the manifold to
another. The orbit through p under the action a is defined by
Gp = {a{g,p)\g € G, fixed p e M}.
B.410)
Induced vector field: Let a Lie group G act on a manifold M from
the left: (g, x) и-» до. Then a left-invariant vector field Xy generated by
V e Te(G) induces a vector field in M. The induced vector field is
_
Jt
t=o
a{t,x)
B.411)
t=o
together with the flow in M

94 Differential geometry, topology and fibre bundles
a(t, x) = exp(tV)z. B.412)
Example: Discussing again our example from Section 2.6.3 we choose the
Lie group G — 50B) acting on the manifold M = R2. For the element V
of the Lie algebra
0 , B-413)
we find immediately from the series B.382)
cos t — sin t
This gives the flow through x e R2
,. . / cost -sint \ ( x\ \
(r(t,x) = I . . *
4 ' ' \ sint cost J \ x2 J
= (я1 cost - x2 sint,x1 sint + x2 cost) B.415)
together with the induced vector field
t=o
Adjoint representation: The following homomorphism is called an ad-
adjoint representation of G
adg-.G -*• G
h .-> ghg-1, g,heG. B.417)
The induced map in tangent space is
adg.:Th(G)-+Tghg-l(G). B.418)
If we restrict this map to the unit of G : h —> e we obtain the adjoint map
(recall the isomorphism Te(G) ~ Lie G)
Adg:Te{G) -* Te{G) or LieG->LieG
V н* gVg-\ geG,Ve Te(G). B.419)
For a matrix group G we just have matrix operations. Let
exp(tV), V e Te(G), be the one-parameter subgroup of G, then the adjoint
representation is
= gexp{tV)g-x = exp(tgVg-1) B.420)

Fibre bundles 95
and we verify the adjoint map by
»* -1
t=0
-1
adg<j)v{t) = —
exp(tgVg *)
t=o
= 9V9-1. B.421)
2.7 Fibre bundles
Fibre bundles are of increasing interest in theoretical physics. They are
the proper modern mathematical framework for discussing physical theories
such as gauge theories or gravitation. We have already considered a special
case—the tangent bundle. The tangent space TP(M) of a manifold M re-
represents a fibre which—when 'glued' together at all points p—establishes
this bundle. Another important bundle we meet in physics is the principal
bundle.
The essential quantities of a bundle are the fibres. The way the fibres
twist—when moving around in space-time—reveals the topological struc-
structure of the physical theory and determines properties like the magnetic
monopole charge, the instanton number or, what interests us most, the an-
anomalous content of a theory.
In this section we shall give an introduction into some basic ideas of fibre
bundles; for further literature we refer to [Choquet-Bruhat, DeWitt-Morette
1982], [Nash, Sen 1983], [Nakahara 1990], [Isham 1989], [Trautman 1984],
[Daniel, Viallet 1980], [Coquereaux, Jadczyk 1988], [Mickelsson 1989].
2.7.1 Bundle set-up
Fibre bundle: We denote a fibre bundle by (E, П, X, F, G). It consists
of
i) a topological space E—the total space,
ii) a topological space X—the base space,
iii) a surjection called the projection П:?->Х,
iv) a topological space F—the standard fibre. It is homeomorphic to
all inverse images П~1(а;) = Fx with x € X; the Fx are called fibres,
v) a group G—the structure group—of homeomorphisms of the fibre
F,
vi) a covering of open sets {Ua} of X together with homeomorphisms
Фа
such that
Пф^1{х,/) = х with x € Ua, f € F.

96 Differential geometry, topology and fibre bundles
The homeomorphism фа maps U~l(Ua) С Е—the union of fibres
above Ua—onto the direct product space Uax F and is called a local
trivialization.
We have illustrated a fibre bundle in Figure 2.38 with help of a Mobius
strip. There the fibre F is a line segment and the Mobius strip represents
the total fibre space which is projected down by П onto the base space X,
a circle S1. The homeomorphism фа untwists the fibre space U~1(Ua) onto
the product space Ua x F, the local trivialization.
Transition functions and global properties: In order to find the glo-
global properties of a fibre bundle we have to study the structure group G.
The group G arises by transition from one set of local bundle coordina-
coordinates {иа,фа) to another set (и0,ф0). Suppose Ua and Up have a nonempty
overlap Ua П Up ф 0. Let us consider the map фа ¦ ф^1:
i , —i
(Ua nU0)xF °-Л {Ua П Up) x F. B.422)
It is continuous and invertible (homeomorphic). Keeping x eUaC\Up fixed
and allowing the fibre point / € F to vary then we have the map фа ¦ фд~х :
F —* F from one fibre point to another (see Figure 2.39) and for this reason
it is named the transition function
9*0 •¦= Ф* ¦ Фр\ B-423)
with
/' = gpaf- B-424)
The letter / denotes a fibre point corresponding to the local trivialization
(иа,Фа) and /' corresponding to (C/^,^).
The transition functions tell us how the fibres must be glued together
in the overlap of two neighbourhoods. Although the local topology of a
bundle is always trivial, its global topology—determined by the transition
functions—might be nontrivial and rather complicated due to the twists of
the neighbouring fibres.
The transition functions satisfy the consistency conditions
gaa{x) = 1, x e Ua
9a0(x) = g?(x), xeUaf)U0 B.425)
The set of all transition functions generate the structure group
G={gnP}. B.426)

Fibre bundles
97
F
UaxF
Фа1
Fig. 2.38. A fibre bundle illustrated by a Mobius strip. F denotes the fibre
and E the total space, П is the projection and X the base space. фа is a
homeomorphic map from the bundle space П~1(С/а) onto the product space
UaxF
Twisted
Fig. 2.39. The transition function gpa from one fibre point / to anothe*
/' = 90af

98 Differential geometry, topology and fibre bundles
Another important property we have to discuss is expressed in the fol-
following proposition.
Proposition:
• The transition functions are not unique!
Let us take two fibre bundles E and E' with the same base X, fibre F and
structure group G. Let (Ua, фа) be the local coordinates of E and (Ua, фа)
those of E'. Next we require that the map
ha(x) := фа ¦ ф~\х) B.427)
is a homeomorphism ha :UaxF —> UaxF belonging to the structure group
G. Then the structure functions gap(x) and g'a0(x) of the two bundles E
and E' are related by
9«p{x) = K\x)ga0{x)h0{x) for xeUanU0 B.428)
since
Kl9<xph0 = фа ¦ ф-1 ¦ фа ¦ ф~01 ¦ ф0 • фр1 = фа • V^1 = g'a0. B.429)
The requirement ha & G implies that as gap varies then the related
ha^Oaphp = д'ад generates all the elements of the structure group G. So
the two bundles E and E' are topologically equivalent; they differ only
in the choice of the local coordinates фа and фа. Actually we always have
to consider the equivalence class of such bundles, which we usually neglect.
For instance, in the case of the Mobius strip the bundles which are twisted
once, three times, five times, ... are topologically equivalent.
Remark: When we return to physics and discuss Yang-Mills gauge theo-
theories we shall see that the transition functions gap represent the gauge trans-
transformations which glue the local coordinates together, whereas the homeo-
homeomorphism ha expresses the gauge freedom on the coordinate.
Trivial bundle: If we can choose for all transition functions the identity
map
{ga0(x) = 1} B.430)
we get a trivial bundle, all fibres are glued together without twist and the
global topology of this bundle corresponds simply to the direct product
E = XxF. B.431)

Fibre bundles 99
The transition functions contain the information as to whether a fibre
bundle is trivial or not.
Proposition:
• If a bundle E is trivial then the transition functions factorize according
to, and vice versa,
9a0(x) = K{x)h-0\x). B.432)
This is in accordance with the choice
9*0 = K^cphp = h^Kh^hp = 1. B.433)
Proof. If the bundle is trivial there is always a global homeomorphism
ф : E —» X x F and together with the usual coordinate choice (Ua, фа) we
take
ha(x) = фа ¦ ф-\х) B.434)
implying
9oc0{x) = фа ¦ Фр1{х) = фа-ф~1(х)'ф- фр1(х) = ha{x)h^1{x). Q.E.D.
Another way to find the triviality of a bundle is to consider the base
space.
Proposition:
• Any fibre bundle over a contractible base space is trivial!
Since the space Rm is contractible the simplest noncontractible spaces we
may choose are the spheres Sn.
Reconstruction of a fibre bundle: Let us return to our definition of a
fibre bundle. It consists of several components.
The question now is:
Which of the ingredients determine already the structure of the fibre
bundle?
The answer is:
Given X, F, G and да0(х) we can reconstruct the whole fibre bundle
(E,H,X,F,G). Thus we can find П, E and фа.
The procedure is as follows:
We start with all direct products
E:=[jUaxF B.435)

100
Differential geometry, topology and fibre bundles
Mobius strip
Cylinder
Fig. 2.40. For the Mobius strip—a nontrivial fibre bundle—we have to
glue together the twisted fibres. For the cylinder—a trivial fibre bundle—
the fibres remain untwisted
and we define for two coordinates (x, /) € Uax F and (x1, /') € Up x F the
equivalence relation ~
(*,/)-(*',/')
B.436)
iff x = x' and gpa(x)f = /'• Then the fibre bundle E is given by the set of
all these equivalence classes
The projection П is defined by
П : E-+X
B.437)
B.438)
when we denote the equivalence class by [(x, /)]. So the projection П is a map
which shrinks each fibre to a point. Finally, we describe the homeomorphism
фа by its inverse

Fibre bundles
101
E
so that
as required.
x X
Fig. 2.41. A section s(x) of a fibre bundle E
ф'1 :UaxF -> IT1^)
(xj) -» [(х,/)]€П~1(х)
tf>-1(T f\ с n-V-ri => Uth-1(T f\ - т
B.439)
B.440)
Example Mobius strip: Let us come back to the Mobius strip to illu-
illustrate the bundle reconstruction. Here the base is X = S1, a circle, and the
fibre F = f, a line segment. The structure group G = {l,g} contains two
elements: 1 the identity and g the twist (or reflection of the fibre about its
midpoint). The covering of the base S1 consists of two open arcs U\ and
Ui with intersection U\ П U2 = 11 U I4 (see Figure 2.40). The transition
functions are
Г
522 (x) = 1
1 if x €
g if x €
B.441)
Now we consider the equivalence classes [(x, /)] in the overlap regions I\
and /2- If x G I\ then [(x, /)] = (x, f) is just one element. If, however,
x € I2 then [(x, /)] = {(x, /), (x, gf)} contains two elements—the fibre and
the twisted fibre. But the equivalence relation B.436) identifies these two

102
Differential geometry, topology and fibre bundles
П
Fig. 2.42. The principal bundle of the Mobius strip
elements; for this reason we have to glue them together and we get the twist
for the bundle E, the Mobius strip (see Figure 2.40).
If we consider in constrast the cylinder the structure group is simply
G = {1}, the fibres remain untwisted, and we obtain a trivial fibre bundle
E = SX xF.
Section: A section of a bundle E is a continuous map
s:X-+E with IIs(x) = x, Vie X.
B.442)
If the section is defined only over an open subset [Iclwe speak of a local
section (see Figure 2.41).
Principal bundle: Given a fibre bundle (E, П, X, F, G) we can always
construct a principal bundle by taking the same base X and the same
transition functions gap, but by choosing the fibre to be identical with the
structure group itself F = G. As before, the transition functions act on the
fibre from the left. In addition, we also define an action of G on F from
the right, see Section 2.7.2. For the construction we follow the procedure
described above. We denote the principal bundle by P(X,G).
The fibre bundles we mainly consider in physics, specifically in gauge
theories, are principal bundles. We discuss several examples separately in
this book.
There is an important theorem on the triviality of a bundle.

Fibre bundles 103
Theorem: The principal bundle P(X, G) and the correlated fibre bundle
(E, П, X, F, G) are trivial iff P(X, G) has a section.
In our example of the Mobius strip we find the following. Choosing the
fibre to be the structure group F = G = {l,g}, which consists of two
elements, we obtain a double covering of the base S1 depicted in Figure
2.42. Choosing a local coordinate в G [0,2тг] on S1 the section s(#) is not
continuous, e.g. s(9) Ф sF + 2тг) due to the double covering. According to
the above theorem the principal bundle and the Mobius strip itself are then
nontrivial.
Remark:
i) Although the principal bundle P(X,G) has no section, the bundle
itself (E, П, X, F, G) may have a section (as in the case of Mobius)! So
the principal bundle is the important one for studying the triviality
of a bundle.
ii) Several bundles with different fibres but with common base, transition
functions and structure group will have the same principal bundle.
Line bundle: A line bundle is a vector bundle, but with a 1-dimensional
vector space as a fibre. For instance, if we replace in the Mobius strip the
finite line segment by the real line R we get a nontrivial real line bundle
over S1. On the other hand, the cylinder S'xR represents a trivial real
line bundle.
Associated bundle: A vector bundle—a bundle whose fibre F is a
vector space—has a certain representation p of the structure group G on
the vector space F. It is termed associated to the principal bundle if its
transition functions are the images under p of the corresponding transition
functions of the principal bundle.
Tangent bundle: Choosing the fibre F at any point x of some base ma-
manifold M to be the tangent space TX(M) = F we construct the tangent
bundle
E = T(M) = (J ТХ(М). B.443)
The projection П is defined by
П : T{M) -> M
V G TX{M) •-> x. B.444)

104 Differential geometry, topology and fibre bundles
For the local trivialization we need the homeomorphism фа on some Ua С
M. Ua is homeomorphic to Rm, Ua ~ Rm and also TX(M) ~ Rm. In local
coordinates the tangent vector V is given by V = Vl(x)-?^ so that the
homeomorphism is
Ua x TX{M) ~ Rm x Rm
(x,Vi(x))GRm xRm B.445)
and locally the tangent bundle has the direct product structure
T(Ua) ~ Rm x Rm. B.446)
The structure group of the tangent bundle is G = GL(m,R), the
group of all invertible m x m matrices since the fibres F = TX(M) are
m-dimensional vector spaces and the action on an m-dimensional vector is
represented by an m x m matrix.
Searching for the transition functions from one fibre to another gap =
фа ¦ фр1 with фа{у) = (x,V*(x)) and 4>0(V) = (x,V*(x)) we find
Ei=9%Vi B447)
(note: i,j are the vector indices and a,/3 refer to the charts) the Jacobian
matrix of vector transformations
9a0 =
P J mxm
Since the fibre is a vector space the tangent bundle also represents a
vector bundle. If we consider all ordered bases of TX(M)—a frame—the
bundle is called a frame bundle. Then the fibre can be identified with
GL\m, R), the structure group itself, hence the frame bundle expresses a
principal bundle.
Cotangent bundle: Analogous to the tangent bundle we can take as a
fibre the cotangent space F =¦ T*(M) and we set up the cotangent bundle
E = T*(M)= \J T*(M). B.449)
хем
T*{M) is the dual bundle to T{M).

Fibre bundles 105
The transition functions, a matrix, are determined by the dual vector
transformations
9a0 —
B.450)
2.7.2 Connection
The 'connection' is of great importance for a fibre bundle (we consider here
only principal bundles). It allows for a parallel transport of the fibres. For
parallel transport or covariant differentiation we have to compare the points
in neighbouring fibres independently of a specific choice of local bundle
trivialization. So we need vectors that lead from one fibre to another. We
already have vectors at our disposal. They arise from the group action on
the bundle—the fundamental vector fields. However, they are tangent to
the fibre. Therefore we have to construct vectors that point away from the
fibre. This we achieve with help of a 'connection'.
In the interplay between the geometry of fibre bundles and physics the
'connection' takes over an important role. Restricting the 'connection' to
a local region represents precisely the Yang-Mills (YM) potential of gauge
theories. Furthermore, the 'curvature' of the 'connection' describes the cor-
corresponding Yang-Mills field strength.
Vertical subspace: Let us start with a principal bundle P(M,G). Let
the fibre F = G be some Lie group G, and the base be some manifold M.
The projection is П : P —> M. We consider the tangent space TU(P) of the
bundle P with и G P some fibre point and we now construct the vertical
subspace VU(P) of TU(P). We take an element A G Te(G) ~ Lie G of the
Lie algebra then we can define a curve in P through и by the right-action
of G on P (recall our discussion of Lie groups and algebras in Section 2.6;
e.g. Rgh = hg; g,heG)
ДехР(«л)" = и exp{tA) = a(t, u). B.451)
This flow a{t, u) lies along the fibre П^) over the base point p G M since
Щи) = П(и exp(tA)) =p G M. The action of the one-parameter subgroup:
t —> exp(t-A) induces a vector field—the fundamental vector field—by
XAf{u) := |
/(uexp(M)) B.452)
"" t=0
with a smooth function / : P —» R. Xa preserves the Lie algebra structure
[XA,XB)=X[A<B]. B.453)

106
Differential geometry, topology and fibre bundles
n
м
Fig. 2.43. The tangent space TU{P) of the fibre bundle P(M, G) is decom-
decomposed into the vertical subspace VU(P) and horizontal subspace HU(P). We
move along the fibre via the right-action of the group G on P, Rgu = ug
Xa is a tangent to the fibre at и and belongs therefore to a space—the
vertical subspace VU(P) of the tangent space TU(P), Xa € VU(P)—which
is defined by
VU{P) := {X e TU(P)\U.X = 0}. B.454)
The map i : А н-» Xa is an isomorphism Lie G —> VU(P), hence dim VU(P) =
dim Lie G = dim G.
Horizontal subspace and connection: The complement of VU(P) in
TU(P)—the horizontal subspace HU(P)—is described by the so-called
connection.
Definition: A connection assigns smoothly to each fibre point и G P a
subspace HU(P) С Ти(Р):
B.455)
such that

Fibre bundles
107
П
M
Fig. 2.44. Parallel transport of fibre point и along the base curve с
ii) Rg*Hu{P) = Hug(P).
B.456)
We have illustrated this assignment in Figure 2.43. Rgt denotes the
differential map (recall Section 2.6.1) of the right-action Rgu = ug of G on
P; g G G, и G P.
From Property i) it follows that any tangent vector X e Tu (P) can be
decomposed into
X = XV +X
н
with Xv e VU(P), XH G HU{P).
B.457)
Condition ii) means that the horizontal subspaces HU(P) and Hug(P)
on the same fibre are linked together by the differential map Rg». For a
given HU(P) all the other Hug(P) on the same fibre are generated by Rg*
(see Figure 2.43). Hence, if a fibre point и is parallel transported, the fibre
point ug is transported too.
Horizontal lift and parallel transport: The projection П : P —* M
induces the differential map П» : Tu (P) —> ГП(„) (М) on the tangent spaces
with the kernel кегП» = VU(P). Therefore the map from the horizontal
subspace into the tangent space П* : HU(P) —> Тщи)(М) is an isomorphism.

108
Differential geometry, topology and fibre bundles
R
П
Fig. 2.45. The bundle with projection П : R —> 17A) has a connection:
Hu = tangent space to R at u. The holonomy group Я = Z = G coincides
with the structure group
For this reason any tangent vector X e TP(M) from the base manifold M
can be lifted to a unique vector X € HU(P) in the horizontal subspace of
the fibre bundle. We have illustrated this in Figure 2.44.
Definition: The vector X is a horizontal lift of X if
i) ГЦХ = X,
ii) Xv = 0 vertical component.
B.458)
Definition: Let с : [0,1] —» M be a base curve. A curve с : [0,1] —» P is a
horizontal lift of с if
i) Пс = с,
ii) tangent vector to c: X € Н-.ЛР).
B.459)
Theorem: Let с : [0,1] —» M be a base curve and let и € П ^(O)), then
there exists a unique horizontal lift с : [0,1] —¦ P such that c@) = u.

Fibre bundles 109
So the parallel transport of a fibre point и = c@) along the base curve
c(t) is provided by the horizontal lift c(t) (see Figure 2.44).
Holonomy: Generally, it occurs that even if the base curve c(t) is closed,
c(l) = c@), the horizontal lift c{t) need not be closed: c(l) = c@)h, h € G.
The set of such elements is called the holonomy group H = {h} of the
connection at the fibre point c@). It is a subgroup of the structure group,
ЯСС?.
For example, the bundle with projection П : R —> 17A) so that
П(«) = ei2nu B.460)
allows for one connection:
Hu = tangent space to R at u. B.461)
Let c(t) = et2nt be the closed base curve, then the lifted curve is c(t) = t,
which is not closed, c@) = 0 and c(l) = 1 (see Figure 2.45). The holonomy
group coincides with the structure group H = Z = G.
Connection 1-form: Next we define the connection in a way which is—as
we shall see—more suitable for physics.
Definition: The connection 1-form w € T* (P) <8> Lie G—a Lie algebra
valued 1-form on the bundle P—is defined by a projection of the tangent
space TU(P) onto the vertical subspace VU(P) ~ Lie G satisfying
i) w(XA) -A, Ae Lie G,
ii) Rg*u = Айд-ш. B.462)
Rg* is the pullback (recall Section 2.6.2) of the right-action Rg: и н-> ид
and Ad means the adjoint map Adg : Lie G —> Lie G given by Ль» gAg-1,
g e G, A e Lie G (recall Section 2.6.6.). So for some vector X e TU{P) we
have
Rg*LJUg(X) = 4JUg(Rg.X) = (T^X^. B.463)
With such a defined connection 1-form ш we can now determine the
horizontal subspace HU(P) by the kernel of the map w : TU(P) —> Lie G
Hu(P) = {X?Tu(P)\u(X) = 0}. B.464)
This leads again to the direct sum separation of TU{P) = VU{P) @HU{P).
The subspace HU(P) also satisfies the previous property ii): Rg,Hu(P) =
Hug(P) (equation B.456)), therefore both definitions B.455), B.456) and
B.462), of the connection are equivalent.

110
Differential geometry, topology and fibre bundles
в
e
un
ua
M
Fig. 2.46. Canonical local trivialization of a fibre bundle
Local connection 1-form and gauge potential: Restricting now to
a local region we find the bridge to physics. There we already know a Lie
algebra valued 1-form denned on the base M. It is the YM gauge potential.
If we pull back the connection ш within a local section down to the base we
will find that it satisfies precisely the properties of the YM potential.
Proposition:
• Locally the connection 1-form и represents the YM gauge potential
A! Be {Ua} ал open covering of M; choose a Joca] section on Ua
then
sa : Ua
Aa = s* w €
Lie G.
B.465)
B.466)
Conversely, we have the following theorem.
Theorem: If we consider a Lie G valued 1-form Aa defined on a patch Ua
together with a local section sa : Ua —» П~1A7а) then there exists exactly
one connection 1-form w on n~1(t/a) such that Aa = «„w. The explicit
expression
u\Ua = д-^АсЯс + д-Чр9а B.467)
is found by construction.

Fibre bundles 111
The operator dp means the exterior derivative on the bundle P and ga
expresses a local bundle coordinate given by the canonical local triviali-
zation (see Figure 2.46) which is denned by the homeomorphism
фа : ГГ^С/а) -» UaXF
и >~> (P,9a), B.468)
thus
фа{и) = {р,да) with и = sa(p)ga. B.469)
Proof. a) First we have to show that expression B.467) gives the YM
potential B.466). Take a tangent vector X € TP(M) on the base, lift
X up to the bundle satX € TSa(P) and note that II»sa» = lrp(M)
and ja = eonsa.
Note furthermore that dpga(sa*X) — 0 since ga = e along the section
satX. Then we get
sa*u{X) = u(satX) = n*Aa{sa*X) + dpga{satX)
= Aa(U,satX) = Aa(X). B.470)
b) Second we demonstrate that expression B.467) satisfies the axioms
B.462) of a connection 1-form.
Axiom i):
Taking a fundamental vector field Xa € VU{P), so П.-Хд = 0, we find
ш(ХА) = g-Udl
ga{uexp{tA))
t=o
exp{tA)) = A. B.471)
t=o
—
Axiom ii):
Take a tangent vector X € TU(P) and a group element h e G, recall
that X is a right-invariant vector field Rh*X\u = X\uh (recall Section
2.6.6) and note that ga(uh) = ga(u)h.
Then we calculate
Rh*u(X) = u(RhtX)
= h-1g-1(u)n*Aa(X)ga(u)h
+ h-1g-1(u)dp9a(u)(X)h
= h~lw{X)h = Adh-iw{X). Q.E.D. B.472)

112 Differential geometry, topology and fibre bundles
Transformation property: The connection 1-form ш is defined uniquely
on the bundle; the decomposition of the tangent space TU(P) = VU{P) ф
HU(P) is unique. Therefore we have in the overlap of two neighbourhoods
Ua, Up the identity
w\Ua = w\Up on иас\и0ф 0. B.473)
As a consequence the YM potential has to obey an appropriate transforma-
transformation law. We now calculate this explicitly. We start with
, 4 4зЧ>Я8- B-474)
In the overlap Ua П Up the transition functions hap are defined by
9p = hpa{p)ga or sp{p) = sa(p)hap{p), peUan Up, B.475)
which we insert into equation B.474) so that we find
п* Аз = KJpTAahct, + hllpdphap. B.476)
This is the compatibility condition the local YM potentials have to fulfil
when pulled back on the bundle P. That means on the base M in the overlap
region Ua П Up we have the following transformation law for YM gauge
potentials
Аз = KpAaha0 + h~ldhap on Ua П U0. B.477)
Remark: A nontrivial principal bundle does not admit a global section, so
the pullback Aa = sa*u exists locally but not to a global extent. Therefore
we have to cover the base M with locally trivialized charts {Ua} and we
get several (at least two) different YM potentials. The YM potentials Aa
on Ua and Ap on Up are then related to each other in the overlap region
Ua П Up by the above transformation condition B.477).
So a single YM potential Aa does not carry any global information. It
is the connection 1-form ш which contains the global feature of the fibre
bundle or equivalently the set of all YM potentials {Aa} (defined over the
covering {Ua} of M) satisfying the compatibility condition B.476), B.477).
Gauge transformation: Let us finally consider two charts Ui, U2 and
let us choose the local sections on the overlap as
s2{x) = si{x)g(x), xeUi<-\U2 B.478)

Fibre bundles 113
(we identify the point x of the manifold with its coordinate x), then the
local YM forms A\, Ai are related by
A2{x) = g-1(x)A1(x)g(x) + g-\x)dg(x). B.479)
In components A\ -» Л, Л -»-49 and A = A^dx, d = d^dx^ we get
Л'„{х) = g-1(x)Ali(x)g(x) + g'l(x)d,ig(x). B.480)
This is the usual gauge transformation we are familiar with in physics.
Example: If the structure group is G = U(l)—the case of electro-
electrodynamics—the transition functions are
haf}:UaDUp -» 17A)
hafi(x) = eiA(x), xeUanUf,. B.481)
So dhap = id\(x)etX^ and the compatibility condition gives the well-known
gauge transformation law
Ax{x) = A(x) + idA(x). B.482)
2.7.3 Curvature
The last concept we need to discuss in our tour of fibre bundles is the
curvature.
Covariant derivative: Let us consider a g-form pona principal bundle
P(M, G) with values in a vector space V
ip : T(P) ® ¦ ¦ ¦ ® T{P) -» V B.483)
q times
(generalizing the previous Lie algebra valued forms) and V is a vector space
of dimension k. Then the form can be decomposed by
tp = <fa ® ea € A"{P) ® V, B.484)
which is a sum over a = 1,2,..., k. The set {ea} represents the basis of V
and (pa € A«(P).
Let X\,...,Xq+i e TU(P) be (q + 1) tangent vectors on the bundle P,
then the exterior covariant derivative of the g-form <p is denned by
¦ ¦ -,Xq+1) = dM*i, ¦ ¦ ¦ .*?i). B-485)

114 Differential geometry, topology and fibre bundles
where X?1 e HU(P) means the horizontal component of Xi € TU(P) de-
determined by the connection (recall equations B.455)-B.457)) and dp<p —
dp(fa <g> ea is the exterior derivative on P.
Curvature: The curvature 2-form fi on the bundle P is the covariant
derivative of the connection 1-form ш
О := JDw € Л2(Р) <g> Lie G. B.486)
Theorem: The curvature 2-form has the property
Rg*Sl = Adg-iCl = д~1Пд, geG. B.487)
Proof. Recall that
(RgtX)H = RgtXH Rgt preserves the horizontal subspace
dpRg* = Rg*dp commutation
dpg = 0 since g € G is a constant element
Rg*w = Adg-iu = g~lwg connection property. B.488)
Let be X,Y e TU{P) then we have
Rg*U{X,Y) =
= dpu((RgtX)H, (Rg*Y)H) = Rg*dvw{XH,YH)
Q.E.D. B.489)
Theorem: fi and ш satisfy the Cartan structure equation
U{X, Y) = dpu{X, Y) + [ш{Х),ш(Х)] I2-490)
for any X,Y € TU(P) and [ , ] denotes the Lie bracket between the Lie
algebra elements of ш(Х) and u>(Y); or in terms of 2-forms on P
B.491)
where
ш2 =шш = \[ш,ш] B.492)
means the wedge product ш of the 1-form ш or half the commutator.

Fibre bundles
Expression B.491) follows from equation B.490) since
) = [Та,Ть]шашь(Х,?)
= [Ta,Tb}(ua(X)u;b(Y)-u;a(Y)u;b(X))
= 2[u(X)MY)}. B.493)
The Cartan structure equation B.490), B.491) between the curvature f2
and the connection w is of fundamental importance and we shall use it
throughout the book.
Proof. To prove this theorem we consider the following three cases.
i) Let X, Y e HU{P) => w(X) = w(Y) = 0 by definition of the connec-
connection 1-form (recall equation B.464)). Prom the definitions B.485),
B.486) we have for the curvature
Sl(X, Y) = Dw{X, Y) = dp{XH, YH) = dp{X, Y). B.494)
ii) Let X € HU(P), Y € VU{P) => U{X,Y) = 0 since YH = 0; and
w(X) = 0 by definition B.464).
So we have to prove that dpu(X, Y) = 0. We remember relation
B.108) for the bundle
dpw{X,Y) = Хш{у)-Уш{Х)-ш(\Х,У\)
= Xw{Y)-w([X,Y]). B.495)
Since Y € VU(P) 3V e Lie G such that Y = Xv, where Xv is the fun-
fundamental vector field. Then we have w(Y) = ш(Ху) = V= constant,
and therefore Xw(Y) = XV = 0. We also note that [X, Y] € HU(P)
since
[Y,X] = CYX = \hn-[Rg-1(t)tX - X]. B.496)
and Rg-nt),X e HU(P). This implies u{[X,Y}) = 0 so that we find
dpu(X, Y) = 0. B.497)
iii) Let X, Y € VU{P) => п(Х, Y) = 0; and we also have
, Y) = Xw(Y) - Уш(Х) - ш([Х, Y\) = -ы([Х, Y\). B.498)
Since [X, Y] eVu{PKAe Lie G such that [X, Y] = XA and we get
for the connection u([X, Y}) = u(Xa) = A. On the other hand, let
X = Xb, Y = Xc, then the commutator is

116 Differential geometry, topology and fibre bundles
[ш{Х)МУ)\ = НХв),ш(Хс)} = [В,С] = А B.499)
since [Хв,Хс\ = X[Btc] = Xa (recall equation B.453)).
So we find
[lj(X),lj(Y)\=w([X,Y}), B.500)
giving
dpw{X, Y) + [w{X), U{Y)] = dpu{X, Y) + W{[X,Y]) = 0. Q.E.D.
B.501)
Local curvature and field strength: Finally we make contact with
physics. In a local region we can identify the curvature with the Yang-Mills
field strength.
Proposition:
• The curvature 2-form П represents locally the Yang-Mills Geld
strength T
F = s*ue A2(U) <g> Lie G, B.502)
where s* is the pullback of a local section s : U —> П"A7) on a chart
U С М.
In terms of the YM gauge potential Л we then find
F = dA + A2 B.503)
since
T = s*?l = s*
= ds*w + s*ws*w = dA + AA, B.504)
where we used s*dpw = ds*u and s*u = A.
Evaluating the 2-form T on the vector fields X,Y e TP(M) we get
) - dA{X,Y) + AA(X,Y)
= dA{X,Y) + [A{X),A{Y)}. B.505)
In terms of components A — Ali{x)dxli, T = \Tllv{x)dxlldxv, expres-
expression B.503) gives
Fu»{x) = д„Аи{х) - д„Ац{х) + [Ац{х), А„(х)}. B.506)
This is the well-known YM field strength.

Fibre bundles 117
Transformation property: Considering finally the YM field strength on
two charts Ua and Up
Ta = sa*u on Ua B.507)
Tp = sp*u onUp, B.508)
then they satisfy on the overlap of the charts the compatibility condition
?fi = Kp~f°ha0 on Ua П U0, B.509)
where hap is the transition function in the overlap as denned before
(equation B.475)). The proof is straightforward by using formula B.503)
together with the transformation law B.477) for the YM potentials.
Remark: For a pure gauge we have
= 0. B.510)

Path integrals, FP method and BRS
transformation
Path integral formalism provides a unified view of nonrelativistic quantum
mechanics and relativistic quantum field theory (QFT). It is particularly
suited to investigating gauge theories and to formulating their anomalies.
In fact, we will use path integrals (PI) throughout the book. For this reason
we shall develop the formalism of PI to the extent we need it in this book;
for a further study we recommend textbooks like [Itzykson, Zuber 1980],
[Ryder 1988], [Cheng, Li 1988], [Lucha, Schoberl 1996] and [Pokorsky 1987].
In Section 3.1 we present Feynman's idea of formulating quantum me-
mechanics as a sum of particle trajectories (paths). We generalize this approach
in Section 3.2 to scalar fields with selfinteraction and we calculate the Green
functions in a perturbative way. In Section 3.3 we introduce the Grassmann
algebra in order to define PI for fermionic systems. In Section 3.4 we explain
the Abelian-, in Section 3.5 the non-Abelian field case where we focus on
the occurrence of the Faddeev-Popov ghosts which play—together with the
BRS transformation of Section 3.6—an important role in our treatment of
anomalies.
3.1 Quantum mechanics
3.1.1 Propagator
In quantum mechanics a state vector in the Schrodinger picture is related
to that in the Heisenberg picture by
\at)s = exp l-^Hti \a)H, C.1)
where H is the total Hamiltonian
H = ^ + V C.2)

Quantum mechanics 119
and a denotes some quantum number. The state vector \at)s obeys the
Schrodinger equation
д , ч
The moving frame is defined by
\xt) — exp — Ht\ \x) C-4)
[ft J
so that we obtain
{x\at)s = (x|exp —-Яп |о)я = {xt\a)H C-5)
or in terms of the wave functions
1 Ф) C.6)
(from now on we suppress the symbols S, H).
The basic quantity we deal with in the PI approach is the propagator.
Quantum mechanically it is the transition amplitude for a state localized at
(xi,ti) to a state at
Г i ]
:= {xNtN\xiti) = (x^|exp \--H(tN -ti)\ \xi). C.7)
L ft J
Inserting a complete system of energy eigenstates
H = i C.8)
gives
K(xN,tN't Xi,t\) = \^.^Pn{XN)^n{xl)exP ~xEn{tN — tl)\ ¦ C-9)
n ^ ^
This propagator, when acting as an integral kernel, connects the final with
the initial wave function
,tN) = / dxK(xN,tN\X,t{)^a{x,t{). C.10)
The time evolution of the wave function is completely determined by the
propagator. The propagator containing the Hamiltonian C.2) supplies the
whole information for the quantum system.

120 Path integrals, FP method and BRS transformation
For t > t\ the propagator satisfies the time-dependent Schrodinger equa-
equation
ih—K{x, t; n, h) = HK{x, t; xu *i), C.11)
which follows immediately by direct differentiation of equation C.7). In the
limit t —> *i the propagator approaches a delta function
lim K(x,t;x1;t1)=6{x-xi) C.12)
t-»ti
due to the normalization
{x\xt) = 6{x - n) = ,2ж%у I dpexp ^p{x - xi) C.13)
{d denotes the dimension). Therefore the propagator represents the Green
function for the Schrodinger equation
^ - H) K{x, t;Xl,t1) = ih6{x - Xl)S(t - h), C.14)
with initial condition
K(x,t;xi,ti)=0 fort<ti. C.15)
3.1.2 Typical examples for the propagator
i) Free propagator
In the case of free motion (V = 0) the propagator is
Р,хиЬ) - (x|exp f-?f-(t - ti)l |xi>. C-16)
Inserting a complete system of momentum states
jdp\p){p\ = l C.17)
and using the plane waves
gives
*о(х,*;*1,*О = 73^52/ dpexphlpix-x^-^it-h)]^ C.19)

Quantum mechanics 121
(again d is the dimension). Evaluating this Gaussian integral with the help
of the formula (for 1 dimension)
dx l\b2l
— expf-ax2 + bx + c] - —== exp he C.20)
2тг y/Wa L4a J
we obtain the well-known result for the free propagator
dl2
ii) Harmonic oscillator propagator
In the case of the harmonic oscillator
or
C.23)
with a and a* being the annihilation and the creation operator we can sum
up the oscillator wave functions and energy levels directly in representation
C.9). The procedure is straightforward leading to the following result for
the oscillator propagator (for imaginary times it = r) [Bertlmann 1986]
Kosdx, r; y) = (x\ exp [-(# - ^)r] \y) C.24)
= ЛГ-1(г)ехр[-^[а(г)(х-г/J+а-1(г)(х + г/J]], C.25)
where
г -,d/2
N(r) = ^(i-e-a-)j C.26)
1 + е
а(т) = ^-^ . C.27)
Clearly, for short times r —> 0 or weak potential ш —> 0 the propagator
C.25) approaches the free case C.21).
Hi) Moments
We call the propagation of a state localized at x (say at x = 0) back to that
point a moment (again for imaginary times r; and % = 1)

122
Path integrals, FP method and BRS transformation
Small tim<
coarse energy
Perturbation theory
Exact
Stability
Break down
All levels Groundstate т
contribute dominance
Fig. 3.1. A typical ratio of moments
М{т) := К(х = 0,г;г/ = 0)=(х = 0|е-Нт
C.28)
Its logarithmic derivative—a ratio of moments—serves as a practicable
quantity to project just the groundstate energy E\
R(r) := -_
For example, in the free case C.21) we have (for d = 3)
В t \ ^ т-*оо .,
П*(т) = Тт — О
C.29)
C.30)
C.31)
C.32)
and for the harmonic oscillator we obtain from formulae C.24)-C.27)
Г
-|3/2
L27rsinho'rJ
3 4, r-oo 3
= -wcothwr —> -u> =
C.33)
C.34)
Such moments have been investigated by Bell and Bertlmann [Bell,
Bertlmann 1981a,b], [Bertlmann 1982] by considering the minimum of the
perturbative result as an approximation to the exact ground state

Quantum mechanics 123
ттД(т) =?1. C.35)
We have plotted the typical feature—the approximation works much better
than expected ('magic moments')—in Figure 3.1.
iv) Partition function
We find another important application of the above discussed propagator,
which can also be generalized to QFT, in thermodynamics. There the basic
quantity is the partition function defined by summing the above moments
over space
Z(C) := f dxK{x,ftx) = I dx {x\e~HI3\x). C.36)
Instead of a time r we work here with the parameter
'-Й C-37)
where T describes the temperature of the system and к is the Boltzmann
constant.
Inserting a complete system of energy eigenstates C.8) and using the
normalization of the wave function we can rewrite equation C.36)
n
<1х\фп(х)\2ехр[-ЕпC}
е-^Н C.38)
so that the partition function can generally be defined as the trace of the
exponentiated Hamiltonian
Z(fi) = tr e~H0. C.39)
The trace is clearly independent of a special representation, say, of equation
C.36) or C.38). For example, we can calculate the partition function of the
harmonic oscillator C.23) explicitly
C.40)
n
Summing up the geometric series gives

124
Path integrals, FP method and BRS transformation
(xN,tN)
Fig. 3.2. Possible paths from (xi,ti) to
1 - exp[—u
C.41)
In the limit of weak coupling ш-»0ог, alternatively, in the high temperature
limit T —+ oo (/3 —» 0) we approach the free case
¦Zosc(/?) —> —ъ ~ — = ^o(/3). C.42)
3.1.3 Feynman's path integral
An alternative formulation of quantum mechanics is based on the propaga-
propagator which is then represented as a sum—integral—over all possible paths of
a particle. This concept was developed by Feynman as a graduate student
and the standard reference for this approach is certainly the textbook of
Feynman and Hibbs [Feynman, Hibbs 1965].
The idea is to insert complete states into the propagator or transition
amplitude C.7). We discretize the time between the fixed t\ and t^
tN
«2
into N — 1 equal pieces
At-t-t- i -
hi - tj ij_i -
C.43)
C.44)
Then the transition amplitude can be decomposed into the following

Quantum mechanics 125
integral—for simplicity we consider just one space dimension—
K(xN,tN;xi,t{) =
= / dXN-ld.XN
...(x2t2\x1t1). C.45)
We integrate over all intermediate states, which means we sum over all
possible paths (see Figure 3.2). So in Feynman's quantum mechanics all
possible paths of a particle play a role.
Now, how can we describe an individual path within a small segment?
There, according to Dirac [Dirac 1933, 1958] and Feynman [Feynman 1948],
the classical action S determines the path. So we may try the following
equation
{Xjtjfa-itj-!) = Aexp \^S(j,j - 1I , C.46)
with A being some normalization constant.
In order to calculate, finally, the classical action S in the limit At —> 0 it
is certainly legitimate to approximate each path segment by a straight line
S(j
U-l) = Г dtLdaM(x,x)= Г dt\^--V(x)]
= At
_ v
C.47)
The normalization A is independent of the potential V. Therefore we
may evaluate it from the free particle case (V = 0), equation C.21),
2. C.48)
Now we insert equation C.46) into the propagator decomposition C.45)
and we refine the time discretization. We consider the limit N —> oo, At —> 0
by keeping the endpoints fixed
hm
= (xNtN\xiti)
Г
hm , Г.л-1
I
N Г ' 1
dxN-idxN-2 dxi П exp \ls{j,j - 1) . C.49)
А— О L J
Defining the multidimensional integral operator (assuming its existence) as

126 Path integrals, FP method and BRS transformation
the path integral
/ d\x(t)\ := lim (n f,A ) / ГГ dxj C.50)
Jxi JV—oo.At—0 \2whlAtJ J ZT2
we obtain Feynman's path integral formula for the propagator
K(xN,tN;xi,ti) = (xNtN\xiti)
= J d[x(«)] expl^jf dtLciass(x,x)\. C.51)
Proposition:
• Feynman's path integral is equivalent to Schrodinger's theory!
Features: Feynman's path integral relies on:
i) the decomposition property C.45) of the transition amplitude,
ii) the superposition principle—summing of all possible paths,
iii) the classical correspondence for % —+ 0.
3.1.4 Connection between Lagrange- and Hamilton formalism
What we finally have to show is that the Dirac-Feynman equation C.46)
for the transition amplitude which involves the classical Lagrangian Lciass is
indeed equivalent to the propagator C.7) discussed at the beginning which
contains the Hamiltonian H. For that purpose we derive the path integral
in the Hamilton formalism.
Hamilton formalism: We discretize the time as before and we decom-
decompose the transition amplitude as we did in equation C.45). But now we
calculate the transition in a small segment in terms of the Hamiltonian
(again we just consider one space dimension)
- t,)J \xj). C.52)
We expand for small Д* = tj+\ — tj
фехр [jp(xj+1 - xj)] - ?Д« (xJ+1\H\Xi). C.53)

Quantum mechanics
127
P
Fig. 3.3. Segments in momentum space
Evaluating the matrix element we begin with the kinetic term
El
and next treat the potential
(xj+i\V{x)\xj) =
where Xj is the mean value
Altogether we have
(xj+1\H\xj) = (a
tern
1
-г/-/— \ /„ 1 \ T/Y« \?/«.
1 f \{ ( J
2тгЙ У [h J+1 3 \ 3
Xj-
Xj + l + Xj
2
? + V(*)
C.56)
~ XJ)\ Я(Р>xj)- C-57)
Therefore we get for the transition C.53) in a small segment of one possible
path

128 Path integrals, FP method and BRS transformation
= ^JdPjexP I jPAxHi ~Xj)\ 11 - \&t HfaxA , C.58)
where we have denoted by pj the momentum between tj+\ and tj (see
Figure 3.3). Including also the higher orders in At implies
(xj+itj+ilxjtj) = ^r J dpjexp I Jfefo+i - Xj) - At H{pjfXj)]\ .
(
C.59)
We insert expression C.59) into the decomposition of the propagator C.45)
and we refine the segments and finally we perform the limit N —» oo, At —> 0
with fixed endpoints. Defining again the multidimensional integral operators
as path integrals (assuming their existence)
N-l
Y[ dxi C.60)
.JV-1 ,
lim / Yl 7p% C-61)
we obtain the path integral for the propagator in the Hamilton
formalism
K(xN,tN;xi,ti) =
\0^^J*N y C.62)
Proposition:
The Hamilton representation is equivalent to Feynman's Lagrange for-
formalism !
Proof.
K(xN,tN;xx,ti) =
/N-l .N-1
TT dxi I TT
о J i
t=2 j = l
. ЛГ-1
C.63)

Quantum mechanics 129
the Gaussian momentum integration, gives (formula C.20))
K(xN,tN;xi,ti) = (xNtN\xiti)
= lim
= f N d[x(t)]exp t f " dtLdvefax)] ¦ Q-E.D. C.64)
Remark: However, this strict equivalence is only valid for Hamiltonians
of the above type. For general Hamiltonians (e.g. with velocity dependent
potentials) the above momentum integration creates an additional term to
the classical Lagrangian which can be subsummed to an effective action
[Lee, Yang 1962] (see e.g. [Ryder 1988] and [Lucha, Schoberl 1996]).
3.1.5 Field theory analogies
Our aim is to generalize the quantum mechanical PI formalism to field
theory. But there we have different boundary conditions. In QFT particles
are created and destroyed. A formalism which describes such features is that
of Schwinger [Schwinger 1969].
Z[J] functional or vacuum-to-vacuum transition: In the spirit of
Schwinger we study a vacuum state at a time t —» — oo which evolves into a
vacuum state at t —> +oo while creating and annihilating a particle through
the action of a source. The source is introduced into the Lagrangian by
replacing
L -> L + HJ{t)x{i). C.65)
Denoting the vacuum state—the groundstate with zero quantum number—
in the presence of the source by |0,?)J we then consider the vacuum-to-
vacuum transition amplitude
@,oo|0,-oo)J~Z[J]. C.66)
This represents a functional Z[J] depending on the source J.
What is an explicit expression for that functional Z[J]7
We imagine the following situation. We split the time into intervals
T<t<t' <T', C.67)
where the source is nonzero in the inner interval

130 Path integrals, FP method and BRS transformation
J{t)^0 for г е [*,*'] C.68)
and vanishes outside
J(E) = 0 for t € [T, t) and t e {f, T'\. C.69)
We represent the transition amplitude of a particle localized in space and
time in the presence of a source by Feynman's path integral formula C.51)
(X'T'\XT)J = f d\x{t)} exp i f dt{L + hJx) . C.70)
On the other hand, we can decompose this transition amplitude by the
following intermediate states
(X'T'\XT)J = f dx'dx (X'T'\x'tf) (x't'\xt)J (xt\XT). C.71)
We express the matrix elements with vanishing current by wave functions
and energy levels in the standard way (recall equation C.9))
(X'T'\x't') = J2 МХ'ЖМ ^Р \-^En(T' - t')] , C.72)
(xt\XT) = Y, *m{xWJX) exp \~Em(t - T)] . C.73)
Now we rotate the time axis by an angle of тг/2—we consider imaginary
times (analytic continuation)—and we take the limits T —»ioo, T' —> —ioo.
Then we just project the groundstate contribution
T'-*-i
^E0T'] (Х'Т'\х'*) = МХ'Ж(х',1г), C.74)
Г i 1
lim exp \--E0T\ (xt\XT) = ipo(x,t)ipZ{X). C.75)
Г-ioo [ П J
T
For the transition amplitude C.71) hence it follows that
lim (X'T'\XT)J
T'-»-ioo,T—ioo ' '
T)\
lT'-.-too,T-»too
Idx'dx Vo(s'it') (x't'\xt)^0(x,t). C.76)

Quantum mechanics 131
The integral now represents the quantity we are finally interested in
Idx'dxip*o{x',t'){x't'\xt)Jipo{x,t)
= Idx'dx @,t'\x') (x't'\xt)J (x\0,t)
= (O,t'\O,t)J t'^°^~o° @,oo\0,-oo)J. C.77)
Since the wavefunction-energy factor in equation C.76) is just a number
which does not matter we get
@,oo|0,-oo)J~ ( lim (X'T'\XT)J C.78)
and inserting the path integral C.70) we find the Z[J] functional explicitly
1 I dt(L + HJx)
C.79)
Alternatively, instead of rotating the time axis we could also add a term
— \iex2 to the Hamiltonian, or \iex2 to the Lagrangian, in order to isolate
the groundstate contribution. Then the Z[J] functional denotes
Z[J] = Id[x(t)]exp U / dt(L + hJx + \iex2)\ . C.80)
We will use this last form for a generalization to QFT.
Derivatives of Z[J] or time ordered products of operators: What
we finally need as analogues for QFT are time ordered products of operators.
Let us begin with the matrix element of the operator x(tx)¦ Discretizing the
time analogously to before (see equations C.43) and C.44)) into JV + 1 equal
pieces between a fixed initial point (ж,,^) and a fixed final point (x/,tf),
we decompose the matrix element
ti) = / dxN ...dxi (xftf\xNtN) (xNtN\xN-itN-!)...
... (xKtK\x{tK)\xK-xtK-x).. ¦ (xiti\xiti). C.81)
We now replace the operator by its scalar eigenvalue which we denote by
the same symbol. We use for each segment transition the Dirac-Feynman
equation C.46) and we perform the continuum limit N —> oo, At —> 0 which
implies the path integral

132 Path integrals, FP method and BRS transformation
(xftflxitK^XiU) = f ' d[x(t)]x{tK)exp \lj'dt bdaee(a;,*)| • C-82)
Note, on the l.h.s. х(Ък) represents an operator whereas on the r.h.s. it is a
scalar function.
Next we consider the matrix element of two operators x(ti()x(tt,). For
tf >tu > tt, > U we get
(xftf\x{tK)x{tL)\XiU)
= / dXN ¦ ¦ •dX\(Xftf\XNtN){XNtN\XN-itN-\) • • ¦
• • • (XKtK\x{tK)\xK-itK-i) . . . (xLtL\x(tL)\XL-ltL-l) ¦ ¦ •
...(xit^XiU) C.83)
which leads by analogous argument to the path integral
(Xftf\x(tK)x(tL)\Xiti)
= f ' d[x{t)]x{tK)x(tL) exp U J ' dtLdass{x, x)) . C.84)
On the other hand, for t/ > tt, > tn > U the operators on the l.h.s. are
reversed whereas the PI on the r.h.s. remains the same (there ж(?ь)ж(к)
are interchangeable scalars). Consequently, the time discretization provides
a time ordered product automatically
= J ' d[x(t)\x(tx)x{t2)exp U J ' <ftLdaee(a;,±)l , C.85)
where T represents the familiar time ordering for the operators, say A
andB,
T[A(t!)B(t2)] = 9(ti - t2)A(ti)B(t2) + e(t2 - ЬЩЬЩЬ). C.86)
The generalization to n operators is straightforward
(xftf\T[x(ti)x{t2)...x{tn)]\xiti)
rxf \i fff 1
= / d[a(«)]i(ti)a;(t2) ¦ • • x(tn) exp - / dtLdass(x, x) .
C.87)
Functional derivatives: Next we want to explain the derivatives of a
functional and, respectively, its Taylor series expansion.

Quantum mechanics 133
Definition: Let / be a well-behaved test function (infinitely differentiable
and decreasing faster than any power of |x|-1) then a functional F is defined
by
Fife C°°(M) -» F[f] e R, С C.88)
By analogy with ordinary functions the expansion of a functional in the
vicinity of an element /
^p + O(?2) C.89)
defines the derivative
=й&i{F[f+eh] -F[f]}- (з-9о)
For the special choice of
h(x)=6(x-y) C.91)
the functional derivative can be written as
^ШМ = lun±{F[f(x)+e6(x-y)}-F[f](x)}. C.92)
Examples:
i)
F[f] = jdxf(x) C.93)
= Yim-?{Jdx[f(x)+e6(x-y)]-Jdxf(x)}
= fdx6{x-y) = l. C.94)
f
F[f](x) = J dzG(x,z)f(z) C.95)
1 = Vm±{JdzG{x,z)[f(z)+e6(z-v)]
- J dzG(x,z)f(z)}
= JdzG(x,zN(z-y) = G(x,y). C.96)

134 Path integrals, FP method and BRS transformation
iii)
F = l, F[f}(x) = Jdy6(y-x)f(y) = f(x) C.97)
= lim i{/(x) + eS{x -y)- /(*)} = 6(x - y). C.98)
We again consider the derivative of a functional as a functional; we
differentiate once more and we obtain the second derivative and successively
all higher derivatives. Then the Taylor series expansion of a functional
is given by
n=0
1 f,
—: I dx\... dxn-
n\J i
6nF[f]
-h{xi)...h{xn). C.99)
l6f(x1)...6f(xn)
Now we return to our Z[J] functional expressed as a PI C.80)
/Г i Г°° 1 1
d[x(t)] exp\- / dt[L{x, x) + hJ(t)x(t) + -iex2{t)\ \ C.100)
l П J-oo 2 J
and we differentiate according to the above rules
6J{h)
¦jexp
- exp
\ Г dt[h(J(t) + ed{t - t!))x(t)
." J-oo
г-j™jt[hJ{t)x{t) + ...]] |.
(З.Ю1)
Expanding the e exponential provides
=iJd[x(t)]x(t1)eXp [\f
HJx + \ie
The higher derivatives then supply the result
C.102)
exp |^ /°° dt \b{x,x) + Mx + \iexA \ . C.
103)

Scalar field theory 135
This PI represents the T product of operators (recall equation C.87))
and the additional term \iex2 just projects the groundstate (recall equation
C.80)) so that we end up with the vacuum expectation value of the time
ordered operators
6J(t1)...6J(tn)
~ in@, oo\T[x(ti).. .x(tn)}\0, -00). C.104)
j=o
The symbol ~ means equality up to a normalization factor. The relation
C.104) represents the important analogue for QFT which we are going to
discuss in the next sections.
3.2 Scalar field theory
3.2.1 Free scalar fields
Instead of trajectories x(t) of a particle we now consider real scalar fields
ф(х) which depend on the space-time point x = (t,x,y,z). The action is
given by
5= IA4хС(ф,д11ф), C.105)
with С being the Lagrangian density. We also suppose that the fields ф(х)
have a source J(x) in the sense of Schwinger. Let us now find a path inte-
integral over these scalar fields. We search the Z[J] functional, an expression
analogous to equation C.80), by proceeding as in the quantum mechanical
case. We discretize the 4-dimensional space-time and we refine the discreti-
discretization considering the limiting case as the PI. So we can essentially take
over formula C.80) by replacing d[x(t)] —» d[<^(a;)] =: йф (we also choose
h = 1) and we obtain the path integral over scalar fields
Z[J}= f <1фещ>и l алх[С{ф,д11.ф) + Зф + -1еф2у>. C.106)
Again, it represents the vacuum-to-vacuum amplitude in the presence of the
source J
Z[J] ~ @, oo|0, -oo)J. C.107)
Now we study the free case
Z0[J]= f#expji (в.х[С0 + фЗ+уеф2]\, C.108)
with Co being the free Lagrangian
Co = i (д^фд^ф - т2ф2) C.109)

136 Path integrals, FP method and BRS transformation
and with the equation of motion, the Klein-Gordon equation
(а + т2)ф = 0. C.110)
For simplicity we write from now on just dx for the 4-dimensional (n-
dimensional in general) volume element. We integrate the kinetic term in
Co by parts and we obtain the Zo[J] functional for free scalar fields
Z0[J}= I dфexpl-i Idx{]-ф{O+m2-ie)ф-Jф}\. C.111)
Note: The ?-regulator term which guarantees the vacuum-to-vacuum tran-
transition also provides the correct Feynman propagator for the field ф where
the mass acquires an imaginary part m2 —> m2 — ie.
Euclidean space: On the other hand, we also achieve the vacuum-to-
vacuum transition by rotating the time axis by an angle of тг/2 (see Section
3.1.5). Defining
ixo = xA and ~~%д° = f4 C.112)
—IKq — /C4
we switch to Euclidean space so that we get
x2 =
k2 = kl-k2 = -{k\ + kl + kl + k\) = -k2E C.113)
D = д2-Ч2 = -{д2+д2 + д2 + д2) = -иЕ
and for the volume elements
id4x — d4XE =: dxE
-idAk = dikE=:dkE. C.114)
Then the Zq [J] functional in Euclidean space is
Zo[J] = Jdфexp{-JdxE \±(дЯфJ + \т2ф2 - Jtfj } . C.115)
The exponent is negative definite so that the path integral can converge.
Alternative representation: When working with the Z0[J] functional
it is more convenient to modify it. In the PI the ^-integration extends over
all possible fields, they do not obey the Klein-Gordon equation. Now we
shift the field

Scalar field theory 137
ф^ф + фо C.116)
by an additional field фо which satisfies the Klein-Gordon equation with
source
Vф0 = J, C.117)
where
V := D + m2 - is. C.118)
The solution is
фо(х) = - JdyAF(x-y)J(y), C.119)
where the kernel AF(x — y) represents the scalar Feynman propagator, the
causal Green function denned by
VAF{x-y) = ~8{x-y) or P4(i) = -AF(i) C.120)
and has the integral representation
Jbn;- <ЗЛ21>
Then the exponential in Zo[J] C.111) changes to
±фЪф - Зф ф-^Хфо Х^фЪф + i J J dyAF(x - y)J(y). C.122)
so that the Zo[J] functioned for the free case can be rewritten
Z0[J] =Afexp \-г- f dxdyJ(x)AF(x-y)J(y)\ , C.123)
with the normalization
N= f dфeщ) \-^ f^хфЪф\ . C.124)
The ф-пеЫ dependence has factorized and can be absorbed by the normali-
normalization. What remains is just an exponential of the sources connected by the
scalar propagator. This expression C.123) is especially suitable for series
expansions and will be needed later on.
Gaussian integration: Another notion which proves important, especi-
especially in connection with gauge theories and anomalies, is that of a determi-
determinant of a differential operator. We will introduce it here. Path integrals of
the above type, equation C.124), are called Gaussian and can be expressed

138 Path integrals, FP method and BRS transformation
in the following way:
A Gaussian formula can be written as the determinant of the matrix A
=(det A)~1/2- (зл25)
Actually formula C.125) is valid for any real symmetric matrix A.
Extending the above formula to the complex space Cn we begin with
n = 2 in equation C.125) and transform the variables to complex ones
z — x + iy, z* = x — iy, then we get
dz* dz _„,., _,
B«)Va B«)
The generalization to n dimensions is straightforward and gives
/
dnz' dnZ -'••,Az) = (debA)-lt C127)
Bтп)п/2 Bтгг)"/2
where Л is a positive definite Hermitian matrix operator.
Shifting the integration by
x
z -» z-A~xu C.128)
we obtain the Gauss-like formulae
/
Bтг)"/
= (det A)-i/2 ei(b,A-4) C 129)
dnz* a z c_(z.t
C.130)
These formulae, valid in a finite, n-dimensional vector space, can be
generalized to an infinite-dimensional function space {0} with the measure
d4> or d4>*. There the inner product with some operator A is defined by
(ф, Аф) = J dxdy ф*(х)А(х - у)ф{у) C.131)
or
(ф, Аф) = f dx ф*{х)А(х)ф(х) C.132)
if the operator A is diagonal
A(x - y) = A(xN(x - y). C.133)

Scalar field theory 139
Returning to our Z0[J] functional C.123) we consider the normalization
Л/* C.124) and notice that it is a Gauss integral
N = ( d<pexp УЛ(ёх фЯ)ф\ = (deti?>)-1/2, C.134)
which can be expressed by the determinant of the Klein-Gordon differential
operator W C.118).
If we assumed complex scalar fields the normalization would be
dф*dфexp - / dx ф*Фф\ = (detiP). C.135)
Note: The Gaussian formula C.129), generalized to infinite dimensions,
serves as a connection between the two expressions C.111) and C.123) for
the Zo[J] functional.
3.2.2 Free Green functions
Now we show that the Zo[J] functional—the vacuum-to-vacuum transition
amplitude—just represents the generating functional for all free Green func-
functions.
First we normalize expression C.107) according to our boundary condi-
conditions. We choose the strict identity
Zo[J] = @,oo|0,-oo)J C.136)
then Z0[J] is automatically normalized to unity for J = 0
Z0[0] = 1. C.137)
Therefore the PI C.111) must be normalized by the constant factor Af
C.124)
Z0[J} = ^Jdфexl>!-iJdx[±фVф-Jф}\ C.138)
so that
= exp j-i fdxdy J(x)AF(x - y)J(y)\ . C.139)
How can we interpret this exponential C.139)? Expanding the exponen-
exponential we obtain the following series
ад . i-i/ (|) [/f (|) [J}
C.140)

140
Path integrals, FP method and BRS transformation
(the integrals are written in an obvious short-hand notation) which we de-
describe diagrammatically as
and
AF(x-y)=
= *
C.142)
So the vacuum-to-vacuum amplitude contains the propagation of 0, 1, 2, 3,
... particles between sources.
Now we turn to the Green functions or n-point functions of the
theory. They are denned by the vacuum expectation value of the time or-
ordered field operators. Generalizing the relation C.104) to field theory we
obtain
=: то(хи...,хп).
C.143)
So the quantization of the fields is given by a path integral.
On the other hand, the Taylor series of the functional Zq[J] is determined
by its derivatives (recall equation C.99))
n=0
oo
8Jn
Jn
j=o
— dx!... dxn
8J{x1)...8J{xn)
j=o
¦J(x1)...J(xn)
C.144)
and with the above definition C.143) for the Green functions Zo[J] becomes
the generating functional for all (free) Green functions
o\J] = у —7 I dxi.. .dxn J(xi)... J(xn)To(xi,... ,xn).
*—' n\ I
n=0 J
C.145)
This series has its diagrammatic correspondence (n = 2 for 1. diagram,
n = 4 for 2. diagram, etc.) in equation C.141).
It is also convenient to rewrite the series C.145) in the following way

Scalar field theory 141
°° ,-n f
Z°[J1 = 52-lJdx1...dxnJ(x1)...J(xn)(O\T4>(x1)...4>{xn)\O)
n=0
= <O|Texp[i ( dxJ{x)<j>(xM |O). C.146)
Examples:
i) 2-point function (for functional differentiation see Section 3.1.5)
0 8*Z0{J]
Ti2 — 77
= iAF(x1-x2)
= * х~Г~Х2 ' C-147)
where we use the short-hand notation Tf2 = tq{x\, x%) and J, = J(xi),
г = 1,2,....
ii) 4-point function
0 8*Z0[J]
T1234 =
J=0
f(x2-x3)] C.148)
or diagrammatically
Ti°234 = - [ = + II + У{ ] = -3 = ¦ C-149)
Remark: We notice that any odd n-point function vanishes
Tb(a:i,...,a:2n-i)=0 C.150)
since there exists a factor proportional to the source J-»0. So we are left
only with even n-point functions which are products of 2-point functions
tq(xi,. .. ,X2n) = 2_^ To{xPl,xPa).. ¦To(xP2n_l,Xp2n). C.151)
perm
This result of the PI formalism relying on the rules of functional differentia-
differentiation corresponds precisely to the Wick theorem in the canonical quantization
formalism.

142 Path integrals, FP method and BRS transformation
3.2.3 Interacting fields
Next we assume that the scalar fields also interact and for simplicity we
choose the ф4 selfinteraction
\ \т2ф2 - |04. C.152)
Then analogous to the free case C.136), C.138), the generating functio-
functional of the interacting fields is
Z[J] = @,oo|0,-oo)J C.153)
= jj f #exp|i f dx[Co + Ci + J<t>]\ , C.154)
with the normalization (numerator for J = 0)
Л/"= /ctyexpji fdxlCo + сЛ C.155)
(or Z[0] = 1). For the vanishing interaction we come back to the free case
which we discussed before
Z[J] 3-^ Z0[J]. C.156)
Perturbation theory: In order to calculate the Green functions pertur-
batively to a certain order in the coupling constant g we have to reformulate
the path integral C.154) in a suitable way.
Proposition:
• The perturb&tive series of Z[J] is supplied by
Z[J] = ±exp![iJdxCI (ij^ZoiJ], C.157)
where Zq[J] is given by expression C.139) and the normalization N
represents the numerator for J = 0.
Proof. For quick reasoning we separate the interaction
Z[J] = ^ fd4>expli fdxCi{4)\expli Idx[C0 + W)\. C.158)
Considering now the first exponential as a power series in the variable ф we
may replace ф by the derivative with respect to J

Scalar field theory 143
*^ir (ЗЛ59)
which will act on the second exponential
C.160)
Recipe: Perturbation theory is established by treating the exponential
as a power series in the interaction, in the coupling constant g. We apply
the functional derivatives to Zq\J] to each order in g. The Green functions
finally are calculated from the derivatives of the total functional Z[J].
Example: To order g we have
1
l--?
We must evaluate the fourth derivative of Zo[J]. This we do successively:
Z0[J] =
+ | / JbF(x-z)\ >exp
J L ^ J
C.162)

144 Path integrals, FP method and BRS transformation
In diagrams the fourth derivative is
(влез)
= {-3 00+
The coefficients 3 and 6 are symmetry factors and arise from joining the
lines of the third diagram in all possible ways.
The normalization Af equals the numerator of equation C.161) for J = 0
so that all diagrams including the sources vanish and we are left with
C-164)
Finally, expanding Л/" to order g we obtain the final result for the gene-
generating functional Z[J) to order g
X ]}[4/]
C.165)
Remark: All vacuum diagrams have cancelled and do not occur in the
properly normalized functional! This is valid to all orders in perturbation
theory.
3.2.4 Green functions for interacting fields
Analogous to the free case we gain the Green functions for interacting
fields by calculating the derivatives of the functional Z[J] C.154)
=: r(a;i,...,a;n) C.166)
and Z[J] represents the generating functional for all Green functions
n=0
~ dxi...dxnJ(xi)...J(xn)T(xi,...,xn)
= @|Texp \i f dx J(x)<t>(x)] |0), C.167)
with <f>(x) an interacting field.

Scalar field theory
145
2-point function: Calculating, for example, the 2-point function, the pro-
propagator, to order g we start from the expanded generating functional C.165)
and we evaluate the second derivative with respect to the source. The first
term corresponds to the free propagator and the second term containing two
sources provides the correction to order g. The third term with 4 sources
does not contribute since after differentiation there always remains a source
factor which vanishes in the limit J —> 0. Then the correction to order g is
T12 =
82Z[J]
J=0
= -2!|Af@) I dz AF(xi - z)AF(x2 - z)
or diagrammatically
Altogether we get for the propagator to order g
712 = i
C.168)
C.169)
C.170)
Mass renormalization: The second term, however, is a quadratically
divergent quantity. It shifts the mass of the scalar particle. Here we meet
the simple example of mass renormalization.
In momentum space the propagator C.170) gives
using the operator expansion
we get
A-B~ A ' A A ' ••¦
ri2(fc) =
k2 — m2 — 8m2 k2 — m2R'
So we have renormalized the mass m2 by the amount
8m2 = -^Af(O)
to achieve the physical mass
C.171)
C.172)
C.173)
C.174)

146
Path integrals, FP method and BRS transformation
C.175)
Similarly other divergent terms which occur in perturbation theory can
be removed by renormalization. We consider the general procedure of renor-
malization, however, beyond the scope of this book and we refer the reader
to the standard textbooks on QFT.
4-point function: The other instructive example is the 4-point function.
Let us calculate it to order g. Starting again from the generating functional
C.165) we calculate each term separately:
Ti234(l- term) =
= T
1234
= -3
ri234B. term)
ie J\i8J2%&JsiSJ4
_
f dz AF{Xl -
or diagrammatically
= -Щ
exp
j=o
[4/H
C.176)
- z)AF(x3 - x4) C.177)
C.178)
П234C. term)
<54 . ,ig f , xx [if,. _]
= : : : (~) / "-z v/V eXP / JAFJ\
= -4!^ / dz AF(xx - z)AF(x2 - z)AF(x3 - z)AF(x4: - z\
or diagrammatically
The factorial 4! results from the permutations of equal terms.
Altogether we obtain for the 4-point function to order g
T1234
j=o
= -3
-"X-
C.181)

Scalar field theory
147
We observe that the first and second diagrams are disconnected. The second
diagram just modifies the propagator. The third diagram is the connected
one and yields the nontrivial part for the transition amplitude. (If we had
not normalized the generating functional C.161) by Л/" we would have to
add an additional diagram
^
00
containing a vacuum diagram^)
n-point function: Generally, we find the n-point function within per-
perturbation theory in the following systematic way:
6nZ[J]
in 6J(x!)...6J(xn)
j=o
Со
г i*dx С1{ф) e*I'dx Co
.ф(Хп)ЕГ=о ё Udz
J=0
J=0
6
i6J(x\) ' '
¦ iSJ(xn) l^k-0 kl
Г г и- г (
s \]k
[i " I\i6J(z))\
( \Лк
J=0
Zo[J]
J=0
C.182)
The functional derivatives are applied to Zq [J] which is given by expression
C.139). The series we evaluate to the order к finding the Green functions
to fc-th order.
3.2.5 Connected Green functions
We are only interested in nontrivial transitions—in connected Green func-
functions.
Definition: The quantum action' W[J] is defined by
W[J]:=-ilnZ[J] or
C.183)

148 Path integrals, FP method and BRS transformation
Proposition:
• The 'quantum action' generates only connected Green functions!
This remarkable feature is true quite generally in quantum field theories.
However, we want to demonstrate it just for our previous examples.
Examples:
i) 2-point function
62W[J]
6 -i6Z[J]
5Ji Z 6J2
62Z[J]
i 6Z[J) 6Z[J]
Z2[J] 6JX 6J2 Z[J]6Ji6J2'
C.184)
In the limit J —> 0 the first term vanishes (odd derivatives vanish)
and recalling the normalization Z[0] = 1 we have
62W[J]
6JX6J2
, 62Z[J)
%
j=o
ig_
2
C.185)
C.186)
Thus we get from W[J] the propagator (to all orders in perturbation
theory) which is clearly a connected diagram.
ii) 4-point function
— г
62Z[J] 62Z[J]
.7=0 lZ2[J\
1
+ 2 permutations -
Z[J]
+ 7-12734 + 7-14T23 - Ti234]-
j=o
C.187)
Considering diagrams up to order д we find from the propagator
C.170) the product
-ig
C.188)
Using also result C.181) for П234 we obtain

6J16J26J36J4
Fermion field theory 149
= i -3 - Згр
j=o
+ 3 = + 3ig
Only the connected (or irreducible) diagram remains! This continues
to all orders in perturbation theory.
Connected n-point function: Generally the connected n-point func-
function is
C.190)
ln 6J(Xij...6J(Xn) J=o
and, conversely, the 'quantum action' (W[0] = 0 by normalization)
v~^ Vх f
iW[J] = }. -7 dxi...dxnJ(xi)...J(xn)Tcoan(x1,...,xn). C.191)
n=i n- J
Green functions which are connected proper vertex functions—Feynman
diagrams that remain connected when cutting an arbitrary internal line—
are called 'one-particle irreducible Green functions'.
3.3 Fermion field theory
In QFT the Fermi field operators obey the familiar anticommutation relati-
relations. In the PI approach, however, we deal with classical fields and classical
sources which are just c-number functions. Nevertheless, in the path inte-
integral over Fermi fields these fields (and the sources) must be regarded as
anticommuting. Such anticommuting functions are elements of the so-called
Grassmann algebra which we will introduce in the following section.
3.3.1 Grassmann algebra
Definition: The generators 6i (i = 1,..., n) of an n-dimensional Grass-
Grassmann algebra are elements which satisfy the anticommutation relation
This implies the important identity
в1 = 0. C.193)
For this reason a function depending on Grassmann variables can be expan-
expanded into a series which contains only a finite number of terms; for example

150 Path integrals, FP method and BRS transformation
in one dimension
/{в)=а + Ъв. C.194)
Differentiation: For the operation of differentiation we can introduce two
types:
the left derivative
= bkOi - OkSa C.195)
the right derivative
dR
61квг. C.196)
For both left and right differentiation we have
T^rAJ = 6ik C.197)
Integration: For the integration over Grassmann variables there occurs
a problem, however. Since
2
for both L and R there exists no inverse operation to the differentiation.
But we can overcome this problem by imposing the following Grassmann
integration rules:
f dOiOi = 1 and f d6i = 0. C.200)
The so-defined Grassmann integration acts like the differentiation. It is de-
defined as a linear algebraic operation.
We shall find finally that a Grassmann PI formalism mimics the boson
PI approach described before and that we achieve a Grassmann integral
representation for the determinant of a differential operator which we are
interested in.
For example, we can define a Grassmann 6-function by
/¦
• 6F - <9')/@) = Г(в') C.201)
j
implying
6F -6') = 6-в' C.202)

Fermion field theory 151
and a Grassmann momentum тг by
dir еш = f dn(l + тб) = i6^ г<5@). C.203)
Both equations C.201) and C.203) simulate the usual result for R- or C-
functions (in equation C.203) up to a normalization constant).
Gaussian integration: Next we evaluate Gaussian integrals which we
will need in our PI formalism.
I
Lemma: The value of a Gauss integral is
d6d6 e~Se = I d6d6 eSe = 1. C.204)
Proof. Let us work in two dimensions; there we have
fl=(fla)' *= (Ma) C.205)
yielding the products
00 = @Ь 02) (вв)= Mi + M2 C.206)
F6J = (Mi + 0202)@i0i + 02в2) = 201010202- C.207)
Higher powers of the product 66 vanish because of identity C.193). There-
Therefore the Gauss function terminates with the quadratic term
@0). C.208)
We then evaluate the Gauss integral by the expansion
Id6d6 e~9e = f d61d61d62de2(l - Mi - 0202 + 6X6\6262)
= I d62d6~id6id62 02010102 = 1. Q.E.D. C.209)
Lemma: For a change of the integration variables like 0 = Xa the Grass-
Grassmann measure transforms with the inverse Jacobian (e.g. in 2 dimensions)
d91d92 = -^—daida2. C.210)
detA

152 Path integrals, FP method and BRS transformation
Proof. Let в = Xa; for the product 0i02 we find
X22CX2)
= detX-aia2. C.211)
Preserving the identity
/ d02d0i 0i6>2 = / da2dai ага2 C.212)
implies for the measure
d62d6i = -—-da2dai. Q.E.D. C.213)
detX
Formulae: Now we transform the Gauss integral C.204) according to the
changes 0 = Xa and 0 = aY
we define
FX =: D C.215)
and recall the properties of determinants
det X det Y = det ХУ = det ?> C.216)
and we end up with the Gauss integration formula for Grassmann va-
variables
[ dada eaDa = f dada e~&D<x = det D. C.217)
So here the Gaussian integral is directly proportional to the determinant
of the operator D in contrast to the usual case, equation C.127). Clearly,
formula C.217) can be generalized to n-dimensional Grassmann vectors. For
D = 1 we reproduce the normalization C.204).
Shifting the integration variables next by
a -> a-D-l(, = a' C.218)
a -> a-lD~1=a' C.219)
the new exponent contains
a'Da' = (&-iD-l)D{a-D-lZ)
C.220)

Fermion field theory 153
and we obtain the Gauss-like formula
/ dada exp[-aDa + fa + af] = det D ¦ expIfjD^], C.221)
which should be contrasted with the result C.130) for C-functions.
Generalization to infinite dimensions: The above formalism can be
generalized to infinite dimensions. Let 6(x) be the generator of such an infi-
infinite dimensional Grassmann algebra obeying the anticommutation relation
O, C.222)
with
62{x) = 0, C.223)
then we have the following rules for differentiation and integration
6L'R6(x)
Щу)
6L
= 6{x-y), C.224)
60(z)
and
0(х)в(у) = 6(х-г)в(у)-в(хN(у-г), C.225)
в(х)в(у) = в{х)8{у-г)-8{х-г)в{у), C.226)
f d6{x) в{х) = 1, I d6{x) = 0. C.227)
The Gaussian integrals give
/ de{x)d6{x) e'{B'De) = detD, C.228)
[ de(x)d6(x) e-&DV+U'e)+(S'V = detD-e«~-D~4 C.229)
with
(в, DO) = fdxdye{x)D(x,y)e(y)
= fdx в{х)Б{х)в{х) if D(x, y) = D(xN{x - y),
(i,0) = Jdx?(xH(x),

154 Path integrals, FP method and BRS transformation
@,0 = J dx 6(x)?(x),
(ZiD-1® = Jdxdy?(x)D-\x,y)Z(y), C.230)
and D some invertible operator.
3.3.2 Dirac fields
We begin with the free Lagrangian for fermions
Co = фг $ф- тфф C.231)
corresponding to the equation of motion, the Dirac equation
(г @ - т)ф = 0. C.232)
We develop the path integral formalism analogously to the scalar case
C.108), C.124), C.138), C.139), and construct the generating functional
for free Dirac fields
Zo[v,v] = jj /##exp{i j dx${i 0 - т)ф + цф + фг])}, C.233)
with the normalization
Я = I d^exp[i I dx ф(г @ - т)ф]. C.234)
The Grassmann functions J?(x), fj(x) represent the Schwinger sources for
the Dirac fields ф(х), ф(х).
We search an appropriate form to evaluate a perturbation expansion.
Defining the Dirac operator by
г ft-m=:D, C.235)
its inverse expresses the Feynman propagator
DSF(x-y) = 6(x-y) or D-\x) = SF(x). C.236)
The fermion propagator Sf is connected with the scalar propagator Ap,
equations C.120), C.121), by
SF(x) = (i @ + m)AF(x) C.237)
and has the following integral representation

Fermion field theory 155
= B^ У ~7^Г = B^ I pi-m> C'238)
(we suppress the ?-term in the Feynman propagator). Applying the Gaussian
formulae C.228)-C.230) we can rewrite the normalization C.234)
Af= I drpdil;ex.pl~(ф,\Dip)] = deb-D C.239)
J L l J г
and we obtain the convenient formula for perturbation expansions
Z0[ri, fj] = exp I-г j dxdy fj(x)SF{x - уНу)] ¦ C.240)
Expression C.240) is the fermionic analogue of the scalar case C.139).
Propagator: For example, the 2-point function—the free propagator—is
determined by the second derivative of Zo
{-) / dx r)(x)SF(x - x2) ехр[-г / i
= iSF(x1-x2). C.241)
Interacting fields: Now we incorporate the interaction described by the
Lagrangian
? = Co + Hi C.242)
and
Ci = -еф^фАц for QED, C.243)
d - гд3ф^ТафАа^ forYM. C.244)
We consider the photon field Лй or the Yang-Mills field (YM) A^ for the
moment as external fields. T° denote the generators of some group.
Then the generating functional for interacting Dirac fields is
with
fhf \ C.246)

156 Path integrals, FP method and BRS transformation
We derive the suitable form for perturbation expansions analogously to the
scalar case (see Section 3.2.3). We separate the interaction term С1(ф,ф)
and replace the fields there by the derivatives with respect to the sources
ф-+т^, Ф-^tjz, C.247)
ior\ lor]
and we obtain the formula for perturbation expansions
Z[v,fj] = т?ехр г I dx Ci[—,ttz\\z0 [77,77], C.248)
N \_ J \wt) ior\)J
with ?0G7,77] being given by expression C.240) and the normalization H
represents the numerator for 77 = 77 = 0.
The Green functions to a given order in the interaction are evaluated
from the functional derivatives of Z[ri,fj]. The connected Green functions
follow from the W-functional
eiW[ri,fi] m Minkowski space,
?[77,77] H _ _ C.249)
e w\^<r^ in Euclidean space.
3.4 Abelian gauge fields
Next we turn to the gauge fields which cause a new complication. Let us
begin with the simple case of Abelian fields, with QED. When constructing a
PI for the generating functional Z we finally want to evaluate the propagator
of the fields. The propagator is the inverse of the operator in the quadratic
field part of the Lagrangian. Now, for gauge fields we also have the liberty
of gauge transformations
A I~.\ . Л („\ 1 Я \(*r\ Ci О^ГЛ
This, however, creates difficulties such that the inverse of the operator might
not exist.
From the previous discussion we would construct the following gauge
field path integral for the generating functional of the Green functions
/й] = / dAp exp < i / dx[C0 + J^A^} > , C.251)
with the free Lagrangian
Co = --.F^F^ C.252)

Abelian gauge fields 157
and the field strength tensor
Fpu = д„А„ - д„А„. C.253)
Here the external current JM represents the source for the gauge field intro-
introduced in the spirit of Schwinger. We consider for our purpose a pure gauge
field theory and disregard the interaction with the fermion fields. The La-
grangian C.252) can be rewritten into a quadratic field term by partial
integration
A) = ^(g^a - д„ди)А". C.254)
Proposition:
• The operator g^O — ЭЙЭ„ has no inverse!
Proof. Choose, for instance, a pure gauge
Av = dvk C.255)
then the operator C.254) has a zero eigenvalue
(а»*° - ад^л = (пам - айп)л = о (з.25б)
and cannot be inverted for this reason. Or consider the momentum space
9^O - ЭЙЭ„ 1д^" -g^k2 + к^ки C.257)
then the inverse must be of the form
agva + bkvka C.258)
and we have by definition
(-g^k2 + k^kv){agua + bkvk") = 8/. C.259)
This gives
-ак26„° + ak^k" = <5/ C.260)
which has no solution for a. Q.E.D.
Thus we cannot find the photon propagator in a straightforward way. The
reason is the following. In the PI C.251) the field integration covers aJJ pos-
possible field configurations, also those which are just gauge transformed fields.
Since the action remains invariant under gauge transformations—constant
along a gauge orbit—the path integral over the fields produces an infinity

158 Path integrals, FP method and BRS transformation
(the volume of the gauge group) for the functional Z and consequently for
the resulting Green functions. So it is the integration over these redundant
gauge equivalent fields which causes the PI to diverge. How can we solve
this problem? Obviously by fixing the gauge.
Gauge fixing: We select only one representative from each gauge orbit
which contributes to the PI.
It is convenient to impose the Lorentz condition
0mj4" = 0 C.261)
then we have the Lagrangian
? = \A»9ilvUAv. C.262)
Proposition:
• The operator g^O has an inverse which is the Feynman propagator
of the photon
DFnV{x) = -д„иАР(х,т = О), C.263)
with AF defined by equations C.120), C.121).
The Lagrangian C.262) can be rewritten
? = —F^F^ - \{d^f C.264)
C.265)
where Cg fix represents the gauge fixing term. Generally we may add
/•„_ (я л^\2 (Ч 9fifi^
~*Gfix — л \"U-** ) ^o.^uui
with a some parameter free to our disposal. Then the Lagrangian is
= Ii4" Ln + (~ - l) 9MaJ A". C.267)
Proposition:
• The operator
\ot ) \ct )

Faddeev-Popov method for non-Abelian gauge fields 159
has an inverse which is the photon propagator
p [ ^г] C-268)
We can check explicitly that
ik) = V- C-269)
In the literature we frequently meet the following parameter selection
а -* 1 : DF^(k) = - /м". Feynman gauge
a->0: DL^(k) = - ^2 ~^~ Landau gauge. C.270)
/С -p ??
The physical quantities do not depend on the parameter a.
3.5 Faddeev-Popov method for non-Abelian gauge fields
3.5.1 Yang-Mills theory
When 1954 Yang and Mills (YM) [Yang, Mills 1954] developed a gauge
theory for non-Abelian fields it was considered rather a curiosity. But nowa-
nowadays non-Abelian gauge theories turn out as the fundamental field theories
due to their rich underlying structure.
Let us start with a free fermion theory
L% = 4>{i$- т)ф C.271)
but in addition we assume that the fermions still have an internal degree
of freedom represented by a vector in some group space G. The famous
example describing the strong interactions is colour SUC) where
/ qp \ pink
ф = qy yellow C.272)
\ qb ) blue
and q denotes the spinor field for a certain flavour kind of quark (q =
u,d, s,c, b, t).
Gauge group: A group element g(x) € G transforms the spinor fields as
ф°(х) = д-1(х)ф(х)
ф°{х) = ф(х)д(х), C.273)
with

160 Path integrals, FP method and BRS transformation
g(x) = ел(х), Л(х) = Ла(:г)Та. C.274)
The matrices Ta are the generators of the group and satisfy the algebra
[Та,Ть] = /аЬсТс, C.275)
where the fabc are the totally antisymmetric structure constants. If
jabc = о we return to the Abelian case.
Here the matrices are chosen anti-Hermitian
= _Ta C.276)
and they are normalized by
tr ТаТь = -~ёаЬ. C.277)
Examples: The special unitary groups in N dimensions SU(N), where
gg] = 1, det# = l C.278)
play an important role in physics. There we have
Га = ^t for SUB) C.279)
Ta = ^r forSC/C), C.280)
сга denote the Pauli matrices a = 1,2,3
'-(Si) '2=(°"o) °*=
and Aa express the Gell-Mann matrices a = 1,..., 8
A1 =
A4=|000)A5=|00 0) C.282)
0 0-2
For SUB) the structure constants represent the antisymmetric e-tensor in
3 dimensions (o = 1,2,3)

Faddeev-Popov method for non-Abelian gauge fields 161
{1 even permutation of 123
0 indices equal C.283)
-1 odd permutation of 123
and in case of 5f7C) the nonvanishing structure constants have the following
values
/123 = 1
/l47 = —/l56 = /246 = /257 = /345 = ~/зб7 = -
/458 = /678 = :y- C.284)
The anticommutator, on the other hand, satisfies the relation
{Aa, A6} = Uabl + 2dabc\c, C.285)
О
where the dabc denote the symmetric structure constants
dabc = j tr {Aa,A6}Ac. C.286)
They follow from relation C.285) by multiplication by Ac and taking the
trace. Analogously we calculate from the commutator C.275)
/аЬс = 1 tr [Aa,A6]Ac. C.287)
The Jacobi identity of the generators
\Ta, [Tb, Tc}} + [Tb, [Tc, Ta]} + [Tc, [Ta, Tb}} = 0 C.288)
implies for the structure constants the relation
jabdfdce + fbcdjdae + fcadjdbe _ q C.289)
One also can construct matrices Ta from the structure constants fabc
according to
(Г°)Ьс = -fabc. C.290)
Such a representation of the Lie algebra, which is of the group dimension,
is called adjoint representation.
Gauge transformations: Now let us return to the Lagrangian C.271).
We want to keep this Lagrangian invariant under a local gauge group trans-
transformation

162
Path integrals, FP method and BRS transformation
)ф. C.291)
Analogous to the Abelian case of QED we introduce a gauge potential
field
A^x) = Al(x)T\ C.292)
which gauge transforms as
The field is brought in via minimal substitution
where the covariant derivative Du transforms as
Then the Lagrangian
= Ф{гР - т)ф
C.293)
C.294)
C.295)
C.296)
remains invariant under the gauge group transformations C.273), C.295).
In particular, for colour SUC), QCD, the gauge potential is a matrix in
colour-group space which contains all 8 gluon fields
\a 1
A - Aa— - —
" " 2i ~ 2i
( A% + -y=Al Al - iA\ A* - и
A*+iA*
- — As
. C.297)
In addition, we have to consider the motion of the YM fields given by
the Lagrangian
=Ьг
where
ц + [лц, Av\
C.298)
C.299)
C.300)
describes the YM field strength tensor. Since the F^v gauge trans-
transforms as
' C.301)

Faddeev-Popov method for non-Abelian gauge fields 163
the YM Lagrangian C.298) remains gauge invariant.
Note: A quantity transforming like equation C.293)—the gauge
potential—represents, geometrically, a connection and the corresponding
quantity C.300)—the field strength tensor—describes the curvature (recall
Section 2.7).
Theorem:
[?>M, Dv] = FliV. C.302)
If, however, the covariant derivative D^ acts on the adjoint group
space—on elements containing the matrix generators Ta—then there oc-
occurs a commutator in the definition of D^.
Definition:
?>M = д^ + [Ац, ] C.303)
D^A = ЭМЛ + [А,,Л], C.304)
with the Lie algebra valued element
A(x) = Aa{x)Ta. C.305)
In components we have
D*a = 0M?6° + fbacAc^ C.306)
(?>МЛ)° = D^Ab = ЭМЛ° + fad>A4Ab. C.307)
Theorem:
[DM, DV]A = [FMI/, A]. C.308)
Proof.
[D^jD^JA = DfiDyA - fj, <-» v
= д^диА + [AV,A]) + [A», {dvA + \AV,A])] -V^v
= [(дцАи - д„Ац), Л] + [Ац, [Av, A]] — [Av, [A^, A]].
Using the Jacobi identity C.288) for the last two terms gives
[Dft, DV]A = [{dftAv — dvAfj), A] + [[A^, Av], A]
= [FMI/,A]. Q.E.D. C.309)

164 Path integrals, FP method and BRS transformation
Lemma:
i)
a C.310)
ii)
Эм tr AiA2 = tr (?>MAi)A2 + tr Л^Лг. C.311)
Infinitesimal gauge transformations: If we just consider infinitesimal
gauge elements
g(x) = l+A(x) C.312)
then the above transformations lead to the infinitesimal gauge transfor-
transformations
вф = —Аф
ёф = фА
8А^ = D^A
SF^ = [i^.A]. C.313)
In contrast to the Abelian case the non-Abelian field strength is not gauge
invariant. It transforms according to the adjoint representation C.301),
C.313) and is therefore termed gauge covariant.
Equations of motion: Next we determine the equations of motion for
the YM field. We consider the total Lagrangian
С = СФ + Сум C.314)
then the Euler-Lagrange equations
Ц-а»«рг° <з-315>
provide the following YM field equations
D^F^ = jv C.316)
with the non-Abelian vector current
j, = fvTa C.317)
C.318)
(Note that we have chosen the coupling strength of the YM fields g, = 1.)

Faddeev-Popov method for non-Abelian gauge fields 165
Theorem: The non-Abelian current is covariantly conserved
Dvf = 0. C.319)
Proof.
= \[F^,F^}=Q. Q.E.D. C.320)
In the absence of matter fields (jv = 0), in the case of a pure YM theory
we have
D^F»" = 0, C.321)
which is still a nontrivial theory with selfinteracting fields.
Finally we mention the Bianchi identity for the YM field strength
tensor
DxF^ + D^x + DuFXli = 0. C.322)
It is a consequence of the Jacobi identity for the covariant derivative and
the commutator relation C.302), C.308).
The Bianchi identity can be rewritten in terms of the dual field
strength tensor
*F^ = \e»va0Fa0 C.323)
(eom _ !) then equation C.322) simplifies to
D^ * F»v = 0. C.324)
3.5.2 Faddeev—Popov determinant and ghosts
Now we are prepared to construct a path integral for non-Abelian gauge
fields. However, in the non-Abelian case we are confronted with additio-
additional problems which have to be mastered. The popular procedure with far-
reaching consequences for physics and geometry has been invented by Fad-
Faddeev and Popov (FP) in 1967 in order to quantize YM fields [Faddeev,
Popov 1967], [Faddeev, Slavnov 1980]. We shall describe it in the following.
As in the Abelian case the PI over non-Abelian fields
Z= f dA^eiS[A"] C.325)
is not well defined for the gauge invariant YM action
„] = J dxCYM{A»{x)) C.326)

166 Path integrals, FP method and BRS transformation
(we neglect the Schwinger sources for the moment) due to the gauge freedom
of the fields. In order to cure this defect we impose a gauge condition
/[Aft = 0. C.327)
We also assume that the functional equation C.327) has a unique solution
g(x) for any given gauge potential А^{х).
We incorporate this gauge condition via the following trick. Since
J
J fffl = 1, C.328)
with
l^ C.329)
we may write
Z = JdA» J' dgAFP^]S(f^])eiS^. C.330)
Now we perform a gauge transformation A9^ —» Ац (or g —» g~1); we use
the gauge invariance of
i) dA9^ = dA^, gauge potential measure,
ii) SlA9^} = 5[j4m], classical action,
iii) dg = d(gg'),g' e G, group measure,
and we obtain v
Z = J dg ¦ J dAn&FplAJSiftA»]) eiS^l C.331)
We observe that this expression is independent of the group element g so
that the group integration factorizes and produces an infinite constant (the
infinite volume of the gauge group). We remove this 'constant' by redefining
Z which we have to normalize anyhow.
Then the correct PI for the generating functional of pure YM
fields is given by
Z = jdA^AFP[A^]e(f[A^}) eiS^l C.332)
This is Faddeev-Popov's equation. We still have to evaluate the FP
determinant.

Faddeev-Popov method for non-Abelian gauge fields 167
Faddeev-Popov determinant:
= det
C.333)
9=1
It is enough to consider just infinitesimal gauge transformations g = 1 + Л,
equations C.312), C.313), then we have
л=о
and
Sf S Sf Sf
— =—-^-&4M = —?>M. C.335)
Choosing for definiteness the Lorentz condition
f[A^\ = д^А^ = О C.336)
we find
_J_ — я/* n to qq^^i
and in components
: - y). C.338)
Gauge fixing: The previous treatment of Z can be immediately extended
to a new gauge fixing function
f[Afl](x)-c(x) = 0, C.339)
with
c{x) = ca(x)Ta C.340)
some function not depending on A^.
Is there any change?
Lemma: The dependence of App(A^) on c(x) can be removed.
Proof: The dependence on с comes from a gauge element go : 3go such
that /[^°] = c, then we have Арр{А9^0). But we may write
= J

168 Path integrals, FP method and BRS transformation
where the second equality is true because the <5-function implies g = go-, and
the dependence of Afp on с has been removed. Q.E.D.
So we can prescribe the following generating functional
Z = J dA^FP^W^) - c) eiS^l C.341)
Next we integrate over the functions c(x) with some, say, Gaussian weight
which changes just the overall normalization constant
Z =
-^ fdxca(x)ca(x)] /"d^
= jdA» det^exp [iS^] - ± jdx/в
C.342)
Faddeev-Popov ghosts: Now we use another Faddeev-Popov trick. We
re-express the FP determinant by an integral over Grassmann fields. Recal-
Recalling formula C.228) of Section 3.3 we introduce the Grassmann fields
FP ghost: v{x) = va{x)Ta C.343)
FP antighost: v{x) = va{x)Ta C.344)
so that we get a Gaussian integral
det|y = / dvdvexp(-i f dxv^-v ) C.345)
6Л J \ J 6A J
(the factor г is absorbed by Euclidean space calculations). The fields va(x),
va{x) are scalar fields obeying Fermi-Dirac statistics. Note that va(x) and
va(x) are not related by Hermitian conjugation but represent two indepen-
independent Grassmann fields.
In this way we arrive at the following generating functional for pure
YM fields
Z = / dA^dvdv
= /
[ijdx [-l^jw - ^
•exp [j [^ ^ ^^
C.346)
If we finally insert the external sources a la Schwinger and include the

Faddeev-Popov method for non-Abelian gauge fields 169
fermions we obtain the generating functional for an interacting non-
Abelian gauge theory
Z[J, w, u>,rj,f}] = -j-f / dAftdvdvdrpdip
¦ exp (i f dx[?tot + Ja*Al + u>ava + vauja + fj-ф + fa]\ .
C.347)
The normalization TV denotes the numerator for vanishing sources J% =
ша = п>а = г) = fj = 0.
The total Lagrangian consists of
Aot = A/, + ?y m + ?g fix+ ?irP, C.348)
where the several parts represent:
Сф fermion term C.296)
Cym pure Yang-Mills term C.298)
^Gfix gauge fixing term C.349)
Cfp FP ghost term C.350).
Resume: We have reached our aim! By restricting the functional measure
with a gauge condition we obtain a gauge fixing term
?Gfix = ~(d»AD2 = - tr (9MMJ C.349)
га ^ а
and a nonsingular gauge boson propagator. We get rid of the unwanted
redundant gauge equivalent fields and the dependence on the gauge group
just drops. But there occurs the FP determinant which creates an additional
term-in the Lagrangian, the FP ghost term
CFp = -^d^V = -^"(Z?^ = 2 tr vd^D^v. C.350)
The ghost and antighost are massless scalar fields which propagate because
of the kinetic part
-vaUva C.351)
and they interact with the gauge potential according to
<. C.352)
Remark: There exists, however, a gauge where the FP ghosts disappear.
This is the axial gauge defined by

170 Path integrals, FP method and BRS transformation
/[A,,] = п^А„ = 0, п"п„ = -1. C.353)
Then the FP determinant does not depend on the gauge potential since the
functional derivative is
due to condition C.353).
In this case the FP ghosts decouple from the gauge fields and may be
integrated out and put into the normalization of the generating functional
Z. But the prize one has to pay is the rather complicated gauge boson
propagator in this gauge. For a field theory in the axial gauge we refer to
Kummer [Kummer 1976] and Leibbrandt [Leibbrandt 1990].
In QED the gauge boson—the photon—is group 'colour' blind and does
not couple to the FP ghosts. Here the FP ghosts disappear without reference
to a special gauge. Therefore the introduction of FP ghosts in QED is not
necessary—and not meaningful.
Physics and ghosts: The importance of the FP (anti-) ghosts for physics
and geometry is the following:
• FP ghosts are needed to preserve unitarity of the 5-matrix elements.
They precisely cancel the unphysical degrees of freedom of the gauge
bosons. For a discussion we refer to Aitchison and Hey [Aitchison, Hey
1989].
• FP ghosts re-establish an invariance of the total Lagrangian. The
gauge invariance (symmetry) is replaced by another adequate inva-
invariance (symmetry)—the BRS transformation. This is essential to esta-
establish the Ward identities (Slavnov-Taylor identities) in order to prove
the renormalizability of a non-Abelian QFT. We shall discuss the BRS
transformation next in Section 3.6.
• FP ghosts play an important role in the differential geometry of gauge
potentials. They are demanded in the study of anomalies in QFT. This
will be elaborated in this book. Together with the BRS operator the
FP ghosts have a geometric interpretation: the BRS operator repres-
represents a derivative, the FP ghost a Maurer-Cartan form on the group
(see Section 8.1).
• Also the FP determinant has geometric roots. Babelon and Viallet
found that it corresponds to a 'natural' Riemannian metric on the
moduli space of gauge potentials [Babelon, Viallet 1979, 1981].

Faddeev-Popov method for non-Abelian gauge fields
171
Spec/
f[A°]=O
/*~ section
Fig. 3.4. The principal bundle in Sp A, gauge orbits and gauge section
Geometry: How can we describe the gauge fixing /[А9] = О, equation
C.327), in a geometric way? Let us consider the afflne space of all
gauge connections Sp A. We denote the space of all gauge elements by
Q — {g(x)}—the gauge group space. Then we can construct a (infinite
dimensional) principal bundle V(A, П, M, Q) with fibre Q and the projec-
projection П : Sp A —> Sp A/Q. The base space Sp A/G—the moduli space—
represents the physical space of the YM theory. The field configurations A9^
determined by the gauge transformations C.293) for all possible elements g
to a given A^ are called an orbit of the gauge group Q. The gauge condi-
condition C.327) then defines a surface in the space of all gauge fields—a gauge
section—which crosses each gauge orbit precisely once (see Figure 3.4).
V
Gribov ambiguity and geometry: In the FP derivation of the gauge
field PI it was crucial to assume that the gauge condition C.327) has pre-
precisely one solution. Gribov [Gribov 1978], however, discovered that non-
Abelian fields—in contrast to Abelian fields—can have values such that the
gauge condition C.327) does not provide a unique solution for the gauge
element g(x)—the Gribov ambiguity. This happens beyond perturbation
theory and the PI representation C.346), C.347) is not meaningful there.
The geometric understanding of this phenomenon has been found by
Singer [Singer 1978] and is due to the nontrivial topology of the principal
bundle V(A, П, M, G) (for M = S4 and SU(N) gauge groups). Gauge fixing

172 Path integrals, FP method and BRS transformation
corresponds to choosing only one representative connection in the bundle
for each element of the base Sp Л/Q, see Figure 3.4. Locally this is always
possible—the delta function in the FP procedure selects from the PI over
the gauge fields only those connections which lie on a section—but globally
no such section exists due to the nontriviality of the principal bundle. So
there exist gauge configurations where a gauge fixing section is not well
defined—ambiguous a la Gribov. (For literature see e.g. [Troster 1994].)
3.6 BRS transformation
We consider again the total Lagrangian for a non-Abelian gauge theory
Aot = Сф + C>ym + Ajfix + ?>fp- C.355)
The fermion term Сф and the YM part Сум are gauge invariant by
construction. However, in the total Lagrangian gauge invariance is ob-
obviously lost because of the gauge fixing term Cqux- But there is still the
freedom to work with the FP ghosts. It was Becchi, Rouet and Stora
(BRS) [Becchi, Rouet, Stora 1974, 1975] who discovered an 'enlarged'
gauge transformation—the BRS transformation—which also involves the
FP (anti-) ghosts and leaves the total Lagrangian invariant.
How is the BRS transformation defined?
BRS transformation: Let us first assign to all fields a ghost number:
ghost number
+1 va{x)
-1 va{x)
О А11{х),ф{х),ф(х).
We define an operator—the BRS operator s—which acts on the usual
fields like a usual infinitesimal gauge transformation with parameter v, but
on the (anti-) ghost fields such that s is nilpotent
s2 = 0. C.356)
This is achieved by the BRS transformations
sv = --[v,v} = -v2
sxj) = —уф
s$ = -фу. C.357)

BRS transformation 173
Examples:
s2Ap — sB^v = d^sv + [sA^, v] + [A^, sv]
— D.Asv H—[гм>1) = О C.358)
s2t; = -svv = v3 - v3 = 0. C.359)
Graded algebra: In fact, the BRS operator s increases the ghost number
by one unit. This happens analogously to the exterior differential d which
raises the form degree by one unit. We can combine the exterior algebra
structure with the BRS structure. We add the form degree and the ghost
degree to get a total degree and we consider the algebra (generated by all
fields) graded by this total degree. Both operators s and d act as antideri-
vations on this algebra and they are required to satisfy
sd + ds = 0. C.360)
Generally, a commutator means
[P,Q] = PQ - (_)degpdeg0gp C.361)
and specifically for the 1-form A = A^dx11, 2-form F = ^F^dx^dx" and
FP ghost v
[A,v] = Av+vA C.362)
[F,v} = Fv~vF C.363)
[v,v] = vv + vv = 2v2. C.364)
Within differential forms we have the BRS transformations
sA = -Dv
sF = [F,v]. p C.365)
JVote: The third BRS equation C.357) reminds us of the Maurer-Cartan
equation on the Lie group (Section 2.6.6). We discuss Stora's [Stora 1986]
algebraic interpretation of the FP ghost separately in Section 8.3.2.
Now we are still left with the treatment of the antighost v. We want to
define its transformation such that the total Lagrangian remains invariant
and that s2 = 0 is respected. We can achieve this with the help of an
auxiliary field.

174 Path integrals, FP method and BRS transformation
Auxiliary field: We reformulate the Lagrangian by introducing a scalar
and commuting auxiliary field (for a review see e.g. [Gieres 1988], [Baulieu
1985])
b(x) = ba(x)Ta C.366)
and we incorporate the gauge fixing condition
/[;4M] = ЭММ = О C.367)
in the following way. We choose as the gauge fixing term
?Gfix = be/e[AJ + ^baba. C.368)
Taking into account the equations of motion for the field b
f=°> C-369)
with its solution
ba = _ifa^ = _±дРДа C.370)
we recover the original gauge fixing
So the auxiliary field has the interpretation of a Lagrange multiplier
for the gauge fixing condition. Due to its canonical dimension 2 it cannot
acquire a kinetic term in a (renormalizable) Lagrangian.
Rewriting the FP part next
T\aafa\A 1 C\ Ч70\
— —" &J I ul io.oi^i
we have for the sum
ol r . ¦.
Lqhx + i-Fp = 0 / [.А„] -) do — v Sf [А„\. (o.o/o)
2
In order to get it invariant under the BRS operator s we choose the trans-
transformations
sv = b
sb = 0, C.374)
which imply

BRS transformation
175
BRS
sA —
sF =
sv =
sv —
sb =
вф =
вф =
S2 =
transformations
-Dv
-[F,v]
-\\v,v]
b
0
—уф
—фу
sd + ds
= ~v2
= 0
anti-BRS transformations
sA = -Dv
sF = {F,v)
sv = — \[v,v] = —v2
sv = — [v,v] - b
sb = — [v,b]
ёф = —уф
Еф = —фу
s2 = sd + ds-0
ss + ss = 0
sv + sv = — [v, v]
Table 3.1.
OL,
= svafa\Ali)-vasfa[Ali) + -
and consequently
«(^Gfix + jCfp) = 0.
Thus we derived the following proposition.
C.375)
C.376)
Proposition:
• The total Lagiangian C.355) is BRS transformation invariant!
Anti-BRS transformation: In analogy with the oKS operator s we can
construct an anti-BRS operator s which decreases the ghost number by
one unit and has similar mathematical properties (for a review see [Baulieu
1985]). We collect the BRS and anti-BRS transformations in Table 3.1.
Remark: The algebra of Table 3.1 is called (anti-) BRS algebra. Its
structure is highly symmetric and holds independent of a Lagrangian. Thus
we can regard the BRS symmetry as a fundamental symmetry that a gauge
theory has to obey. In fact, the principle of BRS symmetry is used
as an alternative approach to construct generalized YM field theories (see

176 Path integrals, FP method and BRS transformation
e.g. [Baulieu 1985], [Baulieu, Thierry-Mieg 1982], [Alvarez-Gaume, Baulieu
1983]).
Outlook: As we have demonstrated the importance of the FP ghosts and
the BRS transformation lies in the restoration of a symmetry of the total
Lagrangian or of the classical action.
What about quantum field theory? Does the 'quantum action' remain
BRS invariant too?
The invariance of W = — i In Z
sW = 0, C.377)
called the Ward identity (Slavnov-Taylor identity) is necessary to get a
non-Abelian gauge theory renormalized [Becchi, Rouet, Stora 1976 a,b,c],
[Stora 1977].
However, it might happen that this invariance is broken. Then an anom-
anomaly appears and renormalizability is lost
SW = G ф 0. C.378)
Thus the BRS transformations are on the one hand a key to showing the
renormalizability of gauge theories and on the other they play an important
role in the determination of the anomalies. We want to discuss this role in
detail in the following chapters.

4
Anomalies in QFT
This chapter is a principal chapter in which we investigate the anomaly
within standard mathematical techniques. That means we work predomi-
predominantly in perturbation theory calculating Feynman diagrams with which we
assume the reader has a working knowledge. We show in great detail the
occurrence of the anomaly in QFT and discuss its meaning.
In Section 4.1 we start with the classical conservation laws and sym-
symmetries for Abelian and non-Abelian fields. Then we describe the quantum
partners—the Ward identities—in Section 4.2 and we discuss their violation
due to the anomaly. We devote Section 4.3 to the calculation of the Adler-
Bell-Jackiw anomaly and we demonstrate in detail how the anomaly arises
in several renormalization schemes. Although this part is somewhat tech-
technical, such a treatment is necessary for an understanding of the anomaly
and will repay careful study. Next (Section 4.4) we describe the anomaly
by means of dispersion relations and, furthermore, as a simple, yet illumi-
illuminating, example, we study the anomaly in the Dirac sea (Section 4.5). In
Section 4.6, making contact with physics, we calculate explicitly the famous
no _>t ^ decay, the milestone of the anomaly. In the next sections we extend
our discussion to non-Abelian fields, we determine the singlet anomaly (Sec-
(Section 4.7) and present the non-Abelian anomaly—Bardeen's result (Section
4.8). A final summary (Section 4.9) on the importance of the anomalies for
physics closes this chapter. ч
w
4.1 Classical conservation laws and symmetries
Let us first discuss the classical conservation laws and symmetries we ac-
actually need in connection with the anomaly.
4.1.1 Abelian fields
We consider QED with the Lagrangian
Cqed = Ф{г $ - m + e ?)ф - ^F^F^ D.1)

178 Anomalies in QFT
and the field strength
D.2)
We construct the currents:
vector jfj. = Фч^ф D-3)
axial jb = ф^цЧьф D.4)
pseudoscalar P = ф^ъф- D.5)
Conservation laws: With the help of the equations of motion—the Dirac
equations—
(г ф - m + e $)ф = О
tjj(i^ +m-efi) = 0 D.6)
we can establish the following conservation laws
= ixj){m — e
= 0 D.7)
дм.?м = гф(т - e
= 2imP. D.8)
In equation D.8) we used the anticommutativity of the Dirac matrices
{75,7м} = 0. D.9)
The vector current is conserved, equation D.7), the axial current too for
massless fermions
d^jl = 0 for то = 0. D.10)
Symmetries: Conservation laws are connected with symmetries, which is
the basic statement of the Noether theorem. Let us consider now as gauge
fields a vector field and an axial field which we denote by VM and A^. Both
fields are assumed to be external fields, and for simplicity we treat just the
massless case m = 0. So we start from the Lagrangian
?(VM) A.) = № 9 + У + -А 1ь)Ф D.ii)
(the coupling constants are chosen unity). This Lagrangian is invariant un-
under the local vector gauge transformation Uy(l)

Classical conservation laws and symmetries 179
i/i -> фе~^х) D.12)
Consequently, the vector current is conserved, equation D.7).
In addition, the Lagrangian D.11) remains invariant under a local axial
gauge transformation
ф -> Vei/3(xO5 D.13)
Л —+ Л 4- 8 8(т)
implying a conserved axial current, equation D.10). Altogether we have the
symmetry Uy(l) x {/дA).
It is also customary to decompose the fields into left-handed (L-) and
right-handed (R-) components. Introducing the projection operator onto
positive and negative chirality fields
P± = -(l±75), D.14)
with properties
P| = P±, P+P- =0, P+ + P_ = 1, D.15)
we have
Фь,п = Ф± '•= Р±Ф, D-16)
with chirality ±1
1ъФ± — ^Ф±- D-17)
We use here the 75-convention [Pietschmann 1983]
75 = «70717273 D-18)
for the identity D.16). / ,
Then the above Lagrangian D.11) can be rewritten-m terms of L- and
Д-handed fields
Лм) = ?l,r := Ыг 9 + 4Ь)Фь + Фя(г ф + ?п)фн, D.19)
where the L- and Д-gauge fields are denned by
D-20)

180 Anomalies in QFT
Obviously we can transform the L- and Д-fields separately in Lagrangian
D.19):
left-handed gauge transformation UlA)
фь ->
% ^ , D.21)
right-handed gauge transformation UrA)
AR - Лй + ЭмЛд(х). D.22)
So altogether the Lagrangian Cl,r D.19) has the symmetry ULA) x UrA).
Note: For Л/, = Лд we recover the vector symmetry f7v(l) and for Al =
—Ar the axial symmetry
Conservation laws: Finally we consider the L- and Д-handed currents
¦l - 1
ff - - 1/ ч /
jjjt = WRl^R — Vm~A ~~ 75)^1 D.23)
which can be expressed by
.L_ 1 .6 _ l R
jR = -(jM — j5) j5 = jL — jR. D.24)
Then due to identities D.7) and D.10) the L- and Д-currents are
conserved separately
QH-L = qh-R = Q ^4 25)
4.1.2 Non-Abelian fields
Our study of fermions interacting with Abelian fields can be extended to
non-Abelian gauge fields, so-called Yang-Mills fields (recall Section 3.5.1).
In this case the Lagrangian
Ф ~ тфф D.26)
contains non-Abelian vector and axial fields

Classical conservation laws and symmetries 181
VM = V?Ta, A» = A^T", D.27)
where the group generators Ta satisfy the algebra C.275) discussed in Sec-
Section 3.5.1. Again, we assume external gauge fields.
Symmetries: The above Lagrangian remains invariant under the local
vector symmetry SUv(N)
Ф -> Ф9=д-\х)ф
ф _+ ф9 = ф(х)д(х)
_д у _дз = д~1(х)А а(х) D 28)
where g(x) denotes the gauge group element
g(x) = ea(x\ a(x) = aa(x)Ta. D.29)
Note that we have chosen the matrices Ta anti-Hermitian, equation C.276),
so that
ol\x) = -a(x). D.30)
For the infinitesimal vector gauge transformation we then have
ёф — —а(х)ф
ёф = фа(х)
ёАц. = [А,а] D.31)
and Vfj. expresses the familiar covariant derivative
Т>» = дц + [VM, ]. D.32)
Considering next chiral transformations
ф —> e р\х>1*ф \ J
ф -» фе-^ъ, D.33)
with
/3(x) = Pa(x)Ta D.34)
we find a local axial symmetry SUa (N) in the Lagrangian D.26) if the
formion mass vanishes, m = 0. Its innnitesimal form—the infinitesimal
axial gauge transformation—is

182 Anomalies in QFT
6ф = -Р(х)^5ф
ёф = -4>f3(x)j5
D.35)
So for massless fermions (m = 0) the Lagrangian D.26) remains invariant
under the combined local transformation SUv(N) x SUa(N)
ёф = (-a(x) - /3(х)^5)ф
ёф = ф(а(х) - f3(x)-y5)
ёА» = Vnp + lA^a]. D.36)
Analogous to the Abelian case we also introduce here L- and Д-fields
with help of the chiral projection operator D.14). Then the above Lagran-
Lagrangian D.26) can be decomposed into separate L- and Д-field terms
AiaD,, Ац) = C^R := $Li РьФь+Фяг РнФн-т(фьфн+фнфь). D.37)
The operators D^'R denote the covariant derivatives containing the L- and
Д-handed gauge fields
?>?•*= дм + [Л?-й, ]. D.38)
Clearly, when applied to the spinors we have
Pl,r = ?+4l,r- D-39)
The L- and .R-fields are denned as in the Abelian case
Л« = УМ-ЛМ Л^ = \{А^-Ап). D.40)
But remember that these fields now express non-Abelian fields
A^R = AlL'RrT. D.41)
In the massless case (m = 0) Lagrangian D.37) is invariant under the
local L- and .R-handed gauge transformation SUL(N) x SUr(N)
Фн - Фдн = 9н1(х)ф„, »я = ел»^
Фн -* Ф~н = Фн9н{х)

Classical conservation laws and symmetries 183
+ д„)дн(х)
К ^gH{x) D.42)
(H = L, R) or in its infinitesimal version
6фн = -Ан(х)фн
6фн = Ф~Лн(х) D.43)
6А% = D»KH{x).
Again, the L- and Д-gauge elements axe non-Abelian
a. D.44)
JVote: Denning the transformation parameters
± i D.45)
we can connect both transformations D.43) and D.36).
Finally we construct several non-Abelian currents (H = L, R)
vector j? =
axial j^a ¦¦
pseudoscalar Pa = ¦
L-, Д-handed jj*a = фнЧцТафн. D.46)
These currents are related in the following way
ja = jLa + jRa jLa = -<ja + 750)
3bf = 3y,a - j%a jf?a = „Од ~ 3»a)- D-47)
Conservation laws: Now we search for classical conservation laws. What
we need are the equations of motion according to Lagraja/gian D.26)
ф{1 ф +m - i У - i / 75) = 0. D.48)
Let us first consider the divergence of the vector current
дцра = ф*р T°V> + фТа ft ф
D.49)

184 Anomalies in QFT
Using the algebra C.275) satisfied by the group generators we obtain
d»fa + /obcVyc + fabcAb^j5fiC = 0. D.50)
Multiplying again by Ta we find in terms of the covariant derivative D.32)
the classical conservation law of the non-Abelian vector current
0. D.51)
Next we study the axial current
D.52)
implying
5
? ^ = 2imP\ D.53)
the classical conservation law for the non-Abelian axial current
z>m f1 + [A«> jm] =2imR D-54)
If the Lagrangian D.26) contains only vector gauge fields (hence Ац = О)
we have
V,j» = 0 D.55)
?W5m = 2imP D.56)
and in the massless case (m = 0) both currents, vector and axial, are
covariantly conserved
ZV" = ^J6" = 0. D.57)
Finally we turn to the laws for the L- and Д-handed currents. From the
vector current conservation D.50) follows when inserting relations D.47)
and D.38), D.40)
DfiJLfl + D*jR» = 0, D.58)
whereas the axial current law D.53) implies
Altogether we obtain as the classical conservation law for the L-
handed (or Д-handed) non-Abelian current
-D*j** = DL^ = imP. D.60)

Ward identities and anomaly 185
In the massless case (m = 0) both currents, L- and Д-handed, are cova-
riantly conserved
DLjLv. = DRjRn = o. D.61)
4.2 Ward identities and anomaly
How are the classical conservation laws for the currents transferred to a
quantized field theory? In QFT we are dealing with Green functions and
with the generating functional of all these Green functions respectively (re-
(recall Chapter 3). The validity of the classical conservation laws induces rela-
relations among various Green functions, so-called Ward identities. They form
the basic equations for a consistently quantized theory.
4.2.1 Green functions
Let us consider the vacuum expectation value of the time ordered product
of a current j'M and some operators O%
@\Тз»{х)О\У1)...Оп{уп)Щ. D.62)
Differentiating such a quantity we obtain
dZ@\Tj»(x)O1(y1)...On(yn)\0) =
= @\Tdx^(x)O\y1)...On(yn)\0) D.63)
- У^О1... O^O*1... On|0>,
where the commutator terms arise from the differentiation of the ^-functions
in the T-product (recall equation C.86) of Chapter 3). When inserting the
classical conservation law for the current and respecting the canonical al-
algebra for the commutator (neglecting so-called Schwinger terms) we call
the above relation D.63) a Ward identity (WI). Such relations between
Green functions have to be satisfied in order to ensure the renormalizability
of QFT. This is the importance of the Ward identities. If they are broken,
renormalizability is lost. V _y
Ward's identity: The simplest example which is naming the above rela-
relation is to be found from the old days of QED. It is the important relation
between the vertex and the propagator established by Ward [Ward 1950]
and Takahashi [Takahashi 1957] which implies the equality of the renorma-
lization constants of the vertex and the fermion wave function.
Let us consider the following 3-point function
т*(х, у, г) = @\Тз*{г)ф{х)ф{у)\0) D.64)

186 Anomalies in QFT
and we shall need the equal-time commutator
= ~{гр(х), ф^(г)}ф(гN(г0 - x0)
= -ф{г)Ь\г - ж), D.65)
where we have used the familiar anticommutation relations for the fermion
fields.
Analogously we have
\j°(z), i>{y)]8{z0 - y0) = ф(г)б\г - у). D.66)
Now we differentiate the 3-point function D.64)
+ @\T\j°(z)^(x)]6(z0 - хо)ф(у)\0)
+ @\Тф(х)[}о(г), ф(у)}6(го - уо)\0) D.67)
and we insert the vector current conservation D.7) and the commutator
results D.65), D.66), then we obtain the vector Ward identity in z-space
д*т»(х,у,г) = -@\Тф(г)ф(у)\0)б\г - x) + @\Тф(х)ф(г)\0)б\г - у)
= -iSF(z-yN\z-x) + iSF(x~zN4(z-y). D.68)
For the more familiar version we transform equation D.68) a la Fourier into
momentum space
(Pm -*>>"(*>,*>') = SF(p) - SF(p'). D.69)
When amputating the 3-point function, the vertex remains
-зжГ
and we arrive at Takahashi's identity
(Pm -PW(P,P') = SF\p) - Spltf). D.71)
Considering the limit p' —* p
PWH'E'M-yw >¦=> ±SM D.T2)
provides the Ward identity

Ward identities and anomaly 187
^^ip). D.73)
Ward identities: In our case the interesting Green functions—which fi-
finally violate the classically expected result—are the following 3-point func-
functions
ь\0) D.74)
\0). D.75)
They correspond to the triangle graphs which we investigate in the next
section. Here we study their features in momentum space
Т„иХ(кик2,я) := i J
¦ {0\Tj^x)jv{y)jl{z)\0) D.76)
Тм„(Ль*2) := i Id4xdiydizeikiX+ik2y-iqz
¦ @\ТМх)ШР{г)\0). D.77)
Actually the above quantities depend on one variable less because of energy-
momentum conservation. We may shift, for instance, the ^-dependence in
the axial quantities to the point z —¦ 0. We ignore these shifts which are
not essential for the moment.
In order to establish a WI we have to differentiate the Green function
D.74). In momentum space there is the relation
-rqz
= f d
*& D.78)
which can be quickly checked by partial integration. When differentiating
the Green function the commutator terms vanish in our case (we ignore
possible Schwinger terms) and we obtain
qXT^x = I dixdiydiz e<*i*+i*av-<e* -~y
$Щ D.79)
Now using the classical conservation law D.8) for the axial current provides

188 Anomalies in QFT
x = 2mi
¦ (О\Т^(х)ШР(г)Щ
= 2тТм„, D.80)
which is called the axial Ward identity AWI
q%uX=2mTliU. D.81)
Analogously when differentiating the vector currents and respecting the
vector current conservation law D.7) we find the vector Ward identity
VWI
к»Т^х = fcT^A = 0. D.82)
Anomaly: On the other hand, within QFT we calculate the amplitudes
Тц„\ D.76) and TMI/ D.77) directly with help of the Feynman rules for the
corresponding triangle diagrams—we work this out in the next section—and
we check if the Ward identities D.81) and D.82) are satisfied. The answer
is no! An additional term—the anomaly—occurs. The actual calculation
(Section 4.3) leads to the following proposition.
Proposition:
• If the VWI is fuffiJJed the AWI is anomalous or vice versa/
So both, the VWI D.82) and the AWI D.81), cannot be satisfied; there
always exists an anomaly.
Anomalous AWI:
gATM1/A = 2mTMI/ + A^, D.83)
with the anomaly
-V = —^рч™»к\14- D-84)
This corresponds to a modification of the classical conservation law for the
axial current.
Anomalous divergence of the axial current:
d»jl = 2imP + A, D.85)
with the ABJ anomaly
Л = ^e^^F^F^. D.86)

Ward identities and anomaly 189
Both expressions D.84) and D.86) are equivalent, which one can verify
quickly by considering the expectation value with 2 external photons (with
momenta ki, k% and polarization vectors ?i,?2)-
So what went wrong in our naive use of classical conservation laws for
the Green functions? Let us return to equation D.78); there we have been
a bit hasty with the integration by parts. The Green functions are singular
quantities which have to be regularized. This creates a 'surface' term which
has been neglected so far and leads to the anomaly. We present the detailed
investigation in Section 4.3.
Ambiguity: Green functions containing operator products at the same
space-time point are singular objects. The question is whether after regula-
rization the Ward identities can be maintained. Let us consider an example
given by Leutwyler [Leutwyler 1986a]—the 2-point function of the vector
currents
= tirSF(x-ybuSF(y-x). D.87)
Sf(z) denotes the free fermion propagator in coordinate space
so that the Green function D.87) has a singular behaviour like (suppressing
indices)
ф)г~°1. D.89)
Therefore the Green function represents a distribution that is well denned
only on test functions fp,v(z) with
/дЛО) = 0. D.90)
The VWI is satisfied on these test functions
д^и(х-у) = д;т^(х-у) = 0. ^ D.91)
However, extending the Green function to arbitrary test functions is not
unique. Two Lorentz invariant extensions obeying the VWI D.91) differ by
n term [Leutwyler 1986a]
%*{*) = r^iz) + ic(g^a - дцд„N*(г), D.92)
with с some constant. So this Green function—the 2-point function—has a
ri'normalization ambiguity.

190 Anomalies in QFT
The higher Green functions are again unique (the 3-point function va-
vanishes because of Furry's theorem) and obey the VWI
^^,...^A1, ...,!„) = dgiOlTJnfa)... JMn(in)|0) = 0. D.93)
Anomaly for L-currents: In the previous section we derived the classical
conservation of the left-handed current (massless, positive chirality fields),
equation D.25). What about the Green functions? Are they conserved?
Up to the 4-point function the Green functions contain ambiguities. Af-
After renormalization the L-currents (or analogously the Л-currents) remain
conserved in the 2- and 4-point functions (a possible choice). These Green
functions satisfy the Ward identities. But then the 3-point function WI
is anomalous
х-у)б\х-г)}. D.94)
Equation D.94) corresponds to the triangle anomaly containing 3 L-currents
and expresses Bardeen's result D.387) (see Section 4.8) when replacing the
gauge group SU{3) by 17A).
Result D.94) we can quickly infer from the ABJ result D.84) when we
re-express the L-currents by the vector- and axial currents, equation D.24).
Suppressing the vector indices we have to consider
3L 3L3L = \(j+ 35K=\(fjj + 33*3 + 333* + fff), D-95)
where we keep only the anomalous contributions. The second term cancels
with the third and for pure axials we have by Bose symmetry
Hence we get
3*3*3* = \з*33- D-96)
3L3L3L=\ (l + 0 3*33 = \fj
3L3L3L=\ (l + 0 3*33 = \fjj- D-97)
The L-handed anomaly amounts to 1/6 of the ABJ anomaly whose coef-
coefficient is 1/2тг2 in coordinate space. Therefore the coefficient in equation
D.94) has to be
For Л-currents the anomalous WI D.94) just changes the sign (since relation
D.97) changes sign).

Ward identities and anomaly 191
4.2.2 Generating functional
In QFT all information is contained in the generating functional for all
Green functions (recall Chapter 3)
= @|Texp[i j dxf{x)Ali{x)\\0)
n=0
D.99)
where A^ represents the external gauge field. The connected Green functions
are determined by the 'quantum action'
M] = 22— dxt...dxnAlll(xi)...AViK(xn)
n=l J
(n)... j"-(in)|0>conn. D.ЮО)
Ambiguity: How do the renormalization ambiguities of the Green func-
functions affect the generating functional?
Proposition:
• The generating functional is unique up to a local polynomial of the
external gauge field and its derivatives!
So different renormalization methods induce different polynomials f[A^] in
the generating functional. In terms of the 'quantum action' we have
D.101)
Proposition:
The renormalization ambiguity induces a U(l) transformation in the
generating functional Z\Ay\
= jWlAJ+iflA,.} = егДА„] eiW[A^\
= eif[A»] г[А„]. D.102)
Example: The renormalization ambiguity of the 2-point function D.92)
is then reflected by the generating functional for the special polynomial
D.103)

192 Anomalies in QFT
Proof. The 2-point function follows from the Z\A^\ functional via
= <O|Tj"(i)j"(j,)|O>. D.104)
v "' i6A»(x)i8Au(y)
Rewriting the polynomial D.103)
D.105)
we differentiate the transformed functional D.102)
i6All{x)i6Av{y)
Au=0
Q.E.D.
D.106)
Ward identity: How are the Ward identities described by the generating
functional?
Proposition:
• The Ward identities for the Green functions are equfvaient to the
gauge invariance of the 'quantum action'
= 0.
D.107)
Proof.
i" f
... dxn
D.108)
and we integrate by parts
n_i)i / dxi...dxnA{x1)Ali2(x2)...Altn(xn)
""(i,,!,...,^). D.109)

Ward identities and anomaly 193
Each term in the sum contains different powers in AM. If 6W — 0 then
each term in the sum vanishes—all Green functions are conserved, equation
D.93)—and vice versa. Q.E.D.
Anomaly: If, however, a Green function obeys an anomalous WI then the
'quantum action' reflects this too.
Proposition:
• The anomalous Ward identity is equivalent to the gauge transforma-
transformation of the 'quantum action'
M] = G(A,A) =: f dxAa{x)Ga[All\{x). D.110)
G(A, A) represents precisely the anomaly in the Abelian as well as in the
non-Abelian case. The index a denotes some group index (a = 0 for GA)).
Note: Equation D.110) is a very compact notation for all WI of the Green
functions.
Example: Let us return to our example D.94) of the previous section.
Then the anomaly for L-fields is given by
G(AL,AL) = --L I dx AL{x)e^a<3dliALu{x)dotAL0{x). D.111)
From equation D.110) follows that all L-handed Green functions are con-
conserved
^On).. .tf (zn)|0> = 0 D.112)
except the 3-point function. For n = 3 the anomalous result D.94) is valid.
Proof. We consider the difference
= 2_] ~in I dxi... da
_, и! J
¦@\TfL>(x1)...Ji"(xn)\0)conn D.113)
niui we integrate by parts and separate the n = 3 term

194 Anomalies in QFT
-| J dxdydz AL{x)A^{y)ALv{z)
i|0>conn. D.114)
Inserting the identities D.112) and the anomalous result D.94) we obtain
after integrating by parts again
= ^ J
¦ 8{x - y)8(x - z)
= G(AL,AL) D.115)
and vice versa. Q.E.D.
Phase and anomaly: In Chapter 3 we have discussed the significance of
the generating functional. It describes the vacuum-to-vacuum transition and
corresponds via a path integral to the determinant of a differential operator,
here the Dirac operator
= @|T exp[i f dxd] |0) = @, oo|0, -oo)
= J d$dil> exp[i / dxiJH Щ\ = det i T/). D.116)
Now performing a gauge transformation, there may occur—as in our pre-
previous example of L- or й-handed fields (we supress the indices)—an anom-
anomaly in the 'quantum action'. This anomaly then is a local polynomial in the
gauge fields and their derivatives. For this reason a gauge transformation
only influences the phase
eiG(AlA) D.117)
<0,оо|0,-оо)|л+м = @,оо|0,-оо)|л eiG(^> D.118)

Ward identities and anomaly 195
k+бД, = detiWA^eiG^'A\ D.119)
We obtain the important result:
Proposition:
• The generating functional, the vacuum-to-vacuum amplitude or the
determinant of the fermionic differential operator are not gauge inva-
invariant quantities if there is an anomaiy. But the anomaly affects only
the phase of these quantities. The modulus or the transition probabi-
probability is gauge invariant again.
We certainly also have renormalization ambiguities in the generating
functional. As discussed before they also induce a local polynomial in the
'quantum action' (see equation D.102)).
Proposition:
• An anomaly does exist if no renormalization polynomial can cancel
the anomaly effect!
The Ward identity remains violated.
The extension to non-Abelian fields is straightforward.
Non-Abelian Ward identities: In the case of non-Abelian fields A^ =
A^Ta the compact notation for the Ward identities is again the gauge
freedom of the 'quantum action'
6W[AJ = W[AM + 6А„\ - W[AM] = 0 D.120)
by the infinitesimal gauge transformation 6А^ = D^K.
As in the Abelian case the identity D.120) implies all WI for the non-
Abelian Green functions. Due to the non-Abelian nature of the fields there
is a slight complication in the derivation. Let us consider the W-functional
and
C'.X" (*i,...,*„) = @\Tj? (n)... j?(*„)|0)COnn. D.122)
We take a gauge variation

196 Anomalies in QFT
::»: (xi,x2,...,xn), D.123)
we integrate by parts and separate the first term
= - j dxAa(x) {ВД(х) + ifacbAl
^ г
ri—2
•'С::С(я;.я;а'---'я;»)}- D-124)
Gauge invariance 6W = 0 implies that all terms to a given power in A^
must vanish. However, here these terms mix in the 2 sums.
Let us demonstrate the quadratic terms in A^. Then from 6W = 0
dx2dx3
-i2faa3bT^(x3,x2N(x-x3)} = 0. D.125)
After symmetrizing the last term
¦zfaaa Tba^3(x2i X3N{x — X2) + ^faa3 T~{?^2 (x3,X2N(x — X3) D.126)
the WI for the non-Abelian 3-point function is
' X3)S(X ~ *2) + ifaa3bT^(x, X2N(x - X3).
D.127)
Calculating, finally, the n-th terms
Jdx2...dxn A%(x2)...All{хпШт^;;;*:(x,x2,...,xn)
-Кп-1)иУ^Х^-1(хп,х2,...,хп^)8(х-хп)} = О
D.128)

ABJ anomaly calculations
p-q
197
Fig. 4.1. Triangle graphs with vector-vector-axial currents and
vector-vector-pseudoscalar currents
implies the WI for the non-Abelian n-point function
2, ¦ ¦ •,x,... xnN(x -Xj). D.129)
j=2
Note: Of course, relation D.129) must follow directly from differentiating
the non-Abelian n-point function D.122) and using the equal-time commu-
commutators of the non-Abelian currents.
4.3 ABJ anomaly calculations
The Adler-Bell-Jackiw anomaly [Adler 1969], [Bell, Jackiw 1969] (ABJ
hereafter) is a true phenomenon of quantum physics. Consequently, ail re-
normalization methods must lead to the same result. We will present tile
customary methods.
4.3.1 Triangle graph regularization
Now let us focus on the famous triangle diagram. In order to understand how
the anomaly unavoidably arises, when regularizing this UV-divergent quan-
quantity, we go through the calculation step by step. Here we follow Jackiw's clas-
classical lectures on anomalies [Jackiw 1985d]. In addition the reader may find

198 Anomalies in QFT
literature like [Cheng, Li 1988], [Hofejsi 1992a,b], [Morozov 1986], [Adam
1990], [Shifman 1991] helpful.
With the help of the familiar Feynman rules (for our. notation see
[Pietschmann 1983]) we find for the amplitudes depicted in Figure 4.1 the
following expressions in momentum space
d4p
) D.130)
f
/j4
B^H tr T
fc;nt2)' DЛ31)
with q = fci + fc2-
Axial Ward identity: In order to establish the axial Ward identity D.81)
of the previous section we use the identity
#75 = 75(j* - # ~ ™) + {$ ~ ™)lb + 2m75. D.132)
Then we obtain the following relation between the amplitudes D.130) and
D.131)
where
_i /"doll 1
У Bтг) |_р — Ц2 ~ тп p — # — m
; 757i/-; r; 7u D.134)
/)—m p — fti — m \
2 /" ^4P
If the rest-terms vanish
Л^,Я'и-»0 D.136)

ABJ anomaly calculations 199
the AWI is satisfied. This happens if we formally shift the integral of the
first term in equation D.134) and in D.135) by
p—>p + k2 and by p—>p + ki. D.137)
However, the integrals are linearly divergent. Note, the seemingly quadratic
divergence disappears because of the trace which gives
ециа^Р0 = О. D.138)
In linear divergent integrals we are not allowed to shift the integral variable!
So the rest-terms may contribute
Я*„, R\v ф 0 D.139)
and the AWI becomes anomalous.
Linear divergent integrals: Let us begin with an example in 1 dimen-
dimension
A(a) = Г dx[f(x + a)-f(x)}
J—oo
f°° a2
= / dx[af'{x) + —f"(x) + ..] D.140)
У-оо 2!
= o[/(oo) - /(-oo)] + |^[/'(oo) - /'(-oo)] + ....
If the integral
I
j—с
dxf(x) D.141)
converges (or is at most logarithmic divergent) then /(±oo) = /'(±oo) =
... =0, and the quantity A(a) vanishes, A(a) = 0. In this case one can
shift the variable x ^ x — a in the first integral.
If, however, the integral D.141) is linear divergent then /(±oo) ф О,
/'(±oo) = /"(±oo) = ... = 0, and Д(а) may create a nonvanishing 'surface
term' f
A(a) = o[/(oo) - /(-oo)] ф 0. D.142)
Generalizing this example to n dimensions (Euclidean) we have
Д(о) - y>z[/(z +а)-/(*)]
= f (Txla^fix) + аРсГд^Нх) + ...]. D.143)

200 Anomalies in QFT
Now we apply the Gauss theorem, we integrate symmetrically (average over
all directions) and obtain
A(a) = o" lim ^-Sn-1{R)f{R), D.144)
R—»oo К
where S'n~1(i?) expresses the surface of an (n — l)-dimensional sphere with
radius R; for instance for n = 4
S3(R) = 2тг2Д3. D.145)
In 4-dimensional Minkowski space we get an extra factor i from x± = ix^
so that we find the formula
A(a) = г2тг2а" lim Д„Д2/(Д). D.146)
R—*oo
We are going to use this formula frequently in this chapter.
Now we return to our rest-terms R^ D.134), D.135) which we have to
investigate. They are precisely of above type
D.148)
so that we can use formula D.146). For the term R1^ we obtain
The trace is calculated with help of the familiar formula
tr 757/37»7а7д = 4ге0„ам. D.150)
(Note that the axial trace containing 1, 2, 3 7M-matrices vanishes.) Then
only the term proportional to ф #i survives leading to
Performing the symmetric momentum limit
gives
lim Щ f D.152)
р-юо p2 4

ABJ anomaly calculations 201
l ^^fcffcf. D.153)
The second term R2^v arising from the interchange fci <-> fc2, ц <-» v provides
the same result
DЛ54)
Altogether the AWI receives the following anomalous contribution
X f. D.155)
But this is not quite the result we finally want to present. Let us investigate
a little bit further.
Ambiguity of the amplitude: For the above calculation we have chosen
the momentum route in the triangle graph quite arbitrarily (see Figure 4.1).
We can certainly shift this internal momentum by some amount then the
linear divergent integral, which is not uniquely defined, also alters its value
when regularized like above.
Note: There is no ambiguity in the amplitude ТД1/ D.131) since it is a per-
perfect convergent quantity. The seemingly linear and logarithmic divergences
disappear because of the vanishing axial trace of 5 7M-matrices or because
of identity D.138).
We shift the internal momentum integration by
p->p + a, D.156)
with
a = afci + (a - /3)fc2 D.157)
and we consider the difference of the amplitudes
= - f ^i-
fc2
v
D.158)

202 Anomalies in QFT
We regularize the linear divergence of type D.143) by formula D.146) which
provides
^ii/A — г2тг2аг lim pTp2
(-) tr (j + mOA75(/* - 4 + mhvit ~ fa + ™)ln
BтгL[р2 - m2][(p - qJ - m2][(p - fciJ - m2] "
D.159)
In the limit p —> схэ only the leading powers in p survive
7) 7>a
-у tr 757,7а7д7л D.160)
and using formulae D.150) and D.152) we obtain
U ^Q- DЛ61)
The second part Дд„л follows from the first by interchanging
and /i <-» z/ so that the total difference of the amplitudes D.158) is
D-162)
(recall equation D.157)).
Considering, finally, the amplitude
D.163)
we establish the AWI by multiplying equation D.163) by the momentum qx
and inserting the results D.155) and D.162). This leads to the anomalous
AWI (for Рф\)
^1 D.164)
Vector Ward identity: The next important relation that we have to
investigate is the vector Ward identity D.82). So we calculate
tr
DЛ65)
Using the identities

ABJ anomaly calculations 203
p — q — m p — q —m
only the first terms remain and we obtain
D.166)
— fi — m
D.167)
Again, we have a linear divergence of type D.143) which we regularize by
formula D.146). Recalling further the formulae D.150) and D.152) we eva-
evaluate
(-fc[) lim pTp2
p—»oo
(-) tr
г2тг2
D.169)
So the VWI of the unshifted amplitude (a = 0) is not satisfied. But also the
shifted amplitude D.163) acquires an anomalous VWI (for 0 Ф —1)
toa/jfcffcf- D-170)
When considering both Ward identities D.164) and D.170) we find the
very remarkable result:
Proposition:
• There is no value of/3 such that both the VWI D.170) and the AWI
D.164), can be fulfilled!
Examples:
i) VWI and anomalous AWI.
We can choose, for instance, /3 = — 1 then the VWI

204 Anomalies in QFT
Fig. 4.2. Radiative corrections to the triangle graph
0 D.171)
is satisfied—the preferable case of gauge invariance—but the AWI
turns out to be anomalous
q^T^x = 2гоТм„ + Aw D.172)
with the ABJ anomaly in momentum space
Л" = -^„„«rffcf fcf. D.173)
ii) Anomalous VWI and AWI.
Or we fix /3 = 1 then the AWI is fulfilled
= 2тТм„ D.174)
but the VWI becomes anomalous
№„а = ~\a,x. D.175)
hi) Anomalous VWI and anomalous AWI.
If we select /3 = 1/3 the anomaly is equally distributed in the VWI
and AWI
\ D.176)
+ \Allv. D.177)
о
This case corresponds to a triangle graph calculation with 3 axial
currents where the Bose symmetry provides a factor 1/3 (Section 4.8).
Note: Clearly, the anomaly result is gauge independent. (The gauge inde-
independence of anomalies, specifically for string theories, has been discussed
by Kummer [Kummer 1990].)

ABJ anomaly calculations 205
Radiative corrections: The anomaly discussed so far stems from a 1-
loop perturbation calculation. Now an important question certainly arises.
Does the anomaly receive contributions from radiative corrections (see Fi-
Figure 4.2) in higher order perturbation theory? The answer is no!
Adler-Bardeen theorem: [Adler, Bardeen 1969] The full structure of
the chiral anomaly is given by the triangle graph!
So higher order radiative effects just renormalize the fields and charges. It
is enough to calculate only the triangle graph for the anomaly result. This
is an important statement and is certainly true in QED and QCD but in
more general theories the Adler-Bardeen theorem needs to be confirmed.
For further literature we refer to [Piguet, Sorella 1992, 1993].
4.3.2 Pauli—Villars regularization
The classical regularization technique for a divergent amplitude is that of
Pauli and Villars (PV) [Pauli, Villars 1949]. How does the anomaly arise in
this scheme? (See e.g. [Jackiw 1985d], [Hofejsi 1992a,b], [Adam 1990].)
Let us consider the amplitude as a function of the fermion mass—the
particle which loops around.
Definition: The PV-regulated amplitude is the difference between the
given amplitude and the amplitude evaluated at a regulator mass M
Wm) - ТД1/А(М). D.178)
The physical amplitude follows from the limit M —> oo
In the case of the ТД1/ amplitude D.131) we already have
= ton T?g = Jim [ТД1,(т) - Т^{М)\ = T^m) D.180)
М—ЮО ^ М—ЮО
since ГД1/(М) ~ -fa as we shall show below. Тм„ is convergent and clearly
no regularization is needed.
Vector Ward identity: The VWI is automatically fulfilled for the re-
regulated amplitude, hence also for the physical amplitude
0. D.181)

206 Anomalies in QFT
The reason is simply that a mass independent term like equation D.170)
just cancels in the PV difference of the amplitudes.
Axial Ward identity: For the AWI however we get
9ЛГ$" = J™/^ = 2тГД1/(т) - Jirn^2MTllv(M). D.182)
Also here a possible massless term like the second term in equation D.164)
cancels in the PV difference D.178).
Theorem: The ABJ anomaly is generated by the limit
lim 2MTuV{M) = -AaV- D.183)
M—юо
So the PV method selects a satisfied VWI and anomalous AWI automati-
automatically.
Proof. We begin with Гд1/(т), equation D.131). For the denominator we
introduce the familiar Feynman parameter integral (see e.g. [Pietsch-
mann 1983], [Pokorski 1987])
= 2 dxx dx2- —— — -, D.184)
010203 Уо Уо \aix2 + «2A -xi- x2) + а3Х1\л
which gives
l~Xl
tr (ф + mO5(y< - j + тO„(^ - ft +
[(p2 - m2)x2 + ((p - qJ - m2)(l - ц - x2) + ((p - fciJ - m2)Xl]3
The axial traces with 2, 3 or 5 7M-matrices vanish. The traces containing 4
7M-matrices cancel or vanish due to identity D.138) except the term
m tr 75 i-iv hi» = m 4is0vailklk^ + ( ^ ^ *2 \ D.186)
so that we get
"" " У (^Vo Г Уо

ABJ anomaly calculations 207
with
к = q(l - xi - x2) +
m2 = m2-q2{l-xi-x2). D.188)
We have already included the terms from the interchange fci <-> k2, Д <-* v
which actually contribute to equation D.187) after the mass limit M —> oo.
The momentum integration is conveniently carried out with the help of the
't Hooft-Veltman integral formula ['t Hooft, Veltman 1972]
T{) (fc2 + 2)ап/2 °"
(p2 - 2pk - m2)a Г {a) (fc2 + т2)а~п/2
D.189)
In our case a = 3 and n = 4, then the integral is
7Г2
with f(xi,x2) a function independent of m. It behaves in the limit of large
masses like
'^
lim J0{M)=^7 lim -i,. D.191)
M-.00 v ; 2г м-гоо М2
So we obtain the relation
lim 2MTllv(M)
M—*oo
f = -Л- Q-E.D. D.192)
4.3.3 n-dimensional 't Hooft-Veltman regularization
Another attractive regularization procedure has been invented by 't Hooft
and Veltman ['t Hooft, Veltman 1972]. There the amplitudes are regulated
by the higher dimensions in space-time; the divergences occur as poles at
dimension n = 4. First all calculations are carried out for n > 4 and after-
afterwards in the final result the limit n —> 4 is performed. Let us see how the
anomaly emerges here (for a review see e.g. [Hofejsf 1992a,b]).
The anomaly involves a 75-matrix. There is, however, a problem of de-
defining 75 correctly in higher dimensions. Choosing
75 = г70717273 D.193)
(note this 7B-definition [Itzykson, Zuber 1980] differs in sign from the con-
convention we used before in Section 4.1) the 't Hooft-Veltman rule is

Anomalies
for,
for
in QFT
/" = 0,1,2,
/u = 4,5,..
3
.,n-l.
D.194)
D.195)
208
All other familiar rules for the 7-matrices persist in higher dimensions.
Triangle amplitude: Now we calculate the triangle amplitude D.130) in
n-dimensions. We split the momenta at dimension 4 and we just consider
the massless case (m = 0) for simplicity
dn~*L 1 1 1
t
- - J B?rL J B7r)n_4 tr
D.196)
fi *-* U J
The internal momenta denote
$ = 47° + ... + erf D.197)
for the first 4 components and
Ц =L474 + ...+Ln_i7n D.198)
for the last (n — 4) components. The external momenta
g",fc?,fcf v,a,0 = 0,1,2,3 D.199)
remain 4-dimensional.
Let us shift the integration momentum for convenience by
e -> e + fci D.200)
then the amplitude changes to
dH f dn~4L
f dH f dn~4L 1
~ J BтгLУ B7r)"-<'tV+^+^i
D201)
Vector Ward identity: The VWI is satisfied
КТ^Х = ЩТ^Х = 0. D.202)
This we can verify in the same manner as before in Section 4.3.1, equations
D.165)-D.168). The analogous equation to formula D.167), D.168) is

ABJ anomaly calculations 209
= - / / -———
1
7A75-
= - / 77m[f(? - fci) - f(t)\ = ° D-203
J B7ГL
since the integration shift ? —> ?+ki in the first term is now allowed. Q.E.D.
Axial Ward identity: What about the AWI? Let us study the expression
D.204)
Note that
##=?2, Ц Ц = -L2, #У + У?=О,
{f'+ #)(f' + Щ = ?2 - L2'. D.205)
Calculating the trace explicitly it happens that all terms cancel or vanish
(ixcept the terms proportional to Ц Ц. Therefore we obtain
/d ? Г dn~L
B^7 -Q^r*
L2
k)* V]' D206)
where we have already included the terms from the interchange fci «-» k2,
/i *-> v which actually contribute after the limit n —> 4. Next we apply, as
usual, the Feynman parameter integral formula D.184) together with the
4 Hooft-Veltman formula D.189) for the I integration. Then we get
wWo dxiL
D.207)
For the L-integration we use the formula

210 Anomalies in QFT
dNL (L2)° _ /e-w-iv/2 Г(а + N/2)T(b -a- N/2)
I
n)N (L2 + f)b B^)N T(N/2)T(b)
so that finally in the limit n —> 4 the integral
jn—4r r2
D.208)
/jn—4r r2
-—г—т-гт г = -1 D.209)
B7r)"-4L2 + /(xi,x2) ^ ;
provides a nonvanishing contribution (from handling (n - 4)/(n — 4)). This
is the origin of the anomaly.
Equation D.207) now yields the familiar anomalous AWI
Remark: Although the 't Hooft-Veltman procedure appears totally dif-
different to the Pauli-Villars regularization technique where a mass cut-off is
introduced, there exists a link between these 2 approaches. Noticing that
the momentum quantity L2 plays the role of a mass squared, Hofejsi, No-
Novotny and Zavialov [Hofejsi, Novotny, Zavialov 1988] have shown that the
dimensional regularization corresponds to a 'continuous superposition' of
PV cut-offs.
4.3.4 Singular current operator
Till now we have considered triangle graphs, their regularization creates
the anomaly which we have calculated in momentum space. But the same
anomaly also occurs in x-space when investigating the current operator.
The canonical commutation relations of field operators teach us that
operator products at the same space-time point are singular objects. So the
axial current as a product of 2 fermion field operators
jjj(z) = ф(х)Ъ15ф(х) D.211)
is necessarily singular. The regularization of this current induces the anom-
anomaly (see e.g. [Jackiw 1985d], [Adam 1990]).
Point splitting method: To regularize the current operator we use the
point splitting method of Schwinger [Schwinger 1962]. There the operators
are separated by a small vector ем in the following way
Г f-x+e/2
, e) = ф(х + е/2)^1ьф{х - e/2) exp ie / dyvAv{y)
Jx-e/2
x+e/2
D.212)
/2
and

ABJ anomaly calculations
e-»0
211
D.213)
The exponential is introduced in order to guarantee gauge invariance.
The rule is:
i) first all calculations are carried out with e > 0,
ii) then the limit e —> 0 is performed in the physical amplitudes.
Now we want to calculate the derivative of the point-split current
= ф(х + e/2) ft 'узф(х — e/2)exp
. fx+
ге I
Jx-e
x+e/2
dyvAv{y)\
/2
¦x+e/2
rx+e/2
- ф(х + e/2O5 ft ф{х - e/2) exp ie / dyu Av{y)
Jx-e/2
l exp
x-e/2
x+e/2
¦ Г
ге
Jx—
x-e/2
dyvAv{y)\.
D.214)
We use the Dirac equations D.6) and the definition of the point-split pseu-
doscalar current
P(x, e) = ф(х
(х - e/2) exp
ге I
Jx-
x+e/2
x-e/2
dyvAv{y)
D.215)
then
we
obtain
,e) = 2imP(x,
гез5{х,е)
/•x+e/2
x-e/2
- A^x + e/2) - Ац.{х - e/2)
D.216)
Я Л (T) — 8 А (тЛ1 ('4 217<\
If we naively take first the limit e" —* 0 we find the classical conservation
law D.8) and the corresponding ordinary AWI D.81) (or PCAC in Section
¦l.G). However, this procedure is not allowed since the operator jg(x,e) be-
11 uves as a singular like
#(x,e)~-. D.218)
Expanding the gauge potentials yields
d^(x,e) = 2imP(x,e) - iej^{x,e

212 Anomalies in QFT
-• + • x • + • к к • +
Fig. 4.3. Fermion propagator in the presence of an external field
We have to follow the above rule. First we carry out all calculations with
e > 0 and afterwards we choose e —> 0. This creates a nonvanishing second
term—the anomaly.
Anomaly: Let us consider now the vacuum expectation value of relation
D.217)
=2im@\P(x,e)\0)
(х,?)|0)[а1/Лм(х) - dpAv{x)] D.219)
and focus on the quantity
e/2)|0) exp
rx+e/2
ie / dyxA\{y)
Jx-e/2
Г fx+e/2 ]
= ev tr 7s7Mr(x - ?/2, x + e/2) exp ie / dyxAx(y) , D.220)
L J'-Ф J
where we have introduced the fermion propagator in the external field A\
т(х-е/2,х + е/2) = @\Тф(х - е/2)ф(х + е/2)|0>. D.221)
We are going to expand this propagator in powers of A\
т(х-е/2,х+е/2) = T0(-e) + ie / dizT0(x-e/2-z) 4(z)T0{z-x-e/2) + ...
D.222)
where tq(x) = iSp(x) denotes the free fermion propagator C.238). Graphi-
Graphically equation D.222) is depicted in Figure 4.3. Only the linear term in A\
is of interest to us since the degree of divergence in the series is decreasing.
The free term vanishes in the trace of equation D.220).
In momentum space the linear term of equation D.222) has the following
representation

ABJ anomaly calculations 213
П /A +/j " У У B7гLB7гLB7гLе
t -m ™ f-m
D.223)
The г-integral supplies a E-function (for momentum conservation at the z-
vertex, I = p — q), which simplifies the ^-integration, and now performing
the p-integration shift
P -> P + 9/2 D.224)
gives
27ГL
1
- Ц2-т
D.225)
We need this result D.225) for the fermion propagator for expression D.220)
tr 757М(^ + 4/2 4- m) ^L(q)(jl — 4/2 4- m)
„2i ^P
лх+е/2 ]
ie/ d»A>lA(») .
Jx-e/2 J
D.226)
In the trace the only surviving term is
-tr 757м tlX 4Ax = -Aie^PaqpAx, D.227)
leading to a linear divergent integral (the quadratic divergence disappears
because of identity D.138)). We also observe that the integral shift D.224)—
iictually not permissible for linear divergent integrals—does not change our
final result because of the special algebraic structure of the trace.
Furthermore we use the identity
ieveiipe = ^-eipe D.228)
OPv
imd integrate expression D.226) by parts

214 Anomalies in QFT
#(z, e)|0) = -e Aie
.eipe.d
BтгL dpv [(p + g/2J - m2] [(p - g/2J - m2]
¦z+e/2
•exp
rx+efi
ie I dyxAx(y)
Jx-e/2
D.229)
Now we may perform the limit e —> 0 and the remaining linear divergent
integral is of type D.143) which we evaluate via the 'surface' formula D.146)
and limit D.152)
^ D-230)
(we re-expressed the Fourier integral). We finally use the regularized result
D.230) for the second term of equation D.217) which now provides the
anomaly
^F^{x)Fal3{x)^A{x). D.231)
So after regularization we obtain an anomalous divergence of the axial
current
@|9Mj?reg(z)|0) = 2im@\P(x)\0) + A(x) D.232)
or in terms of the operator equation
d^ies(x) = 2imP(x) + A(x). D.233)
This modification (see also modified PCAC, Section 4.6) is in accordance
with our discussion on Ward identities in Section 4.2.1. (Note the pseudos-
calar current P(x, e) is already regular for e —> 0.)
4.4 2-dimensional anomaly and dispersion relations
The anomaly originates not only from regularizing the ultraviolet diver-
divergences of an amplitude but occurs, in a different approach, in the infrared
region as well. This approach works by means of dispersion relations (DR).
The use of dispersion relations to calculate the axial anomaly was pro-
proposed originally by Dolgov and Zakharov [Dolgov, Zakharov 1971] (for a
review see HofejSf [HofejSf 1992a,b, 1985, 1986] and Huang [Huang 1982]).

2-dimensional anomaly and dispersion relations 215
p-q
p
Fig. 4.4. Fermion loop containing a vector and axial current
p-q
Fig. 4.5. Fermion loop containing a vector and vector current
Even before, Kummer [Kummer 1970] determined the anomaly in a di-
dispersive way relying on a, at that time fashionable, pion-nucleon model.
We, however, want to demonstrate the dispersion relation approach in a
2-dimensional example, which already contains all features of the method
and is easy to handle. Here we follow closely the work of Adam, Bertlmann
and Hofer [Adam, Bertlmann, Hofer 1992, 1993].
4.4.1 Ward identities
For literature specifically on 2-dimensional QFT we refer to [Jackiw 1985d],
|Adam, Bertlmann, Hofer 1993], [Abdalla, Abdalla, Rothe 1991], [Dittrich,
Reuter 1986], [Ecker 1982], [Grosse 1988], [Lowenstein, Swieca 1971], [Baier,
Pilon 1991].
As discussed before, in QFT we work with Green functions which have
to satisfy the Ward identities to achieve the renormalizability of the theory.
In order to detect the anomaly in two dimensions we consider the 2-point
function
<0|rjM(z)^B/)|0), D.234)
which diagrammatically corresponds to the loop depicted in Figure 4.4.
'I'hen we find for the axial Ward identity (FT stands for Fourier Trans-
Transformation)

216 Anomalies in QFT
<П%, = FT @|^
= 2m FT i(
= 2m P% D.235)
if we insert the classical result
dyfv(y) = 2imP(y) = 2гтф(у)ъф{у). D.236)
Analogously, when considering the 2-point function corresponding to the
loop depicted in Figure 4.5
D.237)
we get for the vector Ward identity
<?%„ = FT @\Т^(х)д^(у)\0) = О D.238)
if we rely on the classical conservation law
х) = 0. D.239)
But now we will calculate the Green functions explicitly, and we do it in a
dispersive way, and we check whether these WFs are satisfied.
In two dimensions we have
7*75 = ?v\7X, D-240)
which implies for the amplitudes
T%, = ^xT^. D.241)
In addition we introduce the tensor structure
. D.242)
So it is enough to consider just the amplitude Тм„, the Fourier transform of
the Green function D.237) containing 2 vector currents. Let us investigate
its structure.
Dispersion relations: The general Lorentz decomposition of Тй„ is
V(<?) = q^uTitf) - g^T2(q2). D.243)
All we require is that it vanishes at large q2

2-dimensional anomaly and dispersion relations
217
Fig. 4.6. Analyticity domain of the amplitudes Tj(i)
0,
D.244)
implying
о
const. ^ 0.
D.245)
We also know that Ti(t) (i = 1,2) are analytic functions on the t-plane with
a cut starting at t = Am2 (see Figure 4.6), and along the cut we have
T(t + it) - T(t - ie) = T{t + it) -T*(t + it) = 2 ImT(t + it). D.246)
Theorem: Let T(t) be an analytic function on the closure D of a domain
D, let be q2 € D, then Cauchy's integral formula is valid
D.247)
So Cauchy's theorem D.247) provides the dispersion relations for the
amplitudes T\(q2) and T2(g2). The unique choice for the amplitude Ti(q2)
is an unsubtracted DR (similarly for P{q2))
dt
- ч*
-г
dt
ImP(i).
D.248)
D.249)
Concerning T2(q2), however, we have the option to subtract the DR once or
not to subtract. The natural choice would again be an unsubtracted DR

218 Anomalies in QFT
D.250)
since the integral exists.
Then the AWI is satisfied
= stiVqvT2(q2)=etiVqv2mP(q2)=2mP^(q) D.251)
since the imaginary parts of the amplitudes do fulfil the WI's in any case
AWI: Im T2{t) = 2m Im P(t)
VWI: 1шГ3(() = tlmTi(t). D.252)
However, considering the VWI we find
Ti(t)
\ n/ Jim'
= Aqu,
with the anomaly
)
*7 Jim'
= A qv, D.253)
A=--. D.254)
In the last step we had to do an explicit calculation; we calculated the
imaginary part of a fermion loop with help of the Cutkosky rules (see at the
end of Section 4.4)
D.255)
Thus gauge invariance is broken, the vector current is not conserved and
the trace of the tensor D.243)
T^(q2)=q2T1(q2)~2T2(q2) D.256)
does not vanish in the limit q2 —> 0
iyV = 0) = -2Г2@) ф 0. D.257)
Now, for physical reasons, in order to restore gauge invariance we have to
subtract the DR although the amplitude itself is finite. This is not peculiar
for 2 dimensions, the same features also occur in 4 or higher dimensions. So
we must use a once subtracted DR for the amplitude

2-dimensional anomaly and dispersion relations 219
? - T2(o) = '- ? ^_ Im Ta(t). D.258)
Then the subtraction constant represents the anomaly
-T2@) = / — Im T2(t) = / dt Im
К Jim* t 7Г J4m2
D.259)
Furthermore we have
-I
= — = A.
7Г
n2 /-oo jf
^ / j^s Im Ti(t) = g2T!(g2) D.260)
and we achieve the familiar gauge invariant tensor structure (with T\ = T)
T»v(q) = (qtfu - q29^)T{q2). D.261)
Now the VWI is satisfied and the trace vanishes in the limit q2 —> 0
as it should
T^(q*) = -q*T(q*)9^?0. D.262)
The AWI, on the other hand, contains the anomaly
= e^qv{2mP(q2)+A). D.263)
For the massless case m = 0 the amplitude P(q2) vanishes
2mP{q2) ->Q for m ^ 0 D.264)
but the anomaly remains
qvT% = Ae^q" D.265)
so that the axial current is not conserved but provides the well-known
anomaly result
dvjl = UvlidvA» = ^e^""- D-266)
Remark: Of course, we could also subtract the amplitude T2(g2) at some
arbitrary point q\\ then the anomaly is distributed on both WI's, on the
AWI and on the VWI. There is no choice to get rid of the anomaly at all
(for details see [Adam, Bertlmann, Hofer 1992, 1993]).

220
Anomalies in QFT
imT
4m2 t
Fig. 4.7. Im T\{t) approaching a «^-function for m —> 0, equation D.268)
Source of anomaly: The source of the anomaly in this dispersive proce-
procedure is the existence of the superconvergence sum rule
Г
Jim2
dtlmTi(t) = 1.
D.267)
The anomaly corresponds to a threshold singularity of ImTi(t) at t = Am2
approaching a «^-function for m —> 0 (IR-region)
lim Im
= lim
t2
Am'
24-1/2
6(t - 4m2) = 6{t). D.268)
The limit must be performed in a distributional sense (see Figure 4.7).
This is a surprising feature indeed, but on the other hand it is the DR
approach for analytic functions which connects the behaviour of the ampli-
amplitude at q2 —> oo with that at q2 —> 0. A special renormalization procedure—
regularizing the other end of the spectrum, the ultraviolet divergences—does
not occur at all. For this reason the anomaly is independent of the renor-
renormalization procedure, a fact which we have demonstrated already in the
previous sections.
4.4.2 n-dimensional regularization procedure
Although the method of DR appears quite different to the methods of regu-
larizations we can reproduce its results in a definite regularization scheme,
say, in the n-dimensional regularization procedure of 't Hooft-Veltman ['t
Hooft, Veltman 1972]. It is instructive to work it out in more detail.
We calculate the tensor amplitude of the loop containing vector currents

2-dimensional anomaly and dispersion relations 221
ЫЯ) = -if |f, tr 7m7J_>_^_. D.269)
For the moment it is sufficient to consider the trace (in n dimensions)
T M, 2s _ 2n/2 • f ^P_ B-n)(p2-pq) 2
iM W )-2 */ Bтг)»(р2-
The first term containing a logarithmic divergence must be regularized, the
second term is already regular.
'Naive' regularization and dispersion relations: Performing the li-
limit B — n) —> 0 in equation D.270) before integration we are left with the
second term
»(a2) - 2n'2i / e — (A 9714
{q)~l lJ B7r)"(p2-m2)[(p-qJ-m2Y ^Л1)
We follow the standard procedure and insert the Feynman parameter
integral
ШГТТТт „Л12' yi.iii)
together with the 't Hooft-Veltman formula D.189) where we have (in our
case, a = 2 and n = 2)
—гтг
~ q2x(l -x)-m2'
Then we obtain
In the limit g2 —> 0
^1)M(«2) -» -1A = Л D.275)
the trace does not vanish but provides precisely the anomaly Л. We repro-
reproduce half of the result D.257). So we pick up just a <7M1/-term
T&b) = -\g»uT2{q2) D.276)
(modulo terms which cancel in the trace, see the 'naive' regularization be-
below) and we have lost gauge invariance or the VWI D.238) is broken.

222 Anomalies in QFT
For the amplitude ^(g2) we then have an integral representation
f
o
«Г** D.277)
Q xi1 -x)-m2'
which we can easily convert into the familiar DR integral. The two trans-
transformations
4m2
y = l-2x and t = т D.278)
1 - У2
provide the result
2 _ 1 [*> dt 2m2/ 4m2y1/2
T2{q) ~ ij1)
1 Г°° dt
= " / 7-^ImT2W' D'279)
* Jim2 t-Q
which is precisely the unsubtracted DR D.250) from before.
Thus the 'naive' kind of regularization corresponds in the DR approach
to the case of an unsubtracted DR for the amplitude Tb^2). However, we
want to explain why we achieve only half of the anomaly result D.257) after
the proper renormalization.
t' Hooffc—Veltman regularization and dispersion relations: Now
we include the divergency term, we renormalize a la 't Hooft-Veltman
properly—performing the limit B - n) —» 0 in equation D.270) after
integration—then the explicit calculation gives
[{pap0 -
[p2 - 2pgf(l - x) + q2(l -x)- m2]2
D.280)
when keeping the tensor structure visible. For the p-integration we need the
t' Hooft—Veltman integrals
= Jofc" D.281)
- 2pfe - m2)
^ = Jo („г + ^^Г] , D-282)
m2)a V n + 2 - 2ay J v '
with a = 2, n = 2 and Jo given by equation D.273), and we obtain

2-dimensional anomaly and dispersion relations 223
rl dx
q2x{\ — x) — m2
)}- D.283)
We observe a cancellation of the divergency B - n)~x by another factor
B — n) coming from the contraction of the 47-matrix trace. It provides the
important contribution to finally achieve a gauge invariant amplitude
Гм„(д) = (дмд„ - q2g^)T(q2), D.284)
with
7Г y0 g2x(l - x) -
Thus the correct n-dimensional regularization provides the gauge invariant
tensor structure automatically, the VWI is satisfied and the trace vanishes
in the limit q2 —> 0
T^{q2) = -q2T(q2) -> --^r у 0. D.286)
On the other hand, the regularized amplitude D.284) satisfies the anoma-
anomalous AWI D.263).
Finally we again convert the integral representation D.285) into a DR
integral by applying the two transformations D.278) and we find the result
2m2 Л 4m2^~1/2
t-q*) t
Im T(t)
2
q ' ""¦ ImT2(i) =Tf(q2), D.287)
which coincides with the unsubtracted DR D.248) for T = T\ and with the
once subtracted DR D.258) for T2.
Therefore the proper n-dimensional regularization a la 't Hooft-Veltman
corresponds in the DR approach to the case of a once subtracted DR for
the amplitude T2(q2).
' Nai've' regularization: Now we rediscuss the 'nai've' regularization from
bofore. It exhibits the interesting feature that both Ward identities, the VWI

224 Anomalies in QFT
2
Fig. 4.8. 'Seagull' term
and the AWI, are equally broken. When regularizing the tensor amplitude
Гм„ D.283) 'naively' we throw away the singular term B - n)^1, in accor-
accordance with our 'naive' procedure from before. Then we have
dx
о q2x(l — x) — m2
и - «V)}- D-288)
Identifying the integrals with the amplitudes introduced before we obtain
+ qtfuTtf) - \g^{q2) D.289)
(note that the second and third terms cancel in the trace and we reproduce
equation D.274)) and we rewrite expression D.289) in a more convenient
fashion by virtue of identities D.287), D.259)
т$ (я) = (ч№ - «Vm (я2) + \лд^. D.290)
Now we study the Ward identities. The VWI involves only half of the
anomaly
q»T&(q) = \Mv D-291)
but the AWI contains the other half
l {q) = e^q"BmP(q2) + \л). D.292)
So the 'naive' regularization is symmetric in the sense that the anomaly is
equally spread in the vector- and axial Ward identity.
'Seagull' term: The term

2-dimensional anomaly and dispersion relations 225
\Ag^ D.293)
is named the 'seagull' term (see Figure 4.8). It corresponds to a local
polynomial in the photon fields which can always be added to the action
(recall our discussion of the renormalization ambiguities m Section 4.2.2).
Subtracting the 'seagull' term now we achieve the gauge invariant am-
amplitude
$ \ ^q2), D.294)
with a satisfied VWI and an anomalous AWL
When adding the 'seagull' D.293)
D.295)
the VWI gets anomalous
<?T$(q) = Aqv D.296)
but the AWI is satisfied
qvl%v = qvevXT^\x = e^q" 2mP(q2). D.297)
So by adding or subtracting the 'seagull' D.293) to the 'naively' regula-
regularized amplitude T^J D.290) we can jump between the 2 extremes of having
the anomaly either in the vector- or in the axial Ward identity. But there
is no 'seagull' term such that both Ward identities, the VWI and AWI,
can be satisfied simultaneously. There always remains an anomaly. (For a
discussion of the significance of 'seagull' terms see Jackiw [Jackiw 1985d].)
Cutkosky rule: Finally we promised to demonstrate how to calculate
the imaginary part of an amplitude. Cutkosky's cutting rule [Cutkosky
1960], which is equivalent to the unitarity relation for the amplitude (see
e.g. [Adam, Bertlmann, Hofer 1993]), is the following:
• Repl&ce in an amplitude each propagator by its discontinuity on mass
shell
^ D.298)
then you get the discontinuity—imaginary part times 2i—of the total
amplitude.

226 Anomalies in QFT
Example: We need the imaginary part of the tensor Тм„ D.269) with the
general decomposition D.243). Since the imaginary parts satisfy the VWI
D.252) it is enough to consider just the amplitude T2(q2). As demonstrated
before, the amplitude T2(q2) corresponds to the m2-term in the trace D.270)
of the tensor
T2(q2) = -T< V(«2) = - T^{q2). D.299)
Consequently the amplitude T2(q2) represents the following integral
T2(q2) = -i [ $L
J B?rJ
J
(p2 - m2)[(p - qJ - m2]
We now calculate the imaginary part according to the abo^e rule
ImT2(?2) = 2m2 (d2p6(p2 - m2N((p - qf - m2)
D.300)
Яо - po). D.301)
By using the familiar formula for the «^-functions
(Pi are the roots here; f(pi) = 0) we first integrate over po
Im T2(q2) = 2m2 f dpi J. 6{q2 - 2pqN{q0 - E), D.303)
with
E(pi) = yjpl + m2. D.304)
We conveniently choose the centre of mass system, q\ = 0, and integrate
next over p\, then we get
D.305)
The Lorentz invariant form is finally (with T(q2) = Ti(q2))
q2 Im T(q2) = Im T2(q2) = ^(l-^j ^ , D.306)
the result which we quoted before in equation D.255).
Resume: The chiral anomaly is completely determined by dispersion re-
relations. We have demonstrated this within 2-dimensional QED. The method

The anomaly and the Dime sea
227
Fig. 4.9. The gauge field values defined on a cylinder
appears rather appealing. All one has to calculate is the imaginary part of
an amplitude, which is an easy task.
Working with an unsubtracted DR for the amplitude TMq2) satisfies the
AWI but breaks gauge invariance. When restoring gauge invariance we have
to use a once subtracted DR for T2(q2) which, on the other hand, supplies
an anomalous AWI. Comparing this DR approach with the n-dimensional
regularization scheme we find that a 'naive' regularization corresponds to
the first case whereas the proper regularization is equivalent to the second.
The nature of the anomaly in the DR approach is different in the sense that it
originates from a singularity of the amplitude in the infrared region whereas
in a regularization procedure it reflects the behaviour in the ultraviolet
region.
4.5 The anomaly and the Dirac sea
The Dirac sea simulates the effects of a quantized field theory. In fact,
the anomaly can also be understood in terms of the Dirac sea (see e.g.
[Jackiw 1985d], [Manton 1985], [Ambj0rn, Greensite, Peterson 1983], [Niel-
son, Ninomiya 1983,1991], [Shifman 1991], [Adam, Bertlmann, Hofer 1993]).
We study the Schwinger model here—QED in 2 dimensions with massless
farmions—with x-space compactifled on a circle S1 of length L.

228 Anomalies in QFT
Configuration space: The fields are defined on a cylinder of space and
time (see Figure 4.9) and have to obey the boundary conditions
D-зо7)
We choose a gauge where A\ is independent of x and Aq (the Coulomb
potential) can be neglected. In fact, we treat A\(t) as an external field
which will be switched on adiabatically. Due to the gauge freedom the values
A\ = 0 and Ai = 2ix/L are gauge equivalent and must be identified. So the
true configuration space of the gauge potential is a circle S1 of length 2tt/L.
Conservation laws: The underlying GA) symmetry (we consider here
constant phases) leads to the conservation of the vector current and electric
charge
0*% = 0 and Q{t) = 0, D.308)
with
Q(t) = fdxjo(t,x). D.309)
The axial UaA) symmetry (chiral symmetry) implies the conservation of
the axial current and the axial charge
dPjl =0 and Q5(t) = 0, D.310)
with
Q5(t)= jdxjl{t,x). D.311)
Introducing the left- and right-handed fermions
*-(*)• ¦-(?)• D312)
with chirality ±1
D.313)
we get the charges
Ql,r = / dxi>L,iaoi>L,R = j dxtl>l<Ril)LiR D.314)
or
Q = Ql + Qr, Q5 = Ql-Qr- D.315)

The anomaly and the Dime sea
229
empty
levels
filled
levels
L
n
T
0
n
~~ L
Зл:
L
<^>
""r
1 L -particle
In Ax
L
Л-hole
Fig. 4.10. Shift of the energy levels in the Dirac sea. The L-levels increase
and the .R-levels decrease producing a L-particle and a .R-hole
So the axial charge for a L- or .R-fermion is
Q
,r = I
,R = ± / йх^ь.л^ь.л =
D.316)
Since Q and Q5 are conserved the numbers of L- and Л-fermions are con-
conserved separately.
The laws D.308) and D.310) are valid classically. On a quantum level,
however, both currents (or charges) D.308) and D.310) cannot be conserved
simultaneously—an anomaly occurs. How can we observe this phenomenon
in a theory a la Dirac?
Energy spectrum: We start from the Dirac equation
(i 0 + 4)fl> = 0 D.317)
and we choose the 2-dimensional Dirac matrices in the following way
7°=<72, 'Y1 = itri, 78 = tV=<73, D.318)
where 04 denote the familiar Pauli matrices (see equation C.281)). Then
the Dirac equation is rewritten by

230 Anomalies in QFT
= 0- D319)
According to the boundaries D.307) the fermion wave function can be ex-
expanded into the Fourier series
1 „ Г.27Г / 1\ 1
which provides the following energy solutions for the L- and .R-fermion
eigenstates
4i D.321)
-Ль D.322)
with fc = 0,±1,±2,....
So the energy spectrum is comfortably discrete because of compactifica-
tion and it depends linearly onii. We have plotted its features in Figure
4.10. At Ai = 0 the energy levels for L- and Д-fermions are degenerate. If
we switch on A\ the levels split; the L-levels increase whereas the Д-levels
decrease. At A\ — 2ir/L we reproduce exactly the original level structure
as it should be for gauge equivalent values. It is precisely this nontrivial
restructuring of the infinitely many fermion levels that is the source of the
anomaly, as we shall see.
IR-behaviour and anomaly: Now we turn from the 1-particle case to
the multiparticle description of QFT by introducing the Dirac sea. We fill up
all negative energy levels and keep all positive levels empty. So at A i =0 we
have a vacuum—the Dirac sea. However, if we increase Ai from 0 to 2ir/L
we produce—by lifting the L-levels and lowering the .R-levels—a left-handed
particle and a right-handed hole (see Figure 4.10).
What does it imply for the charges? The electric charges of the particle
and the hole are opposite. So there is no change in the total electric charge
and the vector current is conserved, equation D.308). The axial charges,
however, are identical for both the L-particle and the Л-hole, so that the
net axial charge changes
AQ5 = 1 + 1 = 2. D.323)
This change compared with the one in the gauge potential Ai gives

The anomaly and the Dime sea 231
AQ5 = 2 = -AAi D.324)
7Г
and per time unit
*? _ k*?. D.325)
At тг At K '
Considering the local change in equation D.325)
j
j^J dxjl(t,x) = ~t j dxA^t) D.326)
we obtain the corresponding relation for the axial current
доЗо = -doAi D.327)
7Г
or finally written in a Lorentz invariant way we arrive at the anomaly
d*jl = \г^А\ D.328)
which coincides with the result D.266) determined in the previous section
via dispersion relations.
UV-behaviour and anomaly: Alternatively we can also study the be-
behaviour at some UV-cutofF which is more directly related to the perturbative
approach in QFT presented before.
In fact, we have to regularize when working with the Dirac sea. The
total energy, the total charge of the vacuum, the fermionic wave function
as superpositions of infinitely many filled states are ill-defined quantities
which have to be regularized somehow. A procedure which preserves gauge
invariance is the point splitting method of Schwinger [Schwinger 1962] which
we have used already in Section 4.3.4. There the gauge invariant regularized
currents are
j;eg = lim^{t,x + ?)j^(t,x)exp\-i f
Г Г+? 1
,z)exp -г / dxAA D.329)
Y Jx J
Г fx+? 1
')(t,x)exp\-i / dxAA . D.330)
L Jx J
The rule is that the computations are carried out with fixed e and afterwards
the limit s —> 0 is taken for the physical quantities.
The regularized charges are defined by
Q(t) = j dxfoes{t,x) D.331)

232 Anomalies in QFT
Q5(t) = Jdxj50re*(t,x) D.332)
Ql,r by equation D.315). Calculating the L- and Л-charges explicitly
we obtain the sum
Ql,r = $>xp |-«e [^ (k + ^j + A,] } , D.333)
with the summation
к = -1, -2, -3,... for L
D.334)
fc = 0,l,2,... for R.
If we first take the limit e —> 0 we are back at the unregularized ill-
defined quantities
Y D-335)
corresponding to the infinitely many filled levels in the Dirac sea.
But summing first and expanding afterwards in s provides for the L-
and Л-sea the result
hM- D-337)
Now we observe the following. The electric charge of the vacuum vanishes
gvac = gvac + qvsc = q D.338)
The electric charge and hence the vector current is conserved. The axial
charge, on the other hand, contains a divergent constant 1/e which will be
subtracted to define the regularized quantity; so we are left with a linear
dependence on A\
gSvac = gvac _ gvac = ^^ D 339)
As A\ increases from 0 to 2tt/L the axial charge changes by 2 units as
before, providing the same anomaly result.
Resume: We consider the Dirac sea, and we switch on a gauge potential.
It shifts the energies in this infinite sea; the L-levels increase and the R-
levels decrease, there is a spectral flow [Atiyah, Patodi, Singer 1976]. It is this
difference of the energy levels—passing through the zero scale (E = 0) and

Decay тг° -> 77 and PC AC
233
e,.*,
Fig. 4.11. Decay тг° —> 77
weighted by the direction of the flow (here +1 and — 1)—which corresponds
to the index of the Dirac operator (see Chapter 11).
Consequently the L- and Д-charges Ql,r change by ±1 so that the
overall axial charge varies by 2 units—an anomaly occurs. On the other
hand, when focusing on the energy scale in the UV-region we find the same
features.
There is no Feynman diagram of perturbation theory needed here, the
anomaly here is a pure topological effect. The Dirac sea is a nice and prac-
practicable picture which reveals the anomalous phenomena of the quantum
world. It provides an intuitive understanding for the different aspects: the
anomaly occurs either as an IR-phenomenon (an index property of the Dirac
operator) or as a UV-phenomenon.
4.6 Decay тг° -> 77 and PCAC
As already emphasized at the beginning we need anomalies to explain ex-
experiments. Anomalies can be experienced in several particle reactions but
the most stringent test for the ABJ anomaly is the neutral pion decay into
2 photons. It is also one of the most prominent decays in particle physics
which created a puzzling situation for some time, whose solution actually
led to the discovery of the anomaly. Let us consider it in more detail; for

234 Anomalies in QFT
literature we refer to [Jackiw 1985d], [Cheng, Li 1988], [Itzykson, Zuber
1980], [Adam 1990], [Caprini, Micu, Visinescu 1992].
The transition matrix element for the pion decay (see Figure 4.11)
is defined by
= B7r)i64(q-k1-k2)e4(ki)ev2(k2Wliv(k1,k2,q), D.340)
where
2j** *" Л D-341)
Because of the negative parity of the 7г° the amplitude D.341) is a pseu-
dotensor with the general Lorentz structure
, k2,q) = Г(д2)е^а/Зк^к1 D.342)
LSZ reduction formula: The pion and the photons can be reduced in
the transition amplitude D.340) according to a procedure invented by Leh-
mann, Symanzik and Zimmermann (LSZ) [Lehmann, Symanzik, Zimmer-
mann 1955].
For scalar fields (analogously for other fields, see e.g. [Itzykson, Zuber
1980], [Lurie 1968]) the 5-matrix obeys the following formula:
Sfi = (cd out|ab in)
= hi + (-04 / d4x1d4x2d4x3dixif^(x4)f:(x3)fb(x2)fa(x1)
Ф(х1Щх2Щхз^{х4){0\Тф{х1)ф{х2)ф(хз)ф(х4)\0), D.343)
with the normalized plane waves
fa(x) = ^ e-'"-1, iV=B7rK/2N/2rf D.344)
and the Klein-Gordon operator
V(x) = Пх + m2. D.345)
The fields satisfy the Klein-Gordon equation with a source
Т>(х)ф(х) = j(x). D.346)
For the pion amplitude D.341) then follows

Decay тг° -> 77 and PC AC 235
Гд„ = e2 Jd4yeik^ Jd'ze-^ia.+ml)
¦(О\Тф„(г)ШШ\О)- D-347)
The variable x has been transformed away due to energy-momentum con-
conservation and the normalization N is absorbed by the definition of the final
scattering matrix element. Extracting the Klein-Gordon operator (integra-
(integrating by parts) gives
Гд„ = e2(-q2 + ml) f d4yd4z eik^-iqz
D.348)
We compare this amplitude with the 3-point function Тд„л which we have
discussed already in Section 4.2.1 (recall equation D.79))
x = Jd*yd*z eik^-^ @|Гд2А^Ы2/Ы0)|0>. D.349)
In fact, we can find a bridge between these two amplitudes D.348) and
D.349): the pion field couples to the nonconserved axial current.
Hypothesis: PCAC (partially conserved axial current)
dXjla = Urnl<fr«(z), D.350)
where /„¦ = 93 MeV denotes the pion decay constant measured in the decay
7Г+ —> /i+^M; and a = 1,2,3 are the SU{2) isospin indices with a = 3 for the
neutral pion 7г°. For massless pions—Goldstone bosons—the axial current
is conserved, the corresponding chiral symmetry is exact.
Using PCAC D.350) now we find the relation
Sutherland-Veltman paradox: Sutherland and Veltman [Sutherland
1967], [Veltman 1967] observed the following behaviour of the amplitudes.
Since ТД„А does not contain any poles in the limit q —v 0 (there are no phy-
physical states between the pion and the vacuum) the l.h.s. of relation D.351)
has to vanish for 9 -»0. Consequently the pion transition element must
vanish too
T(q2 = 0) = 0. D.352)
So the pion should not decay into photons

236 Anomalies in QFT
G7Ю ~ T(q2 = ml) « T(q2 = 0) = 0. D.353)
But experimentally the pion decays!
Resolution: This paradoxical situation is resolved by the anomaly. We
know already that we have to include the anomaly in the axial Ward identity,
equation D.83), or correspondingly in the divergence of the axial current,
equation D.85).
Proposition:
• PCAC has to be modified in the presence of gauge fields by the
anomaly
4 = UmU^(z) + ^e^F^F^. D.354)
Note: In this relation the axial current carries 5GB) isospin
if = ^7д75 ~Ф D.355)
and contains a factor 1/2 compared to the usual axial current definition
D.4) (which entered in result D.86)); accordingly the anomaly coefficient
1 e2 a
Then the anomaly corrected relation D.351) has to be
and we determine the constant с below in the quark model.
Pion amplitude: In the soft pion limit the pion amplitude is now totally
given by the anomaly contribution
and hence
MI/(fci,*2,«f) = ^j-e^kfkl D.357)
9-*° tft Jir

Decay тг° -> 77 and PC AC 237
Quark model: Let us calculate the constant с within the simple quark
model. For the 5GC) flavour group (up, down, strangeness) the quark spinor
denotes
(u(x) \
d(x) 1 D.359)
s(x) j
and the currents are defined by
j^x) = ф{хI1лЯф{х) D.360)
Л3
with the matrices
: 0 0 \
A3 = I 0 -1 0 I . D.362)
The constant с in the anomaly term D.356) accounts for the matrix coup-
couplings in the triangle diagrams, hence we get
с = о - — tr \Q, Q\— = о • — = —. D.obo)
2 2 6 2
The factor 3 in front respects the colour degrees of freedom of the quarks.
Decay rate: Finally we have to evaluate numerically the decay rate of
the pion. We need the formula for the decay width (see e.g. [Pietschmann
1983]
1 Г J3i. j3l._
i8A{q-ki -k2)
°B7rK2fc°v ;
|2. D.364)
polar.
The calculation of the phase space integral is straightforward. In the soft
pion limit with expression D.358) and with the quark model value с = 1/2
D.363) we obtain
Г(тг0 -f 77) = ^fl = 7.63 eV. D.365)
This result is in excellent agreement with the experimental rate
rexperiment(*0 - 77) = 7.37 ± 1.5 eV. D.366)

238 Anomalies in QFT
Fig. 4.12. YM coupling matrices in the triangle diagrams
So it is entirely the anomaly which determines the тг° —> 77 decay. In
addition, the colour factor 3 is necessary to achieve the correct experimental
value. Therefore the anomaly also tests the colour degrees of the quarks.
4.7 Singlet anomaly
Next we turn to non-Abelian fields. There also higher loops contribute to
the anomaly. Considering first the simplest diagram, the triangle, we have
to study the amplitudes
J
D.367)
fei,fe2) = iJd*xd*yeik*+*>v@\T%{x)jl{v)Pc{0)\0),
D.368)
which now carry internal quantum numbers. We have already defined the
non-Abelian currents j*, j^a, Pa in Section 4.1.2, equation D.46). Then we
know from our discussion of Ward identities (Section 4.2) how the anomaly
modifies the AWI
qxT$l = 2mT°Zc + A°ft + commutator terms, D.369)
where
„abc
даЬс _ ^ - aU°4.P Iл V7(\\
denotes the ABJ anomaly for non-Abelian fields. The constant cabc
accounts for the non-Abelianity of the fields. It is the symmetrized trace of

Singlet anomaly
239
лй та
Fig. 4.13. Triangle diagrams for the singlet anomaly
the YM coupling matrices Ta which appear in the triangle diagrams (see
Figure 4.12)
cabc = i tr {Ta,Tb}Tc D.371)
and which we can express by the symmetric structure constants C.286) of
the SU{3) group
D.372)
„abc _ _jabc
4
When we keep the axial current jjj Abelian (as a colour singlet) but the
gauge fields non-Abelian, so when we choose Tc = 1 (see Figure 4.13), we
speak of a singlet anomaly.
Then the matrix couplings change to
cabc
and the anomalous AWI is
D.373)
D.374)
with the singlet anomaly contribution

240
Anomalies in QFT
Fig. 4.14. Quadrangle diagram contributing to the singlet anomaly
27Г2?'
D.375)
As in the Abelian case the expression D.375) corresponds to an anoma-
anomalous operator identity for the colour singlet axial current j\. The only colour
singlet operator, which we can construct from the gauge fields, is
ab
Я^ -5 _ _± l
A ~~ 16тГ2
D.376)
with
F?v = d^Aav - dvA<l + [Ац, Av]a D.377)
being the YM field strength. Then the operator identity D.376) provides
the correct triangle result D.374), D.375); and we have chosen m = 0.
Inserting the coupling D.373) and using the notation .
rewrite the singlet anomaly
= F?vTa we
tr
D.378)
where the trace is taken over the gauge group generators T°.
Equation D.378) is an obvious generalization to the Abelian case D.86).
But now also nonlinear terms are included in the YM field strength and the
singlet anomaly can be re-expressed by the Chern—Simons term

Non-Abelian anomaly—Bardeen's result 241
dxj5x = ^2^"Q/3 tr Эм (AvdaA0 + \а„АоаЛ . D.379)
All anomalous WI's follow from the operator identity D.378). The new
feature for YM fields are the selfcoupling terms. In addition, to the linearized
version proportional to
д„АидаА0, D.380)
which corresponds to the triangle diagram (Figure 4.13) we have a term
дцАиАаА0, D.381)
which leads to a quadrangle diagram (see Figure 4.14).
So we get two contributions for the singlet anomaly, the triangle and
the quadrangle diagrams. But these are not independent of each other. If,
for example, the triangle result vanishes, the quadrangle result is zero too.
Hence it is enough to consider just the simple triangle diagram in search of
an anomaly.
4.8 Non-Abelian anomaly—Bardeen's result
Finally we discuss the non-Abelian anomaly. The classical conservation laws
for non-Abelian fields, which we have derived in Section 4.1.2, are broken
in QFT; the anomaly that occurs has been calculated by Bardeen [Bardeen
1969].
Non-Abelian anomaly: For a Lagrangian like equation D.26) con-
containing vector Vfj, and axial A^ YM fields the underlying symmetry is
S?/yC) x SUa{3) (for m = 0). Then the non-Abelian anomaly is the
following expression (after a laborious work)
U З) J^? tr l
- (Л Л Tv 4- Л Tv A 4- Tv A A \
о V-^V"n""' per ' •fxIL-rvp'no i •' fiv^p-^a)
32 1
—A^AuApAJ , D.382)
with the Bardeen curvatures
rfv = дцАи-диАг + №,Аи] + [Ац,Уи]. D.383)
For result D.382) Bardeen has added suitable counterterms to the action in
order to achieve covariant vector current conservation

242 Anomalies in QFT
x
Fig. 4.15. Diagrams contributing to the non-Abelian anomaly in the diver-
divergence of the axial current, x depicts the axial current, • the vector current
2\dM = 0. D.384)
If we consider just vector fields (VM = A^, Лц, = О) then the non-Abelian
anomaly is simply
tr
This is a straightforward non-Abelian generalization to the singlet case
D.378). The result D.385) transforms gauge covariantly and indeed
equals the covariant anomaly, which we discuss below.
The anomalous Ward identities corresponding to the operator identity
D.382) involve higher order loop diagrams: the triangle-, quadrangle- and
pentagon diagrams (see Figure 4.15). Although the pentagon is a finite dia-
diagram it contributes to the anomaly if VWI is required (because of divergent
'seagull' terms to be added; see Bardeen [Bardeen 1969]).
Triangle anomaly: For example, the anomaly corresponding to the tri-
triangle diagrams is the following. The triangle which couples to 3 axial fields
AxA^Av amounts to just 1/3 of the triangle which couples to 1 axial- and 2
vector fields A\V^VV- The reason lies in the Bose symmetry of all 3 vertices.
Therefore we get

Non-Abelian anomaly—Bardeen's result 243
Anoma(A)
= ^?""'a [tr TaF^(V)Fl™(V) + ^ tr TaF^(A)F^(
D.386)
where F^1" denotes the linearized version of the field strength D.383).
The triangle result D.386) is obviously contained in Bardeen's expression
D.382). The several loops which contribute to the anomaly (a detailed dis-
discussion can be found in the lecture notes of van Nieuwenhuizen [van Nieu-
wenhuizen 1988]) are connected with each other. If the triangle contribution
vanishes then the other loops, the quadrangle and pentagon, vanish too.
Non-Abelian anomaly and chiral fields: On the other hand, we can
consider the Lagrangian in terms of L- and Д-handed fields, or positive-
and negative chiral fields; recall equation D.37). The underlying symmetry
is S?/lC) x SUrC) (for m = 0). Then the classical conservation laws are
broken in a very symmetric way. The non-Abelian anomaly for L- and
Д-fields is (see [Bardeen 1969])
/ 1
ru.VD<r t- грая I дНд дН , x дН дН дН
" J °V* l Л" °PA° + 2 " p "
D.387)
with
+1 fnr и \ L or + chirality ,A ooon
-1 [ R or - chirality v ;
So the anomaly just changes sign for L- and Д-fields. Therefore we suppress
from now on the chirality index H = L,R throughout the book; but we
constantly mean L- or Д-fields.
The anomaly expression D.387) is the important result which we in-
investigate in this book. It is also called the consistent anomaly since it
satisfies the Wess—Zumino consistency condition. We explain this more
precisely in Chapter 8.
But there also exists another type of anomaly, the covariant anomaly
^ivp° tr TaF»vFpa, D.389)
with (+) or (—) for R- or L-handed fields. We study its relation to the
consistent version D.387) in a separate chapter (Chapter 10). In the case of
vanishing axial fields (A^ = 0)

244 Anomalies in QFT
and
Ga ~ -D»jH^ -> i^D^5"
we reproduce the covariant result D.385) for the pure axial current.
4.9 Importance of anomalies
The importance of anomalies for physics is twofold:
Anomalies are good for experiment: This-is'the case if the underlying
symmetry is an external symmetry. Then the anomalies are responsible for
the properties of particles, for the physics of the particle decays or transiti-
transitions.
Examples:
i) Decay тг° —> 77.
This is the most prominent decay which is totally determined by the
ABJ anomaly [Adler 1969], [Bell, Jackiw 1969]. We have already dis-
discussed all details in Section 4.6.
ii) f/(l) problem.
The QCD Lagrangian (in the limit m —> 0) contains a further UaA)
symmetry, besides the chiral SUl{2) x SUrB) symmetry and the
vector ?/y(l) symmetry. This axial symmetry is, however, neither ob-
observed in the hadron spectrum (no parity doubling of baryons) nor
realized as a Goldstone boson (the 7j'-meson is just too heavy). The
resolution of this problem is found in the existence of the anomaly
and of gauge field configurations with nonvanishing topological charge,
called instantons. Then the Ua{1) symmetry is spontaneously broken
without generating a Goldstone boson ['t Hooft 1976a,b], [Kogut, Sus-
skind 1975].
iii) Proton spin crisis.
The ?/дA) axial charge of the proton (nucleon)—the 'spin' of the
proton—departs from its expectation in a naive constituent quark
model due to the anomaly [Altarelli, Ross 1988], [Altarelli, Stirling
1989], [Altarelli, Nason, Ridolfi 1994], [Pritzsch 1991], [Shore, Vene-
ziano 1993], [Narison, Shore, Veneziano 1993].
iv) Reaction 77 —> тг+7г~7г°.
This electromagnetic reaction also contains a part given by the non-
Abelian anomaly [Wess, Zumino 1971].

Importance of anomalies 245
v) Decays K+ —> 7г+тг07, K+ —> w+ir~e+i/e.
In these nonleptonic Kaon decays there occurs in addition to the
weak transition a contribution coming from the non-Abelian anomaly
[Ecker, Neufeld, Pich 1992, 1994], [Bijnens, Ecker, Pich 1992], [Wess,
Zumino 1971].
Anomalies are bad for theory: More precisely, anomalies are harmful
for a quantized perturbative gauge theory. Then the anomalous Ward iden-
identities destroy the renormalizability, and thus the consistency, of the gauge
theory [Gross, Jackiw 1972]. But also the unitarity of the 5-matrix may
be spoiled by anomalies [Korthals Altes, Perrottet 1972] (for a discussion
see [Hofejsf 1994]). This happens if the underlying symmetry is an inter-
internal symmetry and the gauge fields are internal fields and quantized. Ne-
Nevertheless, there are attempts to live within anomalous (nonperturbative)
theories [Eaddeev, Shatashvili 1986], [Jackiw, Rajaraman 1985], [Rajaraman
1985a,b], [Lott, Rajaraman 1985], [Harada, Tsutsui 1987].
In order to guarantee the renormalizability of a theory anomalies must
not occur. This implies important restrictions to the physical content of a
theory.
Anomaly cancellation conditions:
i) Vecior-Ii&e model.
In a vector-like model where all fermions couple symmetrically in the
L- and R-sectors the L-gauge anomalies cancel the Д-gauge ones.
ii) 'Safe groups'.
There are no anomalies if the trace over the gauge group generators
vanishes in a certain representation or even for all representations.
The condition for it is
cabc = i tr {Ta,Tb}Tc = 0 D.390)
resulting from the triangle diagram; then all higher loop contributions
vanish too.
There are 'safe' groups in 4 dimensions like SUB), SOBN + 1) and
SO(AN) with N > 2, or exceptional groups like EF), E(8) where
identity D.390) is valid for all representations; however, unfortunately
not for SU(N) with N > 3 (see e.g. [Georgi, Glashow 1972]).
iii) Standard theory.
The famous anomaly cancellation happens in the standard theory for
electroweak interactions with the gauge group SU{2) x U{\) where
the L-handed fermions are arranged in doublets and the Д-handed
fermions in singlets [Gross, Jackiw 1972], [Bouchiat, Iliopolous, Meyer

246 Anomalies in QFT
1972]. There condition D.390) is satisfied if (see e.g. [Cheng, Li 1988],
[Ecker 1982])
tr QL = 0, D.391)
with Qi, the charge matrix of the L-fermions. Hence for a lepton
doublet we also need a quark doublet with colour factor 3 in order
to achieve anomaly cancellation
D.392)
/9 i\
tr QL : -1
This feature repeats itself in every generation
D.393)
CO.-
So the absence of anomalies restricts the fermionic content of the
theory quite severely. It also tests the colour factor 3 of the quarks
and gives very strong support for the existence of the yet undiscovered
top quark t.
iv) Nonlocal counterterm.
When introducing a nonlocal counterterm of gauge fields into the
action we can compensate the anomaly (see e.g. [Krasnikov 1984,
1985a]). The term is constructed such that a gauge variation just
cancels the anomaly contribution.
For example, in the Abelian case the nonlocal polynomial of L-fields
is the following
<SpO. = g^"*" / dxdy dxA>z(x)D(x - y)F%Fifi(y), D.394)
where D(x-y) denotes the Green function of the d'Alembert operator.
Inserting О-1ПХ = 1 we rewrite expression D.394)
^JWD^F^x). D.395)
Performing a gauge variation

Importance of anomalies 247
6Spol = Л-еГ^ fdx
we obtain after integrating by parts
J
D.396)
which cancels precisely the anomaly D.111).
v) Local counterterm with additional Gelds:
Analogously a local counterterm in the action—but containing addi-
additional fields with appropriate gauge transformations—can also com-
compensate the anomaly (see e.g. [Krasnikov 1985b,c]).
A simple Abelian example is
SPo. = gti^a0 J dx fL(x)F^0(x), D.397)
with a gauge variation
6(pL(x)=AL(x). D.398)
Then the variation of the polynomial D.397) leads again to result
D.396) and cancels the anomaly D.111).
A very sophisticated generalization of this simple example repres-
represents the Green-Schwarz anomaly cancellation mechanism in a 10-
dimensional supergravity and super Yang-Mills theory ([Green,
Schwarz 1984], [Green, Schwarz, West 1985]).
Resume: There exist two types of chiral anomaly, the singlet- and the
non-Abelian anomaly
tr д» (АЛЛ, + \AvApAa\ D.399)
(Note that in the second case the gauge fields and the current are of positive
chirality, for negative chirality we get a sign change, the chirality index is
suppressed.) Although the two expressions resemble each other very closely
they are of different origin. We want to emphasize this different nature of
the anomalies. Let us consider the triangle anomaly part.

248 Anomalies in QFT
• If the axial current is the generator of the external symmetry while
the other two vector currents are coupled to gauge bosons—as in the
singlet anomaly case—then the anomaly signals the breakdown of this
external symmetry in the presence of the gauge fields. Topologically,
the singlet anomaly is completely determined by the Atiyah-Singer
index theorem in 4 dimensions (see Section 11.1).
• If all three currents (say, L- or Л-handed) are coupled to L-, Л-gauge
fields and generate an internal symmetry—as in the case of the non-
Abelian anomaly—then the anomaly causes a violation of gauge inva-
riance, an anomalous Ward identity and the ruin of renormalization.
Topologically, the non-Abelian anomaly is also determined by an index
theorem—but in D + 2) dimensions! (See Section 11.5.3.)
• Nevertheless, the two types of anomaly are related to each other in
a very peculiar way. The non-Abelian anomaly in In dimensions is
related to a singlet anomaly in Bn + 2) dimensions via the differential
geometric Stora-Zumino chain of descent equations (Chapter 9).

5
Path integral and anomaly
As we have discussed already in Chapter 4 the anomaly expresses the break-
breakdown of a classical symmetry by quantum effects. This view of the anomaly
arises quite naturally in the path integral formalism. In quantum field theory
the basic quantity is the generating functional for the Green functions—a
path integral for the classical action—which we have discussed in Chapter
3. How can an anomaly occur when the classical action is invariant under
symmetry transformations? It was the important discovery of Fujikawa that
the path integral measure—the only quantity which contains the quantum
aspect—does not remain invariant under chiral transformations [Fujikawa
1979, 1980]. The associated Jacobian provides precisely the anomaly! In
fact, all types of anomaly emerge in this way. This treatment of the anom-
anomaly is independent of perturbation theory and is, for this reason, called the
nonperturbative approach.
In Section 5.1 we perform a chiral transformation of the path integral
and we find the anomalous Ward identity. In Section 5.2 we regularize the
transformation Jacobian a la Fujikawa and in this way derive the singlet
anomaly; the 2-dimensional case is added in Section 5.3. In Section 5.4 we
show the regularization independence of the anomaly and in Section 5.5
we discuss the conflict between gauge- and chiral symmetry in the light of
an uncertainty principle. The generalization of the path integral method to
non-Abelian fields leading to the non-Abelian anomaly is demonstrated in
Suction 5.6. Finally, in Section 5.7 we carry out the regularization of the
Jacobian by means of the heat kernel method and by the zeta function
procedure.
5.1 Fermionic measure and chiral transformation
1-or the anomaly—and we begin with the singlet anomaly case—we consider
quantized Dirac fermions interacting with an external non-Abelian field A^.
The Lagrangian is
С = ф{г]Р-т)ф, E.1)
with the Dirac operator

250 Path integral and anomaly
= -f(dli + Ali) E.2)
and the gauge potential A^ — A^Ta. The group generators fulfil the com-
commutation relations C.275). For the actual calculations we perform a Wick
rotation into Euclidean space-time
with the metric
So we get
p = j^Dp = g'
We have chosen
ix° ~xA dQ- i— - id4
H° = 74 Aq — 1A4,
«Г = -v».
UI/7i/-DAi = -JuDfj, = -Gi?>i +72-D2 +7з-Оз -\
all 7^-matrices anti-Hermitian
7Mt = _7M> u = 1,2,3,4
E.3)
E.4)
-74D4). E.5)
E.6)
and we recall that the group generators are anti-Hermitian too, T°t = — Ta.
The 75-matrix, on the other hand, remains Hermitian
-?=Ъ, E-7)
with
75 = i7°717273 = 7V7V = -7X727374 E-8)
and
{7", 75} = 0. E.9)
Then the Dirac operator turns out to be Hermitian in Euclidean space-time
ft =P. E.10)
Chiral transformation: Now we perform a local chiral transforma-
transformation
ф _f ф' = е^Ю-пф
ф -у ф'= фе^х)^, E.11)
where /3(ж) denotes some gauge function. For infinitesimal /3 the Lagrangian
E.1) changes to
E.12)

Fermionic measure and chiral transformation 251
and with the usual definitions of the axial current and pseudoscalar density
l E.13)
we get
?'
(Note, for m = 0 equation
The classical action
ql я |
with
remains invariant
P
= ?-
E.15)
jdx?
S
= Ф1ьФ
{д^Р)з1 - 2imj3P.
is also valid for Unite /3.)
](x)\d'1jl{x)-2imP(x)),
- fdxC
E.14)
E.15)
E.16)
E.17)
S' = S E.18)
iff the classical conservation law for the axial current is satisfied
&*jl = 2imP E.19)
in accordance with our considerations in Section 4.1.
Path integral: What happens in a quantized field theory? There we have
to study the generating functional for the Green functions in Euclidean
space
= J dxl)dxj} exp / dx ф{г ф - т)ф = det(i JZ> - m) =: TV. E.20)
It. is a fermionic path integral, a Grassmann integral, serving as a definition
lor the determinant of the Dirac operator (recall Section 3.3 for our notati-
notations). Performing here the chiral transformation E.11) Fujikawa discovered
t hat the path integral measure transforms with a Jacobian containing the
anomaly [Fujikawa 1979, 1980]. The derivation follows in the next section;
Iktc! we just quote the result.
Theorem: The path integral measure transforms chirally as
d^di>' = dфdф J[/3, А„\, E.21)
where the transformation Jacobian

252 Path integral and anomaly
J{C, А„] = exp f- j dx р{х)А[Ац) (i)l E.22)
contains precisely the singlet anomaly in Euclidean space
va& tr F^Fafi. E.23)
The imaginary factor -i is removed in Minkowski space (for ?0123 = 1).
Including also the fermionic sources 77, fj we now expand the chirally
transformed path integral (we suppress the source for the A^ field which
we do not need for our purpose)
Zfo, fj, А„ P] = ^J drP'dft exp || dx[C +
= 77/ ##exp •! / da;[? + # + ^»?] i
• exp I I dx /3(x)[d^jl - 2imP - Л[АМ]
= — dtj)d$ exp ^ / dx[? + j?V + ^»?] [
• 11 + j dx 0(x) \д*з1 - 2imP - Л[А„} + гЩьФ + i^JbV] \ ¦
E.24)
Hence we get the following functional expansion (/3 infinitesimal)
Z[7?,77,AM,/3] = Z[v,fj,A^0}+ fdx0(x)-^Z[r],fj,All,p] , E.25)
with
Z[v,fj,AI,0} = ^ J dxPd^explf dx[C + ^ + rpV}\ E.26)
= j? ##exp i / dx[C + fjip + фт]} \
j* - 2imP - ¦ . r.
E.27)
In the above expansion we transformed only the integration variables ф,
¦ф in the PI. Therefore the generating functional—the 'quantum action'—

Fermionic measure and chiral transformation 253
must remain invariant under the chiral transformation.
Lemma: Invariance of the 'quantum action'
Zh,M^j = Z[i),i),AI,O|, E.28)
which means that
= 0. E.29)
6f3(x)
0=0
This invariance implies the anomalous divergence of the axial cur-
current (finally we have to put r] = fj = 0 in condition E.29))
д*ф = {РзЪ = 2*m<P> + А[АД. E.30)
The symbol () denotes the PI average which equals the vacuum expectation
value of the corresponding operator (recall Chapter 3). Thus we reproduce
precisely our result D.232) of Section 4.3.4.
Ward identity: Condition E.29) determines all (anomalous) Ward iden-
identities of the theory by differentiating with respect to the sources. For exam-
example, differentiating the functional E.24)
6V(x2NP(x)
= -p I dipdtpexp I / dx[C + fji) + iprj] \
0=0
(х-х2)}
1 /• _ Г Г г _ _ ,1
= —. \ dфdфeщ) < / da; ? + пф + фт >
=Q М J U J
and using the invariance condition E.29)
= 0 E.32)
0=O,»7=O,»5=O
6fi(xiNri(x2N/3(x)
applies the anomalous axial WI (in Euclidean space)

254 Path integral and anomaly
= 21т{Р(хЩх1)ф(х2)) + (Л[АI](х)ф(х1)ф(х2))
- г{^ъф{х)ф{х2))Ь{х - xi) - 1(ф(х1)ф(х)'у5N(х - х2),
E.33)
which in the operator formalism corresponds to the vacuum expectation
value of the time-ordered products (in Minkowski space)
= 2гтп@\ТР(х)ф(х1)ф(х2)\0) Щ
- @\ТЪф(х)ф(х2)\0N(х - xi) - @\Тф(Х1)ф(х)Ъ\0N(х - х2).
E.34)
In equation E.33) the factor i in the last two terms and in the anomaly result
Л is removed when we rotate back to Minkowski space. This anomalous
axial WI E.34) is the partner equation to the vector WI D.68) which we
have derived in Section 4.2. Of course, we recover result E.34) when we
follow the standard procedure, differentiating the Green function directly
and inserting the anomalous divergence 9Mj? = 2imP + А[Ац].
Remark: We want to emphasize that for the singlet anomaly
where the axial current represents the generator of an external UaA)
symmetry—it is the invariance of the 'quantum action'
= 0 E.35)
0=0
6P(x)
under a chiral change of the integration variables which determines all (an-
(anomalous) Ward identities.
This is different from the non-Abelian case where all chiral currents are
coupled to gauge fields and where both, the fermions and the gauge potenti-
potentials (A^ = Ац+D^p), are chirally transformed. There it is the noninvariance
of the 'quantum action' under gauge transformations
= G^A^x) E.36)
0a=Q
6/За(х)
—the occurrence of the non-Abelian anomaly Ga [A^] (x)—that determines
all (anomalous) Ward identities.

Fujikawa's method and singlet anomaly 255
5.2 Fujikawa's method and singlet anomaly
We certainly have to prove theorem E.21)-E.23), we demonstrate now that
the Jacobian of the PI measure indeed creates the anomaly. We know already
from Chapter 3 that a Grassmann measure transforms with the inverse
determinant of the transformation operator. However, this determinant is
ill denned in our case. But a correct regularization supplies precisely the
anomaly. Let us explain the procedure invented by Fujikawa [Fujikawa 1979],
[Fujikawa 1980]; additional literature can be found in [Adam 1990], [Dittrich,
Reuter 1986].
Path integral measure: In order to define the PI measure more ac-
accurately we decompose the spinors ф, ф into eigenfu net ions of the Dirac
operator
ф(х) =
ф(х) = 5>У*)Ьт = SH*>bm. E.37)
where the coefficients an, bm denote independent Grassmann elements. The
Hermitian Dirac operator TJ) has real eigenvalues An
>n{x) = \n<pn(x) E.38)
and the set of eigenfunctions {рп(х)} is orthonormal and complete
J dxtplnWipnix) = jdx(m\x){x\n) = (m|n) = 6mn E.39)
\n){n\x) = (y\x) = 6(y-x). E.40)
(Note that in Euclidean space we may choose ф = ^ so that the group
invariant scalar product is positive definite.) So the PI measure and the
action can be re-expressed by
= [det(m|a;)det(a;|n)]-1
= [det(nln)] JJdand6n
П
= UdanSn E.41)

256 Path integral and anomaly
/ dx ф(г Tf) - т)ф = ?(iAn - m)bnan E.42)
and the PI itself, equation E.20), is formally defined by the following Grass-
Grassmann integral
= di/jd-ipexpl dx-ip(i ft-тщ
= Yl dan<S*n exp 5Z(iAn - m)bnan
J n In J
= l[(iK-m). E.43)
Chiral transformation: How do the Grassmann elements an, bn trans-
transform under an infinitesimal chiral rotation E.11)? Let us consider the rota-
rotated spinor
?n?>n(z), E.44)
which is, on the other hand,
ф'(х) = A + Щх)ъL;(х) = A + i/3(x)j5) Yl ат^т(х). E.45)
m
Due to the orthonormality of the eigenfunctions, equation E.39), we obtain
< = J^Cnmam, E.46)
m
with the transformation matrix
Cum = 6nm+if dx р{х)<р\{х)ъЧ>т{х). E.47)
Analogously we get
Ьт = ?СппгЬп E.48)
n
from the rotated adjoint spinor tp.
Since the Grassmann measure transforms with the inverse determinant
Y[da'n =

Fujikawa's method and singlet anomaly
we find for the change in the PI measure
Using the formula
detC = exp[Tr lnC]
and expanding the logarithm
257
E.50)
E.51)
E.52)
we re-express the Jacobian in the following way
J[/3] = (det С)'2 = exp[-2 Tr lnC]
= exp
= exp
-2 Tr ln(Snm +ifdx 0
-2 It i f dx0(x)^n(x)-r6<pm(x]
-2t J dx 0(x) J2 vl
so that
The sum in the exponential, however, is not well defined
x) = tr 75 • 6@),
E.53)
E.54)
E.55)
where we used the completeness E.40) of the eigenfunctions.
Regularization: Fujikawa's idea was to regularize the sum by introducing
я Gaussian cut-off
Л2
E.56)
ilamping the contributions from the large eigenvalues and performing the
limit M -+ oo afterwards. We now evaluate the regularized sum in Fourier
space. Introducing the Fourier components
E.57)
BтгJ

258 Path integral and anomaly
using again the completeness of the eigenfunctions
k) = ът-б(е-к) E.58)
n
and integrating over the momentum ?, we obtain
E-59)
Here the trace has to be taken over the Dirac matrices and over the group
generators Ta which we denote by Tr.
Next we decompose the Dirac operator
= D^ + ^b^YjF^, E.60)
where we used the relation [D^, Dv) — F^. When we insert decomposition
E.60) into the expression E.59) we are aware that the plane waves shift the
differential operator
+ tfcM) E.61)
so that we get
1У75ехр[ ^) -^^1
BтгL 75 P[ M2 2M2 J
= lim M / pKi*K ту
м™»М У BтгL "
E.62)
after rescaling the momentum fcM —> M fcM.
Now we expand the exponential and we use the trace properties of the
Dirac matrices

Fujikawa's method and singlet anomaly 259
tr 75 = tr 757'V = 0 E.63)
tr 757/x7a7/3 = -te»", E.64)
with e1234 = e1230 = 1. Then only the quadratic term in 7/x7"F/xl/ remains
in the limit M —> 00 and expression E.62) becomes
a /
7
мЛ"со2!4М4.
E.65)
We note that our Euclidean convention is
Next we carry out the Gauss integration
= тг2п/2 E.67)
and take the trace over the Dirac matrices. Then the regularized sum a
la Fujikawa provides the result
шссР tr F^Faf,, E.68)
where the trace over the group generators in F^ = FMl/ara remains.
Jacobian: So finally for the Jacobian E.54) of the PI measure we find
J\0\ = exp f- j dx /3(x) j^e1""*13 tr F^F^] E.69)
and recalling our definition E.22)
J\0\ = exp L I dx /9(s)^[i4M](s)] E.70)
we obtain the result for the singlet anomaly
^»1"**3 tr F^Fap. E.71)
In Minkowski space (with the choice e0123 = 1) we have to remove the factor
г due to the Minkowskian change of the trace E.64). Then the result E.71)

260 Path integral and anomaly
agrees with our perturbative calculations D.378) or D.231) in Chapter 4.
5.3 2-dimensional anomaly
In 2 dimensions the calculations turn out particularly simple. For the Jaco-
bian we have to regularize the sum
г \2 
53<pl(xh5<Pn(x) = lire 53Vn^hs exP -"лЙ ^"^
n n L J
- J5./ @*<~
E.72)
where our Dirac operator here is (recall equation D.317))
fl=ft-i4. E.73)
AM denotes the Abelian photon field and as before P is chosen Hermitian
in Euclidean space, Tp =fr. Decomposing the squared operator
E.74)
we obtain for the sum
*,2 f d2fc -it к Г
м у B^e ^fc<-tr ^exp L
E.75
after shifting the differential operator (see equation E.61)) and rescaling the
momentum fcM —> MfcM. We expand the exponential, take the trace of the
Dirac matrices, then only the term 7M7"irMv remains in the limit M -* oo.
The Gaussian integral gives a factor тг (equation E.67)).
In 2 dimensions our conventions in Minkowski space are (recall Section
4.5, equation D.318))
7° = ^2, 71 = iau 75 = 7<V = стз,
eoi = 1 E-76)

Regularization independence of the anomaly 261
and in Euclidean space
9nV = -6^v, ?4i = ieoi = i. E-77)
Then the relations
= -2ед«/ E-78)
are valid in both spaces, which is convenient for our calculations.
So we find for the regularized sum
= ~e F^, E.79)
4тг м
for the Jacobian of the PI measure (equation E.54))
J\0[ = exp | - / dx P(x)—eliVFliV\ E.80)
L J 27r J
and finally for the anomaly (recall equation E.70))
With our definition of the e-tensor, equations E.76), E.77), the expression
E.81) is valid in this form in Euclidean and in Minkowski space. Result
E.81) is in accordance with our calculations in Sections 4.4 and 4.5.
5.4 Regularization independence of the anomaly
As Fujikawa [Fujikawa 1980] emphasized the anomaly result is independent
of the chosen regularization for the large eigenvalue contributions in the sum
E.56). Instead of the exponential damping we could also choose some other
function which is smooth and decreasing sufficiently rapidly at infinity
with
/(oo) = /'(oo) = /"(°o) = ... = 0 E.83)
/@) = 1.

262 Path integral and anomaly
Replacing exp by / we get
(A2 \
2>'
n
/dAk /к к 9ik D^ 4- D D^ kVF \
BтгL 75/ V M2 + M2 + 2M2 ) ¦
E.84)
We expand the function at the point k^k^/M2 =: к2/М2 into a Taylor
series and we use the trace properties of the Dirac matrices, then only the
second derivative contributes to the limit M —¦ oo
E.85)
Inserting the Dirac trace E.64) and rescaling the momentum fcM —> MfcM
gives
tr
1 Г И^к
= л11т ~? / ^rJ"^)^Va
We carry out the integral through integration by parts (/"(fc2) means the
second derivative with respect to the argument)
- f wnfc2)=-i^f
1
167Г2
0
1
167Г2'
E.87)
and we recover the desired result E.68)
)
tr FM1/Fe/,. E.88)
Г
n
For instance, choosing the cut-off function as

Fujikawa's uncertainty principle 263
E"89)
then Fujikawa's regularization corrresponds to the regularization of Pauli-
Villars (Section 4.3.2). For further details we refer to the literature [Fujikawa
1980, 1984, 1985], [Adam 1990], [Delbourgo, Thompson 1985].
5.5 Fujikawa's uncertainty principle
The sum in the Jacobian
^T ft(x)-y5(pn(x) = lim tr 75 ¦ 6(x - y) E.90)
n
can be regarded as a conditionally convergent series in an infinite dimen-
dimensional functional space, very much like +1 + 1-1 — 1 + 1 + 1 — 1-1 +
It has a definite value whenever we specify how to sum the series. Fuji-
Fujikawa's Gaussian cut-off E.56)—and any regularizer with properties E.82),
E.83)—corresponds to a summation that imposes gauge invariance.
Then it happens that the diagonalization of the Dirac operator p implies
a chiral asymmetry and the chiral trace becomes anomalous, tr 75 ф 0—in
contrast to the 'naive' expectation tr 75 = 0 which would correspond to a
summation (+1 + 1 - 1 - 1) + (+1 + l-l-l) + ... = 0.
Fujikawa [Fujikawa 1980, 1989] realized that in QFT the operators 75
and Tp satisfy a kind of uncertainty principle, analogous to the one we have
in nonrelativistic quantum mechanics between the operators x and p.
Proposition: Uncertainty principle
• We cannot impose both symmetries, the gauge symmetry and the
chiral symmetry simultaneously!
• The two operators P and 75 do not commute, their commutator ex-
expectation value gives rise precisely to the anomaly
<ф{х)[р,>уь)ф(х)) = A[AJ{x). E.91)
Hence P and 75 cannot be diagonalized simultaneously. If we diagonalize
the Dirac operator lp—imposing gauge symmetry in the trace E.90)—then
the conflicting symmetry—the chirality 75—becomes anomalous. So the chi-
chiral or axial, or singlet anomaly arises from the conflict of two competing
symmetries: the chiral symmetry and the gauge symmetry.
Proof. Let us calculate the expectation value of the commutator E.91).
Within the PI formalism we first evaluate

264 Path integral and anomaly
= -rj. / dxpdxp ф{хJг^ъ $ф{х) ехр / dx ф{1 $ - т)ф\ ,
E.92)
with the normalization
N = f dtpd$exp f dx$(ifl- тЩ = det(i l/> - m). E.93)
Expanding the Dirac spinors ф, ф in terms of eigenfunctions of ft, equation
E.37), we get
(ф{хJгЪ
= л J П daidbi 5Z bmanipln{xJi'y5
mtn
•exp > (iAfc — m)bkak . E.94)
Lfe J
Respecting now the rules for Grassmann integration, particularly,
ec = l + c, fdc = O, fdcc = l E.95)
(see Section 3.3.1), the only contribution which remains is the following
(ф(хJ^5 ]рф{х)) —
1 /"ТТ j Л V^t +/ ™ ,п,., .г
— T7 / I I daidbi У Ьпап(р1,(хJг/у5 щюп[х) I I («Afe — m)bkak
Jv J LL z-/ Л,А
E.96)
since
Y[{iXk - m) = det(i ф-т)=М. E.97)
For the commutator we then find

Non-Abelian anomaly 265
n
2m y
А„ + zm ^ Лп + г'гга
n
>. E-98)
where we used the regularization result E.68). So we get the anomaly and
the symmetry explicitly breaking mass term
$(*)[#, 75M*)} = A[AJ(x) + 2im(P(x)). Q.E.D. E.99)
5.6 Non-Abelian anomaly
So far we have treated the singlet (or Abelian) anomaly case. But also the
non-Abelian anomaly is determined by the PI measure when performing a
non-Abelian gauge transformation.
We use the following non-Abelian Lagrangian
Cna = фг ]рф, E.100)
where the Dirac operator
Tp=$+y+jfab E.101)
now contains a vector- and ал axial gauge potential
VM = V?Ta, Дм = А1Та. E.102)
We consider the massless case for simplicity, m = 0. This Dirac operator,
however, is ло* Hermitian in Euclidean space (we described our notations
in Section 5.1)
р E-103)
hence
fl(V,A)=P{V,-A). E.Ю4)
So the Dirac operator has no well-defined eigenvalue problem and we cannot
use p2 for the regularization procedure.
One way out is to work with Laplacian operators
E.Ю5)

266 Path integral and anomaly
where
= КФп and pUn = Xn<pn. E.106)
They are Hermitian and have well-defined eigenstates. The regularization
is performed in a gauge covariant way and thus the regularized Jacobian
supplies exactly the covariant anomaly. Again, we describe Fujikawa's
procedure [Fujikawa 1984, 1985]; for further literature see [Adam 1990],
[Banerjee, Banerjee, Mitra 1986].
Path integral measure: We expand the spinor fields with respect to the
eigenfunctions of the Laplacians
ф(х) = ^2фШЬт = '?/(Фт\х)Ьт. E.107)
The function sets {fn}, {фт} form a complete orthonormal system and, as
before, the coefficients an, bm are independent Grassmann elements.
Then the PI measure and the action change to
dil>d$ = [det^nla:) detfcl^)] JJ dan JJdbm E.108)
771
/
dx ^ipxp = 2_^ iXnbnan E.109)
71
so that the PI—the generating functional—or the determinant of the Dirac
operator is formally defined by the Grassmann integral
r _ If- 1
det i p = / dtydty exp / dxipi рф
J \_J J
= [det^t ¦ det ip]'1 / JJdand6nexp
J n
= [det0+ -det^]-1 JJiAn. E.110)
n
The quantities <p and ф^ denote the (infinite dimensional) transformation
matrices
ip := (x\ipn), 0f := (фп\х). E.111)
Their determinants det ip, det ф^ in front of the Grassmann integral play no
role in the calculation of the anomaly.

Non-Abelian anomaly
267
Chiral transformation: Now we perform a non-Abelian axial gauge
transformation—a local chiral rotation
ф'{х) =
E.112)
where /?(х) — /?а(х)Та denotes a non-Abelian gauge element and we choose
/? infinitesimal. Then the Grassmann expansion coefficients transform as
Cnma
Ь'т = 22 Dnmbn,
E.113)
with
D
= 6n
nm = 6nm
- dx ^(х)^(х
- J йхф\{х)р{х
E.114)
Note that ft = @aTa contains the gauge group generator Ta, a matrix
acting on the eigenfunctions ipm or фт. Thus /3 is sandwiched between the
eigenfunctions.
The Grassmann measure transforms with the inverse determinant
E.115)
l[da'n = [
П
l[db'm = [
m m
and we re-express the inverse determinants in the familiar way
[detC]
-l
= exp
= exp
- tr In ( 6nm -
J <
dx <
>)]
[detD] =exp
So we get a change in the PI measure
E.116)
E.117)

268 Path integral and anomaly
#'#' = {detC-detD^dipd
=: J[/3]dipdip, E.118)
with the Jacobian
J[/3] = [detC-detD]-1
= exp \jdxYMiWWwnW + 4(z)/?(zO50n(z)) .
E.119)
Regularization: We regularize the Jacobian, which is still ill-defined,
with a Gaussian cut-off (analogous to the procedure we described before
in Section 5.2)
^
E.120)
Chirality split: Now the trick is to split the Laplacians into positive and
negative chirality parts. We will recall the chiral decompositions which we
introduced in Section 4.1.2:
projection operator
P± = |A ± 75) with Pi = P±, P+P- =0, P++P- = 1, E.121)
chiral spinors
ф± = Р±ф with 75^± = ±V>±, E.122)
chiral gauge fields (or L-, i?-handed)
Л+ = VM + Л. ^ = VM - Л- E-123)

Non-Abelian anomaly 269
Lagrangian
ф-1Р-ф-, E.124)
operators
E.125)
Using the properties
E.126)
we split the Dirac operator and its adjoint into + and — chiral fields
E.127)
E-128)
Then we find for the Laplacians
pty = ф\Р++ф2_Р_ E.129)
фр = ф\Р-+ PlP+ E.130)
and for the exponentials
where we have made extensive use of the projection operator properties
E.121).
Returning to the regularized sum E.120), with this separation E.131)
into + and — chiral operators we obtain
exP —Г
M - j |_ lvl" J /
E.132)

270 Path integral and anomaly
We have reduced the evaluation of the sum to that of the singlet anomaly
for a given chiral field. We have described this evaluation already in Section
5.2 and we just have to take over the result E.68) which represents the
covariant anomaly
(with e1234 = e1230 = l). F^u denotes the field strength corresponding to
the chiral gauge field A^. We will discuss the covariant anomaly in detail
in Chapter 10.
Gauge transformation: However, the covariant anomaly also occurs in
the vector current. Its calculation proceeds quite similarly. We perform a
non-Abelian vector gauge transformation
() ()
ф(х) = ф{х)еа{-х\ E.134)
with a{x) = aa(x)Ta (remember T°t = -Ta, thus a* = -a). Then the
Grassmann expansion coefficients transform with the matrix
Cnm = <5nm - dx у4(х)а(х)у>т(х)
Dnm = 8nm + j dx<t>Ux)a{xL>m{x), E.135)
which leads to the Jacobian
J[a] = [detC-det-D]
= exp N dx 53(^(х)о(х)<р„(х) - фЪ(х)а{х)фп{х))
E.136)
Regularization: We regularize the sum as before
i(x)ot(x)vn(x) - фЪ(х)а(х)фп(х))
E.137)

Non-Abelian anomaly
271
and we separate the Laplacians into positive and negative chirality parts
(recall equations E.129), E.130))
(P+-P_)exp
75
- exp -
Altogether we find for the regularized sum
E.138)
— lim
Jkx\
E.139)
and the evaluation provides the covariant anomaly in the vector current
r a^+Fi - ^-F^). E.140)
Covariant anomaly: If on the other hand we define
J[A] = exp I f dx Аа(х)да[А1Л](х)
E.141)
then the covariant anomaly AaGa[AM] is given by expression E.133) or
E.140).
We get the result for a given chirality—the case we will consider in
Chapter 10—by setting the other component to zero. Accordingly, we find
the covariant anomaly for positive or negative chiral fields
A°LGa[A;
tr K-
E.142)
E.143)
with
A+=a + p, A-=a-f3. E.144)
This result is also the correct expression in Minkowski space for the usual

272 Path integral and anomaly
anomaly definition (see Chapter 10)
6a\A±] = -D^(jf>covar. reg. E.145)
since in Minkowski space the current definition changes by a factor i.
Consistent anomaly: We certainly can also find the consistent anom-
anomaly within the PI procedure but then we have to regularize the Jacobian in
a different way.
For example, achieving a Hermitian Dirac operator by analytic conti-
continuation of Л —> гЛ
one can use as a Jacobian regulator
Г
exp \-
The authors [Andrianov, Bonora 1984a,b], [Balachandran, Marmo, Nair,
Trahern 1982], [Einhorn, Jones 1984], [Gamboa Saravi, Muschietti, Schapo-
snik, Solomin 1984a], [Gipson 1986] essentially follow this route.
Another possibility suggested by Alvarez-Gaume and Ginsparg [Alva-
[Alvarez-Gaume, Ginsparg 1985] is to start with a different differential operator
in the Lagrangian
$
This operator does not change the gauge theory (up to a normalization fac-
factor for the generating functional) and has a well-defined eigenvalue problem.
Then the Jacobian is regularized by
exp [~
The calculations, however, in both cases turn out to be rather tedious,
but as a result the consistent anomaly in the familiar Bardeen form D.387)
occurs—up to a trivial solution (which is the gauge variation of a local
polynomial in the gauge potentials and can be absorbed by redefining the
vacuum functional). We will present a detailed calculation for the latter case
in Section 11.5.
5.7 Heat kernel and zeta function regularization
When working with the fermionic PI there always occurs the problem
of regularization. Fujikawa's Gaussian cut-off procedure is one possibi-

Heat kernel and zeta function regularization
A
273
-X—X-X-X—X—X—X—X—X-
Fig. 5.1. The curve С in the complex A-plane defining a function of the
operator A with spectrum {An}
lity but there also exist other techniques—the heat kernel and zeta func-
function regularization—which are elegant and rest mathematically on sound
grounds. Their connection to Fujikawa's approach is evident. These regu-
regularization techniques have been frequently used in QFT by several authors
[Leutwyler 1985a,b, 1986b], [Gasser, Leutwyler 1984], [Gamboa Saravi, Mu-
schietti, Schaposnik, Solomin 1983, 1984a,b], [Bonora, Bregola, Pasti 1985],
[Adam 1990], [Ball 1989], [Reuter 1985], [Bertlmann, Launer, de Rafael
1985], [Bertlmann, Dominguez, Loewe, Perrottet, de Rafael 1988], [Hawking
1977]. For the mathematical foundations see [Gilkey 1984], [Berline, Getz-
ler, Vergne 1992]. We shall explain these methods for the example of the
finomaly in 2 dimensions where all relevant features occur and the calcula-
calculations are surveyable. We also follow the work of [Adam, Bertlmann, Hofer
1993] here.
5.7.1 Heat kernel regularization
Let A be a Hermitian, positive, semidefinite operator with a well-
posed eigenvalue problem
А^„(х) = \n<pn(x).
The eigenfunctions {ipn(x)} are orthonormal and complete
dxip'n(x)<pm(x)=6nm
J
E.146)
E.147)
E-148)
mid the eigenvalues An are real.

274 Path integral and anomaly
Definition: A function of the operator A is defined by [Gilkey 1984]
Л- <5Л49>
where the curve С in the complex Л-plane encloses all eigenvalues {An} and
the origin, see Figure 5.1. Specifically, we have
е-хД
1 Г е~тХ
= ТГ-т / 4—т«*А. E.150)
2тгг У А - A v ;
Heat kernel: Let us take some squared integrable function (p(x) 6 L2,
then the expansion into eigenfunctions of A gives
«¦/
dy
=: /dyGA{x,y,T)ip(y). E.151)
The operator е~тА has a kernel function—the heat kernel
Сд(х, у, т) = ? e-A">n(zK(y), E.152)
n
which can also be expressed by the propagator
Сд(х,у,г) = (х|е-тД|у). E.153)
The initial condition at r = 0 is
Gb(x,y,T = 0) = 6(x-y) E.154)
due to the completeness of the eigenfunctions, equation E.148). The heat
kernel satisfies the so-called heat equation
AGA(x, y, t) = -^Gu(i, у, т), E.155)
which can be quickly checked by remembering the eigenvalue equation
E.146).
We also assume that the operator Д can generally be decomposed into
(in Euclidean space)

Heat kernel and zeta function regularization 275
A = -D2+X, ?>Й = 9Й + ЛЙ, D2 = DliDli, E.156)
where Ац denotes the gauge potential, D^ the covariant derivative and
X some matrix valued C°°-function which is positive Hermitian. So A is
Hermitian and positive as required.
For the special case of
До = -д2 E.157)
the heat equation E.155) has the solution of the free propagator
E.158)
with d the dimension of space-time (we recover the nonrelativistic result
C.21) for imaginary times т = it in Section 3.1).
We are interested in the behaviour of the heat kernel for small т; there
the following expansion [Seeley 1967] is valid.
Theorem: Heat kernel expansion
±i] E .(*, У)т". E-159)
We shall work in 2 dimensions, d = 2. The coefficients an(x,y) are called
Seeley coefficients and they have the property
a\l{x,y) = an{y,x) E.160)
since A is Hermitian. They certainly depend on the operator A and are
fixed by the heat equation. The coefficient ao is normalized to unity
o0(x,y) = l E.161)
<lue to the initial condition E.154). When we substitute the expansion
E.159) into the heat equation E.155) we find:
Lemma: Recursion relation for the diagonal Seeley coefficients (x = y)
-A an_i(x,x) = n an(x,x). E.162)
'This is immediately evident when we remember that the Gaussian in ex-
expansion E.159) approaches a 6-function for r —> 0.

276 Path integral and anomaly
Jacobian: Considering now the path integral we want to regularize the
Jacobian E.54). Corresponding to Fujikawa's procedure here we choose as
the heat kernel operator
A =#>2, E.163)
with
9=$-%4L E.164)
for the 2-dimensional case (ft is already Hermitian, $)* =W)- Then we get
for the Jacobian
Jdx
-In J[/3] = 2t J
П
= ton 2г / dx /3{x)
— lim 2г / dx C(x) tr
= lim 2i Tr Pj5Gp2, E.165)
which is just the trace of the heat kernel. Here the notation Tr means the
trace with respect to the full operator structure, whereas tr denotes the
trace over the Dirac matrices.
Using expansion E.159) (with d = 2) the Jacobian is determined by the
Seeley coefficient ai
-In J[/3] = 2г Idx/3(x)—- tr 750i(x,x). E.166)
J 4tt
On the other hand, we can calculate the Seeley coefficient ai via the recur-
recursion relation
oi(x,x) = - #>2a0(x,x) = - P2. E.167)
Inserting the Dirac operator E.164) we have (recall relation E.74))
fll(x,x) = -DMD" + J[f,7%, E-168)
where only the second term contributes to the trace via the formula
tr 757/^ = -2е„„. E.169)
Then with our definition of the anomaly
- In J[p] = / dx P(x)AIA,,} (x) E.170)
we find that the Seeley coefficient ai already supplies the anomaly result

Heat kernel and zeta function regularization 277
A[AM](x) = ^ tr 75oi(z,z)
= ^^FilV E-171)
in accordance with our calculations a la Fujikawa in Section 5.3.
5.7.2 Zeta function regularization
Another method to give meaning to the fermionic path integral—the deter-
determinant of the Dirac operator—is the zeta function approach. The determi-
determinant
deti TP = / ##exp / dx фг Щ = Д i\n E.172)
J \-J -In
diverges because of the unbounded product of increasing eigenvalues An of
the operator #>. So we have to regularize the determinant, the zeta function
serves as a convenient technique.
Let A again be a Hermitian, positive, semidefinite operator with eigenva-
eigenvalues An and orthonormal, complete eigenfunctions {ipn}, equations E.146)-
E.148). But this restriction can also be relaxed. What we actually require
is that A is an elliptic and invertible operator (see Section 11.2) defined on
a compact boundaryless manifold. (So we always assume the fields of the
theory to behave at infinity such that a compactification of Rm is possible,
«.g. to Sm.)
Definition: The zeta function of the operator A is defined by the
following sum over the eigenvalues
Л- E-173)
'The sum converges only for sufficiently large positive real part Re s, other-
otherwise it is defined by analytic continuation into the complex s-plane. In
particular, Ca(s) and its derivative are regular at s = 0. Formally we have
s=0
s=0 n
E.174)
so that we can define the regularized determinant or the regularized
path integral

278 Path integral and anomaly
/ йфйфexp \ dxrpiЩ =detRi^ = exp [-<,'p@)] , E.175)
d
where Cms) := ~t~Ca(s)-
Generally, we do not know the spectrum {An} explicitly, hence we cannot
work with the sum E.173). But we can construct another analytic expression
for Ca(s) which we are allowed to continue to s = 0 in order to achieve the
regularization of the determinant E.175).
Laplace transform: Let us search for the Laplace transform of Ca(s)>
generally, for the transform
/(A) = [ dr F(T)e~XT. E.176)
Jo
We remember the integral representation of the Г-function
Г(в) = [ dr Ts-le~T E.177)
Jo
and we shift the integration variable т —+ At
Г(в) =X3 [ dr та-1е~Хт, Re A, Re s > 0. E.178)
Jo
Then the function
/(A) = p E.179)
has the Laplace transform
W = т^гУ1 E-180)
since
1 1 '°°
A- T(s)
For the ^-function we obtain
rOO
f йтт'-1е-*т. E.181)
Jo
Сд(«) = J-t / dr t°-1 Y, e"AnT- EЛ82)
\ > J° ^T
The summation, on the other hand, is nothing but the trace of the ope-
operator
TV с~тД — ^~ч/п1р~тД1п\ — ^~"ч«>~ЛпТ (ч 184^
i.i о — 7 \ II о /4/ — 7 о . IO.J.OOI
п п

Heat kernel and zeta function regularization 279
with \ri) the complete, orthonormal eigenstates of Д
A\n) = An|n). E.184)
In x-space we have to integrate over the diagonal heat kernel
Тге-тД = f dx (х\е-тА\х)
= /dx^2(x\e-TA\n)(n\x) = /dx ?>-*">„ (x)^(x)
= f dxGA(x,x,r), E.185)
which again is clearly the sum E.183) due to the normalization E.147).
Altogether we find for the ^-function the following Laplace transform
1 (SJ Jo
"x dxGA(x,x,r). E.186)
Trace: We can also formulate the ^-function by the trace of the Д ope-
operator
Сд(*) = ?>-« = ТгД-'. E.187)
n
For complex powers of s the operator is defined by (see equation E.149))
A3 = — / r^-rrfA. E.188)
2тгг Jc A - Д v ;
The functional trace then means
ТгД
= / dx^An^n(i)^(i) =: dxKA(x,x,s = l)
sVn{x)v*n{x)=: (dxKA{x,x,-s).
E.189)
I'or an infinitesimally varied operator we may also expand the trace in
die following way
Tr (Д + «5Д)-* = Tr A'3 - s Tr Д-'-^Д + O(FAJ). E.190)

280 Path integral and anomaly
Jacobian: Now we return to the path integral and we perform a local
chiral transformation
ф> = феЩ*Iъш E.191)
Then the determinants are related by
det г If) = / dip dip' exp / dx ф'г Щ'
= / J[/3]d^exp \fdx фе^х)~'Ч Ре^х)ъ ф\
, E.192)
with J[j3] the Jacobian associated to the chiral rotation E.191).
In the C-function approach we work with the regularized determinants
according to definition E.175). So in this case the chiral Jacobian repres-
represents the ratio of two regularized determinants
det4s;^ Eл9з)
and for /3 infinitesimally small
$k EI94)
The Jacobian is for this reason a finite quantity. In Fujikawa's procedure,
on the other hand, we just had to regularize an ill-defined Jacobian (and
there several regularization choices were possible).
Zeta function and Fujikawa procedure: In order to establish the
bridge between the (^-function and the Fujikawa procedure we have to work
with the Hermitian, positive semidefinite operator tfJ.
Generally, for a non-Hermitian but elliptic operator D we can regularize
the determinant as
det RD = exp [-^DtD@)] , E-195)
which we infer from conjugate squaring the determinants
detRD{detRD)m = detRDlD = exp[-CDiD@)}. E.196)

Heat kernel and zeta function regularization
281
The operator D^D is Hermitian and semipositive, which we need to trace
back the ^-function to the heat kernel, equation E.186).
In our case we have D = iTf) (anti-Hermitian) and the determinant is
E.197)
Then the logarithm of the Jacobian E.194) arising from the chiral rotation
becomes
_d_
ds
s=0
E.198)
with
P E.199)
a small correction. Expressing the ^-function by the trace of the operator
and expanding the trace according to equation E.190) we find
ds
ds
(ft2)-* - Tr ф2 + 6 ft2)
6 ft2)-*}
s=0
s=0
±sTr
2S *
2is Tr
E.200)
s=0
where we have used the cyclic property of the trace.
Next we use the equality of the following Jacobians.
Lemma:
The Jacobian Jpz [j3] means chirally rotated with respect to IfJ
E.201)
E.202)
with the determinants regularized according to definition E.175).
Proof: Denoting
wo calculate
E.203)

282
Path integral and anomaly
= lndetfl
d
ds
ds
3=0
3=0
[Tr
- Tr (ft2 + 6 ft2)-3}
2is Tr
s=0
which coincides with equation E.200).
The trace E.200), E.204) explicitly means
E.204)
Q.E.D.
/
= / dx tr Д> (х, x, -s)/3(z)t5.
E.205)
Since the trace of the kernel Kyi (x, x, -s) is regular at s = 0 [Seeley 1967]
we also get
ds
2isTr
= -2* Tr
s=0
E.206)
Now we relate the trace E.200), E.204) to the heat kernel. We insert
the Laplace transform E.181) for X~s into expression E.205) and we obtain
Tr №2
1 \s)
E.207)
The heat kernel Gpz{x,x,T) is falling off exponentially for large т. So
we may cut off the integral at some finite value т = T since the rest-integral
vanishes in the limit s —¦ 0 where we have an overall factor 1/F(s) =
s + O(s2). In fact, only the lower boundary of the integral will contribute
in the limit s —» 0.
Inserting the heat kernel expansion E.159) we find for the Jacobian

Heat kernel and zeta function regularization 283
-lnJpaf/3]
= 2is I dx I dr ts~1 tr -—У~* ап(х, х)тпв(х)^/5
J Л 4тгт L-f
E.208)
Clearly only the Seeley coefficient a\ contributes. The ao-term does not
appear since its trace vanishes
tr oq(x, xO5 = tr 75 = 0.
The higher terms with n > 1 are of order O(s/(s + n — 1)) and vanish too.
The cut-off T may be chosen arbitrarily small so that the expansion E.159)
is valid. Finally we obtain for the Jacobian
— In Jp
tr —oi(x,x)/3(xO5
= 2is / dx-j
J s
= 2i fdxftx)-^ tr 7501A,x). E.209)
J 47Г
This outcome coincides with the result E.166) of the previous heat ker-
kernel procedure which is equivalent to Fujikawa's approach. Of course, result
E.209) represents the anomaly (recall equations E.167)—E.171))
{х) = J- tr 76Oi(x,x) = l-e^Fi". E.210)
Non-Hermitian zeta function procedure: Now we discuss a ?-
function procedure which is valid more generally. It does not rely on po-
sitivity or hermiticity of the associated operator, we only assume ellipticity
mid invertibility. In this case we calculate the Jacobian corresponding to
t Ik; Dirac operator i If)
л\и\ regularize the determinants according to definition E.175). We demon-
demonstrate the procedure with our Dirac operator P =0 — i .Д which is chosen
llcrmitian in Euclidean space, so «^ is anti-Hermitian (see Section 5.3).
Hut the method also works for a non-Hermitian Tf) containing, for example,
nxiiil fields as we had in Section 5.6 where we considered the non-Abelian
anomaly.
We begin the procedure as before. We represent the determinants by the

284
Path integral and anomaly
C-functions associated to the operator i P and the (^-functions we express
by the operator traces. Then the logarithm of the Jacobian becomes (recall
equations E.204))
In Ji
~da
A.
~ds
sTr
3=0
2is Tr (i
s=0
= -2гТг(г^)-3/375|3=0,
where 6 (i Щ is a small correction
E.212)
E.213)
To evaluate the trace E.212) we now use the integral representation E.188)
of the operator (%Щ)~а. In this step we do not depend on hermiticity; this
step holds for any elliptic, invertible operator. We insert plane waves as a
complete system and we get for the trace
Tr (г
,-tfea:
ikx
\-ilp
E.214)
where the plane waves have shifted the differential operator (recall equation
E.61)).
Next we expand the denominator according to the formula
A-B
which gives for the trace
1_
~A
A"A^ A"A"A
E-215)
Л+ Ц

Heat kernel and zeta function regularization 285
/t rPk
E'216)
after rescaling the momentum fcM —¦ AfcM. This corresponds to an expansion
of the Dirac operator into Seeley coefficients [Seeley 1967]. We, however,
want to evaluate the sum in another, more direct way which turns out to
be quite simple. This method is due to Bonora, Bregola and Pasti [Bonora,
Bregola, Pasti 1985].
We integrate in the complex A-plane. Then in the limit s —> 0 only the
third term of the expansion contributes as a residual integral. We observe
that the whole expression is regular at s = 0 as claimed above. Using the
familiar formula
7Г-. f^T = 1 E-217)
2тгг J A
we obtain
dx IШ*tr
/
^УB^(ГТ^*Г^^2. E-218)
We carry out the fc-integration (note that with our Euclidean metric E.77)
we have ft Ц = — fcMfcM =: -A;2)
/
i ii id we find for the trace
Tr (i РГ3Ыа=о = ~Jdx /3(x) ^ tr 75 V2. E.220)
l-'or the Jacobian E.212), which we are finally interested in, we then have

286 Path integral and anomaly
= -2t [ dx p{x) -J- tr 75 TfJ. E.221)
J 4тг
As before, we insert the decomposition E.74) for p2
-In Jito[0\ = ~\Jdx /3(x)^ tr 757д7^д1/ E-222)
and with the trace property E.78) we finally find the anomaly result
-lnJ«p[0]= f dx p(x)^e^F^ E.223)
and
AM = y^^wv E-224)
in accordance with our previous calculations.

6
Physics in terms of differential forms
In this chapter we shall apply the mathematical formalism developed in
Chapter 2—essentially differential geometry and topology—to specific ex-
examples in physics that we are familiar with. The geometry of fibre bundles
describes the physics of gauge theories in a quite natural way. We shall
appreciate the compactness and elegance of the whole formalism.
We begin with the familiar example of electrodynamics (Section 6.1),
turn subsequently to Yang-Mills theory (Section 6.2) and formulate the
anomalies in terms of differential forms (Section 6.3). We discuss the Dirac
monopole (Section 6.4), the Aharonov-Bohm effect (Section 6.5) and the
instantons (Section 6.6) and demonstrate how they arise as effects of a
nontrivial topology of the fields.
6.1 Abelian fields, electrodynamics
Electrodynamics is certainly the most studied and utilized theory. Here we
can follow the development of the theory in the course of history nicely.
Maxwell's original equations formulated for the components of the electro-
electromagnetic fields were simplified at the end of the last century by introducing
a curl and a divergence. At the beginning of this century the equations were
already formulated in a beautiful relativistically covariant way. But in the
last decades it was differential geometry that supplied us the most concise
mid elegant formulation of the theory. A modern treatment of electrodyna-
electrodynamics can be found in [Thirring 1992], [Bamberg, Sternberg 1990], [Curtis,
Miller 1985].
Maxwell's equations: In the standard notation the Maxwell equati-
equations are
Vxl+ — В = 0 VS = 0 homogeneous F.1)
at
$ x В — — Ё = j VE = p inhomogeneous F.2)

288 Physics in terms of differential forms
with Ё and В being the electric and the magnetic field, and j and p the
current and the charge density. We introduce the relativistically covariant
field strength tensor
Ffiv
0
F.3)
B, I ~0° -I I- <">
Ejz —&2 &\ "
where the covariant F^v and contravariant F**" tensors are related by
\iv — у/ш-г дри \"*"/
and we have chosen the metric g^v to be Minkowskian B.146). Next we
define the dual field strength tensor
*Fltv = isnvaeF F.6)
2
in accordance with our dual Hodge star definition B.148) of the differential
forms
F = -Fa0dxadx0 F.7)
In components we find (with ?0123 = 1)
0 -Bi -B2 -B3
Bi 0 — Ez E2 I , x
B2 Ез 0 -Ei I F-9)
B3 — E2 Ei 0
and we notice that the substitution
В -> JS, Ё^-В F.10)
implies
Fpw -»*FP1/ and F -» *F. F.11)
Then the Maxwell equations can be rewritten in a relativistically co-
covariant way

Abelian fields, electrodynamics 289
3^ * FM" = 0 homogeneous F.12)
d,iFliV = f inhomogeneous, F.13)
with the 4-current jv = (p,j)-
Differential geometry: Now we turn to the differential geometric for-
formulation. We first express the 2-form F F.7) in terms of the electric Ei and
magnetic field Bj by inserting the explicit tensor F.3)
F = Eidx°dxi - \Bieijkdx4xk. F.14)
Next we apply the exterior derivative
dF = fdkEi - ^doBjSjik) dx°dxidxk - ldeBieijkdxedxjdxk, F.15)
\ 2 / 2
using dxedxjdxk = eejkdx1dx2dx3 and ?ijk?ejk = 26/ we obtain
dF = -(dkEi - diEk - d0Bjejik)dx°dxidxk - 8iBidx1dx2dx3. F.16)
We compare expression F.16) with the homogeneous Maxwell equations
F.1) in component form
ejimdtEm + d0Bj = 0 and ЗгВг = О, F.17)
which we rewrite by multiplying with e^ and using
?jtm?jik = Sei6mk ~ ^tk^mi F.18)
so that we get
dkEi - дгЕк - doBjSjik = 0 and д& = О. F.19)
Then the homogeneous Maxwell equation in terms of differential
forms is
dF = 0. F.20)
The previous two equations F.1) combine into one.
For the inhomogeneous equations we consider the dual field strength
and its derivative which we obtain from expressions F.14) and F.16) by
substituting the fields (equation 6.10))
F = -Bidx°dxi -^Ei?ijkdxjdxk F.21)

290 Physics in terms of differential forms
d*F = ^(diBk - dkBi - doEjejik)dxodxidxk - diEidx1dx2dx3.
F.22)
On the other hand, we have the inhomogeneous Maxwell equations F.2)
which we rewrite as before
diBk-dkBi-doEjejik=jjejik and ЭгЕг = p. F.23)
When comparing the two equations F.22) and F.23) we need a 3-form—the
dual current—which we calculate explicitly by remembering the dual basis
B.158) in 4-dimensional Minkowski space:
current 1-form
j = j^dx" = jodx° - jidx* F.24)
dual current 3-form
*3 = 3ii * dx1* = jo * dx° - ji * dxl
= pdx1dx2dx3-\jjejikdx0dxidxk, F.25)
with jo = p.
Then we find
d * F = - * j. F.26)
Taking again the dual of equation F.26), noticing that * * j = j, and
introducing the coderivative in 4-dimensional Minkowski space
6 = *d* with 62 = 0 F.27)
(it differs by a minus sign compared to the Euclidean formula B.169),
B.172)) leads to the inhomogeneous Maxwell equation in terms of
differential forms
6F = -j. F.28)
So both equations F.2) are represented by the brief equation F.28).
Gauge potential: Since F is a closed form, dF — 0, Poincare's lemma
(Sections 2.4 and 7.3) implies that, at least locally,
3 1-form A such that F = dA. F.29)
In components we have
F^ = d^Av - <Э„ V F.30)

Abelian fields, electrodynamics 291
A is the gauge potential. It is not uniquely denned; we have the freedom
of gauge transformation (since d2 = 0)
AA = A + dA. F.31)
We may choose the gauge function Л for a given A such that
АЛ = -6A, F.32)
where A = de + 6d represents the Laplace operator B.174). Then
6AA = 6A + 6dA = 6A + AA = O F.33)
is known as the Lorentz gauge. In this case equation F.28) gives
-j = 6F = 6dA = (d6 + 6d)A - AA. F.34)
So in the Lorentz gauge the inhomogeneous Maxwell equation F.28) corre-
corresponds to the set of equations
6A = 0 and AA = -j, F.35)
which the gauge potential A has to fulfil.
Let us calculate the covariant components (recall equations B.154) and
B.158)):
6 A = *d*All,dxfl = *daAlij]eliValidx'rdxudxadx0
j]xadx0 = -dxAx. F.36)
On the other hand, we get
AA = (dS + 6d)A = 6dA = *d* d^Audx^dx"
= \dadliAve'tvafie''afipdx^ = -daaaA/,dx/', F.37)
where we have used the formula
e^ua0e%^ = -2(s"V, - 9%9V°)¦ F.38)
So the differential geometric equations F.35) correspond to the familiar set
of equations in the covariant Lorentz gauge (with D = дада)
dxAx = 0 and uAp = jp. F.39)

292 Physics in terms of differential forms
Current conservation: Due to <52 = 0 the inhomogeneous Maxwell equa-
equation 6F = —j implies current conservation
6j = 0. F.40)
It corresponds to the components
8xjx=0 or J^p + VJ=0 F.41)
since the explicit calculation analogous to equation F.36) gives
6j = -dxj\ F.42)
Action: Searching 4-forms with appropriate components we find
= Apjlfelival)dxf>dxvdxadxf3
= Axjxdx°dx1dx2dx3 F.43)
A*j = Apjlf
and
F*F = \\\llua0pa
Zi Zi Zi
= -F^F^vdx°dx1dx2dx3 F.44)
so that we can establish the Lagrange function of electrodynamics in
terms of differential forms
C = -\F*F-A*j F.45)
giving the action (dx°dxldx^dx* = d*x is the oriented volume element)
S[A) = J
= jd4x [—F^F^ - Axj^. F.46)
The gauge invariance of the action
S[A + dA] - S[A] = - (dk*j= f Ad*j = 0 F.47)
is equivalent to current conservation
d * j = 0 or d\jx = 0 F.48)

Non-Abelian fields, Yang-Mills theory 293
where
d * j = dxjx ^ellvaPdxlidxvdxadx0 = dxjxdx°dx4x2dx3. F.49)
Note that we have ignored the boundary term in equation F.47).
Fibre bundle: Finally, if we want to construct a fibre bundle we have for
the structure group the Abelian U(l), the gauge group of electromagne-
tism; for the base space we take the 4-dimensional Minkowski space-time
and for the fibre the set of all possible U(l) group elements. We find that
the principal bundle is just trivial, P = R4 x U(l), since the base space
is contractible; the base can be covered with a single trivialization. The
connection (pulled back to the base) is the gauge potential A and the
curvature is the field strength F but both multiplied with the Lie algebra
factor i.
6.2 Non-Abelian fields, Yang-Mills theory
In the case of non-Abelian fields we have the quantities (recall our intro-
introduction of YM theory in Section 3.5.1, and [Sexl, Urbantke 1983])
D» = Dp:a = д^ + [Ац, ], F.50)
with Ta the generators of the gauge group satisfying the commutator rela-
relation C.275). In differential forms we get:
gauge potential 1-form
A = A^dx» € Lie G ® A\M), F.51)
field strength 2-form
1
2'
= dA + \[A, A] = dA + A2 e Lie G ® Л2(М), F.52)
F = -F^dx^dx"
covariant derivative
D = d + [A,} F.53)

294 Physics in terms of differential forms
adjoint covariant derivative
t _ f - * D* Euclidean
|_ * D* Minkowskian. *¦ ' '
The covariant adjoint is defined like the simple adjoint (recall equation
B.169); we consider even dimensional manifolds, compact and without
boundaries).
Theorem:
фар_1,/Зр) = (ар_1,?>%). F.55)
Theorem:
D2 = [F,]. F.56)
Proof. Let ap € Lie G <g> ЛР(М) be a Lie algebra valued p-form, then
D2ap = D(dap + [A,ap])
= d2ap + [dA, ap] - [A, dap] + [A, dap + [A, ap}}
= [dA,ap] + [A,[A,ap]]
= [dA + A2,ap} = [F,ap] F.57)
since by definition of the commutator
[A,[AtaP]] = [A2,ap]. Q.E.D. F.58)
However, applied to F we get
D2F = [F,F]=0. F.59)
Clearly, theorems C.308) and F.56) correspond to each other when multi-
multiplying the commutator C.308) with the wedge product ^dx^dx".
Note: The Jacobian identity, when containing forms, also picks up ad-
additional sign factors, e.g.
[A, [A, ap}] + (-У+1 [A, [ap, A]} + [ap, [A, A]] = 0. F.60)
Theorem:
(?»tJ = *D * *D* = (±) • (-)P * D2* = (±) • (-)P * [F, *], F.61)
where (+) corresponds to Euclidean and (-) to Minkowski space.

Anomalies 295
For some p-form ap we have
(PtJap = (±)-(-)*>*[F,*ap]^0
but specifically for F the result is
(?>tJF = (±). (_)P „, [Fj щ = 0 F 62)
since [F, *F] = 0.
YM field equation: Analogous to the Maxwell case F.28) the YM field
equation in terms of differential forms is
= -j, F.63)
where here
j = j«r"da^ F.64)
denotes the non-Abelian current 1-form. If we reduce equation F.63)
to its components we get
F.65)
So the differential geometric equation F.63) corresponds to the familiar YM
field equation in covariant components
D^F^ = f. F.66)
Due to (D^JF = 0 the field equation F.63) implies covariant current
conservation
D*j = 0 or Dxjx = 0. F.67)
For the Bianchi identity we find
DF = 0 F.68)
which follows by direct calculation and corresponds to the familiar expres-
expression C.322) when we multiply by the wedge product ^dxxdx^dxv.
Finally the YM action is given by
S[A]= ftrF*F, F.69)
with the generator normalization C.277).

296 Physics in terms of differential forms
6.3 Anomalies
Now we turn to the anomalies which we are actually interested in and we
write them in differential geometric notation.
Singlet anomaly: Prom Section 4.7 we know the anomaly result (recall
equation D.378))
Ь^а0 tr FiwFaf>- F-70)
(We have a sign change since we work here with definition ?0123 = 1 instead
of ?0123 = 1 for equation D.378).) Defining the axial current 1-form
f = fidx» F.71)
its dual is
*36 = \/>ieliVa0dxvdxadx0 F.72)
and for the exterior derivative we obtain
d*j6 = dxjldx°dx1dx2dxz. F.73)
Multiplying equation F.70) now with the wedge product dx°dx1dx2dx3
and using relation B.154) we immediately find the singlet anomaly wi-
within differential forms
s F.74)
We can re-express equation F.74),
tr FF = tr (dA + A2)(dA + A2)
= dtr (AdA + \A3\ F.75)
(note that tr A* = 0 by virtue of the wedge product) and we get for the
anomaly
5 ^ F.76)
where Q3 is called the Chern—Simons form
Q3 = tr (AdA + |лЛ . F.77)

Dirac monopole 297
Non-Abelian anomaly—Bardeen's result: Taking Bardeen's result
for L- and R-handed fields (equation D.387), D.388) of Section 4.8)
1 . \
vdpAo + -AvApAa F.78)
/
(also here we have a sign change due to our present definition ?0123 = 1)
we find, in the same way as before, the non-Abelian anomaly within
differential forms
Ga[A] = -(D * j)a = ±-L tr Tad (MA + \Л3) . F.79)
(+) for R- and (—) for L-handed fields.
Covariant anomaly: Considering, finally, the covariant anomaly the
previous result D.389) immediately gives
Ga[A] = ±-^r tr TaFF. F.80)
8?r
6.4 Dirac monopole
The Dirac monopole is a simple, but topologically nontrivial example of a
gauge theory that casts light on the fascinating relationship between physics
and geometry. However, in spite of its beauty, the monopole does not seem
to be realized in nature. Nevertheless, its detailed study is well worthwhile;
certain features in non-Abelian gauge theories occur in an analogous way
(see e.g. Section 6.6).
In classical electrodynamics there is a strong symmetry between the
electric field Ё and the magnetic field B, but still a magnetic charge—
magnetic monopole—is absent in the Maxwell equations F.1). Let us
now assume the existence of such a magnetic charge with density рм, then
we would have
VB = pM^0 F.81)
and the symmetry between Ё and В would be perfect. It was Dirac [Dirac
1931] who discovered that, as a consequence, quantum mechanics predicts:
Proposition:
• If a magnetic monopole exists then its magnetic charge is quantized
in units of l/2|e|/

298
Physics in terms of differential forms
This is a very remarkable outcome of quantum physics. We want to un-
understand this phenomenon first within physics, then by means of modern
mathematical techniques—differential geometry and topology. Wu and Yang
[Wu, Yang 1975] were the first to reformulate the Dirac monopole in terms
of charts and fibre bundles; additional literature can be found in [Sakurai
1985], [Hohne 1990], [Eguchi, Gilkey, Hanson 1980], [Felsager 1981], [Sexl,
Urbantke 1983].
Physics: Suppose there is a magnetic monopole with strength g (and no
electric field E = 0) located at the origin
pM(x)=4«g63(x),
then the static magnetic field has the form of a hedgehog
F.82)
r x
= g-r: withf=-.
r2 r
F.83)
We might think that the corresponding vector potential is given by
F-84)
where we have expressed A in terms of spherical coordinates. Generally we
have
A = Arf+A^ + M F.85)
with f, (p, в being the unit vectors in the direction of the radius and the
angles <f, 9.
The curl of A F.84) provides the desired magnetic field В F.83)
В = VxA=
r2 sin в
r r$ r sin 6@
d_ d_ d_
дг дв 'dip
Ay rAe r sin
(Note a sign change in the constant 1 does not affect B.)
F.86)
Problem: However, there is a problem. The vector potential F.84) is
singular on the negative z-axis; so for в — n A is not defined. In fact, it
turns out that it is impossible to construct a single singularity-free potential
which is valid everywhere on R3 \ {0}. Why? Let us see.

Dirac monopole
299
Fig. 6.1. The vector potential A+ is valid on ?/+, A- on ?/_
Integrating the magnetic monopole over a closed surface, a sphere 52,
gives via the Gauss law
/ dfB = f d3xVB = f d3xp{x) = Ang ф 0. F.87)
JS2 ./ball ./ball
On the other hand, if A were nonsingular on R3 \ {0} we would have via
Stokes' theorem
/ dfB= f dxl=0 F.88)
JS2 JdS2
(since dS2 = 0) in contradiction to the above monopole result 4-кд ф 0.
Resolution: When considering the total space we need at least two ex-
expressions for the vector potential. So we construct a pair of potentials which
are valid outside cones (see Figure 6.1)
A- =
rsm0
The potential A+ is valid on f/+ and
correct magnetic field В everywhere.
for <?>?:?/_. F.89)
l_ on C/_; together they lead to the

300 Physics in terms of differential forms
What happens now in the overlap region U+ П ?/_ where both potentials
are legitimate? Since both potentials A+ and A- correspond to the same
magnetic field В they must be related by a gauge transformation
A_ = A+ + VA. F.90)
We can calculate explicitly the appropriate gauge function Л. Using the
gradient in spherical coordinates
д 19-19
V-f— +?—— — +0-752 F.91)
dr r sin 9 dip г дв
we deduce from the difference
A--A+ = VA
2g l д .
-5-А
r sin 0 r sin 9
the result
A = -2gip. F.93)
Quantum mechanics: Next we turn to quantum mechanics. A particle
with electric charge e subjected to the magnetic monopole field В is descri-
described by the Schrodinger equation
[+e+]+=ih§i+ F-94)
where ф represents the scalar potential. This equation is invariant under the
C/(l) gauge transformations
A' = A
ф' = exp [ijr-AJ ф. F.95)
So a particular wave function ф clearly depends on the gauge chosen. In the
overlap region U+ П ?/_ we may use either A+ or A- and the corresponding
wave functions are connected via gauge transformation F.95)
ф. = exp f-t|| J ф+. F.96)

Dime monopole 301
We have already inserted the explicit solution F.93).
Now let us study the behaviour of the wave function. Each ф- and ф+
must be single-valued if it belongs to the domain of definition of Schrodinger
operators. Considering the equator в = тг/2 we increase the angle y> from 0
to 27Г, then both ф- and ф+ must return to their original values
tp = O : ф_=ф+ F.97)
\-i-^-2n\ф+ = ф+. F.98)
This implies the quantization condition
^=±n, n = 0,1,2,..., F.99)
he
hence the magnetic charge must be quantized
9 = n^T F.100)
or in units where —e = |e| = h = с = 1
9=\- FЛ01)
Differential geometry: We can consider the gradient V F.91) on the ba-
basis \~§^i~§--i~§() \ of the tangent space T(M). For differential forms, however,
we need the dual space, the cotangent space T*(M), with basis {dr, dtp, dd}
So we can identify
f = dr
ф = r sin 6dip
в = rde F.ЮЗ)
and we obtain from expressions F.89) the gauge potential 1-forms
A± = |(±1 - cose)dip, F.104)
which are related by the gauge transformation
A+ = Л_ + ndip. F.105)

302 Physics in terms of differential forms
Applying the exterior derivative d which is given by equation F.102) we
find the field strength 2-form
F = dA± F.106)
F = ^ sin 9d9d(p. F.107)
(Note that d6dip means a wedge product.)
It is customary to work with Cartesian coordinates. For that we diminish
the definition domains (e —* n/2) of the potentials
U+ = R3+\{0}
U-. = Ri\{0}, F.108)
which in the overlap represent the plane minus the origin where the mono-
pole is situated
U+ П C/_ = R2 \ {0}. F.109)
With the help of the familiar transformation
x = r sin в cos tp
у = r sin в sin <fi
z = rcos9 F.110)
it is straightforward to re-express the forms F.104)-F.107) by Cartesian
coordinates:
gauge potential 1-forms
with gauge transformation
A+ = A_ + ndarctan - F.112)
x
and (recall example B.187)-B.189) of Section 2.4)
У 1
darctan - = -iy{—ydx + xdy) F.113)
x r
field strength 2-form
F = dA± F.114)

Dirac monopole
303
H+nH_
H
Fig. 6.2. The sphere 52 is covered by two hemispheres H+, H_ with the
equator Я+ П Я_ = S1
n 1
F = --г (zdxdy + xdydz + ydzdx).
2 ra
F.115)
Clearly F is closed
dF = d2A±=0 F.116)
but not exact since dA+ is defined only locally on U+ and cL4_ on ?/_.
Fibre-bundle topology: Now we are going to construct a fibre bundle
(recall Section 2.7) which mirrors totally the physics of the Dirac monopole.
The space where we defined the Dirac monopole is R3 with the origin
{0} extracted. The monopole is sitting there. Topologically this space is
equivalent—homotopic—to the sphere 52. Then the base manifold for
our bundle will be the sphere 52
M = R3 \ {0} ~ S2
F.117)
which we cover by two hemispheres H+, Я_, the charts (see Figure 6.2).
The gauge group C/(l) of electrodynamics supplies the structure group
f/(l) for the bundle. The set of all possible U(l) gauge symmetry elements
establishes the fibre. The fibres are identical to the structure group itself,
so we construct a principal bundle P(S2, U(l)). The wave function ф(х)
corresponds to a section of the associated vector bundle—a line bundle.
Locally the bundle looks like

304 Physics in terms of differential forms
bundle coordinates f^Z] F.118)
and with 0 < в < ж, 0 < у < 2тг.
The transition functions h connect the fibres from one chart to ano-
another
F.119)
or they relate the different sections of the bundle. They represent elements
of the U{\) gauge group and are functions of ip defined along the equator
(в = 7г/2) Я+ П Я_ = S1, h+-(<p) e U(I), and they must glue the fibres
together exactly when we complete a full turn around the equator. Then
the sole choice is
h-+ = eiTUp, F.120)
with n being an integer number. This is the topological origin for Dirac's
quantization condition F.101) of the magnetic charge (—e =\e\ = h = c =
!)•
The Lie algebra valued connection 1-form of the bundle is locally
represented by the gauge potentials (recall Section 2.7.2)
At = iA±, F.121)
which are defined on the hemispheres H±. In our definitions we have a Lie
algebra factor i when A± is given by equation F.104).
The compatibility condition
Л+ = hz\A-h-+ + hZ+dh-+ F.122)
expresses the gauge transformation law and gives for the Abelian case U(l)
Л+- Л-+ indip F.123)
in correspondence with equation F.105).
The Lie algebra valued curvature 2-form of the bundle represents the
gauge field strength
F = iF, F.124)
with F being given explicitly by equation F.107). In terms of the gauge
potential we have
T = dA±. F.125)
So T is closed, dT = 0, but not exact; the Abelian T is gauge invariant.

Dime monopole 305
Winding number and monopole charge: Consider again the transi-
transition functions of the bundle h-+(ip) e U(l), equation F.120). They define
a map from the equator of the base manifold into the structure group
h-+ : Я+ П Я_ = S1 -> 17A). F.126)
These maps are classified by the first homotopy group
n1E1)~{n} = Z. F.127)
The integers n are the winding numbers characterizing the different ho-
homotopy classes which the bundle belongs to (recall Section 2.2). When if
varies from 0 —» 2тг the group U(l) is wound around n times. The winding
number n can be expressed by the integral (recall equation B.20))
1 /ir d
1 ^*ЗД
п = ън1
= ~ I dip-in, F.128)
2m JQ
where here n represents the quantized magnetic monopole charge g = n/2
(in units —e=\e\ — h = c = 1).
These winding numbers—the monopole charges—are typical numbers of
the topology of the bundle and are also determined by the so-called first
Chern number or by the Atiyah-Singer index theorem—which we
shall discuss again later on (see Section 11.4)
Ci = index D+ = ^- / tr T. F.129)
1ъ Js*
The explicit evaluation gives
i / dA++ I dA.]
JH_ J
[ F =
27Г JS2 27Г
= J-17 A++ [ A-]=±.[ (A+-A.)
= IT f dtp-n = n, F.130)
2?r Jo
where we have used Stokes' theorem B.128).
Note: The integral representation for the Chern number C\ or for the
index turns out to be independent of the connection. Only the gauge trans-

306
Physics in terms of differential forms
Electron
Screen
interference pattern
Shield
Fig. 6.3. Aharonov-Bohm effect
formation nd<p enters into the calculation corresponding to the transition
functions hz\.dh |_ which determine the topology of the bundle.
6.5 Aharonov-Bohm effect
In 1959 Bohm and Aharonov wrote a paper investigating the 'significance
of electromagnetic potentials in quantum theory' [Aharonov, Bohm 1959].
They found the striking feature of quantum mechanics—the Aharonov-
Bohm effect—that it is the potential which influences the motion of par-
particles even in regions which are field-free. There is great interest and ex-
excitement for the Aharonov-Bohm (AB) effect, especially in the last years,
since it touches the foundations of quantum mechanics. The topological de-
description of the effect within fibre bundles has been found by [Wu, Yang
1975] and [Tourrenc 1977] (for a review see [Hohne 1990]). An overview on
the theoretical and experimental side has been presented by Peshkin and
Tonomura [Peshkin, Tonomura 1989].
Experiment: What precisely is the Aharonov-Bohm effect? We consider
here only the magnetic version. Take an electron beam and split it into two
parts (see Figure 6.3). The two paths are recombined and form an interfe-
interference pattern on a screen. A solenoid is placed between the two paths which
produces a magnetic field В inside. Outside the solenoid the field vanishes
everywhere, В = 0. A shield prevents the electrons from penetrating into
the solenoid, so an electron never 'touches' the magnetic field.

Aharonov-Bohm effect 307
Question: Does a change of В inside the solenoid influence the interference
pattern on the screen?
Answer: Classically, we expect no influence. In fact, there is no AB effect
in classical physics since only the Lorentz force determines the motion of
the electron and the Lorentz force
К =- xx В = 0 F.131)
с
vanishes on the paths of the beam (where В = 0). But quantum mechani-
mechanically it is the vector potential A which enters into the Schrodinger equation
(or into the path integral formalism) and the vector potential is nonvanis-
hing outside the solenoid. So we shall get an effect.
Even though there is no magnetic field В = 0 outside the solenoid, the
vector potential is nonvanishing there, А ф 0, since the loop integral of A
around the solenoid equals the magnetic flux фм- For an infinitesimally thin
solenoid we have the vector potential
A(r) = ^(-y,x,0), r2=x2 + y2 F.132)
satisfying
I Bdf= /"(V x A)df = фм F.133)
and
B = Vxl=0 for г ф 0. F.134)
Quantum mechanics: Now let us consider the quantum mechanical wave
function. For А ф 0 it is determined by its gauge transformed version (recall
equations F.95))
фА(г) = exp |»^ j A(r)dx\ ф(г), F.135)
where ф(г) denotes the wave function corresponding to A = 0, no magnetic
llux inside the solenoid. The total wave function in the interference region
(>f the screen is the superposition of the wave functions along both paths

308 Physics in terms of differential forms
e
exp \i—
= exp
= exp
he t __
1 Г Г е / - I 1
r < exp г— ф Adx фАхо) +фц(хо) > ,
J I L hc J J J
F.136)
with жо a point on the screen. So we get a relative phase factor
Г e / - I
exp \i— <f> Adx\ F.137)
I he J J
contributing to the interference term.
The relative phase shift is
^Ildx F.138)
nc J
and depends via Stokes' theorem B.128) on the magnetic flux фм inside
the solenoid
AS = ^{Adx=^[ (V x A)df = ± I Bdf
he J he yarea he J&te&
= ^-сФм- F.139)
The probability of finding the electron on the screen depends on the modulus
squared of the total wave function and hence on the phase difference AS
F.138), F.139). If we now vary the magnetic field inside the solenoid we
obtain a periodic (sinusoidal) component in the probability of observing
the electron on the screen. The period is given by the magnetic flux unit
2irhc/\e\. If two values of the magnetic fluxes фгм, ф2и differ by
^с neZ F.140)
we find the same interference pattern.
Resume: We obtain an interference pattern on the screen which arises
on pure quantum mechanical grounds. It has the striking feature of de-
depending on the magnetic field inside the impenetrable solenoid. The phase
factor F.137) describes the AB effect completely and depends on the gauge
potential Л in a nonlocal way. So in quantum mechanics it is the vector
potential A that is more fundamental than the magnetic field В itself.
A series of beautiful experiments has been performed to test this import-

Aharonov-Bohm effect
309
Fig. 6.4. Homotopically different paths in M = R2 \ {D2}
ant quantum effect [Chambers 1960], [Mollenstedt, Bayh 1962], [Tonomura
ot al. 1982]; overall they confirm the AB predictions nicely. For a review see
[Peshkin, Tonomura 1989].
Fibre-bundle topology: Now we turn to the topological description of
the AB effect.
The base manifold M we have to consider here is the 2-dimensional
plane with the origin {0} extracted (or cutting out a disc {D2} correspon-
corresponding to the diameter of the solenoid)
M = R2 \ {0}.
F.141)
The gauge group 17A) supplies the structure group 17A) and all possi-
possible gauge group elements build up the fibre. So we construct a principal
bundle P(M,UA)). The wave function ф(х) corresponds to a section of
i-lin associated vector bundle—a line bundle.
We rewrite the gauge potential F.132) as a 1-form (recall Ao — 0 in our
pure magnetic case)
A = .
< >r in polar coordinates
-ydx + xdy)
F.142)
F.143)
It represents the local connection 1-form of the bundle which determines
t lie parallel transport of the fibre

310 Physics in terms of differential forms
A = iA. F.144)
The field strength F = dA corresponds to the local curvature 2-form
IF = iF = dA. F.145)
In the AB case we have IF = 0, so A is closed but not exact (recall the
example B.187) of Section 2.4). Thus we cannot gauge away the potential
on the whole manifold.
The base space F.141) is a multiply connected manifold which allows
homotopically different paths (see Figure 6.4). They are characterized
by the first homotopy group IIi(M) ~ IIiE1) ~ {n} = Z, where the
integers n represent the winding numbers associated with the paths. All
these paths contribute to the phase factor and hence to the interference
pattern (for a search of higher winding numbers see [Inomata 1986]).
Holonomy: Generally, a fibre point is parallel transported along a
curve a in the base M by the phase factor containing the connection (we
choose — e = |e| = % — с = 1)
[/ i closed curve in M \ ¦ I л\ г • i 1 /•> i же>\
— A\ —> exp -г <t> A\ = exp [-гфм\ • F.146)
Jo J L J J
For a closed curve in M F.141)—a circle around the origin—the lifted curve
in the fibre space ?7A) is not closed anymore. We have a holonomy in the
fibre space. Classifying the curves in M by the winding numbers we obtain
the holonomy group
H= {е~*пфм}, neZ. F.147)
It characterizes a map h of the closed curves in M into the fibre space U(l)
h-.UiiS1) -+ ?7A)
n i-» е~Ыфм. F.148)
In other terms, we find this holonomy by considering the parallel transport
of the wave function—a fibre point—with respect to the local connection A
(see e.g. [Nakahara 1990]). Then we have
Vtp = (d + А)ф = О, F.149)
where V = d + A is the covariant derivative. Calculating with polar coordi-
coordinates F.143)
( ) F.150)

Instantons 311
we easily get the solution
ф((р) = const, exp -г</>мг-| • F.151)
Along a circle S1 in the base given by ц> : 0 —> 2тг (winding number n = +1)
the wave function—a fibre point—is transported as
ф@) >-+ф{0)е~*фм. F.152)
The corresponding fibre curve does not close—a holonomy occurs. So geo-
geometrically the AB effect corresponds to a holonomy in the fibre space.
Remark: We can view the AB effect as a simple example of a nontrivial
topology of the vacuum in a gauge theory. As we shall see in the next
section gauge theories allow for a vacuum which exhibits a rich structure—
instantons—with physical consequences. Here in the AB case there are no
electromagnetic fields (B = 0, E = 0) outside the solenoid. So the energy
density vanishes and we have a vacuum. But still the gauge potential А Ф 0
is nonzero, the vacuum has a structure. In fact, A = ^-d<p is a gauge
transform—a pure gauge—of the 'pure' vacuum A = 0.
6.6 Instantons
Instantons are classical solutions of a non-Abelian field theory which exhibit
a nontrivial topology. They are responsible for the vacuum structure of
QCD. Their effect on physics (confinement, mass contributions, particle
decays ...) has been a hotly debated topic in the last decade; for literature
we refer to [Jackiw 1977], [Actor 1979], [Rajaraman 1982], [Shuryak 1988],
[Felsager 1981], [Shifman 1994]. Here we discuss some basic features which
can be easily understood within the formalism developed so far.
Action: In field theory the fundamental quantity is the vacuum-to-
vacuum transition amplitude, the generating functional from which all
(Sreen functions can be calculated (recall Chapter 3). It is actually well
defined only in Euclidean space (ix° = x4, —ido = З4, —iAg = A\ and the
metric gs = —1)
ZE = e~w* = f dApe-s*W. F.153)
We have formally absorbed the FP ghost part, the gauge treatment of the
fields, into the integration measure dA^. For the actual evaluation of ex-
expression F.153) we need the local minima of the Euclidean action, around
which the quantum fluctuations are calculated.

312 Physics in terms of differential forms
Fig. 6.5. Situation for an instanton
We consider an SUB) gauge theory which is defined on 4-dimensional
Euclidean space ?"* = R4. The gauge potential and field strength in terms
of differential forms are
A = A°(x)Tadx»
F = \F°u{x)Tadx»dxv, Ta = ^-. F.154)
and for the Euclidean action we have
SE
= -f\xF*F.
= jd4xE \F;vFa^ = -| jd4xE tr F^F"»
f\xF*F. F.155)
In the path integral F.153) the Euclidean action SE F.155) has to re-
remain finite. This severely restricts the gauge field configurations which con-
contribute. Clearly the field strength must vanish at infinity
F ix^° 0 F.156)
but not necessarily the gauge potential. A —> 0 for \x\ —> oo is the trivial so-
solution, the 'pure' vacuum. As we already know condition F.156) is achieved
for a pure gauge (recall equation B.510))
A(x)lx^°g-I(x)dg(x). F.157)
So we have the situation—which we sketch in Figure 6.5—that the field
strength F ф 0 is nonzero inside a volume E4, but vanishes, F — 0, on
the boundary dE4 = 53, a 3-dimensional sphere, where the gauge potential
approaches a pure gauge. This is the basis for a nontrivial vacuum.
The gauge elements g(x) 6 SUB), x € S3 represent mappings

Instantons 313
g : S3 - 517B) ~ S3 F.158)
from the sphere S3 into the gauge group SUB), which is again isomorphic
to a sphere S3. These mappings are classified according to the homotopy
classes which are determined by the topological winding number n (recall
Section 2.2)
n3(SC/B))~n3(S3)~{n} = Z. F.159)
Such solutions, discovered by Belavin, Polyakov, Schwartz and Tyupkin [Be-
[Belavin, Polyakov, Schwartz, Tyupkin 1975], are named the BPST instanton.
Topological charge: Let us first introduce a quantity called, in this con-
context, the Pontrjagin index
4х tr f- *Fiiv=-Ь Ltr p2-
We have compactified the Euclidean space E4 —> S4 to a 4-dimensional
sphere. The Pontrjagin index is an integer number
q-±n = 0,±l,±2,...
and—as we shall demonstrate—represents the winding number of the
map S3 —> SUB). The integral F.160) has the remarkable property that it
is onJy determined by the topology of the underlying fibre bundle; for this
reason q is also called the topological charge. The topological charge
density or the Pontrjagin density
Q =* -g^j tr p2 (
is a closed form dQ = 0 because of the Bianchi identity DF = 0
d tr F2 = 2 tr DFF = 0. F.162)
Ну virtue of Poincare's theorem (Section 7.3) it is also locally exact, say on
the upper hemisphere H+ of S4
Q = dK. F.163)
There we have
/ / dK= [ К F.164)
h+ Jh+ Js*
where we applied Stokes' theorem B.128). К is called the topological
current; it is the 3-form

314 Physics in terms of differential forms
\A3] . F.165)
о J
In Q we recover the singlet anomaly d * jb result F.74), in К the Chern-
Simons form Q${A,F) F.77). More precisely, we find
d*j5 = -2Q. F.166)
So we can construct a new current *jnew, a. combination of the axial- and
topological current, which is conserved
*jcomb = *?5 + 2X With d * Jcomb = 0. F.167)
Instanton solution: How do we find explicitly a solution A satisfying the
boundary condition F.157) and classified by a winding number? We start
from the positivity in Euclidean space
I d*x{F°v ± *F^VJ > 0. F.168)
Since
? p ^ ± F^v * Fa»v) F.169)
(note that *F?V * Fa^ = F^vFa^) we obtain
f dix{F^vFa>MV ± F*v * Fa»v) > 0. F.170)
Inserting the Euclidean action F.155) and the topological charge F.160)
we find
SE > 8тг2|д|. F.171)
(Note, if we keep the strong coupling constant gg in the definition of
A = ggAaTa the bound factor 8тг2 becomes 8n2/g2.) Clearly the inequality
F.168) and consequently the lower bound F.171) for the action is saturated
(equality is achieved) if
F = ±*F. F.172)
This means F being selfdual (+ sign) or antiselfdual (— sign). Then the
YM field equation F.63) is obeyed automatically due to the Bianchi identity
DF = 0
^ = 0. F.173)
The topological charge is positive q > 0 for a selfdual field strength, F = *F,
and negative q < 0 for an antiselfdual F, F = — * F. We will consider just
the selfdual solutions in the following.

Instantons 315
Now we construct a field configuration A explicitly which obeys the
boundary condition F.157) and the selfduality F.172) of F. We remember
that the gauge element g € SUB) is an element of the group of unitary
unimodular two-by-two matrices. Such an element g can be written in terms
of the basis A, <7), where a = {a1, a2, a3) are the Pauli matrices C.281)
g(x) = -(x4 + ixa) with x\ + x2 = r2. F.174)
r
This map from S3 —> SUB) ~ S3—the identity map—corresponds to
winding number n = 1. It is the SUB) analogue of the C/(l) case B.21)
which we discussed in Section 2.2. We obtain maps of higher winding
numbers n by substituting g —> gn. Clearly the inverse element is given by
g-\x) = -(x4 - ixa) = g]{x). F.175)
т
For a solution of equation F.172) we make the general ansatz
A(x) = f(r2)g-1(x)dg(x), F.176)
with g, g~l expressed by equations F.174), F.175). Respecting the boun-
boundary condition F.157) for r —> oo and avoiding a singularity at origin r -> 0
the function f(r2) has to satisfy the limits
f(r2) r^ 1
f(r2) ^ const, r2. F.177)
The simplest function which achieves this is
where the parameter Л specifies the instanton size. So the BPST instan-
ton solution is
^, F.179)
with g(x) given by equation F.174). This field configuration minimizes the
action for a fixed winding number, also called the instanton number (here
м 1)—dominating the functional integral F.153)—and it is a solution of
i he field equation.
Topology of instantons: Now we establish the appropriate fibre-bundle
set-up to describe the topological features of an instanton (see e.g. [Eguchi,
(Jilkoy, Hanson 1980], [Nakahara 1990]).

316 Physics in terms' of differential forms
S4
H+n H_=8Z
Fig. 6.6. Compactified base manifold S4 for an instanton
We obtain the base manifold by compactifying the Euclidean space
E4 = R4 to a 4-dimensional sphere S4 (one-point compactification: the
north pole represents the origin, the south pole the points at infinity). We
divide the sphere S4 into two hemispheres H+ and ff_ having the boundary
dH+ — S3 and 9ff_ = —S3, a 3-dimensional sphere S3 (the minus sign
reflects the induced orientation). So the intersection, the 'equator', is H+ Л
ff_ = S3 (see Figure 6.6).
The gauge group SUB) of the non-Abelian field theory provides the
structure group SUB) for the bundle. For the fibres we also take
the elements of the group SUB), so we construct a principal bundle
P(S4, SUB)). The local bundle patches look like
»+*S3o! with bundle coordinates [Х'Ы F.180)
tl- X OU[Z) (X, /_).
The transition functions ft_+ connect the fibres / between the two
patches
/_ = ft—(./+ along the 'equator' ff+ П Я_ = S3, F.181)
they 'twist' the neighbouring fibres. A transition function corresponds to a
gauge element which in our case is given by the map F.174)
h-+(x)=g(x)eSUB), z € ff+П ff_ = S3. F.182)
Running along the 'equator' H+ П ff_ = S3 we wind once around the group
SU{2) ~ S3. Generally, for higher windings, for instanton number n we
have
h-+(x) = gn(x) e SUB), xeH+nH-=S3 F.183)
(note that n must be integer due to S!7B)-matrix multiplication). For n = 0,

Instantons 317
the constant map
g° : x € S3 н+ 1 e 5t/B) F.184)
we obtain a trivia] fibre bundle.
The Lie algebra valued connection 1-form of the bundle is locally
represented by the gauge potentials A+, A_ which are denned on the he-
hemispheres H+, Я_. The compatibility condition
A+ = hz\A-h^+ + hZ\dh-+ onff+n#_=S3 F.185)
describes the gauge transformation law for the YM potentials.
The Lie algebra valued curvature 2-form of the bundle is locally given
by the field strengths F+, F_ which are expressed by the potentials
F± = dA± + A\ F.186)
on the hemispheres H+, ff_. Again, the compatibility condition
F+ = hz\F-h-+ F.187)
corresponds to the gauge transformation of the YM field strength.
= -\ I tr(fclidh-+K. F.188)
3 Js3
Lemma:
/ trF2 \
Js* 3 Js3
So the integral defining the Pontryagin index F.160) is independent of the
connection A. It depends only on the transition functions h (. of the bundle
which determine the 'glueings' of the fibres, and thus the topology of the
bundle.
Proof:
l L( HLd tr
L"Fl - Ld a (F+A+ - H+L
A-dh-+Kl\\ ¦ F.189)
We applied Stokes' theorem B.128) ( dH+ = S3, dH_ = -S3) and we
inserted the compatibility conditions F.185), F.187). The second term va-
vanishes when using Stokes' theorem again, the remaining provides the Lemma
(li.188). Q.E.D.

318 Physics in terms of differential forms
Note: Of course, we could also choose a vanishing connection on the
southern hemisphere, A- = 0 on Я_. Then all topological content about
the bundle is carried by A+ on the 'equator' S3, A+ = hz\dh |_ on
H+ П H- = S3, and Lemma F.188) follows immediately from the term
-gj4+ in the second equation of the proof. (We derive a generalization of
Lemma F.188) to any even dimension within Chern-Simons forms in Sec-
Section 7.5; see equations G.114)-G.117).)
Winding number—instanton number: Next we calculate the integral
F.188) when the transition functions are chosen a la BPST, h |_ = g,
equation F.174)
hZ1+dh-+ = -(x4- ixa)d-(x4 + ix3) and x\ + x2 = r2. F.190)
The integral over 53 simplifies comfortably when choosing as a reference
point on S3 the north pole (я4 = 1, x = 0) of the unit sphere, which we
may do since h-+ maps 53 uniformly onto SUB) ~ S3. Then we have
hz\dh-+ = iakdxk F.191)
and
tr (hz\dh-+K = i3 tr uiuiukdxidxjdxk
= 2eijkdxidxidxk = I2dx1dx2dx3, F.192)
where the wedge product dx1dx2dx3 is an 'area' element at the north pole
of the 3-dimensional sphere S3, and consequently the integral is
/ tr (hzldh-.+K = 12 f dx1dx2dx3 = 12 ¦ 2тг2 = 24тг2 F.193)
Js3 Js3
since the area of the unit sphere 53 is 2тг2 (generally, the area of 5" is
a(Sn) = 2тг(п+1)/2Дп/Г (Ц1)).
Altogether we obtain for the Pontrjagin index—the topological
charge
a value which agrees with the winding number 1 of the map h j. or with
the instanton number 1.
Finally, for maps with higher winding numbers like
Л_+ = g2 : 53 -¦ 5GB) ~ S3 F.195)

Instantons 319
we split 53 into hemispheres H+, Я_. We may always deform homotopically
the map g with winding number 1 as
g+ on Я+ ( 1 on Я+
ITT or g > s Ty ,
on ti- у д_ on ti_
where g+, g_ have winding number 1. Then we get for the map F.195)
[+ ' * ™ ^+ F.196)
Consequently we find (by the homotopic invariance of the winding number)
tr
tr
¦ 24тг2[1 + 1] = 2 F.197)
and iteratively all higher winding numbers.
Proposition:
• Tiie Pontrjagin index—the topological charge—is a topological inva-
invariant
tr {hz\.dh-+K - n F.198)
determined only by the transition functions h |_ of the fibre bundle
and their homotopy classes. It represents the instanton number or
winding number of the map
Л_+ : 53 -> 517B) ^ S3 with Л_+ = дп F.199)
and д given by the identity map F.174). It counts the number of times
the group SUB) covers the sphere S3 under the mapping h-+.
Last, but not least, the Pontrjagin index corresponds to the second
('hern number Ci or to the Atiyah—Singer index theorem, which we
will discuss much later (see Section 11.1)
q = -C2 = index D+. F.200)

320
Physics in terms of differential forms
Resume: There is a remarkable coherence between physics and geometry.
In fact, fibre bundles appear as the natural geometric set-up for describing
the physics of gauge theories. Assuming some internal symmetry group)—
gauge symmetry—the fibre bundle automatically supplies a pure geometric
interpretation for the concepts of gauge theory, specifically for the gauge
potential, the field strength, the gauge element and gauge transformations.
In addition it provides a geometric-topological reason for the quantized
charge of the magnetic monopole and of the instanton, or, as we .shall see
later on, for the anomaly of quantum field theory.
This elegant geometric topology is important for our understanding
of gauge theories, and we also use it to formulate further aspects—the
anomalies—in terms of this fibre bundle technique. Finally, we also find
it very helpful to compose a brief aide-memoire in form of a list which
summarizes the corresponding concepts in physics and geometry.
Aide-memoire
Physics
gauge theory
space-time
set of all possible gauge sym-
symmetry elements
gauge group
gauge element
gauge potential
gauge field strength
gauge transformations
wave function
phase factor of wave func-
function
Aharonov-Bohm effect
magnetic monopole charge
instanton number
anomaly of QFT
Geometry
principal fibre bundle
base manifold
fibre
structure group
transition function
connection
curvature
compatibility conditions
section of fibre bundle
parallel transport of fibre
holonomy in fibre space
winding number classified according to
homotopy group Ui(U(l)) or Chern num-
number Ci, Atiyah-Singer index theorem
winding number according to ПзEТ/B))
or C2, Pontrjagin index, Atiyah-Singer
index theorem, topological charge
Atiyah-Singer index theorem, Bismut in-
index theorem, determinant bundle

Chern-Simons form, homotopy
operator and anomaly
The Chern-Simons form and the homotopy operator play an important
role in connection with anomalies. In fact, we can calculate the anomaly on
pure algebraic grounds from a variation of the Chern-Simons form by using
a homotopy operator.
First we need to discuss a symmetric invariant polynomial of fields (Sec-
(Section 7.1) which is the starting point for the derivation of the Chern-Simons
form and 'transgression formula' (Section 7.2). Next we prove the import-
important Poincare lemma and introduce in this connection a homotopy operator
(Section 7.3). A generalization of the 'transgression'—the Cartan homo-
homotopy formula—follows in Section 7.4. Then we come to the important part
of this chapter; we apply the homotopy formula to a Chern-Simons form
with gauge transformed fields and are able to derive in this way the non-
Abelian anomaly (Section 7.5). A general formula for the variation of the
Chern-Simons form—which expresses the anomaly—is presented at the end
(Section 7.6).
7.1 Invariant polynomials
We discuss the properties of a polynomial function of some matrices. For lite-
literature we refer to [Eguchi, Gilkey, Hanson 1980], [Nakahara 1990], [Alvarez-
Gaume, Ginsparg 1985].
Definition: Let {c^} be a set of complex к х к matrices and consider the
vector space of C-valued functions which are linear and symmetric in each
pair oti
P(ai, ...,ai,...,aj,...,an) = -P(ai, ¦ • ¦, a,,..., au ..., an), G.1)
with 1 < i, j < n. Choose some group G, in practice G = GL(k, C), SU(k),
f'(fc), and Lie G denotes its Lie algebra; then P is called a symmetric
invariant polynomial or characteristic polynomial if

322 Chern-Simons form, homotopy operator and anomaly
Pig^aig, • • •, 9~lan9) = P(ob..., an) G.2)
for all g g G and a» s Lie G.
We now use the invariance of such a polynomial under transformations
ац -> д~1сцд G.3)
by choosing
9 = 9t= etf), G.4)
with /3 S Lie C? and tgR some parameter, and we differentiate equation
G.2) with respect to t
97lai9t, ¦ ¦ ¦,971(-P<*n)9t) + Pig^aM, ¦ ¦ .,9r\otnl3)gt) = 0.
G.5)
For t = 0we obtain the following equation
n
.. ¦,«i-i, [ft on],Oi+i, •.., а„) = 0. G.6)
Definition: If all оц are equal then P is an invariant polynomial of
degree n with notation
Example: The symmetrized trace str represents such a symmetric inva-
invariant polynomial
Р(оь...,о„) =str (ai,...,an) := V — tr api...aPn. G.8)
ТЫ
Perm
For n = 3 we get
Р(ац,а2,аз) = str @:1,0:2,0:3)
= -(tr Q10203 + tr О1О3О2) G-9)
when using the cyclicity of the trace. If all o» are equal then a polynomial
G.7) is given solely by the trace
Pn(a) = P(an) = tr an. G.10)

Invariant polynomials 323
Next we extend the definition of an invariant polynomial of matrices to
that of matrix valued (elements of Lie G) differential forms on a manifold
M. Let a* be a differential form of order d,
at = adiEdi := aMl...Md. (x) -J- dx^ ... da;"* G.11)
where adi and Edi are obvious short-hand notations for the tensor field and
the wedge product. The tensor adi is an element of the Lie algebra of G via
the generators Ta
adi = a°i.Ta e Lie G. G.12)
Definition: An invariant polynomial for Lie algebra valued dj-forms
cti is defined by
u ...,an) = P{adl,..., adn)Edl ...Ed", G.13)
together with equations G.1), G.2).
Note: The product Edl... Edn denotes again a wedge product. Thus we
obtain the following as a rule: extract the form part and treat the matrix
part as an invariant polynomial G.2).
Let /3 be a Lie algebra valued 1-form
/3 = /yaOdz", /?„ = /З^Т0 е Lie G, G.14)
then we can write the following quantity according to the above rules
P(ai,..., <Xi-i, [P, aii}, ai+l, ...,an)
= P(adl,... ,adi_,, [/VadJ,adi+1,... ,adn)(-)d*+-d^dx»Edi ...Ed~
G.15
where the dx^ has been pulled through the wedge products providing the
corresponding minus factor. Applying equation G.6) to the matrix part we
obtain the identity
)*+-+*-ip(ai,...,a«_i, \р,щ],щ+1,...,а„) = 0. G.16)
Choosing a p-form
r, = r,P(xW, r?p = r,apTa e Lie G G.17)
instead of the 1-form /3 just changes the minus factor of the identity G.16)
in an obvious way

324 Chern-Simons form, homotopy operator and anomaly
u ... ,an) = 0. G.18)
n
Now we apply the exterior derivative d to the invariant polynomial of
differential forms
dP{au ¦ ¦ ¦,О = 5](-)dl+-+di-1P(ai,... ,a«_i,dai.ai+i,• ¦ • ,о„)
G.19)
and we add the identity G.16) for /3 = A being a 1-form connection; then
we gain the following frequently used formula.
Formula:
n
dP{au ...,<*„) = JT(-)*+-*-iP(ai,...,ai-i, Da4,ai+b..., an),
G.20)
where D expresses the covariant derivative D = d + [A, }.
Example: Let us choose the symmetrized trace str
i,...,an) = str (аь...,а„) = str {adl,... ,adn)Edl.. .Edn
G-21)
Perm
For instance, for n — 3 we get
str (ai,a2,o3) - (-)d2<J3 str (a1.a3.a2) G.22)
and on the other hand
str (аьа2,аз) = -(tr adiad2ad3 +tr adlad3a
traia3a2). G.23)
Definition: If all differential forms are equal oti = a and a represents a
Lie algebra valued p-form а = apEp then P is of degree n with notation
P(a,..., a) = P(ap,..., ap)E*> ...E» = Pn(a) = P(an) G.24)
and, for example, is just the trace
P(an) = tr an. G.25)

Transgression formula and Chern-Simons form 325
7.2 Transgression formula and Chern-Simons form
Next we are going to construct a certain polynomial in the gauge connection
and curvature—the Chern-Simons form.
Let us consider an invariant polynomial of degree n in the curvature
2-form п ? Lie G ® Л2(М) which satisfies
РЛд-^д) = Pn{U) G.26)
for all gauge elements g ? G. The symbol п will denote either the gauge field
strength tensor п —> F of the Yang-Mills theory or the curvature п —> R of
gravitation. The group G is then either the YM gauge group or the frame
rotation group. The corresponding connection w satisfying
Q, = du + ui2 G.27)
expresses respectively the gauge potential w —> A or the spin connection.
The invariant polynomial G.26) has the following important properties
[Stora 1977], [Eguchi, Gilkey, Hanson 1980], [Stora 1984], [Zumino 1984],
[Alvarez-Gaume, Ginsparg 1985]:
Theorem: Let Pn(fi) be an invariant polynomial, then
i) Р„(П) is closed, dPn(O) = 0,
ii) Р„(О) has topologically invariant integrals, or, equivalently, the diffe-
difference of two invariant polynomials is exact—'transgression'
Pn(fii) - Pn(U0) = dQ2n_1(ujl,uJo) G.28)
/•l
Q2n-i(wi,w0) = n / dfP(w1-w0,Or1)> G-29)
Jo
with
fit = du)t + u\, uit = w0 +t(u>i - w0). G.30)
In fact, any invariant polynomial P(O) being a sum of products of Pn(f2)
satisfies properties i) and ii). The index 2n - 1 indicates the form degree of
Proof.
i) By formula G.20), choosing all a» = fi, we have

326 Chem-Simons form, homotopy operator and anomaly
n
dPn(U) = Y, P(U, ...,U, DU\t, U,..., П), G.31)
where Du stands on the г-th place, and using the Bianchi identity
DO. = 0 (recall Section 6.2) we have proved dPn(Cl) = 0.
ii) Let ojo, ш\ be two connections on a bundle referred to the same system
of local trivializations and fioi ^i denote the corresponding curvatu-
curvatures. As usual we consider a principal bundle (or its associated vector
bundle). We define a homotopic connection
u>t = w0 + tp, P = W!- w0, G.32)
with t e [0,1]; the homotopic curvature is then given by
ut = dwt+ui? = Clo + tDp + t2p2, G.33)
where D expresses the covariant derivative containing uH
D = d + [uH, }. G.34)
The homotopies continuously deform the fields between
wt=o = w0, Ot=o = По = duH + Wq G.35)
and
w*=i = wi, nt=i = fii = dwi + w\. G.36)
We differentiate fit with respect to t
at at
= dp+[ujt,p] = Dtp, G.37)
where Dt denotes the covariant derivative including wt
Dt = d+[u)t, }. G.38)
Next we consider the invariant polynomial Pn(ut) and differentiate
with respect to t
jtPn(ttt) = nP(jtnuSlu...,nt)
= пР(А/3,Пр1). G.39)
The formula G.20) together with the Bianchi identity DtQt = 0 pro-
provides

Transgression formula and Chern-Simons form 327
P(U)
jtr1), G.40)
and integrating from t = 0 to t = 1 we obtain
Pn(fii) - Рп(П0) = dQ2n-i(oJi,ojo), G.41)
with
Г1
= n
Jo
Г
Q2n-i(oJi,oJO) = n dtP^-uJo,^-1). Q.E.D. G.42)
Jo
The 'transgression formula' G.28)-G.30) is a basic relation for our dis-
discussions in the following chapters. It is—as we shall see in Section 7.4—a
special case of Cartan's homotopy formula.
Integrating equation G.41) over a 2n-dimensional compact manifold M2n
without boundary {dM^n = 0) provides
JM2n
= / Q2n-i(wi,wo) = 0 G.43)
>дМ2п
by virtue of Stokes' theorem B.128) and дМъп = 0. Thus the integrals
of invariant polynomials are invariant under changes of the connection a»;
they represent characteristic numbers of the bundle—Chern numbers—and
depend only on the transition functions of the bundle (recall our topology
discussions in Chapter 6). They are also invariant under deformations of the
manifold Min-
Chern-Simons form: Choosing finally in formula G.41) and G.42) wo =
0 and consequently По = 0—which we are allowed to do oniy if the bundle
is trivial, or on a local chart where the bundle is trivial—we find
i(w,n), G.44)
together with the Chern-Simons form
/•l
= n / APfw.fir1) G-45)
Jo
and
uit = tw, ut = dwt+ wt2 = tO, + (t2 - t)u>2. G.46)
Then ojt = tu) is not a connection defined globally on a nontrivial bundle.

328 Chern-Simons form, homotopy operator and anomaly
Note: Equation G.44) also follows from Poincare's lemma, which states
that a closed form like Pn(Cl) is iocaJJy always exact (see Section G.3)).
Examples: We illustrate formula G.45) by some examples. For instance,
for YM fields and n = 2 we recover the formula for the singlet anomaly
F.75) in 4 dimensions
tr F2 = dQ3, G.47)
with
Q3{A, F) = 2 / dt tr A[tF + (t2 - t)A2}
Jo
= tr [AF - ^A3} = tr [AdA + Ia3}. G.48)
3 о
For n = 3 we obtain the analoguous result in 6 dimensions
tr F3 = dQ5, G.49)
with
= tr [A(dAJ + lA3dA + lA5]. G.50)
2, 5
Remark: As noticed by Stora [Stora 1977] formulae G.41), G.42) can
be lifted to the principal bundle P(M, G) and one also arrives at formulae
like G.44)-G.46). But then the formulae contain for ш —> uigiobai the global
connection, for Q —> ?lgiobal the global curvature on P(M, G) and for d —* A
the exterior derivative on the bundle with Пд;оьа1 = Дш51оЫ + global ¦
7.3 Poincare lemma and homotopy operator
We have to discuss an important proposition—Poincare's lemma. It is in-
instructive to introduce in this connection the homotopy operator to gain
more insight into its action on differential forms. Therefore we want to ela-
elaborate the following proof in detail (for literature we refer to e.g. [Flanders
1963], [Singer, Thorpe 1967], [Curtis, Miller 1985], [Nakahara 1990]).
Proposition: Poincare's lemma
• If a domain U in a manifold M is contractible to a point and u) is a
p-form on U С M with вш = 0 then there exists a (p - l)-form a on
U such that ш = da.

Poincare lemma and homotopy operator
329
IxV
M
(a)
(b)
Fig. 7.1. a) Illustration of the maps jt(x) and f(t,x). b) Pullbacks under
these maps in the spaces of the p-forms
Briefly, any closed form (du = 0) on M is locally always exact (w = da),
where locally refers to a coordinate neighbourhood which is contractible to
a point.
Proof. We first specify the contraction of a domain to a point. A topological
space U is contractible if there exists a homotopy (recall Section 2.2)
f:IxU->U G.51)
such that
/A,x) = x = idv(x) and /@,x) = x0 € U, Vx € U. G.52)
We construct a cylinder, the product space (see Figure 7.1a))
IxU = {(t,x)} with t € / = [0,1] G.53)
and let U be contractible according to our presumption. We also define a
map from the domain into the cylinder

330 Chern-Simons form, homotopy operator and anomaly
jt:U->IxU with jt(x) := (t,x). G.54)
The two maps ji(x) = (l,x) and jo(x) = @,x) identify the domain U
with the top and bottom of the cylinder (see Figure 7.1a)).
Let us consider a p-form on I x U such as
w = ap(t, x)Ep + 6p_i(t, x)dtEp-1 G.55)
by working with the short-hand notation of Section 7.1 (equation G.11)).
Then the pullback (recall Section 2.6.2) under the map G.54) is (see Figure
7.1b))
U)^ AP(U)
= u,(t,x)\t=const. = ap(t,x)E" G Ap(U). G.56)
Definition: Next we define an operator К
К-.АРЦхЩ^АР-1^) G.57)
by
Kap(t,x)Ep = 0 G.58)
Kbp-1(t,x)dtEP~1 = (f bp^(t,x)dt\ -EP-\ G.59)
Therefore we have
Ku>= ( f bp^{t,x)dt\ -EP-1. G.60)
Lemma:
K{dw) + d(Kuj) = jjw - JoU) G.61)
valid for any w e Лр(/ х U).
Proof. We prove equation G.61) by direct calculation
K{du) = K(djap(t,x)dx>Ep + —ap(t,x)dtEp)
+ Kdjbp-i(t,x)dxjdtEP-\
K{du)= (f ^ap{t,x)dt\-E*-U djbp-1{t,x)dtYdxjEp-1. G.62)
On the other hand, equation G.60) gives

Poincare lemma and homotopy operator 331
d{Kw) =([ djbp_1(t,x)dt\ -dx^P-1 G.63)
so that altogether we obtain
K{du) + d(Ku) =
Q-E-D. G.64)
We also notice by recalling the contractibility of the domain U that the
product map / • jt : U —> U via U -^-> I x U —> U has the boundary
conditions (see Figure 7.1a)) f ¦ ji = idu, f ¦ jo = Zo, which implies for the
pullbacks of a p-form ш € AP(U)
Jo'-Г" = (/-Jo)'w = 0. G.65)
Now we are prepared to perform the actual proof of Poincare's lemma
and we prove it by explicit construction. The (p — l)-form
a = Kf*uj G.66)
will be the desired result such that the closed form и is also exact on the
contractible domain U
w = da. G.67)
We start from relation G.61) and replace w € Ap(IxU) -> f*u € Ap(IxU).
Then we obtain
K{dfu)+d{Kfw) = Jt-f'w-fi-Гш
= (/•№-(/-Jo)'". G.68)
On the left-hand side the first term vanishes since и is closed (df*w =
f'du) = 0). On the right-hand side w remains due to the pullback properties
G.65). Thus we are left with
u> = d{Kf*cj) and a = Kf*u on U. Q.E.D. G.69)
()f course the solution a is not unique, we always can add an arbitrary exact
form
a^a + d/3. G.70)
Homotopy operator: In proving Poincare's lemma we introduced the
operator К G.57)-G.59) which acts on p-forms defined on I xU. Usually,

332 Chern-Simons form, homotopy operator and anomaly
however, we work with p-forms u) € AP(U) defined on U. So we lift u> to the
cylinder I xU via the pullback /* (see Figure 7.1b)) and apply the operator
К there.
Definition: The homotopy operator is denned by
к = Kf*, G.71)
with К given by equations G.57)-G.59).
Lemma:
kd + dk = 1. G.72)
Proof. Starting with relation G.68) for the К operator, interchanging
df* = f*d, and using the pullback properties G.65) we find
Kf*duj + dKfw = w
kdw + dkuj = u) G.73)
for any из € A?(U). Q.E.D.
Operator kd + dk thus shows that a closed form ш (dw = 0) on a con-
contractible space is an exact form (ui = da)
(kd + dk)u> = ш, d(ku>) = ш and a = kui G-74)
demonstrating Poincare's lemma. Note that the application of к reduces the
form degree of w by one unit (because of property G.59)).
Theorem: The homotopy operator к is nilpotent
k2 = 0. G.75)
Proof. We start with a p-form on a contractible domain U, w = ap(x)Ep €
Ap(f7), and we choose for simplicity the contraction to the zero-point
f(t, x) = tx. Then the homotopy operator к acts as
few = Kf*ap(x)Ep
= Ktpap(tx)Ep + K^-^aji,...^, [tx)dtEp~l
v)-1. G.76)
Repeating the same procedure gives

Cartan homotopy formula 333
k2w = Kfx
= х>х1П dt f
= 0. Q.E.D. G.77)
(Note: xjxl is symmetric, whereas Ojjii...ip_a is antisymmetric.)
Algebraic Poincare lemma: There is also a generalization of Poincare's
lemma, given by Stora [Stora 1977, 1984], which we shall use frequently in
this book.
Proposition: Algebraic Poincare lemma
• Let Рр[ф](х) and <3p_i[0](z) be forms of degree p and p - 1 which
are local functionate in the basic fields—[ф](х) denotes collectively
all fields and their derivatives—then Pp = dQv-\ iff dPp = 0, for
p + 1 < m = dim M, or iff J Pp = 0, for p = m.
7.4 Cartan homotopy formula
In this section we introduce a homotopy operator on pure algebraic grounds
and derive the Cartan homotopy formula. For literature we refer to [Stora
1984], [Zumino 1984], [Alvarez-Gaume, Ginsparg 1985] and [Manes, Stora,
Zumino 1985].
We start again with a homotopic YM connection as in equation G.32)
At=A0 + t(Ai - Ao), t € [0,1] G.78)
and with the corresponding curvature
Ft = dAt + Al G.79)
Definition: We define an operator ?t—homotopy derivation—by
etAt = 0 G.80)
etFt = 6At = dt^-At = dt{Ar-A0) G.81)
at
acting on polynomials in At and Ft; we require the antiderivation rule
+ (-yap(et0q) G.82)

334 Chern-Simons form, homotopy operator and anomaly
for the p-forms ap € Лр and q-forms /39 € Aq. The operator It decreases
the form degree in dxl by one unit and creates a dt. (Here we follow the
approach of [Zumino 1984] and [Alvarez-Gaume, Ginsparg 1985]; in the work
of [Manes, Stora, Zumino 1985] It is defined as an even derivation.)
We can verify explicitly
(ltd + dit)At = ltdAt = et(Ft - A\) = 8At = dt-At G.83)
(etd + det)Ft = et(FtAt - AtFt) + d6At
= 6 At At + At6At 4- d6At
= DtFAt) = 6Ft = dt-Fu G.84)
(the anticommutator of 2 antiderivations is a derivation) which implies for
any polynomial S in At and Ft (possibly noncommuting)
(etd + det)S{At,Ft) = 6S(At,Ft) = dtjtS{AuFt). G.85)
So we find the following operator equation.
Lemma:
ltd + dlt = 6. G.86)
Definition: We define the homotopy operator (the i-integrated version
of the derivation lt) by
fcoi = / It- G.87)
./o
Integrating equation G.85) now with respect to t we arrive at:
Theorem: H. Cartan's homotopy formula
S{Ax, F{) - S(A0, Fo) = (koid + dkol)S(Au Ft). G.88)
A generalization can be found in [Manes, Stora, Zumino 1985].
We can calculate the following term explicitly
koiS{At,Ft)= f ltS{AuFt)
Jo
dt S(At,Ft - (Ay - Ло)|,,Я), G.89)
/

Chern-Simons form, gauge transformations and anomaly 335
where on the г-th place the Ft is replaced by A\ - Ao and the minus factor
arises from the antiderivation rule depending on the form degree up to the
г-th place.
Specializing the arbitrary polynomial now to an invariant polynomial,
S(At,Ft) ~> Pn(Ft), we obtain
koxPn(Ft) = V(-J I dt Pn(Ft -> (Ax - Ао)\,, Ft)
= n f dtP(A1-A0,Fr1) = Q2n-i(Ai,A0). G.90)
Jo
Since the invariant polynomial is closed dPn(Ft) = 0 the Cartan homotopy
formula supplies the 'transgression formula'
Pn(Fi) - Pn(Fo) = dQ2n_x{Ax, Ao). G.91)
Finally, choosing the case Ao = 0, Ax = A—the case of a trivial bundle
or local chart—the homotopy operator is usually denoted by
f It G-92)
./о
./o
and we get from equations G.86) and G.80), G.81) the properties
kd + dk = 1 and к2 = О G.93)
in accordance with our results G.72) and G.75) of Section 7.3.
Applying Cartan's homotopy formula to this special case gives
Pn(F) = (kd + dk)Pn(Ft)
= dkPn(Ft)=dQ2n-1(A,F) G-94)
and equation G.90) provides the Chern—Simons form
Q2n_x(A,F) = kPn{Ft) = nf dtPiA^r1), G.95)
Jo
with
At = tA, Ft = dAt + A2 = tF + {t2 - t)A2. G.96)
7.5 Chern-Simons form, gauge transformations and anomaly
Ли important application of the homotopy operator and Cartan's homo-
homotopy formula occurs in gauge transforming the Chern-Simons form. This

336 Chern-Simons form, homotopy operator and anomaly
procedure provides the anomaly. Again, we follow the work of Zumino [Zu-
[Zumino 1984], Stora [Stora 1984] and Alvarez-Gaume and Ginsparg [Alvarez-
Gaume, Ginsparg 1985].
Gauge transformations: Consider the gauge transformed field
Ae = g-1Ag + g-1dg = g-1(A + V)g, G.97)
with
V = dgg-1 G.98)
and the transformed curvature
F9 = dA^ + {A9J = g~lFg, G.99)
where g = g(x) denotes the gauge element.
Let us choose a homotopy such as
A° = tg-xAg + д~Чд = g-\At + V)g, G.100)
with
At=tA, ie[0,l]. G.101)
The homotopic curvature then is
Ff = dA3t + (A°J = g-lFtg, G.102)
with
Ft = dAt + A2 = tF + {t2 - t)A2. G.103)
Both homotopies G.100) and G.102) interpolate continuously between
Af=1 = A", FU = F* G.104)
and
A9t=0 = g~xdg = g-'Vg, Ff=0 = 0. G.105)
Now we apply the Cartan homotopy formula G.88) to the Chern-Simons
form containing the homotopies G.100), G.102)
^(Af,F^). G.106)
Calculating the gauge transformed Chern-Simons term we get
Q2n_!(>19,F9) = n I dttv A°(Ft9)n~\ G.107)
./0

Chern-Simons form, gauge transformations and anomaly 337
with
F» = tF9 + (t2 - t){A9J = g-lFtg G.108)
and
Ft = tF + (t2-t)(A + VJ G.109)
so that
Q2n-i(i49, F9) = n [ dt tr (A + V)FrX = Q2n-i(A + V, F). G.110)
Jo
Analogously we can calculate
Q2n-i{A9, F?) = n f dt' tr (At + V) [HFt + (t12 - t')(At + VJ}n~l
Jo
= Q2n-i(At + V,Ft). G.111)
On the other hand we know that (recall equations G.90), G.95))
k01dQ2n_1(A9t, F9) = fcoitrF/1 = Q2n-i(A, F) G.112)
and finally we define a Bn - 2)-form
-2 ¦= k01Q2n-i(A9, F9) = fcoiQ2n_i(At + V, Ft). G.113)
Then we can rewrite Cartan's homotopy formula G.106) and we obtain the
following gauge transformation for the Chern-Simons form
Q2n-1(A9,F9) = Q2n-i(A,F) + Q2n-1(g-1dg,0)+da2n-2 G.114)
Q2n-i(A + V,F) = Q2n-x{A, F) + Q2n-i(V, 0) + da2n_2. G.115)
These are important relations satisfied by the Chern-Simons terms. We
shall use relation G.114) again, for instance, for calculating Chern classes
(s<4) Section 11.5.3).
Considering the term Q2n-\{V, 0) we can carry out the i-integration
¦ •xplicitly. Formula G.110) with equation G.109) supplies
Q2n-i(V, 0) = (-)"-xn / dttn-\l - t)n~l tr V2n~\ G.116)
./o
I'lic integral represents the familiar beta-function B(n,n) (see equation
(•». 135)) and we obtain
Qan-i(V,0) = (-)B-l((n2n"iy tr V2n~\ G.117)

338 Chern-Simons form, homotopy operator and anomaly
with V = dgg-1. Note that Q2n-i(V = dgg-^O) = Q2n-i(9~1dg,0) be-
because of the cyclicity of the trace. The form Q2n-i(V, 0) is closed
dQ2n-i(V,0) = const, tr dVV2n~2 = const, tr V2n = 0, G.118)
where we have used dV = V2, which can be verified by direct calculation.
Anomaly: Finally we turn to the term a.in-2- As Zumino [Zumino 1984]
realized, the term dain-г in equations G.114), G.115) arising from a gauge
variation of the Chern-Simons form contains already the anomaly.
For example, for n = 2 we calculate
= f ettr[(At + V)Ft-\(At + VK} G.119)
Jo й
when carrying out the t'-integration in formula G.111). Applying the ho-
homotopy derivation ?t with properties G.80)-G.82) we get
/•i
a2 = / dt tr \-tA2 - VA] = -tr VA G.120)
Jq
since tr A2 = 0. Then the derivative of a2 is
da2 = tr VdA G.121)
when keeping only the term linear in V which we denote by == (recall that
dV = V2). Equation G.121) expresses the non-Abelian anomaly in two
dimensions apart from the normalization.
Analogously we calculate the case n = 3
*t tr [(At + V)F2 ~\(At + VKFt + ^(
= tr F-iv(AF + FA) + ^VA3 + h/AVA + \v3a] . G.122)
For the derivative of 04, when keeping only linear terms in V, there follows
da* = dtr \-\v(AdA + dAA)\
L 2 2
= tr Vd (MA + \A3] . G.123)

Chern-Simons form, variations and anomaly 339
Equation G.123) represents the non-Abelian anomaly in 4 dimensions,
Bardeen's result F.79), apart from the normalization.
Resume: A gauge variation of the Chern-Simons form, more precisely
the linear term in V of the shift A + V, already represents the anomaly
which we are interested in. This is a remarkable result. In the following
chapters we shall elaborate in more detail the relation of the anomaly to
gauge transformations.
7.6 Chern-Simons form, variations and anomaly
Alvarez-Gaume and Ginsparg [Alvarez-Gaume, Ginsparg 1985] have calcu-
calculated a general formula for the first-order variation of the Chern-Simons
form by working with the homotopy derivation. Their procedure is as fol-
follows.
Variation: Consider a variation in the gauge connection
A^A + 6A, G.124)
implying for the corresponding curvature
F-^F + 6F with 6F = d6A + 6AA + A6A = DFA). G.125)
Definition: We define a homotopy derivation t analogous to equations
G.80), G.81) by
Ы = 0 G.126)
IF = 6A, G.127)
obeying the antiderivation rule G.82).
Lemma:
Id + it = 6. G.128)
Now we study an infinitesimal variation of the Chern-Simons form
Q2n-i(A + 6A,F + 6F) - Q2n-i(A, F)
/•i
- ?P(Fn) + din
Jo
Г1
= nPFA,Fn-l) + n(n-l)d / dtP{6A,At,F?~2). G.129)
Jo

340 Chern-Simons form, hornotopy operator and anomaly
(Note that ?Ft = t6A and At = tA.) If we next choose a variation where
6F = DFA) = 0 and draw the derivative d into the polynomial (recall
formula G.20)) we obtain
Г1
= nPFA,Fn-l)-n(n-\) / dtP{6A,D(At,F?-2)). G.130)
Jo
The upper index of the polynomial Q\n_1{8A,A,F) indicates that it is
linear in 6A whereas the lower index denotes the form degree. This is the
important expression for Q^n-i an(^ from experience with equation G.115)
of the preceding section we can already infer that Q\n_i will turn out as
the anomaly in Bn - 2) dimensions!
Formula: Relations G.129) and G.130) can be re-expressed in a very
compact way. Starting with
~l) [
пР^Л,^"-1) = п
JO
f1 d
= n(n-l)/ dtPFA,—Ft,F?~2), G.131)
Jo dt
inserting
jtFt = F + Bt - 1)Л2 = <*Л + [Ле, А] = DtA G.132)
we get
Г1
nP{6A,Fn-1) = n(n-l) dtPFA,Dt(A,F?-2)) G.133)
Jo
by virtue of the Bianchi identity DtFt = 0. Using formula G.20) again for
the derivative of invariant polynomials
dPFA, A,FP~2) = P(DtFA),A,F?-2) - PFA,Dt(A,Fr2)) G.134)
gives
nP{6A, F"-1) = n(n -1) j dt{P(DtFA),A, F?~2) - dP{6A, A, F?~7)\.
Jo
G.135)
Now we insert expression G.135) into relation G.129)

Chern-Sirnons form, variations and anomaly 341
- Q2n_1(A,F)
/•l
= n(n-l)/ dt[P{{d6A + [tA,6A]),A,F?-2)
Jo
?-2)
+ (t-l)dPFA,A,F?-2)} G.136)
and we obtain the formula of Alvarez-Gaume and Ginsparg [Alvarez-
Gaume, Ginsparg 1985]
Q2n-i(A + 6A,F + 6F) - Q2n_i(A F)
/•l
= n(n-l)/ dt(l-t)P{6A,d{A,F?-2))
Jo
/
Jo
?-2)
Г1
Jo
dttP{D{6A),A,F?-2). G.137)
Specializing further to the case where 6F = DFA) = 0 provides the for-
formula
/•l
Ql^SA, A, F) = n(n - 1) / dt(l - t)PFA, d(A, F?~2)). G.138)
Jo
Anomaly: This expression coincides with the formula for the anomaly
originally discovered by Zumino [Zumino 1984,1985a] when 6 A is shifted by
t,he Faddeev-Popov ghost v (see Chapter 9, equation (9.122)). Substituting
the 1-form 6A by the 0-form v (which anticommutes with any odd-form, see
Chapter 8) we obtain an anomaly formula
i,A,FN^vQln_2(v,A,F)
Г1
= n(n-l)/ dt{l-t)P(v,d{A,F?~2)). G.139)
Jo
For n = 2 and n = 3
Q? = trudA G.140)
Qi = tr vd{AdA + ^A3) G.141)
wo recover the non-Abelian anomaly in two and four dimensions up to the
normalization.
The polynomial Q2n-2 is one °f a whole series of terms <22n-i-fc> ^ =
0,1,..., 2n -1, which are linked together by a set of equations—the chain of
descent equations—and is therefore called the chain term. We will discuss
t his topic in Chapter 9.

8
Consistent anomaly
In Chapter 4 of this book we introduced the anomaly as the breakdown of a
classical conservation law. Now we want to investigate the anomaly further
in the light of gauge transformations.
We first introduce an infinitesimal gauge operator, generalize to a BRS
operator and find its representation in the functional space of the gauge
potentials and Faddeev-Popov ghosts (Section 8.1). Next we discuss the an-
anomalous Ward identity in terms of functional derivatives and subsequently
we derive the equation which determines the anomaly—the Wess-Zumino
consistency condition—in the gauge transformation variant and in the com-
compact BRS variant (Section 8.2). Different aspects of the anomaly equation,
like the algebra-, the cocycle- and the cohomology aspect, are presented at
the end (Section 8.3).
8.1 Infinitesimal gauge operator, BRS and geometry
For the calculation of physical processes we need to know the Green functi-
functions of the theory which follow from the generating functional (see Chapter 3;
here we ignore the sources)
= J ##exp[- J с1хС(А»,ф,ф% (8-1)
with
С = фг0ф, 0 = 7"(dM + ЛМ). (8.2)
For the anomaly we only consider the fermionic term in the Lagrangian C,
and we treat only the massless fermion fields as dynamical variables. The
Yang-Mills gauge fields AM are regarded as external fields. Therefore we
just integrate over the fermion fields and the functional depends on A^.
Now we search for an infinitesimal gauge variation of the W-functional
in terms of a differential operator. Following Zumino [Zumino 1984] we find
its representation in the functional space of the gauge fields Л" and the
Faddeev-Popov ghost fields va.

Infinitesimal gauge operator, BRS and geometry
343
Infinitesimal gauge operator: Let us start with the variation of the
W-functional with respect to Aa^
= J
We vary the gauge fields Aa a la BRS (recall Section 3.6)
8Aau = Dbavb,
with the covariant derivative
ba i rbac ac
= djba + fbacA
¦> — ^м
(8.3)
(8.4)
(8.5)
as defined in equation C.306), introducing the Faddeev-Popov ghost vb.
After integrating by parts the variation (8.3) turns into
(8.6)
Definition: We define a gauge operator represented by a functional
differential operator in the functional space of the gauge fields A^
-Xb(x) := Df{x)
(8.7)
Theorem: The gauge operators obey the commutation relation
\Xa(x),Xb(y)} = fabcXc(xN(x - y). (8.8)
Proof.
[Xa(x),Xa(y)} = Xa(x)X~a(y) - X'a{y)Xa{x) (8.9)
du
6Ab,
6A«
6A$
facbдс "
Д 6А\
racb лс
6Ab
¦ гасЬлс "
facb ас
-ГЪА%
6
6A-
facb /
J *
Iе s
Applying the functional derivatives, noting that all second order derivatives
iiuicol, we get

344 Consistent anomaly
¦ '
fadlfacdN(x - у)А1(х)-щ-. (8.10)
For the first line we need the relation (C/M(x) denotes some functional of x)
d?6(x - yW(y) + d?6(x - y)UHx) = 6(x - y)d*U»(x), (8.11)
which follows from partial integration and by virtue of
д?6(х-у)+д»6(х-у) = 0. (8.12)
For the second line we use the Jacobi identity for the structure constants
iacb ibab ¦ radb racd ¦ taad rdbc __ q /q i o\
Then expression (8.10) results in
\Xa(x),Xu(y)} = ~Гаа6(х-у)\д^м + /МсА1(х^
= faudXd(xN(x-y). Q.E.D. (8.14)
Definition: We define the integrated gauge operator by
v-X := f dxvb(x)Xb(x). (8.15)
Then we rewrite the gauge variation (8.6)
eWlAfj] = f dxvb(x)Xb(x)W{Afl] = v-X W[A»\. (8.16)
BRS: In the next step we also include the functional space of the Faddeev-
Popov ghosts va. The variation of a functional with respect to va is
6W[v] = j dx6va(x)-^~^W{v]. (8.17)
We again vary the FP fields va a la BRS (recall Section 3.6)

Infinitesimal gauge operator, BRS and geometry 345
Sva = -^[v,v]a (8.18)
so that equation (8.17) turns into
6W[v] = -
W[v)
2, „ 6ya W[v]. (8.19)
f ь1
= -Jdxv-
Definitions: We define the following diflferential FP operator
уЬ/~л fbca ci \ /o nr\\
— v\ ) '— ~nI \ )~c /—\" \o.?\j)
and the integrated FP operator
v-Xv:= f dxvb(x)Xbv(x). (8.21)
Then we re-express the FP variation (8.19) by
SW[v] = I dxvb(x)Xl(x)W\v] = v ¦ XvW[v]. (8.22)
Definition: We define a generalized gauge operator—the BRS operator
v-Xv iorWlvf ' (8-23)
wliore we know its functional representation in the gauge fields A^ and FP
«hosts va. We add the index v to indicate that its action creates a FP ghost;
Л,, equals v • X when acting on a functional of gauge fields W[^M] and it
equals v ¦ Xv when it operates on FP ghosts W\v].
Proposition:
• The operator 6V satisfies the BRS transformations
6vv = ~^[v,v] = —v2
6VA = -Dv
6VF = [F,v], (8.24)
together with
Si = d2 = 0 , 6vd + d6v = 0. (8.25)

346 Consistent anomaly
Proof. i)
8vv{x) = v ¦ Xvv(x) = - J dyvb(y)^fbcdvc(y)-~^va(x)Ta
= -\fbcavb(x)vc(x)Ta =-l-\v,v](x). (8.26)
integrating by parts gives
6VA = 6»Ац(х)ёх1> = Db?vb{x)Tadxti = -Dv. (8.27)
iii) The third equation of (8.24) is a consequence of the second. Of course,
one can demonstrate explicitly the action of the functional derivatives,
iv) Applying 6V twice to a functional of the gauge fields W[^M] gives
<Wt> = \ v \ v • X — v • Xv • X + v ¦ Xvv • X
{ v-Xv J
= vXvX-^( dxfacbva (x)vc{x)Xb(x)
= -[v,v]-X [v,v]-X — O, (8.28)
where we have used the integrated version of algebra (8.8)
2 v ¦ Xv ¦ X = [v, v] ¦ X. (8.29)
v) Repeated application to a functional of FP ghosts W[v]
6V6V = { V' у \vXv = vXvvXv (8.30)
= - / dxfbcafaeuvbvcv6 —- = 0. Q.E.D. (8.31)
2 / Si)a
J x
The last line vanishes because of the identity
Remark: The FP operator X%(x) (8.20) obeys a commutation relation
with a factor \ on the right-hand side implying the nilpotency of 6V

Infinitesimal gauge operator, BRS and geometry 347
W(x), Xbv(y)] = \jahcXcv{xN{x - y). (8.33)
Functional variation and geometry: We want to equip the functional
variation with a differential geometric set-up intended to interpret the BRS
transformations geometrically and to shed more light on Zumino's geometric
approach of Section 9.3.
Definitions:
Sp A ... The affine space of all gauge connections on
M — S2n. It does not correspond to the physical
configuration space.
Q... The gauge group space, the space of all gauge ele-
elements Q = {g(x)}.
Sp A/Q... The moduli space, where we have identified all
gauge connections which are linked together by
a gauge transformation. It represents the physical
space of the gauge theory.
Whereas Sp A is contractible, hence topologically trivial, the moduli
space Sp A/Q is not. Each gauge connected component of Sp A/Q is clas-
classified by the homotopy of the group; for example by П1(СA)) = Z—Dirac
inonopole bundle (see Section 6.4), by П3EС/B)) = Z—instanton bundle
(sen Section 6.6), or by I15(SU(N)) = Z with TV > 3—determinant bundle
(sec Section 11.5.3).
For a differential geometric setting we recall the functional derivatives
with respect to -4?(x) and define:
< > as a basis for the tangent space TYSp A) in Sp A
LM»(x)J
х)} as a basis for the cotangent space T*(Sp A). (8.34)
Moth spaces are dual to each other via the inner product
Л wedge product is denned by
6Al(xNAl(y) = бЛЦх) Л 6At(y) :=
6A%(x) ® 6Al(y) - 6Abv(y) ® 6A%{x), (8.36)
mill, analogous to the ordinary exterior derivative B.103), we also introduce

348
Consistent anomaly
Spoof
П
A A+SA
Fig. 8.1. Variations in Sp A
an exterior derivative in functional space Sp A
= / dx
6A°Jx)
6AaJx).
A general vector field V on Sp A is given by
and we describe a p-form В on Sp A as
1 /"
— I I axl • • • axP?>ai...ap °^^i Kxl) ¦ ¦ ¦ °-/ifip\J'P)>
where the 6A^. products are wedged.
(8.37)
(8.38)
(8.39)
Note; Here the group indices Oj represent the contravariance (or cova-
riance) in the forms on Sp A and we raise and lower them for this reason.
The Lorentz indices /i< express the tensor structure on the space-time ma-
manifold M.
Bundle: We construct a fibre bundle, a principal bundle with the spaces
Sp A and Sp A/G whose fibre is Q. Space A has the projection П : Sp A ->
Sp A/G, see Figure 8.1 (recall Section 3.5.2; for further literature see [Na-
kahara 1990], [Cotta-Ramusino, Reina 1984], [Falqui, Reina 1985]).
Choosing a gauge potential A(x) = Аа^(х)йх^Та as a representative
of the equivalence class [A] € Sp A/G we consider the gauge transformed
configuration

Infinitesimal gauge operator, BRS and geometry 349
A(x,g) = g'\x)[A{x) + d]g{x) € Sp A (8.40)
being defined on M x Q.
We apply the exterior derivative (8.37)—we now use a bar over the letter
for reasons which become clear below—representing a functional variation
in Sp A denned on M x Sp A (see Figure 8.1) and we obtain
6Л = 6g~1(A + d)g + g~16Ag-g~1A6g + g1
= -V{g-4g)+g-l6Ag. (8.41)
T> denotes the familiar covariant derivative
V = d+[A,]. (8.42)
The variation splits into two parts. The first part represents the deriva-
derivative along the fibre Q, whereas the second term corresponds to the derivative
along the moduli space A/Q, see Figure 8.1.
Comparison: The comparison with Zumino's geometrical approach
shows that the derivative operator 6 of Section 9.3 corresponds to restricting
I lie variation 6 in Sp A along the fibre Q.
We reproduce the structure equations (9.68)
(8.43)
where vmc is the Maurer-Cartan form on group space
vMC=g-16\{.bieg = g-16g (8.44)
mid the equation (recall Section 2.6.6)
= -g~16gg~16g = -v2MC (8.45)
in (tilled the Maurer—Cartan structure equation in group space.
'lVacing back the variation along the fibre to the group parameters of
S.vi.ion 9.3
g(x)-+д(х,0), А(х,д)-*А(х,в), 6 - -^dda (8.46)
we recover
^ (8.47)
vMC = д-'бд = g-1(x,e)-^g(x,9)dea (8.48)

350 Consistent anomaly
^- (8-49)
Note that d6adxfl is a wedge product.
Identification: Now we turn to the geometric interpretation of the ob-
objects used in field theory, the FP ghost vpp and the BRS operator s, or the
operator 6V. They obey an equation, the first of equations (8.24), which is
of the Maurer-Cartan type, (8.45), (9.62). Therefore it is quite natural to
identify the FP ghost with the Maurer-Cartan form (MC) on the group
vFP = vaTa <—> vMc = 9~lbg = vadea. (8.50)
The degree in the 'ghost number' matches the degree of the frame dda
in group space. Thus the FP ghost has a geometric interpretation which
explains the 'wrong' statistics of this scalar field.
The generalized gauge operator 6V—representing by construction the
BRS operator s of Section 3.6—obeys the same algebra relation (8.24) as
the exterior derivative on the group space, when finally the gauged quantities
in (8.43), (9.68) are evaluated at g(x, в = 0) = 1. Consequently we identify
this operator (8.23) with the group derivate (9.43), with the variation in
Sp Л along the fibre Q
dea
These identifications have been proposed by Leinaas and Olaussen [Lein-
aas, Olaussen 1982], Zumino [Zumino 1984], and Stora [Stora 1977]. A ri-
rigorous treatment within fibre bundles and cohomology has been presented
by Bonora and Cotta-Ramusino [Bonora, Cotta-Ramusino 1983]. Somewhat
deviating aspects can be found in [Thierry-Mieg 1980] and [Quiros, de Ur-
ries, Hoyos, Mazon, Rodriguez 1981].
In this book we frequently use the FP ghost synonymously with the MC
form on the group and the extended gauge operator 6V, or the BRS operator
s synonymously with the exterior derivative on the group space.
8.2 Wess-Zumino consistency condition
Again, we start from the generating functional (8.1) where the dynamical
massless fermions are interacting with the external gauge fields. The fields
are chosen as in Section 4.1 either left (L)- or right (i?)-handed but we avoid
labelling the field symbols by L, or R. We assume the theory renormalized in
the one-loop approximation and consider the functional W[j4m] of the one-
particle irreducible Green functions. This is what we need for the discussion
of an anomaly which is determined by loops like the ones of Section 4.8.

Wess-Zumino consistency condition 351
Anomalous Ward identity: Let us perform an infinitesimal gauge trans-
transformation as in equation (8.23)
A» -* An + SvA» (8.52)
then on a classical level the action remains invariant
SySdassiA^) = 6V I dxC(A^, ф, ф) = 0. (8.53)
If we still have gauge invariance on a quantum level—a Ward identity
XaW[AIJ] = 0 , <W[AM] = 0 (8.54)
(considering the differential and the integrated version) the gauge theory
remains renormalizable. If, however, gauge invariance (8.54) is lost we find
the following proposition (recall Section 4.2).
Proposition: Anomalous Ward identity
• The gauge transformed 'quantum action' represents precisely Bar-
Bardeen's non-Abelian anomaly
XaW\A»\ = G44J
6„^[АМ] = G(v,A) - f dxva{x)Ga[All}(x). (8.55)
Proof.
= / d#drl> р>ЧТьфexp[-
(8-56)
with
(8.57)
limn
. Q.E.D. (8.58)
In the last step we used Bardeen's definition of the non-Abelian anomaly
(Section 4.8, equation D.387)) and the fact that G°[j4m] depends only on

352 Consistent anomaly
the gauge fields AM so that the average over the fermion fields ф, ф drops.
Of course, the 'quantum action' W[j4m] is a highly nonlocal and non-
polynomial functional in A. However, its gauge variation—the anomaly
G(v,A)—turns out to be local and polynomial in A (integrated over the
manifold M).
Explicitly, we use Bardeen's result in terms of differential forms (equa-
(equation F.79) of Section 6.3)
G(v, A) = ^lJM tr vd(AdA + ~A3), (8.59)
which is, on the other hand, also the solution of the chain term Q\ (see
equations G.141) and (9.102)), so that we have:
Proposition:
• The integrated Q\ chain term equals the anomaly apart from the
normalization
f
m=s*
We discuss this result later on.
Wess—Zumino consistency condition: Previously we defined the
anomaly by the infinitesimal gauge variation of the 'quantum action'; now
we find a kind of differential equation which the anomaly has to obey, and
we derive it in several versions. One follows straight away when we apply
the commutation relations (8.8) of the gauge operator Xй to the 'quantum
action' W[A,j\
(Ха(х)Хь(у) - X\y)Xa(x))W[A^ = fabcXc(x)W[Ali]6(x - y) (8.61)
and use the anomalous Ward identity (8.55).
Theorem: Wess-Zumino consistency condition
Xa{x)Gb[All]{y) - Xb{y)Ga[All\{x) = fabcGc[AJ(xN(x - y) (8.62)
([Wess, Zumino 1971]). It describes a requirement which a possible anomaly
has to fulfil. There exists, of course, a trivial solution
(х), (8.63)

Wess-Zumino consistency condition 353
where G[AM] denotes a local polynomial in AM and its derivatives.
The anomaly condition (8.62) has now been solved in full generality
[Becchi, Rouet, Stora 1976a], [Brandt, Dragon, Kreuzer 1989, 1990a,b,c] and
it determines the anomaly so that, in reverse, one can use it as a definition
for the anomaly.
Definition: Any solution of the Wess-Zumino (WZ) consistency condi-
condition which is not trivial—is not a gauge variation of a local functional in
the basic fields—represents an anomaly.
Note: If we know the first term of the Bardeen anomaly (8.59), which fol-
follows quickly from the triangle graph calculation (see Chapter 4), the WZ
consistency condition (8.62) completely determines the remaining second
term of the anomaly (8.59).
For the integrated version of condition (8.62) we first work with ordinary
gauge elements.
Let be
va(x) = vaa{x)Ta (8.64)
an ordinary gauge element, v% a commuting scalar field. We mark such
gauge elements by an additional index below: a,C= 1,2,....
The corresponding infinitesimal gauge operator
6Va = J dxvaa{x)Xa{x) =va-X, (8.65)
with Xa denned by equation (8.7) generates ordinary gauge transfor-
transformations. The gauge operators themselves satisfy the group commutation
relations—the integrated version (with vi(x)v2(y)) of Theorem (8.8).
Theorem: Gauge commutator
[SVl,SV2] = 6[VliV2]. (8.66)
Applying these commutation relations to И^[АМ]
f>VlbV2W[A»\ - 6V26V1 W[A^\ = 6[vuV2]W[A^ (8.67)
¦Hid using the anomalous Ward identity
bVnW[A»\ = G{va,A), (8.68)
we arrive at:

354 Consistent anomaly
Theorem: Wess—Zumino consistency condition
6vlG(v2,A) - 6V2G(Vl,A) = G([vuv2},A). (8.69)
In this connection 6Va is also called the Ward operator in the literature.
Of course, we could integrate directly the local form of WZ (8.62) with
vi(x)v2(y) and arrive at the integrated version (8.69).
Descent equation: Now we follow Stora [Stora 1977, 1984] and Zumino
[Zumino 1984], [Zumino, Wu, Zee 1984] and construct a descent equation
involving the ordinary gauge elements.
Starting with the polynomial
P(Fn) = tr F71, (8.70)
it is closed and gauge invariant
dtrFn = 0 (8.71)
6Va tr Fn = 0 (8.72)
by virtue of the Bianchi identity DF = 0 and the gauge commutator for F.
According to Poincare's lemma (recall Section 7.3) the closure (8.71)
implies the existence of a Bn — l)-form Q2n-i sucn that locally
1. (8.73)
Rewriting gauge invariance (8.72)
bvJQln-i^db^Ql^^Q (8.74)
(note that d and 6Va commute) Poincare again assures a Bn-2)-form Q\n-2
such that locally
t>vaQ%n-№) = dQ\n_2{va,A). (8.75)
Equation (8.75) is called a descent equation. It relates the Chern-Simons
form Q2n-i m B" - 1) dimensions to the term Q2n_2, the anomaly in
Bn - 2) dimensions (the upper index denotes the power of va). It is one of
a whole set of equations (see Chapter 9).
Proposition:
• The chain term Q2n_2 represents the anomaly in Bn - 2) dimensions
G(va,A)=[dxvaa(x)Ga[Atl}(x) = N[ Ql2n_2{va,A). (8.76)

Wess-Zumino consistency condition 355
We quote the normalization N later on (e.g. in Chapters 9, 11).
Proof. We prove that expression (8.76) obeys the WZ consistency condi-
condition (8.69).
We suppose that we can extend the gauge potentials from our chosen
Bn - 2)-dimensional sphere S2n~2 to a Bn - l)-dimensional ball B2n~1
whose boundary is the sphere dB2n~1 = S2n~2.
Consider now the following functional
U[A] = N [ Q°2n-i(A), (8.77)
its gauge variation is
SVaU[A] = n[ 6„„䧄_1(А) = Лг/' dQ\n_2{va,A)
= G(va,A) (8.78)
S2T.-2
by virtue of equation (8.75) and Stokes' theorem B.128). Then
6vlG(v2,A)-6V2G(vuA) =
= 6V16V2U[A] - 6Vt6VlU[A] = [bvl,8V2)U[A\
2],A). Q.E.D. (8.79)
BRS: Working finally with the anticommuting FP ghosts v(x)v(y) the
integrated version of condition (8.62) becomes
2v-Xv-G = [v,v]-G, (8.80)
which we use for calculating
6vG(v,A) = 6V f dxva(x)Ga[All}(x)
= fdxv-Xvva{x)Ga[All}(x) + v-XvG
= -l[v,v].G + ±[v,v]-G = 0. (8.81)
Theorem: Wess—Zumino consistency condition
6vG(v,A) = 0. (8.82)

356 Consistent anomaly
This is the most compact and elegant form of the WZ consistency condition
and was found by Stora [Stora 1977, 1984] and Zumino [Zumino 1984].
Of course, condition (8.82) follows immediately from the anomalous
Ward identity (8.55) since 6V represents the nilpotent BRS operator
6vG(v, A) = 61WIA,,} = 0. (8.83)
For the same argument the expression
(8-84)
with G[A] a local polynomial in A is just a trivial solution. Such a local
functional G[A] corresponds to an additional term in the 'quantum action',
which just redefines the regularization scheme and does not alter the an-
anomalous content of the gauge theory (recall our discussion in Section 4.2).
8.3 Algebra, cocycles and cohomology
8.3.1 Faddeev-Popov ghosts and gauge elements
Let us consider the two formalisms discussed so far:
FP ghost formalism gauge element formalism
va FP ghost, anticommuting v% gauge element, commuting
6V BRS operator 6Va ordinary gauge operator
6% — 0 nilpotency [^d<5u2] = ^[vuv2] commutation relation
6vG(v,A) = 0 6VlG(v2,A) - 6V2G(vi,A) = G([vuv2],A)
WZ consistency condition WZ consistency condition
As noted by [Becchi, Rouet, Stora 1976c] and [Alvarez-Gaume, Ginsparg
1985] both formalisms are totally equivalent to each other. Either we can
work with the FP ghost v and with the BRS operator 6V or with the ordinary-
gauge element va and its infinitesimal gauge operator 6Va. Then the role of
the commutation relation for the gauge operator SVa is taken over by the
nilpotency of the BRS operator 6V and the WZ consistency condition on the
right-hand side is given by the concise form on the left-hand side. We can
transfer the gauge element formalism directly into the FP ghost formalism.
How does it work?
We can construct a FP ghost, which we have already identified with the
MC form in group space, by choosing two ordinary gauge functions v\, v%
(see Sections 8.1 and 9.3)

Algebra, cocycles and cohomology 357
v = vide1 + v2d62 = vadea, (8.85)
together with
уа = д-1{х,в)^д{х,в). (8.86)
The BRS operator 6V (8.23), which we have identified with the group deri-
derivation 6 (9.43), creates the BRS transformations (8.24). For instance,
6VA = -Dv = deaDva = dda6Va A (8.87)
implies that
6v = dda6Va. (8.88)
Then we get for the anomalous Ward identity, multiplying equation
(8.68) with dea, the result (8.55) in the FP scheme
f
JM
vaaGa[A]
= f vaGa[A] = G{v,A). (8.89)
Jm
But to achieve in equation (8.88) the full significance of the BRS operator
6V the operator 6Va also has to act upon the gauge functions va. The ordinary
behaviour of the gauge operator 6Va is given up; 6Va must reproduce the BRS
transformation (8.24)
6vv = -v2. (8.90)
Inserting the constructions (8.85) and (8.88)
de1de2FVlv2-6V2v1) = -de1d82[v1,v2] (8.91)
then leads to the BRS-like transformation
-[vi,v2]. (8.92)
In this way we have extended the action of the gauge operator into a
functional space of gauge fields A and gauge elements va. As in Section 8.1
we find its representation
va ¦ X for
va-Xv for W[va] '
where the operator Xv (8.20), is rewritten in terms of gauge elements vp.

358 Consistent anomaly
Finally we can convert the WZ consistency condition (8.69) into its
concise form (8.82). We multiply equation (8.69) with d61d62 and find
v%6VlGa[A) - J v16V2Ga[A] - J[Vl,v2}aGa[A}} = 0
va06VaGa[A} + jFVava0)Ga[A) = 0
- fva6vGa[A}+ fFvva)Ga[A] = 0
6vG(v,A) = 6v fvaGa[A] = 0.
Q.E.D. (8.94)
Of course, when we extend the gauge operator 6Va a la BRS this concise form
of the WZ consistency condition follows immediately from the anomalous
Ward identity (8.89) by virtue of 62 = dBadep8vJVp = 0.
8.3.2 Algebra
The above geometric detour, by identifying the FP ghost with the MC form
on group space, is certainly not necessary to show the equivalence between
the two representations of the WZ consistency condition. Stora's view in his
GIFT lectures [Stora 1986] is purely algebraic.
Stora's view: Stora introduces—in the sense of vector spaces—a dual
algebra *Lie G to the Lie algebra Lie G of the gauge group G. Defining
{va} as a basis for Lie G
{va} as a dual basis for *Lie G (8.95)
both spaces are dual to each other via the inner product
(va,vb)=6ab. (8.96)
Stora then regards the FP ghost v as the Lie algebra valued generator
of the dual *Lie G
v = vava. (8.97)
Whereas the ordinary Lie algebra generators satisfy the usual commuta-
commutation relations
[Va,Vb]=fabCVc (8.98)
the duals obey the BRS transformation
Sva = -\fabcvbvc. (8.99)

Algebra, cocycles and cohomology 359
(For obvious reasons here we raise and lower the indices of the structure
constants in our notation.) Multiplying equation (8.99) by va gives
6v = -hv,v]. (8.100)
(Recall our MC discussion in Section 2.6.6.)
Remembering the familiar 2-form B.111), evaluated on the tangent space
T(M), dxi*dxv(X,Y) = X^YV - УХ", we can calculate a 'Faddeev-
Popov ghost 2-form' vavh analogously
vavb(Z,ri)=evb-rf? (8.101)
on the vector fields (gauge elements)
?,TieUeG; ? = ?<4, V = Vava, (8.102)
where {va} is the basis for Lie G in the spirit of Stora.
WZ consistency condition: Now we are prepared to reformulate the
WZ consistency condition. We start with representation (8.82)
6G(v, A) = j dx8{va{x)Ga\All}(x)} = 0,
where 6 (we drop the index v) means a variation of the fields a la BRS
jdxdy
We integrate by parts and evaluate the 'FP ghost 2-form' va(x)vb(y) on the
vector fields ?, ц
dxdyva(x)vc(y)(Z,V)

360 Consistent anomaly
1
= 0. (8.105)
Recalling the ordinary gauge operator (8.65)
VX, (8.106)
with its commutation relation (8.66) provides the WZ consistency condi-
condition for the gauge elements ?, r] straight away in accordance with equation
(8.69)
6G(v, A)(?, r,) = «fiGfa, A) - 8nG{?, A) - G([?, Ч], А) = 0. (8.107)
8.3.3 Cocycle
We can also consider the WZ consistency condition from a geometric point
of view leading to a cocycle condition.
We return to the space of all gauge potentials Sp Л (Section 8.1). We
denote
I c . , . 1 as the basis for T(Sp A)
\6Afa)}
{6Al(x)} as the basis for T*(Sp A), (8.108)
and introduce vector fields (8.38) and p-forms (8.39). For example, the fun-
fundamental vector field is described by
ix)-—^ (8.109)
and corresponds, in our previous notations (8.65), (8.106), to the gauge
operator 6$
Х(:=?-Х = 6(:. (8.110)
The exterior derivative in Sp A is given by
xsWNA^x)- (8ЛП)

Algebra, cocycles and cohomology 361
Lie derivative: Let us consider the Lie derivative in Sp A. What does it
mean? We calculate its action on a functional /[Лм] along a curve which is
produced by the fundamental vector field (8.109) (recall Section 2.6)
ш t=0
1
1
(8.112)
Thus the Lie derivative in Sp A represents the usual directional derivative
6/6A^(x) along (DIM^)a(x). It gives the fundamental vector field X$ or the
familiar gauge operator 6$ (8.65), (8.106), CXi —> X^,6^.
Analogous to the ordinary space-time manifold M we also have here in
Sp A the formula
(8.ИЗ)
where z;q denotes the interior product of a form on Sp A with the fun-
fundamental vector field X$.
Then we can express the anomalous Ward identity (8.68) geometri-
geometrically by
CX(W[All]=G(t,A). (8.114)
Cocycle condition: Now let us define some 1-form on Sp A
G = JdxGZ[A}(xNAl(x), (8.115)
with G%[A] some functional of the gauge potential A. The inner product
with a vector field X gives a 0-form on Sp A
ixG = G(X) = У dxG?[A](x)Xl(x). (8.116)
Analogous to differential forms on M we can derive here the following
formula for the above 1-form on Sp A-

362 Consistent anomaly
Lemma:
6G{X,Y) = XG{Y)-YG(X)-G{{X,Y})
= CXG(Y)-CYG(X)-G([X,Y\), (8.117)
iyixSG = CxiYG-CYixG-i[xyY]G. (8.118)
If the form is closed, 8G = 0, it is also called a cocycle in the language of
cohomology (Section 2.5.2).
Next we choose G to be the variation—the exterior derivative in Sp A—
of the vacuum functional
. (8.119)
Then the left-hand side of formula (8.117), (8.118) vanishes
= ix4ixFG = 0 (8.120)
and we find the WZ consistency condition written geometrically in
Sp A
CX(G(r), A) - ?XvG(Z, A) - G([?, т)],А)= 0. (8.121)
Proposition:
• The WZ consistency condition corresponds to a cocycle condition in
SpA!
Remark:
i) The logarithm in equation (8.119), however, may not be globally well
denned on Sp A. As noticed by Falqui and Reina [Falqui, Reina 1985]
if there is an anomaly the functional Z[A,j] vanishes at some gauge
orbits in Sp A- Excluding these points from the domain of Z\A^\ the
above functional (8.119) is denned. But then the 1-form G (8.119) is
just closed (and not exact).
ii) Furthermore the 1-form G (8.119) depends nonJocaJJy on the gauge
potential A (since the 'quantum action' W^A^] is a nonlocal functio-
functional). Only if the derivative is restricted along the fundamental vector
field—along the fibre Q in Figure 8.1—
6^гхF = 6^ (8.122)
the functional emerges locally in the fields A and represents the non-
Abelian anomaly. We shall discuss this point further in Chapter 10.

Algebra, cocycles and cohomology 363
Notation: Let us finally list the several aspects and notations:
G = «W[j4m]
^ iG iSWlA] variation along fibre G, 0-form in Sp A
= ^W[^M] ordinary gauge operator
= -X$W[.AM] fundamental vector field in Sp A
— Cx^WlAfj] Lie derivative in Sp A
= G(?, A) non-Abelian consistent anomaly.
8.3.4 Cohomology
The above considerations have been generalized mathematically by several
authors [Bonora, Cotta-Ramusino 1983, 1986], [Dubois-Violette, Talon, Vi-
allet 1985a,b], [Stora 1986], [Kastler, Stora 1986a,b], [Troster 1994]. A prin-
principal fibre bundle P(M, G) is constructed with M the (compact) space-time
manifold, G the gauge group (compact Lie group) and Lie G the Lie algebra
of G. Again, we consider Sp A, the space of all connections in P(M, G), and
we denote the space of all local functionals on Sp A by rioc(Sp A). It is a
representation space for G and Lie G. The representation of the Lie G in
rioC(Sp -4) is achieved by the Lie derivative
--Xzf[A], (8.123)
t=o
with ? € Lie G and f[A] € rioc(Sp A).
Definitions: Recalling the cohomology of Section 2.5.2 we define the vec-
vector space Ck(Lie G, rioc(Sp A)) of all A;-cochains of Lie G with values in
rioc(Sp A) by considering the space of all multilinear skew-symmetric maps
(A;-forms on Lie G)
fk[A] : Lie G x ... x Lie G -+ Tioc(Sp A). (8.124)
к factors
Then we get a coboundary operator
6 : C*(Lie G,rloc(Sp A)) -> C*+1(Lie G,r,oc(Sp A)) (8.125)
on C*(Lie G,rbc(Sp A)) = 0fcC*(Lie G,rloc(Sp A)) with 62 = 0 which
is denned by

364 Consistent anomaly
fc+1
). (8-126)
The symbol ~ denotes the omission of the corresponding element in Lie G.
As in Section 2.5.2 we specify the space of cocycles in Lie G of degree
/с by
Z*(Lie G,rloc(Sp A)) = {fk\6fk = 0} (8.127)
and the space of coboundaries in Lie G by
Sfe(Lie G,rloc(Sp A)) = {fk\fk = «Л-1.Л-1 € С*}. (8.128)
Since 62 = 0 we have
B*(Lie G,rloc(Sp А)) С Zfe(Lie G,rloc(Sp A)) (8.129)
and we can define the cohomology in Lie G
Hk(Lie G,rloc(Sp A)) = Zk(Ue G,rloc(Sp ^))/Sfe(Lie G,rloc(Sp A)).
(8.130)
The cohomology space of the complex (G*(Lie G,rioc(Sp A)), 6) is called
the cohomology of the Lie algebra of the gauge group.
Physics: The physical meaning of the objects involved here is:
• the coboundary operator 6 represents the BRS operator,
• the cochains fk[A] are local functional in the gauge potential A con-
containing к FP ghosts,
• the elements of the cohomology Hk are integrated anomalous terms
with к FP ghosts.
For example, in the case of к = 1 we obtain
«/iMtfbfc) = CXiJx[A)(b) - СХCМА]&) - /i[i4]([&,6]). (8.131)
For fi[A] € Ях(Ые G,rioc(Sp A)) being an element of the first cohomology
class the cocycle condition
Sfi[A] = 0 (8.132)
or equivalently

Algebra, cocycles and cohomology 365
l3Ami,^]) = 0 (8.133)
represents the WZ consistency condition. The cohomology element
) (8.134)
expresses the integrated physical anomaly. It is determined uniquely mo-
modulo trivial solutions
Gtriv = 6fo[A]. (8.135)
We shall rediscuss these cohomology solutions in Section 9.3 in terms of the
views of [Dubois-Violette, Talon, Viallet 1985a,b].

9
Stora-Zumino chain of descent
equations
Now we come to one of the main chapters of the book. We shall show how
the singlet anomaly in 2n dimensions determines the non-Abelian anomaly
in Bn — 2) dimensions via a set of equations. These are part of a whole chain
of equations, which descend in their form degree, thus called the Stora-
Zumino chain of descent equations [Stora 1977, 1984], [Zumino 1984].
We are going to derive this chain on pure mathematical grounds—algebra
and differential geometry—and discuss its meaning in physics afterwards.
We offer the topological aspect of a chain, described by an index theorem,
later on in Chapter 11.
We begin with Stora who approaches the chain from the physical side by
working with the gauge potential Л, the FP ghost v and the BRS operator
s (Section 9.1). Then we extend the formalism to nontrivial gauge bund-
bundles (Section 9.2) and afterwards present Zumino's pure geometric approach
(Section 9.3). Of course, the two approaches are equivalent. Explicit soluti-
solutions for a whole chain are calculated in Section 9.4 and general formulae for
the chain terms are derived in Section 9.5.
9.1 Stora's approach to the chain
Since we are already familiar with the BRS transformation, the anomalous
Ward identity and the WZ consistency condition, we are prepared to follow
Stora's approach [Stora 1977, 1984].
Let us consider a renormalizable field theory given by the vacuum func-
functional W^-A^], equation (8.1), where the chiral fermions are coupled to the
external Yang-Mills fields Ац. The FP ghosts v are introduced via the BRS
transformations (8.24), then the anomalous Ward identity
sW[AIM}=G(v,A) (9.1)
defines the anomaly which we express by the integral

Stora's approach to the chain 367
G(v,A)= f vaGa[A]. (9.2)
Jm
The 4-form t/*Ga[.A] is linear in v, local in A and v, and denned up to a
derivative of a local form dr). In particular Ga [A] denotes Bardeen's anomaly
result F.79) of Section 6.3.
Next we consider the WZ consistency condition (following from the
anomalous Ward identity (9.1) and s2 = 0)
sG(v, A) = a f vaGa[A] = 0, (9.3)
Jm
the equation which determines the anomaly. We rewrite equation (9.3) in
a local way by using the algebraic lemma of Poincare (recall Section 7.3)
which guarantees that equation (9.3) is satisfied by any 4-form for which
s(vaGa) is exact
s(vaGa[A}) = -dQl(v,A). (9.4)
The term Q2 has to be a polynomial in v and A with FP ghost number 2
and form degree 3. What is the solution of equation (9.4)? It can be found
in a very general way.
Now we perform pure mathematics. We consider the symmetric invariant
polynomial P(Fn) (recall Section 7.1). Since P(Fn) is closed (due to the
Bianchi identity DF = 0) it is locally exact (Poincare lemma, Section 7.3)
P{Fn) = dQ2n-i{A,F). (9.5)
We know the polynomial Q2n-b of form degree Bn — 1), already; it is the
Chern-Simons form which is explicitly given by (recall Section 7.2)
Q2n-i(A,F) = n f dtPi^F?-1), (9.6)
Jo
where
Ft=tF + (t2 - t)A2 (9.7)
(geometrically, we consider a trivial fibre bundle). Equation (9.5) together
with equations (9.6), (9.7)—also called 'transgression'—is the starting
point of the following mathematical operations.
Stora noticed that a shift in both, in the gauge connection and in the
derivate
A —* A = A +v
d -» A = d + s (9.8)

368 Stora-Zumino chain of descent equations
leaves the curvature invariant. Stora nicknamed this the 'Russian formula'
[Stora 1984], [Manes, Stora, Zumino 1985].
Theorem: 'Russian formula'
F(A) = AA + A2 = dA + A2 = F{A). (9.9)
Proof: Theorem (9.9) is valid by virtue of the BRS equations (8.24)
A(A + v) = dA + dv + sA + sv
= dA + A2- {A2 + Av + vA + v2)
= F-{A + vJ. Q.E.D. (9.10)
Of course, the Bianchi identity for the shifted fields is also satisfied
DF = AF+[A,F} = 0 (9.11)
so that we obtain the 'transgression formula' for the Stora shift.
Theorem: 'Shifted transgression'
P(Fn) = AQ2n.1(A,F), (9.12)
with
Г1
Q2n-1{A,F)=n dtP{A,F?~l) and Ft = tF + (t2 - t)^2. (9.13)
Jo
Proof. Let us take some infinitesimal variation of the fields
A^A + 6tA => 6tF = DFtA). (9.14)
Then we calculate
.Р"-1). (9.15)
Considering the homotopies At = tA, Ft = dAt + A2 together with DtFt =
AFt + [At, Ft] = 0, and choosing finally 6t = dt-щ, we find after integration
6tP(F?) =
Г1
P(Fn) = пД / dtP{A,F?~l). Q.E.D. (9.16)
Jo

Stem's approach to the chain 369
Applying the 'Russian formula' (9.9) now we can equate the 'transgres-
'transgression' (9.5) with its shifted version (9.12)
AQ2n_1(A + v,F) = dQ2n-i(A,F). (9.17)
We expand the Chern-Simons form in powers of the FP ghost v
+Q22n_3(v, A,F) + ... + Q20n-4v), (9-18)
where the upper index denotes the powers of v and the lower index the form
degree. We insert this expansion into formula (9.17)
(d + s)Q°2n_1 + (d + s)Q\n_2 + (d + s)QL-3
+ ... + (d + s)Q20n-1=dQ°2n_1;
we compare the terms of the same form degree and same power in v then
we obtain the following set of equations:
Stora-Zumino chain of descent equations:
P(Fn) - dQl^, = 0
SQ°n_! + dQ\n_2 = 0
lL3 = 0
= о
Comparing now the chain (9.20) with physics we observe that the third
equation represents precisely the local version of the WZ consistency con-
condition (9.4). Therefore we may identify the chain term Q\, or in general
Q\n-2i with the anomaly uaG°[A] (which we found in Chapter 8).
Proposition:
• The chain term Q2n_2 represents the anomaly
G(v,A) = NJ^ Q\n-2{v,A). (9.21)
The normalization

370 Stora-Zumino chain of descent equations
where the factor (±) stands for positive or negative chirality fields (L- or
Я-fields), however, is not fixed by the chain (9.20) and must be calculated
by other methods:
i) perturbation theory, Feynman diagrams (Section 4.8),
ii) Fujikawa path integral formalism (Sections 5.6 and 11.5.2),
iii) topological analysis, Atiyah-Singer index theorem (Section 11.5.3).
For instance, when using perturbation theory it is enough to calculate the
simplest anomalous loop—the triangle diagram in 4 dimensions—which is an
easy task. (Recall that there are no radiative corrections due to the Adler-
Bardeen theorem.) Then we have the normalization and the total anomaly
expression—which would otherwise involve more complicated calculations—
is given by the chain term Q\. This is the practical value of the SZ chain;
we can easily solve all chain terms Qjjn-i-fc- We postpone the discussion of
the higher order chain terms Qjjn-i-fc to Section 9.3.
Remark: The SZ chain starts with some invariant polynomial. However,
which polynomial we have to choose depends on the physical theory (see e.g.
gravitation, Section 12.8). Of course, in general, the invariant polynomial
may represent a linear combination of trace products like
tr Fptr Fq (9.23)
where F = dA + A2, and A describes some connection (e.g. A = A, or some
gauge connection А ф A, or A = и the spin connection of gravitation).
Then the corresponding chain terms are given by
F» + btr Q^
pF" + 6tr^Qig_2(I), (9.24)
with а + b = 1; e.g. а = ^, b = ^.
9.2 Chain for nontrivial gauge bundles
In deriving the chain of descent equations, we have considered up till now
gauge theories with a trivial fibre-bundle structure. But the algebraic con-
construction of the anomaly and of the whole chain can certainly be extended
to a global validity on nontrivial gauge bundles. This generalization has
been worked out by Manes, Stora and Zumino [Manes, Stora, Zumino 1985]
and we briefly summarize their result.
We start with a gauge theory containing a structure group G, a compact
Lie group, and consider a principal bundle P(M, G) (recall Section 2.7). The
base manifold M is as usual of even dimension, compact and without boun-

Chain for nontrivial gauge bundles 371
dary. The connections on P(M,G) are locally represented by 1-forms, the
gauge potentials, with values in Lie G. The transition functions of P(M, G)
are locally given by the gauge elements g(x) on M into G with suitable
gluing properties. In addition to the gauge potential A a fixed background
field Aq will be introduced whenever the bundle P(M, G) is not trivial.
Such a field Aq also occurs in perturbation theory where one perturbes in
the neighbourhood of Aq.
We introduce the BRS transformations (8.24) but we do not transform
the field Ao
sA0 = 0. (9.25)
The anomaly appears via the anomalous Ward identity
sW[A,Ao] = G{v,A,A0) = / vaGa[A,A0], (9.26)
JM
and from the nilpotency s2 = 0 follows the WZ consistency condition
sG(v, A, Ao) = s f vaGa[A, Ao] = 0. (9.27)
Jm
Again, the solution vaGa[A,Ao] is ambiguous; the replacement
vaGa[A, Ao] -» vaGa[A, Ao] + sG[A] + d-q (9.28)
also solves equation (9.27) where G and 7] are local polynomials of the fields.
For deriving the chain we consider now the 'transgression formula'
G.28)-G.30), valid for nontrivial bundles
P(Fn(A)) - P(Fn(A0)) = dQ2n_1(A,A0)
/•i
= nd dtP{A-AQ,F?~l), (9.29)
Jo
with the homotopies
Ft = dAt + A\ (9.30)
and
At = tA + A - t)A0. (9.31)
Shifting the gauge field A and the derivative d a la Stora-Zumino (9.8) and
applying the 'Russian formula' (9.9) to the 'shifted transgression' we obtain
the relation
dQ2n-i(A,Ao) =
= «Д / dtP{A + v- Ao^p-1), (9.32)

372 Stora-Zumino chain of descent equations
with the homotopies
Ft = AAt + % (9.33)
and
At = t(A + v) + (l-t)A0. (9.34)
Expanding the Chern-Simons form as usual in powers of v
2n-l
Q2n-i(A + v, Ao) = J^ QP2n-i-P(v, A, Ao) (9.35)
p=0
the relation (9.32) provides the desired chain of descent equations when
comparing terms of the same order:
Chain of descent equations:
P(F"(A))-P(F"(A0))-dQ!L-i(AA>) = 0
SQp2n_1_p(v,A,A0)+dQp2+1_2_p(v,A,A0) = 0
eQJj"^) = 0 (9.36)
andp = 0, l,...,2n-2.
In particular the chain term Q2n_2(v, A, Ao) solves the WZ consistency-
condition (9.27) and therefore represents the anomaly up to the normaliza-
normalization
NQL-2(v, A, Ao) = vaGa[A, Ao]. (9.37)
The set of equations (9.36) containing a fixed background connection Ao
is globally valid on a nontrivial bundle P(M,G). Only for a trivial bundle
one may choose Ao = 0 and one recovers the usual local chain formulae
which we have considered before.
9.3 Zumino's approach to the chain
In a Yang-Mills gauge theory the element g(x) of a gauge group G is usually
just a function of x, the space-time manifold M. Zumino [Zumino 1984]
extends this space; the gauge element
д = д(х,в) (9.38)
should depend on additional parameters
в = {ва}, (9.39)
variables which form a manifold in a space, the group parameter space
M$ with dimension dim Me = p.

Zumino's approach to the chain
373
G
д(х,в)
g+Sg
M
M
(a)
(b)
Fig. 9.1. a) Gauge group elements on M x Me- b) Gauge group space Q
Definitions: Analogous to the base space M we define:
1 ~яоа f as a basis f°r *be tangent space T{Me) in M#,
{d^a} as a basis for the cotangent space T*(Me). (9.40)
Both spaces are dual to each other via the inner product
and dimT*(Me) = dimT(Me) = di
A wedge product is defined by
deade0 = dea Л dd0 := dda <g> dd0 - dd0 <8» d6a,
(9.42)
and we also introduce an exterior derivative in the group parameter
space Me
8 = —dea with <52 = O, (9.43)
together with the antiderivative rule.
Now, the x-space M with dimension dimM = 2n is enlarged by consi-
considering the Cartesian product
M x Me (9.44)
and again we define wedge products in this enlarged space (9.44), e.g.
..dx^-dO ...dO"" =
= dx Л dx111 Л ... Л dx"a" Л de°" Л d6a* Л ... Л d0V (9.45)

374 Stora-Zumino chain of descent equations
An extended gauge group element defines the map (see Figure 7.1a)
g{x, в): М x Me —> G
i I (9.46)
S2n x Sp „ Q
where we finally work with compactified manifolds. The parameter depen-
dependence of g(x, в) actually means that we also vary g at each point x € M; so
we consider the space of all gauge elements {g(x)} = Q, the gauge group
space (see Figure 9.1b).
An infinitesimal gauge transformation involves an extended gauge ele-
element v given by
v = д-'бд, (9.47)
or, more explicitly,
v = v(x, в) = va{x, в)Aва, (9.48)
with
va(x,e)=g-1(x,e)—g(x,e). (9.49)
Thus v acts as a 1-form in gauge group space Q (or in Me) but as a 0-form
on the base manifold M, the x-space, and again as a 1-form in the product
space M x Q (or in M x Me) (see Figs. 7.1a,b).
Gauge transformations: Let A(x) be some Yang-Mills reference gauge
potential over the base manifold M. We carry out a gauge transformation
over the product space M x Me (or M x Q)
A -^ А9(х,в) = д-\х,в)[А{х) + d + 6}д(х,в) = g-1[A + A}g, (9.50)
where we introduce a new exterior differential operator Д acting in the
product space M x Q (or M x Me)
A = d + 6, with A2 = 0 (9.51)
since d2 = <52 = d6 + 6d = 0.
Considering, on the other hand, just a ^-family of gauge transformed
configurations like
Л(х,в) =g-1(x,e)[A(x) + d}g(x,e) (9.52)
we rewrite the gauge potential A9 (or connection on M x Me) by
A9 = A + v. (9.53)
The corresponding field strengths, or curvatures, are

Zumino's approach to the chain 375
F = dA + A2 = F(A), (9.54)
f = dA + A2 = Jr{A), (9.55)
with the gauge transformation
f = g-1Fg; (9.56)
and
F JU F9 = AA9 + {A9J = F9{A9) (9.57)
= A{A + v) + {A + vJ=:?(A + v), (9.58)
with the gauge transformation
F9 = g~lFg. (9.59)
The gauge transformation formulae (9.56) and (9.59) follow quickly by
explicit calculation.
Structure equations: Next we apply the 6-operator to v, A and T:
6v = б^-Чд) = 6д-Чд, (9.60)
by virtue of 62 = 0, and using
Sg-1g + g-16g = 0: (9.61)
which follows from д~гд = 1, we get
6v = -д-Чдд-Чд = -v2 = ~[v,v]. (9.62)
Kquation (9.62) represents the Maurer—Cartan structure equation in
group space Q and the element v = g~16g is called the Maurer—Cartan
form on group space Q (recall Section 8.1).
The application of 6 on the connection A is
SA = elg-^A + d)g] = ёд-\А + d)g - д^Аёд + g'Hdg. (9.63)
Using equation (9.61) and
dv = -g-ldgg-4g - g~4dg (9.64)
we obtain
6A = -dv - Av-vA= -Vv, (9.65)
wlicrn

376 Stora-Zumino chain of descent equations
T> = d + [A,} (9.66)
is the covariant derivative with respect to the gauge field A.
Clearly, for the curvature T we find
ЬТ = Tv-vT=\T,v\. (9.67)
Summarizing, we gain the following structure equations
6v = -v2
6A = -Vv
ЬТ = [JP>]. (9.68)
We certainly recognize the BRS algebra in equations (9.68) and we iden-
identify, as before in Section 8.1, the MC form on the gauge group with the FP
ghost and the exterior derivative on the group with the BRS operator
vmc = 9-^9 = vadea <—> vFP = vaTa ¦
6 = ^dea <—> s- (9-69)
Chain of descent equations: Now we can derive the chain of descent
equations in an analogous manner as we did before.
We start with the 'transgression'
tr Fri = dQ2n-1{A,f), (9.70)
where
/•1
Q2n-i{A,F) = n dttr AT?~X and Tt = tT + (t2 - t)A2. (9.71)
Jo
We shift the fields and the operator
A^A = A + v , d-*A = d + 6 (9.72)
so that we obtain the 'shifted transgression'
tr^" = AQ2n-i(^ + u,^), (9.73)
where
/•1
Q2n-i{A + v,F) = n / dttr {A + v)??-1 and .Ft = tf + {t2 -t){A + vJ.
Jo
(9.74)
Now using the 'Russian formula'

Zumino 's approach to the chain 377
F{ A) = АЛ + A2 = dA + A2 = ?{A) (9.75)
(notice, this was evaluated before at A —* A, g —> 1; within a fibre-bundle
set-up identity (9.75) is also called the 'horizontality condition', see our
discussion in Section 11.5.3) we find the relation
,-1(A + v,Jr) = dQ2n-i(A,F). (9.76)
We expand the Chern-Simons form in powers of v
... + Q2n~2(v,A) + Q2)n~1(v). (9.77)
The upper index labels the powers of v, the Maurer-Cartan form on group
space—it counts the degree of the form in group space Q (or Me)—and the
lower index indicates the degree of the form on the space-time manifold M.
We insert the expansion (9.77) into relation (9.76), and compare the
terms of the same order in space-time M as well as in group space Q—note
that d increases the form degree in M by one unit and so does 6 in Q—then
we arrive at the Stora-Zumino chain of descent equations
tr^1-<*<&,_! = 0
«Йп-2 + <«Э2п-3 = О
8Qln~l = 0. (9.78)
Before discussing its meaning we observe that the several chain terms
Q2n-i-k are no* determined uniquely. For instance, we can replace
Qln-l -»• Q2n-l+da
C^2rj 2 —* ^&2n 2 * ^^ "t" u/y
(9-79)
and the above tower of descent equations (9.78) remains invariant. These
different choices for the chain terms—when interpreted in terms of physics—
иго equivalent.
Again we emphasize that the above chain of equations (9.78) is only
valid for gauge fields with vanishing topological number—the trivial bundle

378 Stora-Zumino chain of descent equations
Cohomology: What is the meaning of the descent equations? On the
mathematical side there is the Dubois-Violette, Talon and Viallet [Dubois-
Violette, Talon, Viallet 1985a,b] interpretation that chain (9.78) corresponds
to a cohomology problem.
Generally, an anomalous term В is defined by its consistency con-
condition
6B = 0. (9.80)
В is a polynomial in the fields and their derivatives, integrated over the
space-time manifold M. However, solutions like
В = 6B, (9.81)
where В represents a local functional in the fields and their derivatives, are
trivial since <52 = 0. Indeed, for В = G anomalies and В = S Schwinger
terms these trivial solutions can be absorbed by redefining the 'quantum
action'.
For example, in case of the anomaly when we set
= N f Q\n_2 (9.82)
JM
В
1м
the condition eG = 0 implies, by virtue of the algebraic Poincare lemma
(Section 7.3), the descent equation
«?2n-2 = -ЛЙп-з- (9-83)
This means Q2n-2 is a <5-cocycle modulo d.
For the trivial solution G = eG when we write
G
= NJL%n_2 (9.84)
JM
Poincare again ensures the existence of a polynomial L2n_3 such that
Qln-2 = SL°2n_2 + dLln_3. (9.85)
Then Q2n_2 is called a 6-coboundary modulo d. The task is now to
calculate the space of 6-cocycles modulo d quotiented by the 6-coboundaries
modulo d. This is H2n_2(e/d), the 6-cohomology modulo d in FP ghost
degree 1 and space-time degree Bn - 2) (recall Section 8.3.4).
As another example, Faddeev [Faddeev 1984] discovered that certain
Schwinger terms obey an analogous cohomology problem
В -» 5 = N I Q22n^. (9.86)

Zumino's approach to the chain 379
The consistency condition 6S = 0 implies the next descent equation
leading to the cohomology -Нг
Dubois-Violette, Talon and Viallet have calculated all cohomologies
H2n_1_kF/d) (k — 0,1,..., In -1)—all possible anomalous terms—for the
BRS algebra of differential forms which are generated by exterior products
of A, v and their differentials d, 6. (For further literature see [Kastler, Stora
1986a,b].)
Physics: What about physics? There the research of the last years has
come up with fascinating results. For the moment we just summarize some
results which we explain further in this book:
• 6, v
The derivative 6 on group space is interpreted as the BRS opera-
operator. The Maurer-Cartan form v is identified with the Faddeev-Popov
ghost field (Chapter 8).
• tr .F1
The trace of the curvatures J- expresses the singlet anomaly in 2n
dimensions. The normalization, found first in perturbation theory
(Chapter 4), can also be calculated by Fujikawa's path integral method
(Chapter 5) or by the index theorem (Chapter 11).
• QL-i
The Chern-Simons form serves as a term to redefine the vacuum func-
functional, the 'quantum action', leading to so-called topological field theo-
theories, specifically to the Chern-Simons theories (for literature see e.g.
[Deser, Jackiw, Templeton 1984], [Dunne, Jackiw, Trugenberger 1989],
[Witten 1989a,b]).
• QL-2
The chain term to order v represents the non-Abelian anomaly in Bn—
2) dimensions. It is the consistent anomaly obeying the Wess-Zumino
consistency condition (Chapter 8). Again, besides perturbation theory
(Chapter 4), Fujikawa's method (Chapter 5) or the Atiyah-Singer
index theorem (Chapter 11) supplies the correct normalization.
• Qln-S
The chain term to order v2 has been identified by Faddeev [Faddeev
1984] with the Schwinger term in an equal time commutator of Gauss-
law operators for a Yang-Mills theory.
• QL-4
The chain term to order v3 has an interpretation within ordinary quan-
quantum mechanics. It is related to the failure of the Jacobi identity for

380
Stora-Zumino chain of descent equations
velocity operators at a magnetic monopole [Jackiw 1985 a,b,c], [Gross-
[Grossman 1985], [Wu, Zee 1985], [Mickelsson 1985]. For attempts to inter-
interpret this term within field theory see [Jo 1985], [Levy 1987], [Zhang
1987].
For the higher order chain terms Qjjn-i-fc w^^ ^ > 4 no physical in-
interpretation is known up to now. But there is a common belief that all
solutions of the chain will have a counterpart in physics.
Comparison of Stora's with Zumino's approach:
Stora Zumino
basic fields and operators
A
F
VFP
S
A =
vMC = g
6-
9~4A + d)g
= dA + A2
~x6g, g = д(х,в)
algebra
svFP
sA =
sF =
= -vFP
—Dvpp
[F, vFP)
Svmc = ~v2mc
6A = -Vvmc
8T = [T, vMc]
A
—j
d-
A + i
-* d +
if
s
shift
p
A-
d
¦* A + vmc
'Russian formula'

Explicit solutions for the chain terms
381
tr Fn
SQ2n-2
sQ2n-2
chain of descent equations
- Я&.-1
+ dQ\n_2
+ dQ02n-1
= 0
= 0
= 0
= 0
= 0
tr T71 — dO"
SQL-i + dQL^
SQL-2 + dQ22n_3
6Q2n~2 + dQlT-1
= 0
= 0
= 0
= 0
= 0
identification
vFP
s
VMC
6
anomaly and WZ consistency condition
NQln-2 = vaGa[A]
sG(v,A) = sfvaGa[A} = 0
9.4 Explicit solutions for the chain terms
With the operator rules, we have learned so far, it is possible to solve expli-
explicitly the whole chain of descent equations (9.78). Let us demonstrate this
t crm-by-term for the simple case of n — 2.
Beginning with the first equation the singlet anomaly in 4 dimensions,
we insert J7, equation (9.55),
tr T2 = d tr [AdA + Ia3} (9.88)
О
= dQ%, (9.89)
so that we can read off the Chern-Simons term in 3 dimensions
Q% = tr [AdA + -A3} = tr [AT - -A3}. (9.90)
О О
Пнч1 we apply S and use the structure equations (9.68)

382 Stora-Zumino chain of descent equations
6Q°3 = 6t±3
A}
= tr [SAT - AST - \ Z8AA2)
О
= -dtvv(T-A2). (9.91)
Comparison with the chain
6Q°3 = -dQ\ (9.92)
gives
Q\ = tr v(T - A2) = tr vdA. (9.93)
Solution (9.93) represents the non-Abelian anomaly in 2 dimensions (besides
a normalization factor).
Next we apply the «5-operator to result (9.93)
6Ql = 6tv vdA = tr [6vdA + vd6A] = d tr v2A, (9.94)
compare with the chain
8Q\ = -dQ\ (9.95)
and find explicitly
Q\ = _tr v2A. (9.96)
Solution (9.96) is the result for the 1-dimensional Schwinger term occurring
in a 1 + 1-dimensional Yang-Mills field theory (note that time is fixed in
the corresponding commutator relations).
Finally, using the above recipe for the term Q\ we get
8Q\ = -Str v2A=-d tr v3 (9.97)
О
6Q2 = -dQ30 (9.98)
Ql = ~\tvv\ (9.99)
Clearly the last equation of the chain is satisfied
6Ql = -\б tr v3 = -tr 6vv2 = tr v4 = 0. (9.100)
О
Set of solutions:
i) For n = 2, we collect all terms

Zumino 's formulae for chain terms 383
Q\ = tr v(T - A2) = tr vdA
Q\ = -tw2A
Ql = -\tvv3. (9.101)
ii) For n = 3, the case which we are actually interested in, we gain the
following explicit solutions analogously
Ql = tr [AT2 - \a3T + ^-45]
Q\ = tr u[^2 - \{A2T + ТА2 + ATA) + ^
= tr vd{AdA + i.43)
= trK^
= - о tr К*4 + u-^u + -Av2)dA + v2A3}
= - tr [-v3dA + AvAv2}
f)i _ I t_ „4 л
Vi — 2
Go = ^try5- (9Л02)
Term Q5 is the explicit solution of the Chern-Simons form in 5
dimensions—dQ% is nothing but the singlet anomaly in 6 dimensions—
and term Q\ provides the familiar result for the non-Abelian anomaly
in 4 dimensions, equation F.79) (apart from the normalization fac-
factor). Q\ represents the 3-dimensional Schwinger term occurring in a
1 + 3-dimensional Yang-Mills field theory (see [Faddeev 1984]).
Alternatively, the chain of descent equations can also be solved in a
reversed way, beginning with Q2]n~1 and ending with <5°n-i (see [Sorella
1993], [Sorella, Tataru 1994]).
Although the recipe for finding the explicit solutions of a chain is quite
simple the actual calculations can become rather cumbersome in higher
dimensions. But fortunately we can find simple, general formulae for all
chain terms. We present these in the next section.

384 Stora-Zumino chain of descent equations
9.5 Zumino's formulae for chain terms
As we have discussed in Section 9.3 the chain terms <52n-i-fe belonging to
the chain of descent equations are not denned uniquely. Zumino [Zumino
1985a] takes advantage of this ambiguity to simplify the explicit solution.
His choice for <52n-i-fe—*ne simplest choice—is characterized by having
i) the smallest power in the gauge connection Л,
ii) the largest power in the number of derivatives dv.
As before, only gauge connections with vanishing topological number are
considered which correspond to trivial fibre bundles. In order to create a
chain of descent equations we have to start with some kind of 'transgression',
which we obtain by considering some homotopy of the fields.
Homotopy: For the chain terms Qjjn-i-fc wi*n Zumino's simplest choice
we consider the homotopic connection
where
The associated
At--
Ai =
homotopic
= tA + v,
A + v,
curvature
t
is
G[0,
Ao
1].
= V.
(9.
(9.
103)
104)
+A2, A = d + 6, (9.105)
together with
ft=1 = A{A + v) + {A + vJ = f{A + v)= F(A)
ft=o = {d + 6)v + v2 = dv, (9.106)
where we have also used the 'Russian formula' (9.75).
We rewrite Tt by inserting again the structure equations (9.68)
= Tt + A - t)dv, (9.107)
with
Tt = tT + {t2 - t)A2. (9.108)
From expression (9.107) we also reproduce equations (9.106).
Next we differentiate Tf, equation (9.105), with respect to t

Zumino 's formulae for chain terms 385
we introduce the homotopic covariant derivative
Vt = A + [A, ] (9.110)
and obtain
Of course, the Bianchi identity is satisfied
VtTt = 0. (9.112)
Considering now a symmetric invariant polynomial P{F?) we find
(9.113)
(we used the property G.20) for invariant polynomials together with the
Bianchi identity (9.112)). The integration in the interval t e [0,1] leads to:
Theorem: 'Transgression'
Р(Я*) - P((dv)n) = AQ2n-1, (9.114)
where
Q2n-i =n I dtP(A, (Tt + A - t)^)"). (9.115)
Jo
This is the basic relation for creating the desired chain of descent equations.
Chain of descent equations: Analogous to Section 9.3 we expand the
('hern-Simons term (9.115) in powers of dv
n-l
\n-l\ =
Tt + (l-t)dv)n-l) = \^
fc=0
(9.116)
with
fcl(n-fc-l)!'

386 Stora-Zumino chain of descent equations
(Note that dv behaving as a 2-form on the underlying manifold M x Mg
(or M x Q) can be interchanged freely in the symmetric polynomial P.)
Separating each term of a given power of dv with the obvious notation
<?2»-l = QL-l + Qln-2 + Qln-S + ¦¦¦+ Qn~' (
we rewrite relation (9.114)
dvY) = (d+6)Q°2n_1+(d+6)Q12n_2+.. .+(d+6)QZ-\ (9.118)
We compare all terms of the same order and obtain the first part of the
chain of descent equations
! = о
6Q°2n_1+dQln_2 = 0
8Ql~l + P((dv)n) = 0. (9.119)
Formulae: The expansion (9.116) for the Chern-Simons form (9.115) sup-
supplies a formula to order (dv)k for each term of the chain (9.119)
Qtn-i-k = Щп _П| _ fc), fo dt(l - t)kP((dv)k, A, JFT1-"), (9.120)
with 0 < к < n - 1.
i) For к = 0 we recover result (9.71) for the Chern-Simons form <52n-i-
ii) For к = 1
/•l
Q12n-2(dv,A?r) = n(n-l) / dt(l-t)P(dv,A,F?~2) (9.121)
[
o
I dt(l-t)P(v,d(A,F?-2)). (9.122)
o
The second formula is equivalent to the first since Q2n-2 is defined
only up to a total derivative of a local form, which we can neglect.
For instance, for n = 2 and for n = 3 we recover the result, we found
before (equations (9.101), (9.102)).
JVote: In Section 7.6 we also derived a formula for the anomaly with
the help of the homotopy operator. We observe that both results
G.138) and (9.122) agree when we substitute A + 6A -> A + v.

Zumino's formulae for chain terms 387
iii) For к = 2
Q22n_3(dv, A, T) = П(П " ff" ~ 2 J^
(9.123)
iv) For fc = n — 1
Qrx(dt;M) = j-^-гт-, f dt(l-tr-lP((dvr-\A)
\n ~ *¦)'• Jo
= P((dv)n-\A). (9.124)
So far we know the chain terms up to к < n - 1 in the power of dv.
In order to find expressions for the higher powers we have to consider a
different homotopy of the fields.
Homotopy: Zumino's choice is [Zumino 1985a]
At = tv, (9.125)
where
Ai =v, Л = 0, (9.126)
together with
Tt = ДА + A\ = t(d + 6)v + t2v2 = tdv + (t2 - t)v2, (9.127)
where
Tt=i = dv, Tx=q = 0. (9.128)
Then equations (9.111)—(9.113)—the basis for the 'transgression'—are also
valid for this homotopy choice (9.125) and (9.127) leading to:
Theorem: 'Transgression'
P((dv)n) = AQn-i, (9.129)
where
Qn_i = n [ dtP(v, (tdv + (t2 - t)v2)n~l). (9.130)
Jo
Chain of descent equations: Again we expand in powers of dv
fc-o

388 Stora-Zumino chain of descent equations
and separate in expression (9.130) each term of a given power in v
Qn-i = Q?_! + Qnnt\ + Qnn±\ + ... + QT-1. (9.132)
Splitting the relation (9.129) according to expansion (9.132)
P((dv)n) = (d + 6)Qnn_x + (d + b)Qnn+_\ + ¦ • • + (d + ejOg" (9.133)
and comparing all terms of the same order we find the second part of the
chain of descent equations
n)-Aft_i = 0
_1+dQ^\ = 0
SQgn-i = 0 (9134)
Formulae: Now we look for a formula for the chain terms. In the expan-
expansion (9.131) the i-dependence can be factorized generating integrals—beta
functions—such as
(-)kB(n, к + 1) = (-)fc f dttn-\l - t)k = H^"^' (9-135)
JO [Jl + K)\
so that we find the formula to order (dv)n~1~k or vn+k
(n _ 1
(9Л36)
with 0 < к < n - 1.
All these terms (9.136) contain neither the curvature T nor the connec-
connection A but only v, the Maurer-Cartan form on the group!
For example, for к = 0 we have
n-\v), (9.137)
and for к = n — 1, the last term of the chain
Qln~\v) = (_)«-i^i|p(^-i). (9.138)
Formulae (9.120) and (9.136) contain all chain terms. Both formulae
have been derived from different homotopies but they belong to one and
the same chain. Clearly, both parts (9.119) and (9.134) fit together.

Zumino 's formulae for chain terms 389
Examples: Finally, we present the examples which we can compare with
the solutions of Section 9.4. For the symmetric invariant polynomial P we
choose the trace (see Section 7.1). In the case of n = 2 formulae (9.120) and
(9.136) give
Ql = tr {AdA + ^A3)
Ql = tr dvA
Ql = tr dvv
Ql = ~\trv3- (9-139)
In the anomaly term Ql the derivative can be shifted from v to the gauge
potential A, then it agrees explicitly with (9.101). The difference is just a
total differential which can be neglected. Furthermore we observe that the
Schwinger term Qf in its simplest choice is independent of the gauge connec-
connection A\ Whereas the equivalent solution (9.101) exhibits a linear dependence
on A.
For the example n = 3 the formulae (9.120) and (9.136) provide:
Q\ — — tr dv{AdA + dAA + A3)
Ql = tr{dvJA
Ql = tr (dvJv
Q\ = -^tvdvv3
Ql = ^<*<A (9.140)
In comparison with expressions (9.102) we observe the economical choice of
solutions (9.140). Again, the anomaly term Q\ agrees explicitly with expres-
expression (9.102) after shifting the derivative to the second factor. The Schwinger
term Q\ appears linear in the gauge potential A, and this is the minimal
choice. Thus the Schwinger term—as a solution of the chain—must contain
a dynamical variable like A in this dimension. The equivalent expression
Ql (9.102), however, is much more complicated; there the dependence on
A is quadratic and cubic. The next terms Q3, and Q\ of the solution set
(9.140) are already independent of the gauge connection A, in contrast to
the equivalent terms (9.102).

10
Covariant anomaly
The non-Abelian anomaly, Bardeen's result, which we have discussed so far
is called the consistent anomaly because it is a solution of the Wess-Zumino
consistency condition. However, it is not gauge covariant since the corre-
corresponding current does not transform covariantly under gauge transformati-
transformations. But there exists another type of non-Abelian anomaly—the covariant
anomaly—whose corresponding current does transform gauge covariantly
and consequently the covariant anomaly itself. These two types of anom-
anomaly reflect two different regularization procedures—the ambiguities in the
regularization—when calculating anomalous terms. In one case gauge inva-
riance is lost in the regularization, in the other it is maintained no matter
whether we work with Feynman diagrams (see e.g. [Leutwyler 1985a,b]) or
with the path integral measure (see e.g. [Fujikawa 1984, 1985], [Banerjee,
Banerjee, Mitra 1986], in Chapter 5).
In Section 10.1 we introduce the Bardeen-Zumino polynomial which
relates the consistent- to the covariant anomaly. We discuss the features of
the covariant type. This discussion will be reformulated by using differential
forms on the space-time manifold M (Section 10.2), as well as differential
forms on Sp Л, the space of all gauge connections (Section 10.3). Finally,
we extend the anomalies to nonlocal forms over Sp Л (Section 10.4).
10.1 Bardeen-Zumino polynomial and covariant anomaly
Naively we expect the non-Abelian current j? (8.57) to transform covari-
covariantly under gauge transformations
%? = M°- (ЮЛ)
However, in the presence of an anomaly this is not so. Let us find the correct
gauge transformation law for the consistent current. Here we follow the work
of Bardeen and Zumino (BZ) [Bardeen, Zumino 1984].
Definition: We introduce a general variation by

Bardeen-Zumino polynomial and covariant anomaly 391
6YA = Y, A0.2)
together with
6YF = DFYA) = DY, A0.3)
where D denotes the covariant derivative D = d + [A, ].
We can view this variation as a derivative in the space of gauge connecti-
connections Sp Л (recall Section 8.1). The variation 6y represents a derivative along
an arbitrary (but fixed) vector field Y
*)^, (Ю.4)
where ё denotes the exterior derivative (8.111). Analogously the gauge va-
variation 6% expresses a derivative along the fundamental vector field (8.109)
6t = iXi6 = J dx(D^nx)-^-y A0.5)
Theorem: Commutation relation
[6y,6^]=6iy,q. A0.6)
Proof. Recalling the transformations
6(A = Dt, a€f = [f,?], 6(Y = 0 A0.7)
we verify explicitly
]A = FY6t-6e6Y)A = 6Y(dZ+[A,?])
A0.8)
A0.9)
Relations A0.8) and A0.9) imply validity for any polynomial in A and F
so that the operator equation A0.6) is satisfied. Q.E.D.
Gauge transformation law: In order to derive the transformation law
for the consistent current we start with the vacuum functional (8.1) and
apply the commutator relation A0.6)
Using the anomalous Ward identity (8.68) we get

392 Covariant anomaly
6Y I dxe{x)Ga\Ali]{x) - 81 jdxY^j^
. A0.11)
Definition: The consistent current is defined as a variation of the va-
vacuum functional (recall equation (8.56))
№)> = ЩдЩАА' A0Л2)
Then we find
J
+ J dxe(xNYGa[Ali}(x), A0.13)
and extracting the vector field Y? provides the following theorem:
Theorem: Gauge transformation law
J A0Л4)
The naive transformation law (first term) is broken by the occurrence of the
anomaly (second term). Only if the anomaly vanishes does the consistent
current also transform covariantly.
Bardeen-Zumino polynomial: Bardeen and Zumino [Bardeen, Zumino
1984] discovered that one can construct a gauge covariant current leading to
the covariant type of non-Abelian anomaly. The trick is to add a polynomial
to the consistent current which gauge transforms in an opposite way in order
to cancel the unwanted anomalous term.
Proposition:
• There exists a local polynomial in the fields Ац—the BZ polynomial
—with the gauge transformation property
Jj^ A0.15)
so that the covariant current

Bardeen-Zumino polynomial and covariant anomaly 393
ra=ti + vz (юле)
obeys the correct covariant transformation law
Using Bardeen's result D.387) for the consistent anomaly (for negative chi-
rality) BZ find a unique result for the polynomial
J tr Ta(AvFpa + FpaAv - AvApAa). A0.18)
Using expression A0.18) one can reproduce equation A0.15).
Definition: Analogous to the consistent anomaly the covariant anom-
anomaly is defined by the covariant divergence of the covariant current
A0.19)
Inserting the covariant current A0.16) provides the relation
= С^] - (ZV*1)»- (Ю.20)
Now we just use the results for the consistent anomaly D.387) and the BZ
polynomial A0.18), then we find for the covariant anomaly
^ tr TaF^F^. A0.21)
For the integrated covariant anomaly we obtain
tvTaF^Fp<J(x). A0.22)
Actually the covariant anomaly is expressed only by field strength tensors
F, as expected by the covariance. Nevertheless we keep the gauge potentials
A in the argument.
Covariant anomaly condition: Whereas the consistent current (jg) is
given by the variation of the vacuum functional A0.12)—leading to the WZ
(•(insistency condition—the covariant current C?) cannot be obtained as a
variation of a functional. So it is no surprise that the covariant anomaly

394 Covariant anomaly
does not satisfy the WZ condition. Instead, the following condition can be
derived [Adam 1990], [Banerjee, Banerjee, Mitra 1986]:
We start from the covariant gauge transformation law
6nG(Z,A) = G([r],(;],A). A0.23)
We give a general derivation in Section 10.2; here we can check equation
A0.23) easily by using the explicit expression A0.22).
Then the difference of two gauge variations yields the theorem:
Theorem: Covariant anomaly condition
], A). A0.24)
Equation A0.24) differs from the WZ consistency condition (8.69), (8.107)
by the factor 2!
Alternatively, we may view the condition for the covariant anomaly
6tG(ri,A) - 6nG(Z,A) - G(K,t,], A) = G{[Z,ri\,A) A0.25)
as a WZ consistency condition broken by an 'inhomogeneity' which is the
covariant anomaly itself.
Resume: The non-Abelian anomalies may occur in two different types-—
the consistent or the covariant anomaly—corresponding to the two different
currents—the consistent or the covariant current—which the anomalies de-
depend on.
Consistent anomaly:
• The consistent current is defined as the variation of the vacuum func-
functional.
• As a result the covariant divergence of this current—the consistent
anomaly—satisfies the WZ consistency condition.
• The consistent anomaly is part of the Stora-Zumino chain of descent
equations.
• However, the consistent current does not transform covariantly under
gauge transformations. The covariant transformation law is broken by
the anomaly.
Covariant anomaly:
• By adding the BZ polynomial to the consistent current one can obtain
a covariantly transforming current.

Covariant anomaly and differential forms 395
• However, the covariant current cannot be obtained from the variation
of a vacuum functional.
• For this reason the covariant divergence of the covariant current—the
covariant anomaly—does not satisfy the WZ consistency condition.
The covariant anomaly obeys a modified WZ condition containing an
additional anomaly term as an 'inhomogeneity'.
• Consequently the covariant anomaly does not fulfil the Stora-Zumino
chain of descent equations but it is part of more complicated alge-
algebraic relations (see [Tsutsui 1989], [Abud, Ader, Gieres 1990], [Bonora,
Cotta-Ramusino 1986]).
• The anomaly cancellation conditions remain the same for either type—
covariant or consistent.
• The covariant anomaly may have its physical significance when the
covariant current is coupled to other nongauged external fields.
10.2 Covariant anomaly and differential forms
What we have discussed in the previous section—the BZ polynomial and the
covariant anomaly—we can also formulate, which we prefer, in the compact
notation of differential forms on the space-time manifold M (dim M — m).
Definitions: Current 1-form (consistent current)
j=#k"T«, with U?) = jfc- (Ю-26)
Dual current (m — l)-form
(*j) = j?, with *j = ^_L^eMlM...MmjMidaJ" ... dx^Ta. A0.27)
Treating
6A = SA^dx^Ta A0.28)
as a 1-form on the manifold M we consider
6 1 6
6A ~ (m-1)!
A0.29)
as a dual form. Then the scalar product of such forms can be written as
(recall Section 2.4, equation B.164))

396 Covariant anomaly
SA-± =
6 A
= /
J
м *ЬАа^т\
T^- (Ю.30)
Gauge transformation law: Now we look for the gauge transformation
law of the consistent current by working with differential forms. Applying
the commutation relation A0.6) to the vacuum functional again
[6Y, h\W[A»\ = (Me - WWW = 6[y,aW[AM], A0.31)
using
6W
6YW = 6YA-—-=Y- (*j) A0.32)
6A
and the anomalous Ward identity (8.68) we obtain
6Y ? • G[A] -S(Y- (*j) = [Y, ?} ¦ <*j), A0.33)
which is rewritten:
Theorem: Gauge transformation law
Y ¦ e€(*j> = Y ¦ [{*j),t] +eY?. G[A]. A0.34)
In deriving equation A0.34) Y does not change under gauge transfor-
transformations 6%Y = 0. If instead we also allow Y to transform as
A0.35)
then we find the theorem more compact.
Theorem: Gauge transformation law
6z Y ¦ (*j) = 6Y ? • G[A]. A0.36)
Here the variations act on all factors to the right but ? remains unchanged
6Y? = 0. In components both versions A0.36) and A0.34) correspond to
equation A0.13).
Bardeen-Zumino polynomial and covariant anomaly: As explained
before, we introduce the BZ polynomial in order to obtain a covariantly
transforming current.

Covariant anomaly and differential forms 397
Proposition:
• There exists the local BZ polynomial V with transformation pro-
property
S^Y ¦ V = -6Y? • G[A] A0.37)
so that the covariant current
*J=*j + V A0.38)
transforms as
ё? Y-(*J)= 0. A0.39)
The transformation law A0.39) can be regarded as a gauge invariance rather
than a covariance. (Note, V is certainly the dual of a 1-form.)
Definition: The covariant anomaly is defined by
G(?,A) = -?.?>*J. A0.40)
Inserting the covariant current A0.38) we get the relation
= G{Z,A)-Z-DP. A0.41)
Lemma: The anomaly G—a covariant divergence of a covariant current—
also transforms in a covariant way.
Proof. By definition we have (we choose the case 6^Y = 0)
6v*3=[*3,ri\> A0-42)
which implies covariance for
6nD*J = FriD)*j + D6ri*j
= [D*],r]} A0.43)
and consequently covariance for
6r,G(t, A) = -Z-6,,D*J=-t-[D*3,Ti]
, ?], A). Q.E.D. A0.44)
Formulae: Bardeen and Zumhio have found explicit expressions for the
BZ polynomial and covariant anomaly in any (even) dimension. The proce-
procedure is the following. Since the consistent anomaly is known we first solve

398 Covariant anomaly
equation A0.37) for the polynomial V and then relation A0.41) for the
covariant anomaly. The actual procedure, however, is rather technical (for
details see [Bardeen, Zumino 1984]), so we just quote the results given in
Bn - 2) dimensions:
BZ polynomial
Y-V=-Nn(n-1) f dtt f P(Y,A,F?~2), A0.45)
Jo Jm
with Ft=tF + (t2 - t)A2.
Covariant anomaly
G{v,A) = Nn f P{v,Fn~l) A0.46)
Jm
in components
^..Ftln_ll/n_1. A0.47)
P denotes a symmetric invariant polynomial and we have already quoted
the normalization N in Section 9.1, equation (9.22).
Examples:
i) in 2 dimensions (n = 2, N = —-fa)
Covariant anomaly
G(v,A) = ~ ftvvF A0.48)
Z7T J
—e^trTaF^. A0.49)
BZ polynomial
integrating the formula
f txYV = Y -V = — 2 / dtt fstr(Y,A) A0.50)
J 4тг Jo J
gives
V=~A. A0.51)
4тг
We obtain the components in conformity with notation A0.16) from
the dual (recall J = j + *V)

Covariant anomaly and differential forms 399
*V = e^Vdx* A0.52)
then
^v. A0.53)
ii) ш 4 dimensions (n = 3, N = jh
Covariant anomaly
G(v,A) = ^2 hrvF2 A0.54)
L^F^. A0.55)
BZ polynomial
integrating the formula
Y-V = ^3-2/ dtt fstT(Y,A,(tF + (t2-t)A2)) A0.56)
24ttj Jo 7
gives
/bJ 13 (io-57)
and therefore
V = ^Vvp<rdx4x»dx° = --^(AF + FA- ~A3). A0.58)
Again, we find the components from the dual
jj.'T) — ? "ту1'porrfx^1 =• "P dx^1 A0 59^
then
AvFpa + FpvAv - AvApAa), A0.60)
which is the result A0.18) quoted before.
Remark: The chain term Q\n-2{vi^iF) represents a polynomial in the
variables v (linear), A, F where the leading term in powers of F is
Qan-aK A, F) = P(v, F"-1) + .... A0.61)
For the consistent anomaly this implies to leading order

400 Covariant anomaly
G(v, A) = N I Qln_2(v, A,F) = N I P(v, Fn~l) + .... A0.62)
Jm Jm
We observe that the covariant anomaly A0.46) is a factor n—a factor 3
(n = 3) in 4-dimensional YM field theory—larger than the leading term in
the consistent anomaly. This reflects the Bose symmetrization factor when
calculating the anomaly with Feynman diagrams. It also shows that the
anomaly cancellation condition, as discussed in Section 4.9, remains the
same for both anomalies, the covariant and the consistent types.
Recipe: We find a recipe to obtain the covariant anomaly from a singlet
anomaly:
• The covariant anomaly in Bn — 2) dimensions follows from a singlet
anomaly in In dimensions by substituting F —> F + v and selecting
the linear term in v!
The singlet anomaly is given by
Gsinglet = -27И / Ch F = -27h \ f tr .F"
JM2n Bтг)"п! JMin
= N f tr .F", A0.63)
JM2n
where — 2тгг is a phase factor arising from topology and will be explained
later on in Section 11.5.3; the integral corresponding to an index theorem
contains the Chern character in In dimensions (see Section 11.4) and gives
the normalization N (9.22). Then the following substitution provides the
covariant anomaly
Nn
f trvF"-1 =G(v,A). A0.64)
This illustrates that a singlet anomaly cancellation in In dimensions also
guarantees a cancellation of non-Abelian anomalies in Bn - 2) dimensions.
10.3 Geometry in the space of gauge potentials
Finally we also formulate the anomaly equations as differential forms over
Sp A, the space of all gauge connections (see e.g. [Bonora, Cotta-Ramusino
1983, 1986], [Zumino 1985b], [Kelnhofer 1991]). We have already described
the differential geometric set-up in Sections 8.1 and 8.3.
Again we start from the vacuum functional (8.1)

Geometry in the space of gauge potentials 401
W\A] = -\aZ\A\. A0.65)
Definition: We define a nonlocal current 1-form on Sp Л by
^ A0.66)
where
<i?№)> = jj^ ЩА] (Ю.67)
coincides with the usual definition (8.56) for the consistent current.
Anomalous Ward identity: The interior product of the current with
respect to the fundamental vector field X%
iX((J[A]) = iX(SW[A] = (iX(S + SiX()W[A] = CXiW[A] A0.68)
provides the Lie derivative on W[A] (note: the interior product of a 0-form
vanishes and we used relation (8.113)). Since the Lie derivative CX( along
the fundamental vector field represents the gauge operator <5^ geometrically
(recall Section 8.3.3), we can formulate the anomalous Ward identity by
CXiW[A] = G{tA). A0.69)
Then the consistent anomaly is given as a restriction of the current 1-form
along X$
A0.70)
In local components we have
dxb(x)Gb[A}(x) = I dx{fa[A]{x)){D^)a{x). A0.71)
J
Wess—Zumino consistency condition: The consistency condition fol-
follows from the commutator of the Lie derivatives (see Section 2.6)
,] A0-72)
and from the commutator of the fundamental vector fields (8.66)
lXt,Xv} = X[i}V] A0.73)
implying the identity

402 Covariant anomaly
?{хе,хч]=Схк„у A0.74)
Applying the commutation relation to the vacuum functional
({CXvCXri}-Cxl(JW[A} = 0 A0.75)
and using the anomalous Ward identity A0.69) we find:
Theorem: WZ consistency condition
) - G([t,rj\,A) = 0. A0.76)
Gauge transformation law: For the gauge transformation of the consi-
consistent current we start with the identity
CXi{J[A}) = (iXF + 6iX(){J[A}). A0.77)
The first term on the r.h.s. vanishes because the current 1-form is closed
(recall our discussion in Section 8.3.3)
6(J[A})=0 A0.78)
and for the second term we insert the anomaly definition A0.70). Then we
immediately find the following theorem:
Theorem: Gauge transformation law
). A0.79)
The gauge invariance of the current is broken by the anomaly. Equation
A0.79) represents the geometric realization in Sp A of Bardeen-Zumino's
formulation of the current transformation law A0.36).
For a strict equivalence to expression A0.36) we have to confine the
variations in equation A0.79) along an arbitrary but fixed vector field Y[A]
iYCXi (J[A}) = iY8G(t A). A0.80)
Rewriting equation A0.80) with the help of the commutator property
[?xt,iY]=i[xe,Y] A0-81)
we obtain:
Theorem: Gauge transformation law
). A0.82)

Geometry in the space of gauge potentials 403
This equation describes the anomalous covariant transformation property
of the consistent current closest to the spirit of Bardeen-Zumino.
Local components: In local components relation A0.82) coincides with
expression A0.14) which we had in Section 10.1. It is instructive to carry
out this little exercise. We first calculate the l.h.s. (note that the interior
product of a 0-form vanishes)
CX(iY(J[A}) = (iXF + 6iX()iY(J[A})
^ j dy(fb[A]{y))Yb{A){y)
= J dyYt[A}(y) J'
+ J
A0.83)
For the r.h.s. we need a Lie bracket in the space of gauge potentials.
Definition: The Lie bracket in Sp Л is defined by
[X,Y] = J dy[X,Y}l(y)-^-y A0.84)
with
The X, Y are vector fields
= J
г 1й?^) A0-86)
Then we can calculate the interior product of the current
Чх^)(АА}) = j dy(fb[A){y))[XbY)l{y)
= f йхйу{з1[А]{у)){О^)а^)^щ
+ JdyYub[A}(y) Jdx[(r[A}),t;}b(xN(x - y). A0.87)
The last term in equation A0.82) gives

404 Covariant anomaly
iY6G(?,A) = J' 4ууЪ[А)(у)щ^ J dx^(x)Ga[A}(x). A0.88)
Collecting our results A0.83), A0.87) and A0.88) we formulate relation
A0.82) in local components
J dyYt[A}(y) J dxiD^Mj^—
- JdyYf[A]{y) jdx[{j"[A]),Z]b(xN{x - y)
+ j dyYb\A\{y) j dx?{x)j^L—Ga\A]{x)- (Ю.89)
Since the vector field 1^[^4] has been chosen arbitrarily we can drop it here
and we recover precisely the local transformation law A0.14).
Bardeen-Zumino polynomial and covariant anomaly: As pre-
previously we introduce the BZ polynomial to achieve a gauge invariant current.
Proposition:
• There exists a JocaJ polynomial 1-form in Sp Л—the BZ polynomial
V[A] = J dxV?[A]{xNAl(x), A0.90)
with transformation property
CXtV{A] = -6G(t,A) A0.91)
so that the covariant current 1-form
J[A] = J[A] + V[A\ A0.92)
obeys gauge invariance
CX((J[A])=0. A0.93)
Restricting equations A0.91) and A0.93) along an arbitrary but fixed vector
field Y[A] we obtain the transformation property
CX(iYV[A] = i[Xe,Y]P[A] - iY6G(t, A) A0.94)
and the covariant transformation law
CXiiY(J\A}) = ilXt,Y](J[A}). A0.95)

Nonlocal extensions of anomalies in Sp Л 405
The local components Vg[A](x) are exactly the BZ solutions in four dimen-
dimensions A0.60) or in two dimensions A0.53).
Definition: Analogous to the consistent case A0.70) we define the cova-
riant anomaly by restricting the covariant current 1-form along X%
A0.96)
In local components we have
J dxtb{x)Gb[A\(x) = j dx{Ta[A]{x)){D^)a{x) A0.97)
which coincides with definition A0.19) (after partial integration).
Then the two types of anomaly—the covariant and consistent types—
just differ by ___
G{?, A) = G(?, A) + iX(V[A], A0.98)
where
iX(V[A] = V[A]{Xt) A0.99)
denotes the BZ polynomial evaluated at the fundamental vector field X$.
10.4 Nonlocal extensions of anomalies in Sp Л
We consider again the vacuum functional W[A]. It is highly nonlocal in A.
Z[A] = e~wW - fdipdipexp[- /"dz#/(dM + А^ф). A0.100)
Therefore the current 1-form A0.66)
(J[A\) = 6W[A] A0.101)
also represents a nonlocal functional in Sp A. Restricting, however, the
current to a gauge orbit—evaluating along a fundamental vector field X$—
iXt(J[A]) = G(t,A) A0.102)
yiolds the anomaly which turns out locally in the gauge connection A. This
locality is a fact (and a consequence of the anomaly calculation with Feyn-
nian diagrams). The same is also true for the covariant case.
On the other hand, the difference of the consistent and covariant
current—the BZ polynomial V[A]—does represent a local functional of A
(J[A])-(J[A])=V[A]. A0.103)

406 Covariant anomaly
Since the BZ polynomial 1-form is a local functional in the whole space
Sp A, it clearly remains local along a gauge orbit, characterizing the two
types of anomaly ___
iX(V[A] = G(?, A) - G(?, A). A0.104)
The anomalies themselves (consistent or covariant), however, cannot be
expressed as local forms on Sp Д but, as emphasized by Zumino [Zumino
1985b] and Stora [Stora 1991], nonlocal extensions do exist.
Nonlocal extensions: We present two examples in 2 dimensions.
i) Consistent anomaly
The functional 1-form
G
= N ftr dAD-1Dx6Ax A0.105)
represents a nonlocal extension on Sp A of the consistent anomaly.
The nonlocality is caused by the nonlocal operator D, where ? =
DXD\. The operator D-1 means the Green function T>(x - y) denned
by
OxT>(x-y) = 6(x-y). A0.106)
In components the 1-form A0.105) denotes
G = N I dx»dxv tr дцАифа-1 DxTa6Aax{x)
= N [ dxdye»v tr д^А„{х)Щх - y)DxTa6Aax{y).
A0.107)
The last line shows explicitly the nonlocal correlation of the two terms
d^A^x) and Dx6Ax(y).
Restricting, however, the nonlocal 1-form A0.105) along a gauge orbit
6A\ —* sA\ (we use the BRS formalism now)
G(sA) = isAG = N /tr dAa~1DxsAx
= N ftvdAa-1DxDxv
= N ftr dAv = G{v, A) A0.108)
supplies us the Iocai functional G(v, A)—the consistent anomaly.

Nonlocal extensions of anomalies in Sp Л 407
ii) Covariant anomaly
Similarly, the nonlocal functional 1-form on Sp A
G = N -2 /tr FO~1DX6AX
/tr FD~1Dx6Ax A0.109)
represents a nonlocal extension of the covariant anomaly. The re-
restriction along the gauge orbit
G{sA) = isAG = N-2 ftr FD~1DxsAx
= N-2 /tr Fv = G{y,A) A0.110)
reduces to the local functional G(v, A)—the covariant anomaly.
Recipe: In order to construct a nonlocal extension of an anomaly into the
space of all gauge connections Sp A we find the following recipe (for the
covariant case, for example):
• Start with the anomaly
G{v,A) = N -n IstxFn~1v A0.111)
insert the identity D^D = 1 with О = DXD\
G(v,A) = N-n fstv Fn-1O-1DxD\v A0.112)
and use the BRS transformation sA\ = D\v
G{v,A) =N-n fstr Fn-1D-1DxsAx. A0.113)
Then, by extending the BRS variation sA\ —* 6A\ to a general varia-
variation, the nonlocal extension on Sp Л is given by the 1-form
G = N-n f f\tr Fn-1D-1DXSAX. A0.114)
Restricting, conversely, the 1-form G A0.114) to the gauge orbit 8A\ —> sA\
G(sA) = isA6 = G(v, A) A0.115)
brings back the local anomaly G(v, A).

11
Index and anomaly
In this chapter we study one of the most fascinating aspects of the
anomaly—its role in the topology of gauge theories. In Chapters 2 and 6 we
discussed the fibre bundles as the proper geometric set-up for the physics
of gauge theories. Here we shall see that the anomaly also has a 'natural'
explanation; it occurs as an obstruction in certain nontrivial bundles and it
is determined completely by a topological quantity—the index.
First in Section 11.1 we discuss the relation of the singlet anomaly to the
Atiyah-Singer index theorem and then, after some introduction into index
theory (Section 11.2), we describe the geometric-topological character of
the non-Abelian anomaly in the context of index theorems. In particular,
we show the connection between the index of the Weyl operator and the
heat kernel of the Laplacian in Section 11.3, shedding much more light on
Fujikawa's regularization procedure. In Section 11.4 we present the Atiyah-
Singer index theorem for the case of YM fields. Introducing, in Section
11.5.1, a special Dirac operator which is equivalent to the Weyl operator,
we calculate, in Section 11.5.2, the non-Abelian anomaly, Bardeen's result,
by the path integral method. Finally, in Section 11.5.3, we explain the pro-
procedure of Alvarez-Gaume and Ginsparg, how to determine the non-Abelian
anomaly by a generalized index theorem.
11.1 Singlet anomaly and index
It has been discovered by several authors [Jackiw, Rebbi 1976, 1977], [Niel-
[Nielsen, Schroer 1977], [Nielsen, Romer, Schroer 1977], [Romer 1979, 1981],
[Ansourian 1977], [Kiskis 1977], [Brown, Carlitz, Lee 1977] that the singlet
anomaly is related to the index of a differential operator. This discovery
triggered a new era of topological investigations for the anomaly. We pro-
proceed here within our path integral approach of Chapter 5 and study the
Jacobian of the PI measure.
As Fujikawa [Fujikawa 1979, 1980] noticed, the evaluation of the Ja-
Jacobian in the PI corresponds to the calculation of the index of the Dirac
operator for a given chirality. Why is this so?

Singlet anomaly and index 409
Let us consider the eigenvalue equation for the Dirac operator in Eucli-
Euclidean space
0<pn(x) = ХпЫх), (П.1)
with {ipn(x)} being a complete orthonormal system of eigenfunctions. Then
-f5<pn satisfies the equation with negative eigenvalues (—An)
(H.2)
due to {7^,75} = 0.
Lemma: The eigenfunctions ipn and j5<pn are orthogonal for An ф 0
VnJ-75Vn or (^n,75Vn) =/ dx ipl(x)y6ipn(x) =0. A1.3)
Proof. The orthogonality holds because of TJ)^ =#>. Let us consider
which is, on the other hand,
implying
G5Vn,Vn)=0 for An ^0. Q.E.D. A1.4)
Zero-modes: However, this is not so for zero-eigenvalue An — 0. In this
case y>° and 7sy° are eigenfunctions to the same eigenvalue An = 0
x)=0 and ^75^(x)=0. A1.5)
This is also true for positive or negative chirality eigenfunctions
<p°n+ = Р+1роп, <p°n = P-<pon, A1.6)
with P± = 5A ± 75). We clearly have
^(l±76)v>=0. A1.7)
The zero-modes уз°± are also eigenfunctions of 75 with eigenvalues ±1

410 Index and anomaly
Jacobian: Now we return to the Jacobian of the PI, equation E.54),
which arises from the infinitesimal chiral transformation E.11)
J[/3] = exp -2i j dx /3(x
A1.9)
We have evaluated the sum in the exponent and it supplies the anomaly
result (recall equation E.68))
with ?1234 = ?1230 = 1. Considering, on the other hand, the eigenfunctions—
we keep the transformation parameter /3 constant—then we find that due
to the orthogonality condition A1.3) only the zero-modes survive
n
= П+-П-. A1-11)
n+ and n_ denote the number of positive and negative chirality zero-modes.
Index: The difference A1.11)—as we shall learn later on—is a charac-
characteristic integer number attached to a differential operator. It is called the
index. In fact, it represents precisely the index of the Dirac operator
projected to the positive chirality subspace
index ?>+ =n+-n_, A1.12)
with
D+ =V>P+ = 0\Wn+}. A1.13)
Using the anomaly result A1.10) we obtain an explicit expression for the
index
index D+ = -^2 j dx e^a?> tr F^,Fa0, A1.14)
which represents the famous Atiyah-Singer index theorem. Rewritten within
differential forms we have:

Singlet anomaly and index 411
Theorem: Atiyah—Singer index theorem
index D+ = --^ / tr F2. A1.15)
Sr Js*
We study this relation in more detail later on, when we work on a compact
manifold like the sphere S. But also in our Euclidean space R4—where
we are actually working in the PI formalism—the relation A1.14) is true;
but then we have to assume that the gauge potential approaches a pure
gauge configuration A^ —> g~1d/J/g at infinity x —> oo (as found by [Jackiw,
Rebbi 1977]). We discussed this situation in our previous study of instantons
(Chapter 6). In fact, we met the integral A1.14) in Section 6.6. It was
equation F.160), named the Pontrjagin index or the topological charge
q. Hence we find the identity
n+-n_=q. A1.16)
We also remember that the topological charge, and therefore the index
A1.12)—A1.15), is determined only by the topology of the bundle (recall
(equation F.198)).
The regularized sum of eigenfunctions represents an index density
dx ¦ J2 Ы*O5?>»(*) = ~g^2 tr p2 = Index D+- AL17)
So Fujikawa's regularization procedure (Section 5.2) corresponds to a local
«•valuation of the index.
Anomaly: The singlet anomaly Л (or axial-, or chiral anomaly), which
is the anomalous divergence of the chiral current, occurs in the Jacobian via
J\fi] = exp f- jdx p{x)A[A^{x)\ , A1.18)
I hcirefore we find the following relation between the anomaly and the
index (in Euclidean space)
dx d*jl(x) = jdx
= 2i-(n+ -n_) =2i- index D+ A1.19)
or
iii terms of differential forms we have:

412 Index and anomaly
Theorem: Anomaly and index
d* f = Д[ЛД] =2i- Index ?>+. A1.20)
11.2 Predholm and elliptic operators and index theory
We want to give an idea of the concept of an index as it is used in the ma-
mathematical literature (see e.g. [Gilkey 1984], [Booss, Bleecker 1985], [Berline,
Getzler, Vergne 1992], [Lawson, Michelsohn 1989], [Nash 1991]). Let us be-
begin with some definitions.
Operator: A linear operator Г : E —> F acting between the Hilbert
spaces E, F is said to be bounded if the operator norm
||T||:=supH!<oo A1.21)
хфО INI
is finite, where the norms in E, F are induced by the inner product
IN| = (*,*I/2- (П-22)
Kernel: The kernel, or null space, of the operator T is denned by
kerT = {es?|Te = 0}. A1.23)
Range: The range, or image, of the operator Г is given by
range Г = {/ € F\Te = f for some e € E}. (П-24)
Note that kerT is always closed, whereas range Г need not be closed.
Orthogonal complement: Let А С F, then the complement of Л is
Ac = {ac € F\ac ? A}. We have A U Ac = F, А П Ac = 0. The orthogonal
space—orthogonal complement—is denned by A± = {a± ? F\(a,a±) -
0 V a € A}. Note that A± is closed.
Adjoint operator: Next we consider the adjoint operator T* : E *— F,
adjoint with respect to the inner product
(/, Те) = (TV, e) for e ? E, f € F. A1.25)
Then we find for the orthogonal complement of the range:

Fredholm and elliptic operators and index theory
413
ker T
range T
(range T)x=ker Т*
Fig. 11.1. The operator T : E -> F, the adjoint T* : E *- F, kerT (dotted
urea), range T (hatched area), (range TI = kerT* (dotted strip)
Lemma:
(range T)x = kerT*.
Proof. Take the elements / € range T, g € (range T)x, then
0 = (<?,/) = (
A1.26)
--(T*g,e) A1.27)
We dom T (domain of T). Consequently we have T*g = 0 or g € kerT*
mid vice versa, thus (range T)x = kerT*. Q.E.D.
('okernel: We define the cokernel of the operator T
coker T:=F/ range T ~ (range T)x = kerT*, A1.28)
where the symbol ~ denotes isomorphic.
We have illustrated the above concepts in Figure 11.1.
I-Yedholm operator: We speak of a Fredholm operator
/•< fred(je;,f)
if T : E -> F with T bounded
dim kerT < oo, dim coker T < oo. A1.29)

414 Index and anomaly
This means that the homogeneous equation Те = 0 has a finite number
of linear independent solutions and to solve the inhomogeneous equation
Те = f the function / has to obey a finite number of conditions (g, f) = 0,
where g € kerT*; thus T*g = 0.
Note that if Г € FRED(?, F) then range T and range T* are closed.
Lemma:
i) if T € FRED(?, F) => T* e FRED(F, E),
u> if ?
Now we are prepared to define the index of a Fredholm operator.
Index: Let Г € FRED(.E, F), then we define the index of Г
index T := dim kerT - dimcoker Г. A1.31)
Because of the finiteness of the dimensions of the kernel and the cokernel
this definition makes sense and, using isomorphism A1.28), we also have
index Г = dim ker T - dim ker T*. A1.32)
This kind of index is actually named the analytic index.
Example: Let {фп} be a complete orthonormal basis for L2, the space of
square integrable functions. We define the displacement operator by
{фп-i if n > 0
0 if n = 0 A1.33)
фп if n < 0.
Then T is surjective (onto), hence kerT* = (range T)x = 0. The space
kerT = {</>o|T</>0 = 0} A1.34)
is 1-dimensional, therefore we find for the index
index T = dim ker T - dim ker T* = 1. A1.35)
Considering next the operator squared for the n = 1 basis vector we get
Т2фг=Тф0^0 A1.36)
in addition to

Fredholm and elliptic operators and index theory 415
Т2ф0 = 0. A1.37)
Hence the space
kerT2 = {фо^г^фо = Т2фг = 0} A1.38)
is 2-dimensional and the index is
index T2 = 2. A1.39)
Generally we obtain
index Tn = n € Z. A1.40)
This shows that the index is a (here surjective) map from the space of
Fredholm operators FRED into the integer numbers Z
index: FRED(?, F) -> Z. (П-41)
Lemma:
i) index T = -index T*
ii) indexT = 0 if T = T* selfadjoint. A1.42)
Both are evident from the index definition A1.32).
On the other hand, for a vanishing index we may have
either dim ker Г = dim ker Г* = О
or dim ker T = dim ker T* ф 0 A1.43)
I hen the statement
index T = 0 A1.44)
m equivalent to the more familiar Fredholm alternative:
cither the inhomogeneous equation Те = f has a unique solution e e E
for a given / € F;
or the homogeneous equation Те = О has finitely many, linear in-
independent solutions {en} and just as many solutions {gn} as
the adjoint equation T*g =\ 0. The inhomogeneous equation is
solvable whenever (gi, /) = ... = (gn, f) = 0.
I.otnma: Direct sum
i.H. T € FRED(?, F), S € FRED(?', F'), then
i) S®T(EFBED(E'®E,F'®F)
ii) index E ф Г) = index 5 + index T. A1.45)

416 Index and anomaly
Proof. Property i) is evident; property ii) is quickly verified:
index E © Г) = dim kerE © T) - dim ke^S" © T)*
= dim ker S + dim ker Г - dim ker S* - dim ker T*
= index 5 + index T. Q.E.D. A1.46)
Attention! The ordinary sum of two Fredholm operators is not in general
Fredholm-like again, S + T <? FRED. For instance, we could set Г = -5.
Lemma: Product
Let Г € FRED(?,F), 5 6 FRED(F,G), then
index ST = index S + index T. A1.47)
We postpone the proof until after our homotopy discussion.
Theorem: The index
index: FRED(jE;, F) -> Z A1.48)
is locally constant or homotopic invariant.
This homotopic invariance is an important property of the index—which
we need when we work with the index theorems later on.
Homotopy: (in analogy with the homotopic functions discussed in Sec-
Section 2.2)
Two Fredholm operators TO,TX € FRED(jE;,F) are homotopic (To can
be continuously deformed into T\)
Г0~Г! A1.49)
if there exists a continuous family of Fredholm operators Tt para^
metrized over the product space E x I
Ex I ^U F with I = [0,1] A1.50)
such that
ri- A1-51)
Example: Let us consider the group of invertible operators on a Hil-
bert space 7i, specifically the group of complex N x N matrices GL(N, C).
We study the group of invertible operators on the product space H ®ti,

Fredholm and elliptic operators and index theory 417
specifically matrices in GLBN,C) which can be written as 2 x 2 block
matrices.
We choose two such operators S, T € GL(N, C), then the following two
operators defined on the direct sum space
are homotopic
To ~ %/2, A1.53)
which means that there exists a continuous family of Fredholm operators Tt
GLBN, C)x7^ GLBN, C) with / = [0, тг/2] A1.54)
such that
Tt=0 = T0 and Tt=V2 = TT/2. A1.55)
The explicit solution—the homotopy—is
with the rotation matrices
/ cost sint\ 1=ГсаЫ -sinjy
\_ -sin* cos* y' * \_ &mt cost J K '
I'roperty A1.55) is easily verified. So the deformation here involves a rota-
rotation around an angle t € [0, тг/2] of the space where S is operating.
Proof of Lemma A1.47): The homotopy also allows us to prove Lemma
A1.47). What we need is the direct sum of the index, lemma A1.45), the ho-
motopic invariance of the index, theorem A1.48), and the example A1.52),
A1.53) of two homotopic operators. Then we quickly prove
index ST = index ST + index 1
= index (ST © 1)
= index (T © 5)
= index T + index 5. Q.E.D. A1.58)
KUiptic differential operator: In physics we mainly work with diffe-
differential operators. Mathematically we can describe a differential operator D
in ti very general way as acting between the spaces of sections of complex

418 Index and anomaly
vector bundles E, F over a compact manifold M without boundary. In this
sense D is a linear map on the sections Г of the bundles
D:T(M,E)->T(M,F), A1.59)
which has the following form
D = \ aa(x) , A1.60)
where a denotes a multi-index
a = (ai,...,am), a, e Z and сц > О
H = ai + ... + am. A1.61)
The derivative means
and N is the order of the differential operator D.
For a particular section s(x) € Г(М, .E) of the bundle E—all sections in
dom D are actually smooth functions, Г(М, E) = C°°(E)—the local action
of D on s is
Note that x denotes the point in the manifold M and its coordinate as
well. The coefficients o?- (x) are smooth к х к matrices depending on x and
1 < i,j; < fe with fe = dimE = dimF the bundle dimensions.
We are mainly interested in the case of the Dirac operator, N — 1, and
in the Laplacian, N = 2. For example, if E is the spin bundle over M, the
(massless) Dirac operator is
A1.64)
and acts on a section ф(х) of E as
WW)* = «GV^jD (n-65)
Symbol: The symbol a of the differential operator D is defined by
а(*)Г- A1.66)

Fredholm and elliptic operators and index theory 419
It is a polynomial of order N in the dual variable ?, where ?Q means
e = &*$*...%r. (n.67)
The pair (x, ?) is regarded as the coordinates in the cotangent bundle
T*{M).
It is also customary to describe the symbol by the Fourier transform of
the operator; in this case it is more convenient to replace
t; > "A—, (H-68)
dxa i\a\dxa K '
then we have
Ds{x) = /
\a\<N
= Jd^a[D}(x,0s@eixi. A1.69)
The leading symbol is the highest order part of the polynomial A1.66)
aL[D](x,O= ? aa(x)C A1.70)
\a\=N
and is homogeneous and of order N in ^.
Ellipticity: The differential operator D is termed elliptic if its leading
symbol is invertible for all (x, ?) with ? ^ 0.
For example, the Laplace operator defined by
has a symbol
= e2 = # + ...+& (n.72)
and is clearly an elliptic operator. More generally, let D be a differential
operator of order 2 with real coefficients, acting between two trivial line
bundles; then the leading symbol is
aL[D}(x,0 = о„(х)Й + (a12(x) + a21(s))fcfc + a22(z)?f. A1.73)
The operator D is elliptic if Vx € M the leading symbol ctl [D] is invertible—
t his is the case if the curve
Z) = l A1.74)

420 Index and anomaly
describes an ellipse in the ?1, ?2 plane V x e M.
Another example is the Dirac operator A1.64). Its leading symbol is
=*• (П-75)
FYom
e=e-i (ц.76)
we find that ? is invertible in Euclidean space
Г1 = 4- (П-77)
In contrast, the d'Alembert operator
д V ( д ^
is not elliptic—but hyperbolic—since its (leading) symbol
vanishes for
& = # + ¦•¦+&-! (П-80)
and is therefore not invertible on this cone.
The same is true for the Dirac operator in Minkowski space (see the
previous example).
Note: Elliptic differential operators, like the Laplacian A1.71), are clearly
unbounded with respect to the usual L2-norm but are bounded with respect
to Sobolev norms (see e.g. [Gilkey 1984]).
Theorem: An elliptic differential operator over a compact boundaryless
manifold is a FVedholm operator (in the sense of a suitable Sobolev norm).
This is an important theorem since now the whole previous index discus-
discussion is applicable to elliptic operators which we are interested in in physics.
The elliptic operators play a central role in the topological view of index
theory, which we shall see in the next sections.
Weyl operator: The former Dirac operator P E.2) in Euclidean space,
equation E.5), is an elliptic operator. Thus ф has an index but
index P = 0 A1.81)

Fredholm and elliptic operators and index theory
421
Fig. 11.2. The Weyl operators D+, D- and the Laplacians A+, A_ acting
on the positive and negative chirality spinor spaces S+, S-
since ip is selfadjoint, ft = ft^. Therefore ft itself is not interesting for index
theory and we have to consider a different operator. We introduce the Weyl
operators
D+:=ftP+, D_:=PP~, A1.82)
with the chirality projection operators P± = ^A ± 75). The adjoints are
Dl=pP- = D~, Dl=ftP+ = D+. A1.83)
Separating the space of positive and negative chirality spinors
S+ = {v+ = P+y}, 5_ = {v>- = P-4>}t A1-84)
where the v^'s are the eigenfunctions of ft, then the Weyl operators act on
the spaces (see Figure 11.2)
D
A1.85)
since
D+tp+ = ftP+P+tp = ftP+tp = P- ftp = \P-<p = \<fi- € SL. A1.86)
The Laplacians, on the other hand,
A+ = D\_D+ = D-D+
A_ = DlD- = ?»+?»_ A1.87)
net on (see Figure 11.2)

422 Index and anomaly
-(- * '-'-Ь) *^ — О—. {1L.00)
The index of the Weyl operator D+
index D+ = dim ker D+ — dim ker D'+
= dim ker D+ — dim ker D_
= n+-n- A1.89)
is the difference of the number of (linearly independent) zero-modes of po-
positive and negative chirality, which we have already met in the Jacobian of
the Fujikawa procedure, equation A1.11).
11.3 Heat kernel and index
There is an interesting connection between the index of a differential ope-
operator and the heat kernel of its Laplacian (see e.g. [Gilkey 1984], [Berline,
Getzler, Vergne 1992]).
Let D be an elliptic differential operator A1.60) of order N and D*
its adjoint
D
C°°(E) ^ C°°(F) A1.90)
acting on the sections C°° of the vector bundles E, F over a compact,
boundaryless manifold M.
Laplacians: We construct two Laplacians
Д+ := D*D
A_ := DD* A1.91)
acting on
Aj_ Д
/"ЮО/1Т1\ ~К /"»ОО / E1\ /"tOO / T7IЛ /"IOO / TTi\ /1 1 f\O\
О \ft) > t-* K^li ^ \?) * О (-г )¦ \i.l.i)Z)
They are elliptic too and Hermitian (selfadjoint)
A; = (D*D)* = D*D = Д+
AI = (DD*)*=DD* = A_. A1.93)
Since the manifold is compact the spectrum of A± is discrete and the dege-
degeneracy of each eigenvalue is finite. So the Laplacians A± have a well-defined
eigenvalue problem. Suppose the eigenvalue equation
= D-Щх = Хфх, A1.94)
then

Heat kernel and index 423
Офх = Фх (П-95)
is an eigenfunction of A_ with the same eigenvalue
= Хфх. A1.96)
The spectrum for the nonzero eigenvalues (Л ф 0) of A+ and A_ is identical.
Denoting the spaces of the eigenfunctions of A+ and A_ by
Е+(\) = Ш, Е_(Л) = {Ы (П.97)
we have (for Л ф 0)
dim?+(A)=dim?_(A) A1.98)
and the operators act on
E+(X) 7=1 E_(A) A1.99)
E+(X) -^ E+(X), E_(A) ^ E_(A). A1.100)
However, this is not true for the zero-modes, A = 0. But the zeros of Д+
and D, respectively A_ and D*, are the same.
Lemma:
kerA+ = kerD
kerA_ = kerD*. A1.101)
Proof.
kerA+ =
kerD = {f\Df = 0} A1.102)
if Df = 0 => A+/ = 0 A1.103)
ii) if / e ker A+, A+/ = D*Df = 0
=> 0 = (D*Df,f) = (Df,Df)
=> Df = 0, /ekerD. Q.E.D. A1.104)
Now we are prepared to establish the following important theorem.

424 Index and anomaly
Theorem: Index and heat kernel
index D = TrE+e-tA+ -TrE_e~tA- , Vt > 0. A1.105)
Proof. We compute the traces
e-tx
Фх Фх
= ]TVtA[dim?+(A)-dim?_(A)]. A1.106)
А
Due to equality A1.98) only the zero-modes remain, with the property
A1.101) leading to the index A1.32)
Тг?+е-(Л+-Тг?;_е-(Л- = et0[dim?+@) - dimE_@)]
= dimker Д+— dimker Д_
= dim ker D — dim ker D*
= index D. Q.E.D. A1.107)
Weyl operator: Specifically, for our previous Dirac operator projected
to a given chirality—the Weyl operator D± A1.82)—we obtain:
Theorem: Index and heat kernel
index D+ = TrS75 e~tp\ V* > 0. A1.108)
Proof.
index D+ = Trs+e-tD-D+-Trs_e-tD+D-
-^2. Q.E.D. A1.109)
Remembering Fujikawa's anomaly procedure with path integrals (Sec-
(Section 5.2)—where the regularized sum in the PI Jacobian describes the
index—we notice that the index-heat kernel formula A1.108) corresponds
precisely to Fujikawa's Gaussian regularization choice! So we just have to

Atiyah-Singer index theorem 425
expand the heat kernel into Seeley coefficients, equation E.159), pick up the
^-independent part and perform the limit t —* 0.
Example: In 2 dimensions we get
Trs75 e~1^ = TrS75Gp2 = f dx tr -y5GP2(x,x,t)
= j dx-^ Ы ъа1(х,х) = —^ J dx e^F»», A1.110)
where we have used equations E.167)-E.169) together with the convential
?-tensor definition ?14 = ?^ = 1 for Euclidean space.
So we find
index D+ = --J- f dx e^F^ = ~ I F, A1.111)
4тг J 2n JS2
which represents the Atiyah-Singer index theorem in two dimensions.
We learn this in the next section.
11.4 Atiyah-Singer index theorem
Now it is time to discuss more precisely Atiyah and Singer's index theorem
(AS) [Atiyah, Singer 1968a,b,c], [Atiyah, Singer 1971a,b], which we have
met already in the preceding sections. Up to now we have studied the so-
called analytic index A1.32) of an elliptic differential operator. Atiyah and
Singer have shown that this analytic index equals an index which is fully
determined by topology—called the topological index for this reason.
This topological index is a topological invariant depending only on the fibre
bundles considered and on the manifold. Moreover, this topological invariant
can be expressed as an integral over certain characteristic classes. These
characteristic classes can be found explicitly for a given differential operator.
Chern character: In our case of the Dirac operator containing the YM
gauge potential A = A^dx^ Ta the characteristic class is given by the
Chern character
ch(F) = trexp — F
where r is the dimension of the group and F is the curvature 2-form,
F = dA + A2. . A1.113)

426 Index and anomaly
Theorem: Atiyah-Singer index theorem
index D+= f ch(F). A1.114)
JM2n
D+ is the Weyl operator A1.82). The integral is taken over the compact
manifold Min with dimension 2n so that we pick up the n-th term only
+ = —t (— )
index D+ = —t (— ) / tr Fn. A1.115)
n\ \2wJ JM2n
We have given a proof via the heat kernel or Fujikawa procedure before.
So the AS index theorem is an interesting connection between the lo-
local properties of a differential operator—the local information about the
solutions of a differential equation—and its global properties on a manifold.
Physics: Let us return to physics.
i) Instantons
The YM fields compactified on a sphere S4—the instantons—
correspond geometrically to an SUB) principal bundle P(SUB), S)
over the sphere S (recall our discussion in Section 6.6). The index
of the P(S77B),S4) bundle is then
index D+ = k(iItr p2 = -i Itr p2 s *
We recover the Pontrjagin index or topological charge q (or the in-
stanton number) which we have discussed already in Section 6.6.
ii) Anomaly
Recalling the relation between the anomaly and the index, equation
A1.20), we obtain the anomaly from the index density
d * j5 = A[AU] = 2i - Index D+ = -i—^ tr F2 in 4 dim
= -i-F in 2 dim. A1.117)
7Г
(The factor — г is removed in Minkowski space.)
iii) Dirac monopole
The Dirac monopole that we studied in Section 6.4 corresponds geo-
geometrically to a C/(l) principal bundle P(UA), S2) over the sphere S2.
Remembering that the curvature in the U(l) case is pure imaginary
F -+• iF, the index of the P{UA), S2) bundle is given by

Non-Abelian anomaly and generalized index theorem 427
index D+ = ^- f tr iF = --!- / F = -n. A1.118)
It represents the quantized monopole charge n.
Remark: We again emphasize the pure topological content of the AS in-
index theorem. The integral A1.115) is independent of the chosen connection
A, it depends only—which we have demonstrated in Chapter 6—on the
transition functions of the underlying bundle.
We find the remarkable result that basic features of QFT, like anomalies
or instanton- and monopole charges, are determined by a global, topological
quantity—the AS index theorem.
11.5 Non-Abelian anomaly and generalized index theorem
Again, we return to the fermionic path integral. Alvarez-Gaume and Gin-
sparg (AGG) [Alvarez-Gaume, Ginsparg 1984, 1985] have introduced a spe-
special Dirac^operator D in order to calculate the non-Abelian anomaly. This
operator D can be used either for regularizing the Jacobian of the fermionic
PI or—what interests us most—for determining the anomaly from an index
theorem.
11.5.1 Differential operator D
Let us start with the usual non-Abelian Lagrangian (see e.g. equation
E.124))
Cna = Ф+г Р+Ф+, A1.119)
where we consider only the massless (m = 0), positive chirality part. The
Dirac operator
P+ = ? + 4+ (П-120)
contains only the positive chirality gauge potential (recall equation E.123));
from now on we suppress the indication + in the gauge potential. We rewrite
t he Lagrangian A1.119) in terms of the Weyl operator A1.82) D+ =
Cna = фРЛ фР+ф = фг фР+ф = фШ+ф. A1.121)
The Euclidean convention we choose here is the following
7Mt = 7M Hermitian
In
75 = in
ll = 76, G5J = 1 (П.122)

428 Index and anomaly
with the metric
g»v = S^. A1.123)
(Note that this choice differs from our convention in Chapter 5.)
The gauge group generators Ta satisfy, as before, the commutation rela-
relations C.275) and they are anti-Hermitian Tat = — Ta. Hence the operator
i If) becomes Hermitian
(i#)f = i# A1.124)
and
ft = у ?>M = gi^^Dfi = 7MJDM. A1.125)
Now we consider the generating functional
dфdф<sx.p - dx ф1В+ф\ = detiD+. A1.126)
L J J
The fermionic PI (again, the notation here differs from that in Chapter 5)
formally gives a determinant. However, this determinant is not well-defined;
we cannot identify the determinant with the product of the eigenvalues
detiD+=Y[i\n A1.127)
since the eigenvalue equation
iD+фп l Хпфп A1.128)
is meaningless. The Weyl operator iD+ has no well-defined eigenvalue
problem—in contrast to the Dirac operator i If)—since the operator iD+
maps positive chirality spinors ф+ into negative chirality spinors ф-
S+ <g> V —i S- <g> V. A1.129)
S± = {ф± } are the sections of the spin bundle and V is a section of a vector
bundle associated with the gauge group.
The way out proposed by Alvarez-Gaume and Ginsparg [Alvarez-Gaume,
Ginsparg 1984, 1985] is to define a new differential operator.
Definition: Differential operator
3 := 7"(dM + AMP+) = d_ + D+ A1.130)
or in a 75-diagonal basis
is ( ,+ J

Non-Abelian anomaly and generalized index theorem 429
0 d_
ч D+ 0
with d± = ftP±.
Then the generating functional, the path integral, defines a determinant
/Г С - 1
#<tyexp - / dx ijjiDipl = detiD. A1.132)
L ** J
The operator iD acts on Dirac fermions mapping the combined space of
positive and negative chirality spinors into itself
®v. A1.133)
Hence iD has a^ well-defined eigenvalue problem and the determinant exists.
Furthermore iD = г(д_ +D+) has gauge couplings only to positive chirality
spinors and it has only positive chirality zero-modes which coincide with
those of iD+. So a gauge theory arising from iD is equivalent to a gauge
theory from iD+—the difference lies in an overall normalization constant
that is independent of the gauge potential. In addition, the absolute value
of the determinant turns out gauge invariant so that only the phase of the
determinant may receive an anomalous contribution under gauge variation.
This fact will become important when we relate the anomaly to the index,
in Section 11.5.3.
Properties: Altogether we find the following iD operator properties:
i) well-defined eigenvalue problem,
ii) gauge coupling only to positive chirality spinors,
iii) only positive chirality zero-modes contribute,
iv) | detiD| is gauge invariant.
Proof. We will prove property iv), which is important for our anomaly
discussion. Let us consider
| detiD\2 = detiDdet{iD^ = det(iD(iD^); A1.134)
we decompose the operators as
/
0 id. \ ( 0 г?>_ \ ( id-id+ 0
iD+ 0 ){id+ 0 ) = { 0 iD+iD
A1.135)

430 Index and anomaly
so that the determinant becomes
A1.136)
= det{id-id+)det{iD+iD-).
We relate the second determinant to the Hermitian Dirac operator (in a 75
diagonal basis)
then we have
\detip\2 = det{ipJ
JD_). A1.138)
Altogether we find
|deti5|2 =
\detiD\ = const. -{detipI/2, A1.139)
where const, is some (irrelevant) gauge potential independent constant. The
Dirac operator i ]fi has only real gauge invariant eigenvalues A^irac (see our
discussion below) so that its determinant
A1.140)
is gauge invariant. This also implies gauge invariance for |deti?>|. Q.E.D.
Lemma: The operator iD is elliptic but not Hermitian.
Its eigenvalues are therefore complex satisfying the eigenvalue equations
iD(pn(x) = \n<Pn{x)
(t?)tXn(x) = \*nXn(x) or xUD = XnXl (И.141)
We choose the eigenfunctions to be orthonormal
dx xl(x)vm(x) = 6nm. A1.142)
j

Non-Abelian anomaly and generalized index theorem 431
Proposition:
• The spectrum ofiD—the eigenvalues Xn—is not gauge invariant!
Proof. Under a gauge transformation the operator changes
= г7м(дм + АМР+) + ig fa'1 + i fag~lP+
= W{A) + ig fa'1 - ig fa~1P+
Ф iD(A) A1.143)
(note: fag'1 + g fa'1 = 0 from gg'1 = 1). Q.E.D.
In contrast, we have:
Proposition:
• The spectrum of the Dirac operator i ft—tie eigenvalues Л°1Гас—is
gauge invariant!
Proof.
= г7^(9д + A^) -f- ig fa — ig fa
= ift{A). A1.144)
('onsequently every eigenfunction фп of i
ИР{А)фп = \™™фп A1.145)
I uis an associated eigenfunction g-ll4>n of г ft (A9) with the same eigenvalue
ip(A°)g-^n = \™racg-^n. Q.E.D. A1.146)
Theorem: The classical action remains invariant
Sagg = fdx &П-Ф = S'AGG A1.147)
under gauge transformation on the positive chirality components
A» -» А'ц = A» + D^v
¦ф —» ip' = ip - уф+
ф -, ф' = ф + ф_у, ¦ A1.148)

432 Index and anomaly
where v denotes the FP ghost and ф+ = Р+ф, ф- — фР-.
Proof.
фЧЗ(А')ф' = ф'^дц + А'^Р+М'
= фгВ{А)ф. Q.E.D. A1.149)
11.5.2 Anomalous Jacobian
As we discussed in Chapter 5 it is the PI measure which changes under the
above gauge transformation A1.148)
dфdф -> (fy'dip1 = J[v]drpd^. A1.150)
The Jacobian J[v] provides precisely the non-Abelian anomaly, here the
consistent version (8.59) satisfying the Wess-Zumino consistency condition
(8.82).
Path integral: The explicit calculation is a variant of Fujikawa's method
described in Section 5.2. We expand the spinors -ф, ф' in terms of the eigen-
functions <pn, Xn of the operator iD (recall equations A1.141))
The Grassmann elements an, bm can also be written by the inner products
ап = (х1Ф), 5m = (&?>m) A1.152)
since, on the other hand, we have
ф(х) = (х\ф)
/ y Ыу){у\Ф)
^2ф). A1.153)
n
We notice that
(x\n) ~ <pn{x) and (n\y) ~ xl(y) A1.154)

Non-Abelian anomaly and generalized index theorem 433
are not Hermitian conjugate due to the non-Hermiticity of iD.
The PI measure and action are re-expressed by
= Yl dbndan A1.155)
= fdx $%Щ = J2 ЬпКап. A1.156)
J n
Sagg
(Note that the gauge noninvariance of the eigenvalues An is compensated
by the noninvariance of an, bn to finally achieve a gauge invariant action
Sagg-) Then the generating functional is given by the following Grassmann
integral
J
/ Y[
exp -
A1.157)
This infinite product of eigenvalues makes sense—as usual—in terms of the
C-function regularization (recall equation E.175))
detHi5 = exp[-C8'g@)]. A1.158)
Jacobian: We calculate the Jacobian J[v] a la Fujikawa. We study
t he Grassmann elements under the above restricted gauge transformation
A1.148)
an = / j Cnmam
m
b'm = J2DnmSn> A1.159)
n
with
Cnm = 6nm- dx Xn(x)v{x)P+<pm(x)
Dnm = 6nm + Jdxxl{x)P-v{x)<pm{x), A1.160)
which leads to the Jacobian

434 Index and anomaly
J[v] = [detC-detD]-1
= exp
= exp
dx Y, xl(x)v(x)(P+ - P-)ipn{x)
I
A1.161)
Regularization: We regularize the sum with a Gaussian cut-off
Л2
A1.162)
= lim limTru75exp --
6{x-y).
Noting that we have in the 75-diagonal basis
ar+*+«.) <11Л63)
so that the trace with respect to this chiral basis is
Tvchiral(i5J = i2(d-D+ + ?>+a_) = -(?pp++ p pp-), (n.164)
we rewrite the total trace Tr A1.162)
Tr tr^exp —
M2
= Tr v I P+ exp
M2
As usual we evaluate the trace in the plane-wave basis
(р-
J B7ГL
M2 J
A1.165)
A1.166)
= hm M
4 f d*k
у BтгL
~k к
exp
M2
+
M
where the plane waves have shifted the differential operators (recall equation
E.61)) and we have also rescaled the momentum k^ —> Mk^\ Tr now means
the trace over the Dirac matrices and over the gauge group generators. We

Non-Abelian anomaly and generalized index theorem
435
treat the second term of the trace A1.165) analogously.
Next we expand the exponential; we use the trace properties of the 7-
matrices in our Euclidean convention A1.122), A1.123)
tr 75 = tr
tr M"
7" = 0
A1.167)
(with e1234 = e1230 = 1); then only terms proportional to 1/M4 survive.
We need the integrals
d4ke~k
d4ke k
= 7Г
7Г
T
A1.168)
In the expansion we keep only the negative parity terms, the terms propor-
proportional to ?И"а0. The positive parity terms can be absorbed by renormaliza-
tion of the generating functional.
Anomaly: Then we find after a little calculation (for computational de-
details see e.g. [Adam 1990]) the non-Abelian anomaly vaGa[Ali]
lim lim Tr vj5 exp
М—юоу—>х
M2
S(x - y)
= -vaGa[AM].
So the Jacobian J[v] contains precisely the anomaly G(v, A)
A1.169)
J[v] = exp
/
dx lim lim Tr vys exp —
M2
6(x-y)\
= exp
= ~ exp[-G(v,A)}.
tr
A1.170)
On the other hand, the Jacobian relates the generating functionals
(whore we can now use Z[A^.\ A1.126) instead of Zagg[-^] (H-157) since
both differ only by an overall constant which cancels)

436 Index and anomaly
Z[AD = АуЩА^], A1.171)
which we can rewrite in terms of the 'quantum action' W[AM] = — In
= -J[v] + 1
= G(v,A). A1.172)
So we produce the consistent, non-Abelian anomaly (note again, 6W =
<5Wagg)—Bardeen's result F.79)—or in other terms the anomalous Ward
identity defined by
= - jdxvaDf{fb), A1.173)
with
О"') - т^ЩАг]
J dфdф ф'у^1ТьР+фехр [— J dx фЮ+ф\
J с1^ф ехр [- J dx фгИ+ф]
The result A1.172) is also correct in Minkowski space, since there the current
is defined without the factor i (see equation D.46)). If we consider negative
chirality fields the sign is reversed in result A1.172).
11.5.3 Alvarez-Gaume and Ginsparg's index procedure
Atiyah and Singer [Atiyah, Singer 1984] and Alvarez, Singer and Zumino
[Alvarez, Singer, Zumino 1984] (in case of gravitation) have discovered that
the non-Abelian anomaly is related to a more refined index theorem—the
family index theorem (see also [Sumitani 1984]). However, to establish
this theorem one has to know the mathematical discipline of /f-theory and
the theory of index bundles. For a review of this field, see e.g. [Berline, Getz-
ler, Vergne 1992], [Lawson, Michelsohn 1989], [Nash 1991], [Troster 1994].
A 'physicist's approach' to determining the non-Abelian anomaly from an
index theorem, avoiding these further mathematical techniques, has been
proposed by Alvarez-Gaume and Ginsparg [Alvarez-Gaume, Ginsparg 1984,
1985]. They relate the anomaly in 2n dimensions to the AS index theorem in

Non-Abelian anomaly and generalized index theorem 437
Fig. 11.3. Topological equivalence: Sl x S2n ~ S2n+1
a Bn+2)-dimensional space. We are going to explain this procedure—which
somehow mimics the family index theorem.
Space of gauge potentials: We consider the 2n-dimensional Euclidean
space R2n compactified to a sphere S2n. We choose a semi-simple gauge
group G, a simply connected compact Lie group; e.g. SU(N) where the
first homotopy group vanishes Hi(SU(N)) = 0. We extend the space by
working with a one-parameter family of gauge elements
g{e,x):S1xS2n->G V0 6 S1, x 6 S2n A1.175)
with boundary condition
g@,x0) = gBir,xo) = l. A1.176)
Due to this boundary condition the space S1 x S2n looks topologically like
Uio sphere 52n+1, see Figure 11.3,
S1 x S2n ~ S2n+1. A1.177)
So <7 is regarded as a map
g(9,x):S2n+1^G A1.178)
< liissified by the homotopy group (recall Section 2.2)
Z. A1.179)
Specifically, we have G = SU(N) with N > n + 1.
Next we choose, without loss of generality, some reference gauge connec-

438
Index and anomaly
Щ1)
Fig. 11.4. Disk D2: homotopy А*'в, х ... points where the fermion deter-
determinant vanishes, U(l) map defined by the determinant phase
tion A in the zero instanton sector (vanishing Pontrjagin index, recall Sec-
Section 6.6) so that the Dirac operator i I/) has no zero-modes and we consider
the one-parameter family of gauge potential transformations
= д-1(в,х){А{х) + d)g{e,x). A1.180)
The infinitesimal variation
A9W _> Ад(9+бв) A1.181)
involves an infinitesimal gauge transformation with the gauge element
v = g-Hg = va(e, x)dea, A1.182)
with
d
vaF,x)=g-1(e,x)—gF,x)
A1.183)
and a = 1 in our case. The element v A1.182) is nothing but the MC form
on group space (and it is identified with the FP ghost, recall Section 8.1).
Then we have for the infinitesimal gauge potential variation
6Ад(в) = Ag(e+sey _ Ад(в) = _PU) A1.184)
with the covariant derivative
V = d+[AgW,]. A1.185)
The gauge potential A3^—geometrically a connection—describes a circle in

Non-Abelian anomaly and generalized index theorem 439
the space of gauge connections Sp Л. To construct a disk we introduce a ho-
motopy. We consider the two-parameter family of gauge potentials—
the homotopy
A*'° = tA9<e\ *e[0,l] A1.186)
interpolating between
At=o,e = Q A1.187)
Note that here our assumption that A defines a trivial gauge potential is
crucial, otherwise we cannot interpolate as in equation A1.186). Geometri-
Geometrically this homotopy is a 2-dimensional disk D2 with boundary dD2 = S1
in Sp A; see Figure 11.4.
Fermion determinant: Next we consider the fermion determinant
detiD{Ab'e), A1.188)
with the differential operator a la AGG
Ъв) =0 + А*>вР+. A1.189)
This determinant is a complex functional of the gauge potential At>9 on
the disk D2. We may choose the reference gauge potential A such that the
determinant does not vanish on the boundary 3D2 = S1
detiD(At=1<e) ф 0. A1.190)
Remembering that the real part of the determinant is proportional to the
Dirac determinant, equation A1.139), and is thus gauge invariant, we have
along the boundary
= exp [-[]
= detiD{A9W) = (detHP)V2eiw(A'e\ A1.191)
Proposition:
• It is only the imaginary part of the determinant which may receive an
anomalous gauge variation/
The determinant restricted to the boundary—the phase—defines a map-
mapping, see Figure 11.4,
f(9) := eiw(A'e) : dD2 = S1 - U(l) = S\ A1.192)
which is characterized by the winding number (recall Section 2.2)

440 Index and anomaly
m =
It counts the net winding of the phase of the determinant A1.188) as we
move along the boundary of the disk.
On the other hand, the anomaly is the gauge variation of the 'quantum
action'. Therefore we find
= i6w(A,e)=ide—w(A,e). A1.194)
do
Proposition:
• The anomaly G(v, A9^) is given by the ioca] winding of the determi-
determinant phase!
Strategy: The strategy according to Alvarez-Gaume and Ginsparg is now
the following:
i) We find a situation for which the winding number A1.193) is nonva-
nishing, m Ф 0, for pure topological reasons. Then the phase of the
determinant must change under a gauge transformation and we have
an anomaly.
ii) We find an appropriate Bn + 2)-dimensional Dirac operator whose
index is equal to this winding number. Then the index density, given
by the AS index theorem, supplies an explicit expression for the non-
Abelian anomaly.
Winding number: First we relate the winding number m A1.193) of the
determinant phase to the local behaviour of det iD(At<e) on the interior of
the disk D2. We note that we have chosen the Dirac operator i J/) not to
contain any zero-modes. Since we have (recall relation A1.191))
|deti5(A9(9))| = (deti^I/2 ф 0 A1.195)
the operator iD(A9^) also does not contain zero-modes. But the operator
iD(Ab'e) defined on the interior of the disk D2 may receive zero-modes be-
because At>e is not the gauge transformation of A. These zeros occur at isolated
points. There the determinant—as a regularized product of eigenvalues—
vanishes. Then the winding number of the phase of det iD(At>e) around the

Non-Abelian anomaly and generalized index theorem 441
boundary 3D2 = S1 equals the sum of the local winding numbers of the
phase around small circles enclosing the zeros, see Figure 11.4. This follows
from a continuous deformation of the loop 3D2 = S1 into small contours
around the zeros. So we have
m = y^mi, A1.196)
where m^ — m(t», #») is the local winding number and ti, 9i denote the zero
points in the disk D2.
Now Alvarez-Gaume and Ginsparg have shown via an 'adiabatic appro-
approximation' that these interior zeros of det iD(At>e) correspond one-to-one to
the zero-modes of a specific Bn + 2)-dimensional Dirac operator i P2n+2-
The winding numbers at these zero points turn out to be m» = ±1, the chi-
rality of the Bn + 2)-dimensional zero-modes. Therefore the total winding
number m around 3D2 is equal to the difference of the positive and nega-
negative chirality zero-modes n+, n_ of the operator i Ръп+2—this is precisely
the index
m = n+-n- = index i |?2n+2- A1.197)
Recalling equation A1.193) we find the relation
Г
IT Г de lb
= index • #»n+a. (П.198)
Then, for i Р2П+2 given, the AS index theorem A1.114), A1.115) enables
us to identify the anomalous local variation -^w{A,в) with characteristic
classes, up to a total derivative.
Construction of a Dirac operator in Bn+2) dimensions: The bundle
we consider is defined on the base manifold
S2 x 52n, A1.199)
where S2 is a sphere in Sp Л/Q, the moduli space, and S2n is the usual
rompactified space-time manifold. In order to construct a noncontractible
sphere S2 we actually take two disks in Sp A and glue them together at
the boundaries. (We avoid such a complicating boundary term; compare
with Figure 11.6.) The disk discussed before with parameters (t, в) gives
t ho upper patch
H+ x S2n A1.200)
and some trivial disk with parameters (s, в) serves as the lower patch
H~ x 52n. A1.201)

442 Index and anomaly
t=0
н~
s=0
Fig. 11.5. The disks with parameters (t, в) and (s, в) are glued together at
t = s = 1 to form a sphere S2 in Sp A/Q
The two patches are glued together to form a sphere; see Figure 11.5.
We choose as local gauge connection 1-forms in Bn + 2) dimensions
A+(t,6,x) = tg-1{A{x) + d + 6)g on H+ x 52n
A~(s,e,x) = A(x) on H~xS2n, A1.202)
with ё = dO d/dO. Specifically, we have
A+ = Atdxa , A- = A~dxa, A1.203)
with a = (^, t, в) = A,..., 2n, 2ra + 1,2n + 2) and
dx2n+1 = dtoids
dx2n+2 = d6 A1.204)
and the components
A+- At,6 Л+—П Л+ —tn~1 — n
А-=А», А; =0, Ag=0. A1.205)
On the overlap, on the 'equator' (t = s = 1) the local connections fulfil the
compatibility condition (recall our bundle discussion in Section 2.7)
A+ = д~\в,х){А- + A)gF,x) V@,s) € S1 x S2n. A1.206)
The operator
A = d + 6 + 6t A1.207)

Non-Abelian anomaly and generalized index theorem 443
means the exterior derivative in Bn + 2) dimensions with
or 6t -> 6S = ds — j A1.208)
and the gauge element д(в,х) represents the transition function. Note
that here on the equator (t = 1) we have
A -> A = d + 6 A1.209)
since 6t g@, x) = 0. Therefore the pair
4~} A1.210)
defines a global connection on the principal bundle over the manifold
S2 x S2n characterized by the transition function gF, x).
The curvature 2-form T in Bn + 2) dimensions is given by
f=AA + A2. A1.211)
The local curvatures «F+ and T~ ¦, defined on the upper and the lower patch,
are related on the equator by the compatibility condition
F+ = д-1{в,х)?-дF,х) VF,x)eS1xS2n. A1.212)
Finally we construct the Dirac operator in Bn + 2) dimensions by
* #>2n+2 = i(da + Aa)Ta. A1.213)
Aa are the local gauge connections A1.205) and Г° are Bn+2)-dimensional
Dirac matrices which are conveniently chosen in a Weyl-like basis
Г" = <ti®V
Г2п+1 = a2®l
Г2"+2 = CTl® 75
2n+2
Г5 = in+1 Д Г" = <t3 ® 1, A1.214)
o=l
with 7** and 75 being the usual Dirac matrices in 2n-dimensional space.
Index theorem: We have prepared everything: the bundle, its connection
iwul curvature, and the Dirac operator. Now we can apply the AS index
theorem A1.114), A1.115) in Bn + 2) dimensions

444 Index and anomaly
1 / i \n+1 Г
index i $>2n+2 = -. -r-. — / tr ^n+1- A1.215)
(n + 1)! \2irJ ys:
We show that the integral representation indeed gives the winding number
we are interested in. Let us evaluate the integral. From our previous studies
(recall e.g. Section 7.2) we know that tr Т"+1 is closed, hence locally exact
(Poincare's lemma, Section 7.3). This means
3 <92n+i • • • Chern-Simons form
such that
) A1.216)
is valid on a local patch. Then the integral becomes
= / AQ2n+1(A+,f+) + f AQan+iCA-.-F")
JH+xS2n JH--X.S2"-
= f [Q2n+i(A+,f+)\t_1-Q2n+i(A-,f-)\3=1}.
Js1xS2n
A1.217)
In the last step we applied Stokes' theorem B.128). So we obtain the diffe-
difference of the Chern-Simons forms evaluated on the 'equator' S1 x 52n where
A+ and A~, F+ and T~ are related by the gauge transformations A1.206),
A1.212). We have already considered such a difference in Chapter 7 (recall
equations G.114)-G.117)) and we find the result
9,0) + Aa2n,
A1.218)
with
^1)! (n.219)
The second term a2n—a local polynomial in the gauge connection—vanishes
upon integration via Stokes' theorem. Then we arrive at the following index
formula
index» ^+2= ^")""!;, f^:)" / tr 0Г1Дя)а»+1. A1.220)
Bn + l)! \2nJ Jsixs3»
Here on the 'equator' we have A —» Д and we emphasize again the topologi-
cal invariance of the index which is determined only by the transition func-

Non-Abelian anomaly and generalized index theorem 445
tions д(в, x) of the bundle. As we know (recall our discussion in Section 6.6),
this integral formula represents the winding number of the map д(в, х) and
counts the number of times the group G covers the sphere S2n+1 ~ S1 x S2n
under the mapping; it is homotopically classified by Il2n+i(Gr) = Z. So the
net winding of the determinant phase which equals the index of the Dirac
operator (recall equation A1.198)) is traced back to the winding of g@, x).
Anomaly: Finally we determine the explicit expression of the non-
Abelian anomaly from the index density given by theorem A1.215). We
first observe that
= 0 A1.221)
s=l
since due to our chosen A = A(x), equation A1.205), there is no volume
element d6 to integrate over the manifold S1 x 52n. Then the index theorem
A1.215) together with calculation A1.217) provides
index i ?>2n+2 - 7—тту E- / Q2n+i(A+,
(n + 1)! VW Js'xs^
'(П-222)
where we rewrite A+, F+ in terms of Ая<~в), v (equations A1.180), A1.182))
index i p2n+2 = т гтт ( ¦?-) / Q2n+i(A9<-e) + v,
(n + 1)! \2nJ Js*xs2"
A1.223)
We expand the Chern-Simons form <22n+i in powers of v and recall that v
is the only quantity that carries dO. Therefore only the linear term in v
ГI (it AgW r//i#)\l A1 994^
contributes to the integral over S1 x52n. Of course, this linear term A1.224)
represents the chain term Q\n of the SZ chain of descent equations (recall
equations (9.77), (9.78)). Then we obtain
1 / ,• \ n+1
index i.
i JW = 7-^T ( ? ) / QL(v,
(n+l)! \2ttJ Jsixs*"
A1.225)
H(!membering~now the relation: anomaly—local winding of the determinant
phase—index density (equations A1.194), A1.198))
-G(v, A9(e)) = ide-^w(A, в) = i6w{A, в) A1.226)

446 Index and anomaly
— / 6w(A,e) = index i #>2n+2 =: / Index i #>2n+2 A1.227)
2n Jsi Jsi
we find the anomaly result
-G{v,Ag<-e)) = 2тгг Index г #>2„+2
A1.228)
Finally, we may choose в = 0, </@, x) = 1 and shift n —> (n — 1), then we
have proved via the index theorem the following proposition which we have
used frequently in the past.
Proposition:
• The anomaly G(v, A) is given by the chain term <2гп-2
G{v,A) = N I Qln_2(v,A,F), A1.229)
JS2n-2
with the normalization
N = -2m-r^~.—:. A1.230)
Bтг)"га! v '
For negative chirality fields we obtain an overall (—) sign factor.
We already know the chain term Q\n-2 (see formula (9.122)),
/•l
Qln_2(v,A,F) = n(n-l) dt(l-t)P(v,d(A,F?~2)), A1.231)
Jo
with Ft = tF + (t2 - t)A2. In 4 dimensions (n = 3) we find Bardeen's result
At the end of this chapter we want to summarize the several topological
aspects which are important in our index discussion.
'Russian formula': Let us concentrate again on the 'equator' of the
sphere S2 (t = s = 1) in Sp A/G- For the connections we have
A+\t=1 =
A-\..i = A, A1.233)

Non-Abelian anomaly and generalized index theorem 447
with v = g~l6g the MC form, and for the curvatures we get
(A+J\t=1
+vJ
= dA + A2 = F. A1.234)
Then the curvature compatibility condition A1.212) provides
F+(A+) = g-x{9,x)F-gF,x) = g-1(e,x)Fg(e,x) = F(A°W), A1.235)
with
F(A«W) = dA9W + (А9^J. A1.236)
But the identity
v)= F(A°M) A1.237)
and the version restricted to в = 0, g@, x) = 1
A1.238)
we know already—it is the 'Russian formula' found by Stora and Zumino
in a quite different context. Equation A1.237) represents Zumino's identity
(9.75) and equation A1.238) Stora's version (9.9). So in our fibre bundle
construction the 'Russian formula' A1.237), A1.238)—which is the basic
relation for the differential geometric derivation of the anomaly and of the
whole chain of descent equations—emerges quite 'naturally' by the compa-
compatibility condition A1.212) of the curvature on the 'equator' S1 x 52n. It
represents the Maurer-Cartan horizontality condition of the curva-
curvature (see e.g. [Baulieu 1984], [Baulieu, Thierry-Mieg 1984], [Bonora, Cotta-
R-amusino, Rinaldi, Stasheff 1987], [Troster 1994]).
Topology: The topological spaces we consider are
Sp A.. ¦ The affine space of all gauge connections. This space
is topologically trivial.
Q ... The space of all pointed gauge transformation func-
functions
g = {g(x) : S2n -> G\g(xo) = 1},
where Xo corresponds to oo in the uncompactified theory.
Sp A/Q.. ¦ The moduli space obtained by identifying the points in
Sp A which differ by a gauge transformation. This is the
physical space. Generally, its topology is nontrivial.

448 Index and anomaly
Spos/
Fig. 11.6. The deformed disk A*'e in Sp A projected down to a sphere S2
in Sp A/0
The topological origin of the non-Abelian anomaly is that the fermion
determinant represents a section of a nontrivial line bundle over Sp A/g—
the determinant bundle. In other words, the variation in the determinant
phase along dD2 = S1 does not allow the determinant to be a single-valued
functional over Sp A/g. This happens because the variation in the de-
determinant phase under gauge transformations provides an obstruction to
projecting the gauge orbits in Sp A down to the points in Sp A/G.
Specifically, we project the disk D2 = {Af'e} in Sp A down to a sphere
S2 in Sp A/g. We continuously deform the disk D2 and we identify the
boundary dD2 = {А9(-вЦ with a point in Sp A/g obtaining a 2-dimensional
sphere S2 embedded in Sp A/g; see Figure 11.6. To show the nontriviality
of the determinant bundle—the occurrence of an anomaly—we just have to
consider the bundle over such a noncontractible sphere S2 in Sp A/g.
Determinant bundle DET: So we construct a bundle with the fermion
determinant detiD(j4*>e). It assigns a complex number to each point of the
sphere S2 in Sp A/g and defines a complex line bundle—the determinant
bundle DET—over this two-parameter subspace S2 of Sp A/g. A more
abstract definition is given by Quillen [Quillen 1985] (for literature see e.g.
[Mickelsson 1989]). The structure group is clearly U{1). The transition
function of DET—the phase function e™(A<e) which represents the ima-

Non-Abelian anomaly and generalized index theorem 449
ginary part of the 'quantum action'— is defined on the equator S1 of the
sphere S2 and connects two fibre points, or two sections
detiD(A9{e)) = eiw^eUeti3(A).
Homotopy: In terms of homotopy groups (recall Section 2.2) we have
the following situation. Complex line bundles, U(l) bundles over S2n, are
classified by the homotopy group Il2n-i(^(l))—the homotopy class of the
transition functions on the 'equator' 52"". This group is nonvanishing only
for n = 1
IIi(Z7(l))=Z. A1.239)
The gauge elements gF, x) on the boundary of the disk dD2 = S1 describe
a loop in Q classified by
Ili(a) = Z. A1.240)
Since Sp A is contractible—which means topologically trivial IIn(Sp .4) = 0
Vn, specifically for n = 2—there exists the isomorphism
Ы$? A/9) ^n^Q). A1.241)
Thus the 2-dimensional sphere S2 in Sp A/G is noncontractible iff the one-
parameter family g{9,x) A1.175) generates a nontrivial homotopy class
A1.240) in Q (compare with Figure 11.6). On the other hand, we obviously
also have
П2п+1(С) A1.242)
so that we finally find
n2(Sp A/Q) ~ U^Q) ~ П2п+1(С) ~ Z. A1.243)
The gauge orbit space Sp A/G—the physical moduli space—has a noncon-
t ractible sphere S2 if and only if the homotopy group Пгп+i {G) is nontrivial;
for instance for G = SU(N), N > n + 1, with N > 3 in our case. Then
the determinant bundle over this noncontractible sphere is nontrivial—an
iinomaly occurs (for more elaborate work see e.g. [Blau 1988, 1989]).
Resume: We have reached our aim to determine the non-Abelian
iinomaly—the solution of the Wess-Zumino consistency condition—on pure
topological grounds. The non-Abelian anomaly characterizes the nontrivial
topology of the determinant bundle over the manifold S2 x S2n. More pre-
precisely, the anomaly represents the local winding of the fermion determinant
phtuse. This winding around the boundary of a disk in Sp A—which is fa-
fabricated to a sphere S2 in Sp A/G—corresponds to the index of a Dirac

450 Index and anomaly
operator in Bn + 2) dimensions. Then the index density given by the in-
index theorem provides the explicit anomaly expression. An anomaly occurs
whenever the homotopy group of the gauge group G is П2п+1 (G) = Z.
Note: The condition Il2n+i(G) = Z, however, is only a sufficient condition
for the occurrence of an anomaly!
For example, in four dimensions a [7A) gauge theory coupled to Weyl
fermions gives U5(U{1)) = 0 but the theory is anomalous, which we know
from perturbation theory. Such a situation is actually not puzzling. The
determinant phase may have a local variation although the net winding
number is vanishing when we integrate the phase over a complete loop—
that means we consider the global topology.
To pick up this anomaly by topological techniques it seems we again
need a more refined index version—the local index theorems as developed,
for instance, by [Bismut 1991], [Bismut, Freed 1986a,b], [Knecht, Lazzarini,
Stora 1991a,b].

12
Gravitation
In a field theory for elementary particles we also have to account for the
effect of gravitation. It plays an important role, particularly for the anoma-
anomalies in quantum field theory. Due to the basic principle of the equivalence of
gravitation and inertia we can describe the effect of gravitation entirely by
a curved space-time manifold. So we need to study Riemannian geometry.
We regard gravitation as a gauge theory, where the gauge transformations
are given by the coordinate- and by the Lorentz transformations. As in
the case of YM fields, only the matter fields—the fermions—are quantized
but not the gravitational field itself. In our presentation of gravitation we
certainly rely on the classical books like [Sexl, Urbantke 1983], [Thirring
1992], [Weinberg 1972], [Misner, Thorne, Wheeler 1973]; additional litera-
literature close to our interest is [Nakahara 1990], [Visconti 1992], [Gockerler,
Schucker 1987], [Birrell, Davies 1982], [d'Inverno 1992], [Wald 1984], [Sachs,
Wu 1977], [Choquet-Bruhat, DeWitt-Morette 1982].
We want to explain the gravitational formalism to the extent that we
specifically need for our description of the anomalies. So we begin with the
Riemannian geometry (Section 12.1) where we introduce the metric, the
Christoffel connection and curvature, and the general coordinate transfor-
transformations. Next we consider the orthonormal tangent frame (Section 12.2)
where we discuss the vielbein, the spin connection and the local Lorentz
transformations. We derive the equations of motion from the action princi-
principle (Section 12.3), and we introduce the energy-momentum tensor here. We
discuss the fermionic action in a separate section (Section 12.4) which is fol-
followed by the study of the invariances of the fermionic action, the Einstein-,
the Lorentz- and the Weyl invariance (Section 12.5). Then we come to our
main interest, the broken symmetries—the gravitational anomalies (Sec-
(Section 12.6). We shall distinguish between the Einstein-, Lorentz- and Weyl
anomalies and we determine the corresponding consistency conditions. We
shall find the equivalence between the Einstein- and Lorentz anomalies, and
the type of covariant gravitational anomaly will be explained too. Our in-
interest in gravitation is as in the gauge case. We introduce anticommuting
Grasemann fields—ghosts—and the BRS algebra, then we are able to derive

452 Gravitation
the SZ chain of descent equations (Section 12.7). Finally, we calculate the
gravitational anomalies with help of the AS index theorems (Section 12.8),
where we also include the mixed case of gauge and gravitational fields.
12.1 Riemannian geometry
A manifold can be endowed with a metric tensor in order to achieve a further
structure. It generalizes the inner product of two vectors in flat Euclidean
space to that in curved Riemamiian space. Let us study the differential
geometry and the analysis of tensors in this curved space.
12.1.1 Metric
Let M be a differentiable manifold, then the inner product of two vectors
X, Y € TP{M) is denned in the tangent space TP{M). We introduce the
following metrics:
Riemannian metric: A Riemannian metric g is a tensor field of type
@,2) on M with the subsequent properties at each point p € M
i) gp(X,Y)=gp(Y,X) A2.1)
ii) gp(X, X) > 0, equality only for X = 0.
So gp = g\p, which means the tensor g evaluated at point p is a symmetric
positive definite bilinear form.
The metric gp is called a pseudo-Riemannian metric if we have
iii) gp(X,Y) = 0 VXeTp(M) => Y = 0 A2.2)
instead of property ii).
Let us recall our discussion in Section 2.3, the inner product of a vector
X € TP(M) with a dual vector, a 1-form u> € T*(M). It represents the map
;R, A2.3)
where
(w,X) = (uvte",*"^) = ^*" e R. A2.4)
When we have a metric we define the inner product between two vectors
X, Y e TP{M) by the map
gp(X, Y) : TP(M) ® TP(M) -> R. A2.5)
Then we may define a linear map
gp(X, ) : TP(M) -» R A2.6)

Riemannian geometry 453
by
Y~gp(X,Y) A2.7)
and we identify this map with a 1-form
gp(X, ) «-> ш € Т;(М). A2.8)
On the other hand, the 1-form u) € T*(M) induces a vector X € TP(M) by
n A2-9)
where X is unique due to property iii). So the metric gp establishes an
isomorphism between
тр{М) ~ т;{
On a given chart (V, <p) of the manifold M with coordinates x^ we can
express the metric tensor g € T°2(M) as
ffP = <Ms)dsM ® *Л A2-И)
where
^=*>{??>¦??)¦ <12Л2>
We regard <?м„ as a matrix and g^v as its inverse according to
9^9ua = «5/. A2.13)
We denote the determinant of the metric by
g:=detgfiU. A2.14)
Clearly, we have
detg'lu=g-1. A2.15)
The isomorphism A2.10) between vectors and covectors is supplied by the
metric
wM = 9»uXu, X" = дГиЬ A2.16)
and
gp(X, Y) = (W> У) = u^ = g»vXvY». A2.17)
So we raise and lower the indices with the help of the metric tensor д^и,
v Generally, a mixed tensor is obtained analogously by
. A2.18)
The norm ||X||, or length, of a vector X e TP(M) is denned by

454 Gravitation
\\Х\\2=др(Х,Х)=д^Х»ХГ A2.19)
We find the infinitesimal distance squared—the line element—by in-
inserting the infinitesimal displacement dx11 d/dx^ € TP(M) (here dx^ is an
infinitesimal vector component)
d'2 - *("?•*"?)-*(?•?
dxv A2.20)
• dxv actually denotes the symmetrized tensor product). This quantity
is often called a metric in a sloppy way.
The quadratic form of g^v can be diagonalized. If g^v is Riemannian all
diagonal terms are positive; in the case of a pseudo-Riemannian metric д^и
may have positive and negative diagonal terms g —> diag(—1,..., 1) with
-1 occurring t times and 1 occurring s times. The pair (?, s) is called the
signature of the metric. If t = 1 we have a Lorentz metric. In this way
we obtain
Riemannian metric —> Euclidean metric
Lorentz metric —> Minkowski metric
7/ = diag (-1,1,..., 1)
(Note that here we choose a convention for т]^и which differs in sign from
the previous use.) Let M be a differentiable manifold with a Riemannian,
a pseudo-Riemannian or a Lorentz metric g; then the pair (M, g) is called
a Riemannian, a pseudo-Riemannian or a Lorentz manifold. Consider a
Lorentz manifold; then the vectors X € TP(M) are classified by
g(X, X)>0 X is spacelike
g(X, X)=0 X is lightlike (X ф 0)
g(X, X)<0 X is timelike.
Induced metric: We begin with some definitions.
Let / : M —> N be a smooth map with dim M < dim N, then
i) / is an immersion of M into N if the tangent map /» : TP(M) —»
Tf(p)(N) (recall Section 2.6.1) is an injection (one-to-one),
ii) / is an embedding if / is an injection and an immersion. The image
f(M) is called a submanifold of N.
Now, if the map / is an embedding of M into N the pullback induces the
metric on M

Riemannian geometry 455
9м = f*9N A2.21)
or in coordinates
^ A2.22)
where x and j/ are the coordinates of p and /(p).
Examples: Let us consider the unit sphere S2 embedded into the 3-
dimensional space with Euclidean metric (R3,<5). Let (в, <р) be the polar
coordinates of S2; we define the embedding / as the inclusion
/ : {в,ip) >->¦ (j/j = sin в cos ip, j/2 = sin в sin <p, уз = cosO). A2.23)
Then we obtain for the induced metric
9s* = 9\iv dx^ (8) dxv
в + sin2в dip® dip. A2.24)
If we embed the circle S1 into the plane (R2, S) by
/ : (r,<p) !-> (yi = rcostp,y2 = rsin<p) A2.25)
we find for the metric of the plane in polar coordinates
g = dr®dr + r2dip ® dip A2.26)
and for the induced metric on the circle S1 (with constant radius r = L)
gSi = L2d(p ® dip. A2.27)
Isometry: Spaces of constant curvature like the sphere admit a special
symmetry group—the isometry group.
Let (M, g) be a (pseudo) Riemannian manifold, the diffeomorphism / :
M —> M is an isometry if it preserves the metric
Г9/(р)=9р, A2-28)
which means
gm(f*XJ.Y)=gp(X,Y) A2.29)
for the vectors X, Y € TP(M). In coordinate language—x is the coordinate
of p and у of f(p)—we have

456 Gravitation
Isometries form a group—the isometry group—preserving the length of
vectors. For instance, in Rm we have the isometry group
f :xi->Ax + b, A2.31)
with A e SO(m), b e Rm.
Conformal transformation: Let (M,g) be a (pseudo) Riemannian ma-
manifold; the diffeomorphism / : M —> M is called a conformal transfor-
transformation if it preserves the metric up to a scale
), A2-32)
that is
A2.33)
for X, Y € TP(M) and a(p) € F(M) (the set of smooth functions). In terms
of coordinates we write
f?§^. A2-34)
The conformal transformations on (M, g) also form a group—the confor-
conformed group. They preserve the angles between vectors, but not the lengths.
Let g, g' be two metrics on M; if we have
g'p = e2°Mgp A2.35)
we say that g'p is conformally related to gp and the transition gp —> g'p is
named Weyl rescaling. The Weyl rescalings form the Weyl group.
If there is an atlas where we find for each chart (V, <p)
„ A2.36)
with p G V, the manifold (M, g) is called locally conformally fiat.
Theorem: All spaces with constant curvature are conformally flat!
Example: Let us consider the unit sphere S2 with the standard metric
ds2 = de2 + sin2 9 dip2. A2.37)
Then the map
/:(*,?>)-(«,«) A2.38)

Riemannian geometry
denned by
u = log
в
tan —
= <p
457
A2.39)
generates a conformally flat metric. We calculate
du = -de = — log
a" a"
в
tan-
2 sin | cos |
sin0
so that
ds2 = sin2 6»
sin
+
= sin2 0 {du2 +
A2.40)
A2.41)
is indeed conformally flat.
Theorem: Any 2-dimensional Riemannian manifold is conformally flat!
Proof. Let (x, y) be the local coordinates of M with metric
ds2 = gxxdx2 + 2gxydxdy + gyydy2. A2.42)
The determinant of the metric is given by
9 = 9xx9vv ~ gly A2.43)
We rewrite the metric A2.42) in the following way
ds2 = д
+
y/9xx
( / л , 9x
\
dy
J \
9xx
, 9xy-i^fg \
+ "—У dy .
J
A2.44)
Now the theory of differential equations tells us that there exists an inte-
integrating factor A(x, y) = Ai(x, y) 4- i\2(x,y) such that
+
y/9xx
9xv Zl
y/9xx
= du + idv
= du-idv.
A2.45)

458 Gravitation
Defining
-i-= |Л|-2=:е2сг A2.46)
we get for the metric
ds2 = -^z(du2 + dv2) = e2a(du2 + dv2) A2.47)
which is indeed conformally flat. Q.E.D.
12.1.2 Equivalence principle
The basis of gravitation, or general relativity, is the principle of the equiva-
equivalence of gravitation and inertia. It rests upon the equality of gravitational
and inertial mass as demonstrated experimentally by Eotvos and Dicke. It
was Einstein who concluded that we do not detect an external static ho-
homogeneous gravitational field in a freely falling elevator. The reason is that
the observer and the elevator itself react to the gravitational field with the
same acceleration.
Equivalence principle:
• At every space-time point in an arbitrary gravitational Geld it is pos-
possible to choose a locally inertial system such that within a sufficiently
small region the laws of nature are determined by special relativity.
If there is no gravitational field we find an inertial system in the whole
space; the line-element is of the form
ds2 = ¦q^dx» ¦ dxv = -dt2 + dx2. A2.48)
If there exists a gravitational field, then the line-element can be chosen in
the above form only in some small region; in a neighbourhood the system
will be accelerated and the line-element is determined by
ds2 = g^ix) dx» ¦ dxu, A2.49)
where the tensor g^v(x) describes the gravitational field.
The mathematical technique to implement the equivalence principle is
the analysis of tensors, which we will study next. We want to construct phy-
physical equations that are invariant under general coordinate transformations.
So we must know how the quantities appearing in these equations behave
under these transformations.
Coordinate transformations: Assuming a general coordinate transfor-
transformation x —> x'(x) the following quantities transform as

scalar Ф
Riemannian geometry
Ф'(х') = Ф(х)
contravariant vector V*
covariant vector Vu
mixed tensor
tensor density
ca dx0
459
A2.50)
A2.51)
A2.52)
A2.53)
_
dx
w
дха
where 4^r represents the Jacobian of the coordinate transformation x —+ x'
and W is the weight of the density.
Considering, for instance, the metric tensor
дха
dx'" dx'v
we find for the determinant
dx
dx1
_
dx
-2
A2.55)
A2.56)
which is a scalar density of weight W — —2. Hence we can re-express a
tensor density #*" by a tensor T^" and an appropriate determinant factor
" = \g\~w/2
A2.57)
The quantity-
\9'\W/2
dx
-w
dxl
dx'
dx
w
dx'" dx'v
lg\W/2
dxa
A2.58)

460
Gravitation
indeed transforms as a tensor.
The volume element occurring in the integration transforms as
_
dx
Hence the element
A2.59)
A2.60)
—named the invariant volume element—remains invariant under coor-
coordinate transformations
dV =
dx
-l
dx
A2.61)
12.1.3 Christoffel connection and curvature
In Riemannian geometry we also introduce a connection and curvature.
Christoffel connection: For a Riemannian manifold M with metric g^
there exists a unique connection 1-form—the Christoffel connection—
Га т^а j—H (лп f-r)\
0 — * ц0ах i \Ia.O6)
where Г%^ denotes the familiar Christoffel symbol. It is determined by
the metric tensor via formula
A2.63)
A2.64)
and it is symmetric in the lower indices
Га ра
ив — i I-
(we shall demonstrate that it transforms as a connection, theorem A2.82),
later on).
For example, in 2-dimensional Euclidean space (R2,6) all components
of the Christoffel symbol ГаД/д clearly vanish if we refer to Cartesian coordi-
coordinates. But if we consider polar coordinates the metric is given by equation
A2.26); hence
9v» = ( I r°2 ) • A2-65)
Then formula A2.63) provides the Christoffel symbols explicitly

Riemannian geometry 461
Tvrip = ¦z94>4> (~
2
rrw = -г A2.66)
and all other components vanish
T\r = Trrtp = rV = T"rr = Г%„ = О. A2.67)
Analogously, when embedding the unit sphere S2 into Euclidean space
(R3,i5) we have the induced metric (recall equation A2.24))
and we calculate from formula A2.63)
rew = - sin в cos в
Г% = r?!?;e = cot^; A2.69)
all other components vanish
в в % r*w = 0. A2.70)
Covariant derivative: We define the covariant derivative as we did
in gauge theories
V = d+[T,], A2.71)
with
V = VMdi't, d^d^dx», Г = Т^х» A2.72)
and Г is the matrix notation of Га^. Applied to a tensor-valued p-form
l ^Ч^^ A2-73)
we have
dT + [T,T], A2.74)
where the commutator means (analogous to definition B.353))
[T,T] -> [Г,Т]|;;; =
— Та„Тр" + all upper indices
- (-)pV:"Tv0 - (-)P a11 lower indices A2.75)
(the factor (-)p stems from interchanging the 1-form Г with the p-form T).

462 Gravitation
Definition: A connection is called a Riemannian connection if
i) the metric is covariant/y constant [
Veff^, = 0 A2.76)
ii) the connection has zero torsion
Ta = Vdxa = 0. A2.77)
A Riemannian connection is sometimes also called the Levi—Civita
connection in the literature.
Torsion: Let us consider the torsion introduced above. The torsion 2-
form
Ta = \та ^dx»dxv A2.78)
2
Ta = Vdxa = ddxa + [Г, dx]a
= Tavdxv = V^dxTda?. A2.79)
is defined by
So the torsion-free condition implies
Т%„ = rV - rV = 0; A2-8°)
this is the symmetry of the Christoffel symbol
Г%„ = TV A2.81)
Note: Here dxa is thought of as the component of a vectorial 1-form
The index a is a space-time index, therefore we can contract the indices as
in equation A2.79) and we get a 2-form. For this reason we find a concept
such as torsion in gravitation but, in contrast, not in YM gauge theories.
Theorem: Conditions i) and ii) determine the Christoffel connection Г
uniquely in terms of the metric g^ and its derivative д\д^„ via formula
\ A2.82)

Riemannian geometry 463
Proof. Using the constancy of the metric
= Q A2.83)
we have for the components
a0v = 0. A2.84)
Adding and subtracting cyclic permutations, respecting the symmetry of
the Christoffel symbols, we find
= 2gmTav0, A2.85)
which is
Tav0 = \gaX{-dxgv0 + dug0X + dpgx,,). Q.E.D. A2.86)
Curvature: The curvature 2-form R associated with the connection is
given by the Cartan structure equation
pa ,-VPa i ра т~»А /in q*7\
zt 0 = Я1 0 + 1 \l p [1.Z.OI)
or briefly in matrix notation
J"P I 1~*2 /in QQ\
= (XI + 1 . A.Z.OO)
On the other hand, we write for a 2-form
Да „ _ ра //т^/^т" (Л 9 8СЛ
where Ra0^v denotes the Riemann tensor. Then equation A2.87) corre-
corresponds to
The Riemann tensor is antisymmetric in the following indices
Da "not
doc p a /in Q-i \
imd satisfies the cyclicities^—the Bianchi identities
Я<Х I DOC i pO f\
«"/^m = 0. A2.92)

464 Gravitation
We rewrite the last equation with differential forms
VR = 0. A2.93)
We emphasize that Ra0 represents a matrix valued 2-form and it must not
be confused with the Ricci tensor
Щи = Ra0«v A2.94)
The Ricci tensor is a symmetric (in E and u) 0-form which we need rather
rarely. Its contraction gives the Ricci scalar
П = g0vn0v. A2.95)
Note: These contraction operations are not possible in YM gauge theories
since there we have mixed indices—space-time and gauge group indices—
whereas in gravitation there are only space-time indices.
The Riemann curvature tensor is also determined by the commutator
of the covariant derivatives analogously to the gauge case (recall theorems
C.302), C.308) and our discussion of differential forms in Section 6.2).
Theorem:
[V^V^^F^. A2.96)
Applying the commutator to vectors we obtain
[VM,V,,]Ae = [V^,Vl/}a0A^ = Ra0^
[VM,V,,]A0 = \yit,Vv\fiaAa = RpaltvAa
= -AaRa0tlv A2.97)
and to tensors we get
= -TX0Rxatlv-TaXRx0tlv. A2.98)
So applying the commutator to tensors gives:
Theorem:
[V^V,,] = [R^, ]
[V^.V^T = [ДД„,Т] A2.99)
or in components

Riemannian geometry 465
[V^, V,,]^";;' = RP'xpvTp" + all upper indices
- Txa-Rx0liV - all lower indices. A2.100)
Within differential forms we have:
Theorem:
V2 = [R, }. A2.101)
Of course, we can quickly verify relation A2.101) by direct calculation (recall
the gauge case, equation F.56)).
Remark: However, in the case of a nonvanishing torsion
we would get
[V^, Vv]Aa = Ra0^ + (ГАм„ - Гл„д)УАЛа. A2.103)
12.1.4 Variations and derivatives
Variations: Later on, when we study the action principle, we need to
know the variations of fields and tensors when the metric is varied
a (r\ _> a (rL-f>a (r\ ('12 104'»
Here we consider the change of the quantities j**", g and TZ^. Let us start
with the relation
9 gv\ — o^\. (lz.105)
Let 6 be some variation, then we have
bg^gvx + д^бд^х = о A2.Ю6)
yielding
a\9 ¦ A2.107)
For the determinant we vary the formula
g — ucl gin/ — с , t iz.iuoj
which gives
eg = дд^бд^и A2.109)

466 Gravitation
f>V\9\ = \у/\9\аГбд1и,. A2.П0)
We use formula A2.63) to reformulate the contraction of the ChristofFel
symbol
A2.111)
Choosing as a variation the ordinary derivative 6 —> d^ we find
Yaaii = Talia = ^=dli^\. A2.112)
Vlffl
Theorem: Palatini identity
Ortfiv = VaOl уд — Vv0L aii. A2.11o)
(Note that 6Tavll is a tensor; whereas the analogy in the YM gauge case
6Aa is a vector.) ^
Proof.
Щи = Ranau = даГаи„ - d^V
¦ pa -пЛ _ pa pA /-jo 11Д\
hence
д. A2.115)
On the other hand, we calculate the expressions
д A2.116)
„ A2.117)
and we find
Q.E.D. A2.118)

Riemannian geometry 467
Derivatives: There are important cases where the covariant derivative
simplifies to an ordinary derivative.
i) Take some scalar Ф; then we clearly have
V^$ = д„Ф A2.119)
since [Г„,Ф] =0.
ii) For a covariant vector Vv we get
VMVL, = d»Vv + [Г„, V]v = d»Vv - VxTx^ A2.120)
so that the difference is
ЧуУ» - VVV» = d»Vv - д„У„. A2.121)
iii) For a contravariant vector V* we observe the following important
property
V^F" = d^V» + [TMIVY = d^V» + T^^XVX. A2.122)
Inserting the Christoffel contraction A2.112) we obtain
or
vlffl
A2.123)
A2-124)
As a consequence we find a covariant version of the Gauss theo-
theorem
Jd4x,/\g~\ V^V = Jd4xd^V\9\V^) = У ^мЛ/Ы ^ -* 0
A2.125)
if V* -+ 0 at the boundary. Note the appearence of the determinant
i/\g\ to achieve an invariant volume element,
iv) For a contravariant tensor we analogously have
-^^ШП+Г"^. A2.126)

468 Gravitation
If the tensor is antisymmetric
^A-^ra = -T^m A2.127)
the last term vanishes (since Г"^\ is symmetric) and we get
VMT?m = -—. d^yM 72?m). A2.128)
v) A covariant tensor gives
м - ТмЛГАа1/. A2.129)
Again, if the tensor is antisymmetric all Г-terms cancel in the sum of
cyclic permutations
у n-asym , у /т-asym . ^7 /т-asym _ q /т-asym . a /т-asym , q /т-asym
A2.130)
vi) For a mixed tensor we find
A2.131)
A2.132)
In the difference all Г-terms cancel since the F's are symmetric
A2.133)
This is an ordinary derivative and vanishes upon integration by virtue
of Gauss' theorem A2.125) if we assume T —* 0 at the boundary.
12.1.5 General coordinate transformations
We have to study more precisely the transformation laws of tensor quan-
quantities. We consider a general coordinate transformation x —> x'(x); then a
tensor-valued p-form transforms as
T(x) x-=3' T'(x') = Л-1(х)Т(х)Л(х). A2.134)

Riemannian geometry 469
The coordinate transformation matrix
Ju.
A2.135)
acts on aJJ upper coordinate indices of the tensor-valued p-form and the
matrix
acts on all lower indices. Clearly, we have
(Л-уЖа = 7ГТ~гГ^ = 6lt<T' A2.137)
In components we write
T'(X') -> -J't:. a,...u (
A2.138)
The Christoffel connection transforms as
Г(х) *^? Г'(х') = Л-1(х)[Г(х) + d]A(x) A2.139)
(here d acts only on Л), which follows either from formula A2.63) or from
I he requirement that VT transforms as a tensor.
The curvature transforms again, tensor-like
Д(х) x^ R'(x') = Л-1(х)Д(х)Л(х). A2.140)
Diffeomorphism group: The group of coordinate reparametrizations—
I he diffeomorphism group—actually allows locally for two representations:
i) passive coordinate transformation: all tensor-valued forms
change according to
T(x) x^ T'(x')
and the integration measure changes too
dV(x) x^i dV(x').

470 Gravitation
ii) active coordinate transformation: all tensor-valued forms vary as
T(x)-+T(x).
So the forms are shifted back to the point x and the integration mea-
measure dV(x) remains unchanged.
We work with an infinitesimal coordinate transformation
ia^i'a=ia-f(i). A2.141)
Then the transformation matrices are infinitesimal as well
^Wjdx) ' = «¦/> + («*)¦* A2Л42)
where л
Щ^ A2-143)
In matrix notation we write
А~г(х) = l-vfa)
A(x) = 1 + v^x), A2.144)
with the infinitesimal coordinate transformation tensor
ve = d?. A2.145)
Passive coordinate transformation: We find the infinitesimal passive
change in the following way
T(x) x-=3' T'(x') = A-\x)T(x)A(x)
= Т(х) + 6сщТ(х). A2.146)
The index с stands for coordinate transformation and the prime for passive.
Inserting expressions A2.144)
T'(x') = (l-Vc(x))T(x)(l+t;c(x))
= T(x) + [T,«c](x) A2.147)
(the commutator contains contractions of the tensor indices only) we get the
infinitesimal passive coordinate transformation of a tensor form
Sc4T(x) = T'(x') - T(x) = [T, t/€](x). A2.148)

Riemannian geometry 471
Specifically, for the curvature tensor 2-form we have
6*(R(x) = R'(x') - R(x) = [R, uc](x) A2.149)
and for the Christoffel connection 1-form
^Г(х) = Г'(х') - Г(х) = V«c(x)
= d«€(x) + [r,«c](x). A2.150)
Lie derivative: The active change, on the other hand, is closely related to
the Lie derivative which we have introduced in Section 2.6. Here we need the
Lie derivative along the flow of the vector field ?a given by the infinitesimal
coordinate change A2.141). We can take over the formula B.356), derived
in Section 2.6.5, the Lie derivative along ? of a tensor-valued p-form
W." = К"^::,м1...м1>)
- ^Г^;;;М1...Мр(х) - all upper indices
+ V!";^...^(x)d^v + all lower indices
+ TJ3.::,v...vP(x)dnAv + a11 form indices]
~ dx ... dx"". A2.151)
pi
In matrix notation we write
Ci = ?-д-[д?,], A2.152)
where here the commutator definition aJso includes the contraction of the
form indices!
In terms of the interior product, however, we have the formula (recall
equation B.359))
- [ve, T)
[«c, ], A2.153)
where now the commutator contains only the contraction of the upper and
lower tensor indices as in definition B.353) (remember that ? ¦ д together
with the contraction of the form indices gives precisely the interior product
t<-rm, equation B.358)).
Let us collect the necessary properties of the Lie derivative
ii)

472 Gravitation
iii) [?Cl,tfo]=tKlifo]
iv) [?«..?&]=%,&]. A
where the Lie bracket is denned by (recall equations B.340), B.341))
A2Л55)
The Lie derivative depends only on the structure of the manifold and
not on the connection. For a Riemannian connection, which is torsion-free,
we can replace the ordinary by the covariant derivative in formula A2.152)
?c = ? ¦ V - [V6 ] A2 156)
since
] = ГС A2.157)
(recall that torsion-freedom implies symmetric Christoffel symbols).
Note: However, in the case of a nonvanishing torsion
Тах0 = ГаХ0-Га0хфО A2.158)
we find instead
% = Z ¦ V - [У0Г + TaX0e, }, A2.159)
which can be quickly verified
] = Се. A2.160)
Active coordinate transformations: Now we turn to the active change^
of a tensor field. We perform a Taylor expansion to first order in ?
^;;m...(x). A2.161)
We subtract the initial field
= T'g;;;,M...(x) - 7^;;;/1...(x) - ?A9AT^;.^...(x) A2.162)

Riemannian geometry 473
and insert for the l.h.s. the commutator result A2.148). In this way we find
the infinitesimal active coordinate transformation of a tensor field
= ?cT?;;M...(z). .A2.163)
Here the commutator also includes the contraction of the form indices [i
If we multiply equation A2.163) by the wedge product ^dx^1 .. .dx^ we
obtain the change of a tensor-valued p-form
«?T(x) = T'(x) - T(x) = ?cT(x) A2.164)
(written in matrix notation) and using formula A2.153) further we get
6%T(x) = (tcd + dtc)T(x) - k,T](x). A2.165)
Specifically, the change of the curvature 2-form is
b\R{x) = R'{x) - R(x) =
= (ied + diz)R(x) - [ve, R](x); A2.166)
t he change of the metric is
A2.167)
I>ct us calculate result A2.167) explicitly. Using the covariant form A2.156)
and the covariant constancy A2.76) of the metric we get
= VM^ + V^M. Q.E.D. A2.168)
Expanding the Christoffel connection instead of a tensor field
Г'Х*') - T%0(x) = T'a^0(x) - T%0(x) - edxT%0(x) A2.169)
we find analogously the infinitesimal active coordinate transforma-
transformation of the Christoffel connection
A2.170)

474 Gravitation
The multiplication with dxM gives the transformation of the Christoffel
connection 1-form (written in matrix notation)
^Г(х) = Г'(х) - Г(х) = С^Г(х) + dv^. A2.171)
So the connection picks up an additional term dv^. Re-expressing the Lie
derivative in terms of formula A2.153) gives
A2.172)
These active coordinate transformations are also named Einstein trans-
transformations. N
Let us eventually collect the passive coordinate transformations I
6%' T = [T, v^\ tensor-valued p-form
E=' Г = V«? Christoffel connection 1-form A2.173)
6!* и = 0 scalar-valued p-form
and compare with the active ones
CCar / ¦ j - j- \rr- . cc' rr
Ofl = \l?U + Ul?)J + 0v 1
CCX^ t' Л i J' \T^ i CC T"l
дЛ = (l?U + Ul?)L + 0v 1
A2.174)
then we find for the difference
A2.175)
Killing vector: Let ? = ^д/дх* 6 Tp(M) be a vector field on a Rie-
mannian manifold (M, g) and let the infinitesimal displacement s? generate
an isometry—the isometric map
/:/-t/ + eC. A2.17&)
This means that the metric is preserved (recall equation A2.30))
У-...(,)¦ (ШП)
The expansion to order ? gives
v + ed^^u + eduf) = 9lu,(x)

Riemannian geometry 475
which implies the Killing equation
d^ = 0. A2.178)
The solution ? is called the Killing vector. It plays an important role in
the study of the symmetries of manifolds.
The Killing equation can be rewritten in terms of the Lie derivative
(recall equation A2.168))
х) = VM&, + У„ем = О; A2.179)
more explicitly
d^u + dvZp - 2?ЛГЛМ1/ = 0. A2.180)
For example, let us consider the familiar Minkowski space (R4, rf), which
is flat, hence all Christoffel connections Глм„ vanish. Then we get for the
Killing equation
дц?и + dv^ = 0, A2.181)
with the general solution
?M = V,x" + Ь„, а„и = -avix. A2.182)
This is precisely the infinitesimal transformation of the Poincare group con-
containing 10 free parameters (m(m + l)/2 in m dimensions).
Finally, if we allow the map / A2.176) to be conformal instead of
isometric then the metric is conserved only up to a scale (recall equation
A2.34))
and the conformal Killing vector ? is determined by the equation
?<: 9liV = 2ад^. A2.184)
Consider, for instance, the Euclidean space (Rm,6) with x^ the coordi-
coordinates, then the dilatation vector
^s«^ A2.185)
(•'presents a conformal Killing vector since we have
= 26^,. A2.186)

476 Gravitation
12.2 Tangent frame
Later on we also introduce matter fields, where we are mainly interested
in fermions which are described by spinor fields. The spinor, however, does
not 'know' how to transform under a general coordinate transformation—
the diffeomorphism group. The mathematical reason is that the GL(m, R)
group has no spinor representations. (By a spinor representation we mean
a finite dimensional double-valued representation, for an infinite double-
valued representation, see [Ne'eman, Sijacki 1985].) So how can we incorpo-
incorporate spinors into general relativity? Spinors obey Lorentz transformations,
hence we must work with the group of local frame rotations SO(m) or
SO(lym — 1)—the Lorentz group. This change from diffeomorphisms to
Lorentz transformations can be achieved by the vielbein.
12.2.1 Vielbein
For spinors we need a tangent frame, and we establish a tangent frame
which is orthonormal everywhere in the whole space. Due to the equi-
equivalence principle we can find at every point xo a set of coordinates ??0>
a = 1,..., m = dimM, which are locally inertial at xo. Then the metric
in any noninertial system is given by .
= W\{x)e\{x), A2.187)
where
e\(x) = d&^x). A2.188)
The Minkowski metric г]аь can also be replaced by the Euclidean metric
Sab. Of course, at a point different from xo we have to choose a new, diffe-
different set of coordinates—unless space-time is flat—in order to achieve eoM
be orthonormal. The quantity eaM is named vielbein—zweibein in two
dimensions, vierbein in 4 dimensions. The Greek letter index fj, refers to
the original coordinates; the Latin letter index a refers to the new, locally
inertial ones and is the index of the orthonormal tangent frame.
Introducing the vielbein 1-form
ea(x) = e%(x)dx" 6 Т;(М) A2.18$)
the metric can be rewritten by
9P = 9^(x)dx" ® dxu = vabea(x) ® e\x). A2.190)
The vielbein eaM(x) is a hybrid object. Under general coordinate transfor-
transformations x —> x' it behaves like a covariant vector
e\{x) x^5 e'V*') = ^e\(x) A2.191)

Tangent frame 477
but under local frame rotations it transforms as
e\(x) - e'%(x) = (L-^z^VV*). A2-192)
The matrix Ьаь(х) € 50A, m — 1) or SO(m) represents the local Lorentz
transformation satisfying
r,abLac(x)Lbd(x) = Vcd. A2.193)
FYom the metric A2.187) we see that the vielbein eaM—the orthonormal
tangent frame—is defined only up to local frame rotations; the rotation
A2.192) satisfies the metric A2.187) as well.
Definition: Denoting the inverse vielbein by E = e, it is denned by
ES(x) = ПаЬ9'мЧх)е\(х). A2.194)
It has the following coordinate and Lorentz transformations
Ea»(x) -> E'a»{x') = —Eav{x) A2.195)
?."(*) - E'a»(x) = Eb»(x)Lba(x) A2.196)
and the vector field
Ea(x) = ЯЛ*)^ 6 TP{M) A2.197)
is the dual quantity to the 1-form ea A2.189). The vielbein and its inverse
Nutisfy the equations
ea ¦ Eb = е\Еъ>* = 6ab A2.198)
Еа»еа„ = 6»и A2.199)
)Еа»{х)Еъ"{х). A2.200)
('onsidering the vielbein determinant
e:=|dete%| A2.201)
»r find from relation A2.187)
g = det^ = ±(deteaMJ, A2.202)
with ( + ) for a Euclidean metric and (—) for a Minkowski metric; hence
e = y/\g\. A2.203)

478 Gravitation
Returning to the oriented invariant volume element for a manifold M
we then have
= edx1 ...dxm = e1 ...em, A2.204)
where clearly the dots denote a wedge product.
Variations: For a general variation of the vielbein we get (analogous
to the metric equations A2.107)-A2.110))
<5e% = -e°v8Ebl'el'й A2.205)
be = еЕ^Ье*^ -т-%- = еЕа» A2.206)
which implies for the contracted ChristofFel symbol (recall equation
A2.112))
TV - Г%а = - 0Me. A2.208)
In particular, we need the infinitesimal active coordinate transforma-
transformation of the vielbein (we thereby lose the orthonormality)
^e%(z) = e'%(z) - e%(z) =
= ^dveati{x) + eal/(x)dti^ A2.209)
or replacing d —* V in ?%
<5|e%(z) = TV,,e%(z) + е°„(х)УйГ- A2.210)
For the vielbein determinant we then find
6\e = еЕа»6\еац = ivdve + edv?v. A2.211)
We shall use all these formulae frequently in our anomaly calculations later
on.
Reference system: Now, we can take a contravariant vector field
and use the vielbein е°й(х) to refer its components to a coordinate system
which is locally inertial at x
Va - e%(xO"(x). A2.212)

Tangent frame 479
Analogously we can refer covariant vectors V^(x) to a locally inertial system
and, generally, also tensor fields
Vb = ^(я)Я"ь(я) A2.213)
Tab = еац(х)Т»„(х)Еъ"(х). A2.214)
So we use the vielbein ea^ and its inverse Еа^ to refer a tensor quantity
from a coordinate basis to an orthonormal tangent frame and vice versa.
Note: Actually the coordinate system (CS) is also a tangent frame, but it is
not orthonormal in the whole space (unless space-time is flat). In contrast,
what we call the tangent frame (TF) is the frame which is an orthonormal
tangent frame in the whole space.
12.2.2 Spin connection and curvature
Spin connection: Certainly, we can also define the connection in refe-
reference to the tangent frame e°—in fact, when including spinor fields into
gravity we have to. Then we call the connnection: spin connection 1-
form
шаь = шаЬсес = ujaby.dx)i € Lie SO(m) <8> Л1 or
€ Lie 50A, m-^^A1. A2.215)
This is the connection which is contained in the Dirac operator, in the
fermionic action—as we shall see in Section 12.4—and which interacts with
the fermions, the spin 1/2 particles. In the tangent frame the gravitational
formalism appears quite analogous to the YM gauge case.
Covariant derivative: The covariant derivative is denned by
D = d + [u,}, D = Dcec = D^&x» A2.216)
and acts on tensor-valued p-forms which are referred to the tangent frame.
Curvature: The curvature 2-form R associated with the spin connec-
connection w is, again, given by the Cartan structure equation
R = duj + uj2 matrix notation
Rab — dujab +u)acwcb component notation. A2.217)
On the other hand, as a 2-form the curvature is expressed by
\d, A2.218)

480 Gravitation
where eced denotes the wedge product-^ind Rabcd the Riemann tensor re-
referred to the tangent frame. Clearly, with the help of the vielbeins we can
change to the coordinate basis
1 а с d
2
= ]:RabnVdxtidxv A2.219)
and
jr\ ot jja A) nc „d na /1 о ооп\
"a ¦"- bcd& 0° Ц.& v — ¦"- @)iv \i-&.ZiZ\l)
As we discussed before, the curvature is also determined by the commu-
commutator of the covariant derivatives.
Theorem:
[JJ^jUn] ;, = К b/iv A2.221)
[D^jD^b — [Rfiv, }аь for tangent frame tensors A2.222)
D2 = [R, } within differential forms. A2.223)
The Bianchi identity can be written as in the YM case
DR = 0. A2.224)
Torsion: The torsion 2-form is defined by
Г = De = de + we matrix notation
T° = (De)° = dea + шаьеь component notation. A2.225)
Applying d gives
dT - dDe = d?e + due - wde
= Re-шТ A2.226)
which is called the torsion consistency condition.
Again, we use the vielbeins to establish the bridge to the coordinate
basis
T° = ^ Tabcebec = ^ Tabcebllecvdxtidxv
and

Tangent frame 481
rr>Q, ППО, ?.Ъ С
VV A2.228)
Definition: As in the coordinate basis the connection is called the Rie-
mannian connection if
i) the metric is covariantly constant
DcVab = 0 4=> U)ab = ~ШЬа A2.229)
ii) the connection has zero torsion
Ta = 0. A2.230)
Theorem: The Riemann conditions i) and ii) determine the spin connec-
connection ш uniquely in terms of the vielbein e° and its derivative dea via the
formula
= 2 i^bc + tbca ~ Scab), A2.231)
with
ebc = dea(Eb,Ec). A2.232)
Proof. First, we notice that the covariantly constant metric
+ [ш, rj\ab = 0
Г)сЬШса + T)acU)Cb = 0 A2.233)
implies the antisymmetry of the spin connection matrix
= -Wbo. where ujab = r)acujcb, A2.234)
which is equivalent to шаь being an element of the Lie algebra Lie S0(m).
Next, we define a quantity ?°ьс by
dea =: \ ebcebec, A2.235)
i lien the torsion-free condition gives
T° = dea + шасес = О
^€,аЪсеьес + шасЪеъес = 0; A2.236)
||1ЧКЧ!
Uc = bJabc ~ Uacb- A2.237)

482 Gravitation
We raise and lower the tangent frame indices with the metric tensor r]ab.
Next we add and subtract the cyclic permutations of equation A2.237) and
we find the explicit expression for the spin connection
Uabc = gteoic + &» - Scab)- A2.238)
We now rewrite the quantity ?°ьс, denned via equation A2.235). We evaluate
the wedge product ebec at the vector fields Eb, E5 (analogous to equation
B.111) of Section 2.4)
ebec(EhEc) = (eb® ec - ec ® eb)(Eb,Ee)
= 6b-b6ce-6c-b6be; A2.239)
then we get
dea{E-b, E-c) = i Zabcebec{E-b, Es) = ?abE. A2.240)
Note that ?°ьс is antisymmetric in be. So finally in terms of components we
have
E & —
' с dxV
e dxdx (Eu
dx» v \b дх<*
со., -
4 bc ~
= {d^eav-dveatt)Eb^Ecv. Q.E.D. A2.241)
Remark: The zero torsion implies by virtue of condition A2.226)
Re = 0 A2.242)
which is the first Bianchi identity of the curvature tensor
Rabcd + Racdb + Radbc = 0. A2.243)
12.2.3 Examples in 2 dimensions
Now let us illustrate the concepts introduced on the 2-dimensional sphere.
Metric: Generally, the metric is expressed by
9p = 9^{x)dx't ® dxv = r)ab ea{x) ® eb(x); A2.244)
here, on the sphere S2 with radius L we have the standard metric A2.24)
= L2dd ®de + L2 sin2 6d<p <8> dip
= e1®e1+e2®e2 A2.245)

Tangent frame
or, written as a line-element
ds2 = L2d92 + L2 sin2 9d<p2
Zweibein: So the zweibein 1-form is
with
and the zweibeins are
e1 =
a = 1,2,
e2 = L sin 6«<p,
= Lsm9
e\ = L e2e=0.
The exterior derivative as a 1-form is
д д
dtp 89 '
so that we get
de1 =0
The inverse zweibein
de2 = Lcos9 d9dip.
p _ p v ® , p в 9
^a — &а -^ г Ilia -^
dtp 89
luis to satisfy the system of equations
е\Еь^ + еавЕьв = 6аь,
implying the components
2 ~ Lsin0
iiikI the vectors
л —
Ел = -
483
A2.246)
A2.247)
A2.248)
A2.249)
A2.250)
A2.251)
A2.252)
A2.253)
A2.254)
A2.255)
Spin connection: The spin connection 1-form шаь = uabcec is given
l.v formula A2.231), A2.232).

484 Gravitation
Let us first calculate the quantities
?Ъс = de1(^b,^c)=O
?2ъс = de2{Eb,Ec)=Lcos9d9d<p{Eb,Ec). A2.256)
Generally we have (analogous to equation B.111) of Section 2.4)
dedp(Ea, Eb) = EjEb* - ЕьвЕа* A2.257)
and particularly
dedy(Ex,E2) = -dediP(E2,E1) = L2\n0, A2.258)
so that we get
2 2 cosfl
* *
Then formula шаьс — (?аЬс + ?ьса ~Ссаь)/2 provides the (nonvanishing) spin
connection components
^ A2.260)
L sin 9
and the spin connection 1-form
-ш12 =ш21= w2i2e2 = cos 9dip. A2.261)
Curvature: For the curvature 2-form given by the spin connection
Rab = dtjab + шасшсъ A2.262)
we find specifically in 2 dimensions
-Д21 = Д^ = dui12 = smBdedip = -^ еХ<?
R11 = Д22 = О, A2.263)
so that
Rab = jp eaeb. A2.264)
So the Riemann tensor itself (recall Rl2 = Д1212б1е2)
-R}221 = Rl212 = jp A2.265)
expresses the constant curvature of the sphere S2.

Tangent frame
485
JVote: Of course, in this simple 2-dimensional example we can deduce the
spin connection even more quickly from the zero torsion condition A2.230)
(which in fact implies the formulae we used)
T° = (De)a = dea + шаьеъ = О.
From the antisymmetry of шаь we know that
Шц = u>22 = 0,
then we get for a = 2
2 21e1 = 0
ш2\йв = cos 6
or2! = cos 6d<p.
A2.266)
A2.267)
A2.268)
5"-sphere: If we generalize the above calculations to the n-dimensional
sphere Sn using stereographic projection coordinates, we find for the infini-
infinitesimal line-element
A2.269)
with
and for the vielbein
The invariant volume element is
0=1
dxa
A2.270)
A2.271)
A2.272)
giving the 'volume' over Sn
A2-273)
Tin; spin connection 1-form turns out to be

486 Gravitation
(А = Х 7 Х Л\л , A2.274)
2^2 [l + Ш]
supplying the curvature
Rab = -L e°eb, Rabcd = ^{SacSM - SadSbc) A2.275)
and the Ricci quantities
L2 L2
R2-plane: Finally, let us consider the R2-plane; we express the line-
element by polar coordinates
ds2 = dr2 + r2dip2 = (e1J + (e2J. A2.277)
Then the zweibeins
e° = eardr + eavdip A2.278)
follow as
e1 = dr e2 = rdip
The exterior derivative
gives
de1 = 0,
so that we find the connection
T2 = (DeJ =
hence
The curvature clearly vanishes
-Rl2-
as it should be on a plane.
e2r = 0
e2^ = r.
— dr + — dip
dr dip
de2 = drdip,
1-form from the zero torsion
de2+a;21e1 = 0
u}2\dr = dip dr;
= ш21 = dip = —.
r
= R2! = dw2i = 0
A2.279)
A2.280)
A2.281)
condition
A2.282)
A2.283)

Tangent frame 487
12.2.4 Relation between Christoffel- and spin connection
For a Riemannian manifold with conditions A2.76), A2.77) or A2.229),
A2.230) there exists one unique connection which is given by the Chri-
Christoffel connection Г if we refer to a coordinate system (CS) or by the spin
connection ш if we refer to the tangent frame (TF). The associated curvature
is then determined by
Д(Г) = dV + Г2
in CS
in TF.
A2.284)
A2.285)
Again, we emphasize that Д^Г) and Rab{w) describe one and the same
geometrical object—namely the curvature—which is either referred to CS or
to TF. With the help of the vielbeins we can shift the curvature—in general
any tensor—from its CS reference to the TF reference and vice versa
Ra0 =
A2.286)
In order to ensure this relation for the curvature, the connections Г and ш
must transform into each other in a definite way. Alternatively, the connecti-
connections Г and ш depend on the metric g^u and on the vielbein e°^ respectively
(recall formulae A2.63) and A2.231), A2.232)), g^ and e% are closely
related to each other, consequently Г and ш must be related too.
Let us find this relation. First, we notice that
bu = 0
(recall the covariant constancy of the metric); hence
?b"Vae% + ?°"Уаеьй = О.
Secondly, by using
dea = Ve° =
A2.287)
A2.288)
A2.289)
we rewrite the torsion
T° = dea
The zero torsion condition then gives
°cb - uabc}ebec. A2.290)
% % + шасЪ - uabc = 0 A2.291)
to which we subtract and add cyclic permutations. Using identity A2.288)
iiiui the antisymmetry of ы„ь we obtain

Gravitation
VcebM = -Eb»Vce\.
488
Noting that
we find the desired relation:
Theorem:
Here the covariant derivative V acts only on the vielbein.
Next we turn to the covariant derivative
A2.292)
A2.293)
A2.294)
A2.295)
and we find:
Theorem:
Dab =
Here the derivative also acts further to the right.
A2.296)
A2.297)
Note: As emphasized before, there is only one covariant derivative. If we
refer to a CS it is V containing the Christoffel connection Г, if we refer to the
TP it is D containing the spin connection w, and both can be transformed
into each other via the vielbeins.
For instance, if we apply the derivative to a vector we have
DVa = {DV)a = DabVb =
To reverse the above relations we calculate
we use
dEb>*eb0
b0 + Eb^de^ = 0
following from E e = 1 and we obtain the relation:
A2.298)
A2.299)
A2.300)

Tangent frame
489
basis
metric
connection
covariant derivative
curvature
torsion
tensor
coordinate system
Г = T^dx»
V = d + [Г, ]
R = dT + Г2
Ta = Vdxa
tangent frame
{еа},Ш
Vab, Va = r)abVb
w = u)cec
D = d+[u>,]
R = dw + <J2
Ta = De°
4::
Ikble 12.1.
coordinate system
dx»
д
дх»
9„Л
Г =
V =
Ta _
= ?о^(х)е°(х)
= е%(х)?о(х)
i) = »?аье%(х)<
E{De)
EDe
= EaaRabeb0
= EaaTa
/Га q-a...
transformations
'К(х)
eb0...
tangent frame
e»(x) = e\{x)dx»
p ( \ p ц( \ &
%ь=5м^И^ом(а;)?ь1/^)
w = e(V?)
ft b — e ari 0Hib^
Ta = eaaTa
тъ.::=еа<*---т0а::Еъ0---
Table 12.2.
Theorem:
¦ 0 = ba ae 0 + tia w ье /3
Va^ - EaaDea0.
= EaaDe%
A2.301)
A2.302)
Again, in equation A2.301) the derivative acts only on the vielbein, whereas
in equation^ A2.302) the derivative acts further to the right.
Let us summarize in our practical matrix notation:

490 Gravitation
Theorem:
w = e(VE) = edE + еГ?
Г = E(De) = Ede + Еше
D = eVE = e{d + T)E
V = EDe = E{d + ui)e. A2.303)
Finally, we collect the discussed quantities for the two kinds of reference—
CS or TF—in Tables 12.1 and 12.2. What we observe is the following.
We may regard Гар as a gauge transformation of ыаь with the viel-
bein ea0 € GL(m, R) as a gauge element. Similarly Vap represents a
GL(m,R) gauge transformation of Ваь, the curvature Rap a transforma-
transformation of Rab, the torsion Ta a transformation of T°, generally a tensor T^y
a gauge transformation of T?~.
12.2.5 Local Lorentz transformations
As we have discussed already, the vielbein is determined only up to a frame
rotation—a local Lorentz transformation Ьаь{х) € 50A, m — 1) or S0(m)
E -+ EL. A2.304)
Under such a rotation the following Lorentz transformations hold
oj —> L~ (oj ¦
D -+ L~lDL
R -* L~lRL
Л. —У ±J A.
T -> L~lTL, A2.305)
where we use the convenient matrix notation; it is understood that L acts
on all upper indices and L on all lower indices. All equations follow when
using expression A2.294) as a definition for the spin connection. These trans-
transformations are quite analogous to the gauge transformations in YM gauge
theory—the Lorentz transformation L(x) plays the role of the gauge group
element g(x)—and we regard the equations A2.305) as gauge transfor-
transformations, the Lorentz group as a 50A,m - 1) or S0(m) gauge group.
Mainly we consider an infinitesimal Lorentz transformation
A2.306)

Action principle 491
then the 'rotation angle', the matrix ааь, is antisymmetric
o-аь = -aba- A2.307)
This is a consequence of the definition of the Lorentz transformation
rjabLacL d = Vcd
hence
Vabaac8bd + r)ab6acabd = 0
adc + acd = 0. A2.308)
For our purpose—the study of anomalies—we work with the infinitesimal
Lorentz transformations
oae —
bLaE =
6%ш =
%R =
—ae
Ea
Da
\R,a
= -aT. A2.309)
Again, we regard these equations as gauge transformations with the
gauge element a{x) analogous to the YM case.
12.3 Action principle
In order to derive the dynamical equations for a physical system we use the
principle of least action. It means that the action remains stationary
with respect to small variations of a field. This view provides the important
bridge between symmetry and conservation law.
12.3.1 Einstein-Hilbert action
The theory of general relativity describes the dynamics of geometry. The
corresponding action—the Einstein-Hilbert action—is given by
= y I (Tx у/\д\П, A2.310)
where (c = 1)
к = 8nG, A2.311)
<>' denotes the gravitational constant, and TZ is the Ricci scalar A2.95).
I'lie factor 1/2k is chosen to reproduce the correct Newtonian limit in the

492 Gravitation
presence of matter. Clearly Seh 1S a scalar under general coordinate trans-
transformations.
We vary the metric arbitrarily, where the variation should vanish at the
boundary of the integration domain (including infinity)
9^{x) - 9lu/{x) + %„(*), 69llv{x) W-^™*" o. A2.312)
We consider the metric gliv{x) as an external gravitational field. Then we
apply the action principle
° A2.313)
to gain the equations of motion, the Einstein equations. We have to vary
A2-314)
= 6уДд~\ д^П^ + v^f бд^П^ + у/\д~\ д^бП^.
For the variation of the Ricci tensor we use Palatini's identity A2.113);
multiplication with the metric gives
V\9~\ g^STl^ = y/\g~\ VaOrarV) - y/\g~\ Ъ„(д^6Гаа11), A2.315)
which we convert into an ordinary derivative via formula A2.133)
y/\g~\ д^бП^ = da(y/\g~\ g^ST*^) - dv(y/\g~\ g^6Taafi). A2.316)
This expression vanishes upon integration by virtue of Gauss' theorem since
<5Г"М —> 0 at the boundary (that means we also postulate that 6д\9ци —* 0
at the boundary, besides <5<7М„ —> 0).
For the remaining variations we use formulae A2.107), A2.110), we find
6{y/\g\ K) = V\g\ (\<ГЪ - к^ 69llu, A2.317)
and from the action principle
~ (\ A 9llV = 0 A2.318)
we obtain the Einstein equation without matter
W - \tTTl = 0. A2.319)
The symmetric tensor
i A2.320)

Action principle 493
is usually called the Einstein tensor. So without matter, for a free gravi-
gravitational field we just have
G"" = 0. A2.321)
12.3.2 Energy—momentum tensor
Next we include matter by considering the action
Sm = f dTxy/\g\ CM, A2.322)
where См is the Lagrange density of matter—scalars, fermions, gauge
fields—as discussed in Chapter 3.
The change of the matter density under a variation <5<7М„ defines the
energy—momentum tensor TM" of the matter system
SSm = -
J
^~ A2.323)
The following properties hold:
i) Г"" = V» symmetry A2.324)
ii) УдТ'"' = О covariant conservation. A2.325)
Property i) follows from the symmetry of <5<7М„; we prove property ii) later
on.
Now we consider a gravitational field which is coupled to matter; the
corresponding action is
S = Seh + 5M A2.326)
and the action principle
provides the Einstein equation with matter
A2.328)
So energy and momentum influence the structure of space-time represented
by the Einstein tensor.
Sometimes an additional constant—the cosmological constant Л—is
included in the action without destroying the invariance of the action

494 Gravitation
But we do not pursue this case further.
Proof. Now we prove the important property of the covariant conservation
of the energy-momentum tensor (equation A2.325)). We start with the
Bianchi identity A2.92) and respect the constancy of the metric A2.76)
= 0. A2.330)
We contract the indices a and fj, which yields the Ricci tensors
Van0v - VvU&a - VaRpava = 0,
contracting again between /3 and v gives
van - vvnve - vanaa - о
= 0. Q.E.D. A2.331)
In the last step we used the Einstein equation A2.328).
Examples:
i) Scalar fields
Let us first consider scalar fields given by the action
= -\j ^хуДдЦд^д^фд^ф + т2ф2}. A2.332)
The variation gives (remember formulae A2.107), A2.110))
\ A2.333)
so that we find for the energy-momentum tensor
\ тп2ф2). A2.334)

Fermionic action
ii) Photon fields
Varying the photon action (recall equation C.252)) provides
495
A2.335)
Vielbein formalism: When working with fermions we have to express
the action in terms of the vielbein e°M. Then we vary the vielbein
e\(x)^e\{x) + 6e%(x), 6e\{x) |x^^ndary 0 A2.336)
and the energy-momentum tensor of the fermionic system is defined by
6SM = f dmx eT"a<5eoM
|^ = eT\. A2.337)
We extract the vielbein determinant e = | dete°M| to achieve a true tensor
<|iiantity instead of a tensor density. The tensor TMO is a quantity which
represents a contravariant coordinate vector and a covariant Lorentz vector.
It- is related to the previously introduced symmetric tensor TM" by
1 A2.338)
Alternatively, we could also vary the inverse vielbein
Eav{x) - Eav{x) + 8Ea"(x), 6Eav{x) lx^
i lien the energy—momentum tensor is defined by
6SM = - / dTxeTav6Eav
0, A2.339)
(i2'340)
in itf-fordance with definition A2.337).

496 Gravitation
12.4 Fermionic action
In order to describe the anomalies we need the fermionic action. Fermions
are represented by spinor fields which are subjected to the Lorentz group
(and not to the diffeomorphism group). So we must refer the Dirac opera-
operator in the action to the tangent frame, which we achieve with help of the
vielbein.
According to the equivalence principle special relativity is valid in a lo-
locally inertial system and it does not matter which inertial frame we choose
at a point x. Therefore the action must be invariant under local frame
rotations—Lorentz transformations—and clearly also under a general coor-
coordinate change.
Covariant derivative: To construct such an invariant action we need
a covariant derivative of a spinor which transforms locally like a Lorentz-
vector spinor (like a tensor product of a Lorentz vector and a spinor). This
we achieve via the vielbein
Da ф = Ea" О„ф. A2.341)
Under a Lorentz transformation L(x) the spinor transforms as
ф{х) - р-\Ь
ф{х) -> ф(х)р{Цх)), A2.342)
where p(L(x)) denotes the spinor representation of L(x). For an infini-
infinitesimal Lorentz transformation A2.306), L = 1 + a, we have
p{L{x)) = l + \aab{x)aab
p-\L{x)) = 1~Х-ааЪ{х)ааЪ. A2.343)
The matrices \aab are called the generators for the spinor represen-
representation of the Lorentz group. A matrix aab is antisymmetric and can be
represented by
aab = i[7°,7b], aab = -aba. A2.344)
It fulfils the frequently used relation
[aab, аы] = r)adabc - r)acabd + r]bcaad - rjbdaac. A2.345)
We construct the covariant derivative as
A2.346)

Fermionic action 497
and specify the additional connection term wM later on. For the moment we
just require the transformation property
wM-> p~1ullp + р~1дц,р\ A2.347)
then the covariant derivative A2.346) satisfies the desired transformation
(we use д^р~1р + p~1dfip = 0 following from p~lp = 1).
. A2.348)
In the spinor formalism, where we refer to TF, we have a constant metric
T]ab and constant Dirac matrices 70. Whereas referring to CS, where the
metric <?м„(х) is x-dependent, we get x-dependent Dirac matrices given by
-f{x) = Ea"(x)ja. A2.349)
Then we also have
{-fix), -f{x)} = 2^ (*), {70, /} = 2Vab. A2.350)
Fermionic action: Denning the Dirac operator
P = 4»{x)D^ = KM(xOa^ A2.351)
1 Ik; fermionic action is given by
S^ = IdmxeC^= I(ГхефИРф
(Гхеф iEa"-yaD^. A2.352)
Vote: Whenever we work with spinors (in this section and in the following
"lies) we choose for convenience the metric signature (+1,-1,—1,-1) as in
1 lie tffiuge case. Therefore we have in the Lagrangian the operator i TJ) which
г. also Hermitian for a Euclidean signature, and the Euclidean signature is
liimlly used in the index and anomaly calculations.
The action A2.352) is invariant under both the general coordinate trans-
IcMunitions arid the Lorentz transformations. Eati{x)Dll describes a coordi-
imlc scalar and ф{х), ф(х) are coordinate scalars too. A Lorentz transfor-
transformation, on the other hand, leaves the Lagrangian invariant

498 Gravitation
rbD^. A2.353)
Spin connection: Now we determine the explicit form of wM. Under an
infinitesimal Lorentz transformation it changes as
l-r<
aab
b-z-+a>ab |«V —I
z L z J
= wM + Df.a A2.354)
when defining
ab
a(x):=aab(x) — . A2.355)
So we get
<5?u;M = D^a. A2.356)
This clearly reminds us of the transformation behaviour of the spin connec-
connection which we have introduced in the previous sections (e.g. in Section
12.2.5). Indeed, we identify wM of equation A2.346) with the spin connection
given by equation A2.294)
<*V — ш bfi~7T- = e i/VMEb ——-. A2.357)
Then expression A2.357) satisfies the transformation property A2.354). To
show the converse we have
Uabu— > kW—— + диааЬ-— + -acdiJabJ(Ta ,<т° ]\ A2.358)
2 2 24
for the last term we use the commutator formula A2.345), then we recover
precisely the spin connection transformation law A2.309)
(uaciiacb - аас^сьм)
= u;abli + D^aab. Q.E.D. A2.359)
Equivsdent actions: In the literature one can also find other types of fer-
mionic action which are, however, equivalent to our expression A2.352) (see

Fermionic action 499
e.g. [van Nieuwenhuizen 1988], [Alvarez-Gaume, Ginsparg 1985], [Alvarez-
Gaume, Witten 1983]). Since these types are convenient for our anomaly
discussions we will show their equivalence.
Let us first note some identities which are very useful.
Recall the vielbein eaM; it represents a tensorfield—the unit tensor in а
mixed tangent frame and coordinate basis—and not 4 covectorfields. When
applying the covariant derivative one certainly has to contract all the indices
(which we indicate by the subscript tot); the result then is:
Theorem: Vielbein condition
fieK - eaArV = 0. A2.360)
Proof. The condition A2.360) holds due to the equivalence of the
Christoffel- and spin connection, equation A2.301), which guarantees that
= д^е\ + uabfiebu - eaxEbxdfie\ -
= 0. Q.E.D. A2.361)
Generally, the covariant derivative for a hybrid frame-coordinate
tensor is defined by
- ТаахГ\/з. A2.362)
Hence we obtain for the inverse vielbein the condition:
Theorem: Inverse vielbein condition
D**Eav = д»Еа» - .Е^сЛд + Г%А?аА = 0, A2.363)
specifically, for [л = v
D™Ea" = д^Еа» + Г\хЕах - .ЕУсЛм = 0. A2.364)
Note that condition A2.360) or A2.363) is a result and not an independent
condition.
Using furthermore the formula A2.208) for condition A2.364)
D^ES = -d^eES) - ?(АЛМ = 0 A2.365)

500 Gravitation
we find the relation
/V A2.366)
Finally, what we also need are the following commutation relations
[aab,jc] = 7VC-7VC A2.367)
{.До*} = {^y for«'b'.cdifferent A2.368)
1' ' J (^ 0 otherwise. v '
Lagrangian reformulation I: Now we are prepared to reformulate the
Lagrangian A2.352)
'сф. A2.369)
We split the second term into
7<Vbc = _[7a)<T<*] + _{7a)<7bc}; A2.370)
we use the commutator formula A2.367) and obtain
A2.371)
A2.372)
Next we replace the derivative in the Lagrangian
which does not change the equations of motion. In this case, however, the
derivative d^ (acting to the left) acts on the product еЕа^ф. Therefore we
get a term which is
-\dll{eEa»)iHk1> = -\еЕь^ьа^ф7аф A2.373)
when using formula A2.366). This term cancels the second term in equation
A2.371) precisely and we are left with the Lagrangian
A2.374)

Fermionic action 501
Here дц is denned by
and acts only on the spinors ф, ф. Denning a left-acting covariant deri-
derivative
A2.375)
nt deri-
deriA2.376)
we finally find the Lagrangian
еСф = \ erpi^D^- D» ^)ф. A2.377)
On the other hand, a quick way to arrive at expression A2.377) is to use
the following theorem.
Theorem:
[Яд, 7*1=0. A2.378)
Proof. We calculate
xiV A2.379)
\'> A2-380)
w<! insert the equivalence of the connections, equation A2.301), and the
commutator result A2.367), then we find
- 7<Леа„ - 7„и/вьдеь„ = 0. Q.E.D. A2.381)
I 'sing the commutator identity A2.378) now we decompose the Lagrangian
A2.352), we integrate by parts and recover result A2.377)
Ф^О^ + Б^^ф
^ A2.382)
Lagrangian reformulation II: Another possibility of reformulating La-
K.inngian A2.352) is the following. We can substitute
Од -» \ D^ A2.383)
with ^ _ «
5 ^ A2.384)

502 Gravitation
already at the beginning so that we have the Lagrangian
1 - *-
еСф = -еф11рф. A2.385)
But then the Dirac operator
p = -f(x) Dy. = Ea^x)^ Dp A2.386)
acts left on both, on the spinor ф and on e?aM. Again, the term д^(еЕа^)
cancels with a commutator term leading to expression A2.374).
Remark: Considering expression A2.374) in two dimensions a = 1,2
(where the 7-matrices change into Pauli matrices, recall Section 4.5) we
notice that
a12 ~ Ьъ<?2] ~ &з
the anticommutator term vanishes and we find the important result:
Proposition:
• In two dimensions there is no spin connection term in the fermionic
action!
Resume: Finally we consider massless fermions in a given chirality P+ =
^A+75). We can construct fermionic actions which are invariant under Lor-
entz transformations and general coordinate transformations. These actions
are equivalent to each other and are given by
= [еГхеф
/1 - *-*
сГх - ефгТР Р+ф A2.387)
12.5 Invar iances
Now we want to study the classical symmetries of the fermion action, and
we choose the following expression
A2.388)
= J

Invariances 503
Calculating the energy-momentum tensor
e T"° = 11Г = Ъ е ^ ^Gа-Сд- D» 1а)Р+ф A2.389)
we find for the symmetric energy momentum tensor
^ v- Dv 7M)P+V + A*«-» v. A2.390)
The antisymmetric part vanishes as we shall see. In the derivation of ex-
expression A2.390) we have used the equations of motion (as a consequence
the term proportional to 8e/Seali vanishes)
ёф
only then does a general variation of the action
¦f = 0 =>¦ РР+ф = 0
= 0 =» ^#>Р+ = 0, A2.391)
+ 7^^J A2.392)
V ^ ' ёф
agree with the definition A2.337) of the energy-momentum tensor (since
the first two terms drop).
12.5.1 Lorentz invariance
The fermionic action is invariant under infinitesimal local Lorentz
transformations
Йе" = -а\еь
be\
a
= 0
„" = Eb»ab
«^ = \ааЬгРсгаЬ. A2.393)
The Lorentz-like variation of the action A2.388) is then
<S? S+ = f cTxe T»a 6% e% = - f сГх е Т"а ааь еьм A2.394)

504 Gravitation
supplying the following proposition.
Proposition:
• The Lorentz variation of the action is given by
= f dmx e ааЬ ТаЬ. A2.395)
Theorem: The energy—momentum tensor is symmetric!
8%зф = 0<==> ТаЬ = ТЬа A2.396)
(since ааь = —аъа)- Clearly, if ТаЬ is symmetric, TM" is symmetric too.
12.5.2 Einstein invariance
The fermionic action A2.388) is also invariant under Einstein transfor-
transformations
й|^ = r<9.,V>- A2.397)
(Geometrically, we consider the trivial fibre-bundle case.) Then the Einstein-
like variation of the action becomes
dmx e T»a 61 e%
SIS* = I
= f dmx(e\Vli^ + ^^ие%)еТ\, A2.398)
which we are going to calculate further. We integrate the first term by parts
e\VMreT"a = eVMr^Mv —¦ -eCV^T",, A2.399)
and for the second term we use the connection formula A2.295)
?,vVve\eT\ = ? EfV vf „.О* а = Г^аЬ,еТаЬ. A2.400)
Altogether we obtain:

Invariances 505
Proposition:
• The Einstein variation of the action is given by
= - [(ГхеС(УцТ»и-иаЬ„ТаЬ). A2.401)
The second term, however, vanishes here since, due to Lorentz invariance,
the tensor ТаЬ is symmetric whereas и)аъи is antisymmetric. So we find:
Theorem: The energy—momentum tensor is covariantly conser-
conserved!
Ь\ вф = 0 and 6% SV = ° ==*• V" = 0. A2.402)
(Notice that we need both Einstein- and Lorentz invariance.)
Alternative approach: We also arrive at the above conclusions by split-
splitting the vielbein variation into a symmetric and an antisymmetric part.
Lemma:
ёе% = \ ёд^Еа» - ааье\, A2.403)
with
ааъ = ±{ЕаЧеъи-ЕъЧе%). A2.404)
We insert the splitting A2.403) into the variation of the action
= /
^ jh; A2.405)
due; to Lorentz invariance the tensor Tab is symmetric whereas ааь is an-
i isymmetric; that implies the vanishing of the second term and we are left
with
^J fii A2.406)
Thus due to Lorentz symmetry we can choose the action to depend only on
</„„. The Einstein-like variation gives
^Wnb + V^). A2.407)
lull-uniting by parts, using again the symmetry of Т>*и we finally find:

506 Gravitation
Proposition:
• Due to Lorentz symmetry the Einstein variation of the action is given
hy r
8ISi, = - <Гху/\д\ ^VMr"v. A2.408)
This is clearly in accordance with result A2.401).
Note: If we choose, on the other hand, a Lorentz-like variation 6 —> 6% in
equation A2.405) the first term vanishes and we recover result A2.395). In
this case the ааь of equation A2.404) represents precisely the frame 'rotation
angle' A2.306).
12.5.3 Weyl invariance
Finally we discuss the conformal or Weyl transformations (recall Section
12.1.1). Let us consider the Weyl rescaling of the. vielbein
eV(z) = P(x)e%(x)
Еа*'(х) = р~1(х)Еа"(х) A2.409)
e' = pm(x)e,
which alters the metric by
9?ЛХ) = Р2(х)р„и(х)
д' = Р2т(х)д, A2.410)
where
р(х) = еа(ж). A2.411)
Spin connection and Dirac operator transform conformally according to
jj' tx\ _ p~r(x)D (x)pr(x) A2.412)
together with the Fermi fields
ф\х) = р-Т{хЩх)
ф'{х) = р~т{х)ф{х), A2.413)
where
г = W~ , m = dim M. A2.414)
These scaling transformations are determined by the required scaling i
riance of the Lagrangian.

Gravitational anomalies
Choosing the transformation parameter infinitesimal
p(x) = 1 + a(x)
we have the Weyl transformations
6™e\ = ae\
507
A2.415)
Ал>
cW
e = m a e
= —г а ф
= —г а ф.
ь ~ еъцЕ )
A2.416)
Note: We want to emphasize that the Weyl transformation of the connec-
connection is already determined by the rescaling of the vielbein independent of
the construction of the Lagrangian. The reason is that the connection is
uniquely determined by the vielbein and its derivative. Remember formu-
liic A2.238), A2.241), a Weyl variation gives (we keep only the derivative
il/tcr-terms, all сг-terms cancel)
av - dx<rebliEax + dvaeay.Ebv)
= dva{eaVLEbv - ebfiEav). Q.E.D. A2.417)
So it turns out that, our Lagrangian A2.388), constructed to be Lorentz-
uiid Einstein-invariant, is Weyl-invariant in addition.
Finally, what does the Weyl symmetry imply for the energy-momentum
tensor?
Proposition:
• The Weyl variation of the action is given by
cW
о
-I
сГх е
ThiHirem: The energy—momentum tensor is traceless!
6?
= 0
= 0.
A2.418)
A2.419)

508 Gravitation
12.6 Gravitational anomalies
Now we come to our main interest, the anomalies. We consider the quantum
case—the 'quantum action'. Then the invariances—the Einstein-, Lorentz-,
Weyl invariance—which we have discussed previously might be broken. In
this case we speak of gravitational anomalies.
12.6.1 Einstein-, Lorentz-, Weyl anomaly
In the quantized theory we work with the generating functional for the
Green functions. We quantize only the Fermi fields; the gravitational field
represented by the metric or by the vielbein we still consider as an external
(nonquantized) field. Then we have for the generating functional
/Г /• I
dtydip exp - / dx е?ф , A2.420)
L J J
where the Lagrangian еСф is given by equation A2.388). The variation of
the 'quantum action'
6W[e%] = J
dx ^VW^WfeM A2.421)
provides the energy—momentum tensor
= Jdxe{T^a)8e\
Щ^ = e(T\) A2.422)
analogous to the classical case A2.337). The bracket ( ) denotes the mean
value with respect to the path integral. Choosing in particular
6 —> 6% Lorentz transformation
69 Einstein transformation
6™ Weyl transformation
and inserting the corresponding transformations of the vielbein we obtain
the several variations of the 'quantum action'. These variations might not
vanish as in the classical case, then we have anomalies.

Gravitational anomalies 509
Proposition: Gravitational anomalies
• Lorentz anomaly
«?W4e%] = fdx eaab(Tab) =: GL{a). A2.423)
• Einstein anomaly
x er(VM(r^) - u;abl/(Tab)) =: GE(?)- A2.424)
• Weyl anomaly
6™W\f?A = fdx е<т(Г%) =: Gw(ff). A2.425)
Note: Now (ГоЬ) means the antisymmetric part of the (total) energy-
momentum tensor (TM°)
(ТаЬ) = 1(е%(Т"ь) - еЬц(Т»а)). A2.426)
In equation A2.423) only the antisymmetric (Tab) contributes, the symme-
symmetric part of the energy-momentum tensor also exists but does not contribute
here.
Physical significance: The gravitational anomalies mean the following:
• The Lorentz anomaly is equivalent to the existence of an antisym-
antisymmetric part of the energy-momentum tensor.
(Tab) = -(Tba). A2.427)
• The Einstein anomaly involves the nonconservation of the energy-
momentum tensor
УМ(Т^") ф 0. A2.428)
• The Weyl anomaly expresses the nonvanishing tensor trace
(Г%) ф 0 A2.429)
(and is sometimes called trace anomaly for this reason).
Remark:
• Even in the absence of an Einstein anomaly, GE(?) = 0, the energy-
momentum tensor is not covariantly conserved!

510 Gravitation
This nonconservation is a consequence of the Lorentz anomaly, implying the
existence of an antisymmetric part {Tob), and does not signal the breakdown
of general coordinate invariance. Physically, it means that there is an energy
flow into the degrees of freedom which specify the direction of the vielbeins—
the metric is not the only external field that the system interacts with
[Leutwyler 1985b, 1986a].
Pure Einstein anomaly: However, it is possible to make the energy-
momentum tensor symmetric [Alvarez-Gaume, Ginsparg 1985] so that Lor-
Lorentz symmetry is preserved—no Lorentz anomaly occurs—then the second
term in equation A2.424) vanishes and we are left with a pure Einstein
anomaly
6\W = - f dx e&V^TH = GE(?). A2.430)
• Then the nonconservation of energy and momentum is equivalent to
the breaking of coordinate invariance!
In this case the 'quantum action' depends only on g^ (recall our discussion
in Section 12.5.2) and we have
\fl A2.431)
with
Щ^^ A2-432)
The insertion of the metric transformation A2.167) leads directly to the
anomaly equation A2.430).
Resume: The anomalies in gravitation are the analogues of the anomaly
in Yang-Mills gauge theory as discussed in Chapter 8. Particularly, the
pure Einstein anomaly A2.430), which represents the nonconservation of
the energy-momentum tensor, is the precise analogue to the gauge anomaly,
which expresses the nonconservation of the gauge current (compare with
equation (8.58) of Section 8.2).
As in the gauge case there exist different methods of evaluating the gra-
gravitational anomalies. The anomalies are given by a local polynomial in the
connection and curvature. This polynomial was first found within perturba-
perturbation theory. The authors [Alvarez-Gaume, Witten 1983], [van Nieuwenhui-
zen 1988], [Langouche 1984] have calculated Feynman diagrams where the

Gravitational anomalies 511
external gravitational field—the metric (or vielbein) in a linearized version—
couples to a fermion loop via the energy-momentum tensor. Another way
to find the gravitational anomalies is the heat kernel method of [Leutwy-
ler 1985b], [Leutwyler, Mallik 1986] or Fujikawa's path integral approach
[Fujikawa 1983], [Fujikawa, Tomiya, Yasuda 1985]. Finally, one can deter-
determine the anomalies algebraically and by topological techniques; we want to
present this in the forthcoming sections.
12.6.2 Consistency conditions
The gravitational anomalies have to satisfy certain consistency conditions
analogous to the gauge case. We neglect the Weyl anomaly for the moment.
What we need first are the commutation relations for the generators of the
infinitesimal symmetry transformations which are denned on the space of
local functionals of e, ui or Г. For convenience we use a matrix notation, e.g.
e = e°M denotes the vielbein here (and not the determinant as before).
The generator of a variation acting on some functional U[e, ш\ or U[e, Г]
is represented by
or
6 = Idxri+6r-k}- A2-434)
Considering the Lorentz- and Einstein transformations we have (recall equa-
equations A2.309), A2.171), A2.172), A2.209))
6%e = -ae б^ш = Da A2.435)
and
5 5 A2.436)
= i^de + ev^ = (цс1 + Aц)Г + Vv^.
Then we obtain for the generator of a Lorentz transformation
6°=Idx[-aei+Dai] A2-437)
and for the generator of an Einstein transformation
61= fdx li^de + ev^ + d^d + di^r + Vv^-^] . A2.438)
They satisfy the following commutation relations [Bardeen, Zumino 1984],

512 Gravitation
[Stora 1984,1986], [Langouche, Schiicker, Stora 1984], [Alvarez-Gaume, Gin-
sparg 1985]. Here we consider only the case of trivial fibre bundles; we dis-
discuss the formalism for nontrivial bundles at the BRS level later on (Section
12.7.3).
Theorem: Commutation relations
rcL sL i cL
ld6 = °
[««,«!) = %э«- A2-439)
These operators are sometimes also called Ward operators. The Lorentz
part behaves analogously to the gauge case (remember the proof (8.9)-
(8.14)); we demonstrate the correctness of the Einstein part below. The
Lorentz- and Einstein generator 6^ and 6| do not commute but give 5f da
due to the i^d = ? • д terms in equation A2.438); we show the correctness
again below.
Now, applying these commutation relations to the 'quantum action' to-
together with
^a^Ie>4 = GL(a,uj) Lorentz anomaly
6\W\e, Г] = GE{v^ Г) Einstein anomaly
(analogously to the gauge case of Section 8.2) we find:
Theorem: Consistency conditions
,ш)-6^Сь(а1,ш) = Сь([а1,а2],ш)
^GE(^,r)-5|GL(a,w) = GL(?-da,uj). A2.440)
These equations determine the structure of the anomalies—the consistent
gravitational anomalies—in terms of the connection and curvature. We
have to take the normalization, however, from somewhere else, e.g. from per-
perturbation theory, from the path integral method or from the index theorem.
Of course, there also exist the trivial solutions
| A2.441)
where G[uj] or G[F] represents a local polynomial in the fields a» or Г and
their derivatives.

Gravitational anomalies 513
On the other hand, we could also work with the passive coordinate trans-
transformations. There we have (recall equations A2.148), A2.150))
«?е = ег/€ e?r = Vt^. A2.442)
Then the generator for a passive coordinate transformation is
and satisfies, analogous to the gauge case, the relations:
Theorem: Commutation relations
$;«,.$;«,] = 6ivtl,v(a)
[<,#] = 0. A2.444)
The passive generator of the difFeomorphism group certainly leads to the
same Einstein anomaly as the active one [Alvarez-Gaume, Ginsparg 1985],
[Bardeen, Zumino 1984]
<W[e,r] = СВ(«с,Г) = / trt/€GB[T]. A2.445)
The reason is that the difference of the generators—when acting on
functionals—yields a total derivative (recall equation A2.175)) which va-
vanishes under suitable boundary conditions
/
Mm
= / d(UQm) = I HQrn - 0. A2.446)
JMm JdMm
(Qm is a form of maximal degree m.)
Using the commutation relations A2.444) gives:
Theorem: Consistency conditions
6*'(Оъ(а,и)-ё%СЕ(У1,Г) = О. A2.447)
The second equation, however, is trivially satisfied since
6счи> = ё%Г = 0. A2.448)

514 Gravitation
Clearly, the commutation relations of the active and passive diffeomorphism
generators are not independent of each other.
Proposition:
• The commutation relations of the active diffeomorphism generators
A2.439) are equivalent to the passive ones A2.444) and as a conse-
consequence the active consistency conditions A2.440) are equivaJent to the
passive ones A2.447).'
Proof. Using formula A2.175) we calculate
GE(vb,r) A2.449)
(note that v^ is a 0-form and GE(F) a form of maximal degree). Then by
using the 'passive' relation A2.447) we get for the commutator
= / trH^dwfo+i&dwfo + ^.w&DG^tT]. A2.450)
Jm
We evaluate the bracket (recall v% = d?)
= («fe.C.])% A2-451)
and we recover the desired 'active' commutation relation A2.439)
e,T\ = I trv[btil]GE[T)
J м
For the mixed relations we first calculate the 'active' relation A2.440) (re-
(remember the generators act on functional)
e^GE(vbT)-SIGL(a,cj)
= - / tr adi^GL[bj) = / tr i^daGL[uj]
Jm Jm
= GL(Z-da,u>) A2.453)

Gravitational anomalies 515
and then we use this result to show the 'passive' relation A2.447)
(,u) A2.454)
= - / tr adijGL[w] + / tr adijGb[w] = 0. Q.E.D.
Jm Jm
12.6.3 Equivalence of Einstein- and Lorentz anomaly
We have introduced two kinds of gravitational anomaly—the Einstein- and
the Lorentz anomaly—depending on whether we perform a general coordi-
coordinate transformation on the 'quantum action' or a Lorentz transformation
(again, we disregard the Weyl anomaly). These two anomalies are, howe-
however, not independent of each other. In fact, Bardeen and Zumino [Bardeen,
Zumino 19,84] discovered the following feature:
Proposition:
• There exists a local, nonpolynomial, counterterm S for the 'quantum
action' W that shifts the gravitational anomaly from pure Einstein-
type to Lorentz-type and vice versa!
So the Einstein- and Lorentz anomalies appear as two different aspects of
one and the same anomalous phenomenon.
Let us choose the 'quantum action' W[g, Г] such that it depends only on
the metric g and the connection Г (this is always possible). Then the action
is Lorentz invariant and we get a pure Einstein anomaly
6lW\g,T] = GB(t/c,r) = / tr vzGE[r) =: - f <&,(««,Г). A2-455)
JM2n JM2n
Here we introduce a polynomial in v% and Г—the chain term Q\n(vZ^)—
analogous to the gauge case; we shall discuss its significance in the chain
of descent equations later on (Section 12.7). For convenience we further use
the GLBn) matrix notation for the vielbein, e = eaM, and we introduce the
following t-parameter family—the homotopy
et = 1 + t(e - 1) «?/ = [0,1], A2.456)
with
i et=1 = e et=0 = 1. A2.457)
Bardeen-Zumino counterterm: The BZ counterterm can be con-
constructed as [Bardeen, Zumino 1984]

516 Gravitation
S= I Q2n+i(wet +vt), A2.458)
where
щ = e;16tet = et1—etdt A2.459)
at
and Qin+\ means, as usual, the Chern-Simons form. The homotopic
connection ojet interpolates between the spin connection, for t = 0,
Ljet=°=uj A2.460)
and the Christoffel connection, for t = 1,
l = T. A2.461)
Г is considered as a GLBn) gauge transformation of из. So S is a local
functional—it contains a finite number of vielbeins e, de and connections из,
dw—but it is nonpolynomial.
Next we define a homotopic metric (again, in matrix notation)
9t = ej v et, A2.462)
with
9t=i = eT r] e = g gt=o = Ц- A2.463)
For a variation with respect to the parameter t we find
Tr)et + efr)etet
A2.464)
with
i / — I ? \JL ? JL ( — 1 \ JL / 1 О Л С С\
furthermore
Stu3et = Dtvt = dvt + [wet,vt], A2.466)
St
W[guu;et} = - f Qlnivuu"). A2.467)
JM2n
To evaluate the counterterm A2.458) we expand the Chern-Simons form
in powers of Vt
Q2n+i K< + vt) = Q°2n+1(u;et) + Q\n{vt, ^) + ..., A2.468)
where only the linear term Q\n contributes to the integral (here v\ = 0 and
the series stops). Then we obtain

Graxntational anomalies 517
S = f Q2n+i(uet+vt)= [ [ Qln(vt,ue<)
JM2nxI JO JM2n
= - I StW[gt, аЛ] = W[V, u) - W[g, Г] A2.469)
./o
and we find that S satisfies the variations
6%S = 6%\?[г,,ш) = Сь(а,и) = - f Qln(a,u). A2.470)
¦/м2„
Anomaly shift: So we can use the counterterm S to shift the anomaly!
For instance, adding S to W
W-^W + S A2.471)
shifts the anomaly from Einstein-type to Lorentz-type
e%(W + S) = 0
6%(W + S) = GL(a,u). A2.472)
Choosing, on the other hand, the 'quantum action' such that only the Lor-
entz anomaly occurs (which is possible)
6%W = 0
6%W = GL(a,w), A2.473)
then the subtraction
W -* W - S A2.474)
shifts the anomaly back to pure Einstein-type
SI(W-S) = GE(vt,r)
8^{W-S) = 0. A2.475)
Physically, this means that we have a choice of the energy-momentum ten-
tensor. We can choose between a symmetric tensor which is not covariantly
conserved and a conserved tensor which is not symmetric!
Remark: Calculating, finally, the gravitational anomalies by regularizing
the determinant of the Weyl operator
6W = -6\ndetipP+,

518 Gravitation
with the help of the heat kernel method Leutwyler [Leutwyler 1985b],
[Leutwyler, Mallik 1986] found out that there exists only the Lorentz
anomaly—coordinate invariance remains intact! To shift the anomaly now
from Lorentz-type to Einstein-type involves the nonpolynomial BZ coun-
terterm. This term, however, lies outside the class of counterterms which
occur in the regularization procedure!
12.6.4 Covariant gravitational anomaly
The gravitational anomalies discussed so far obey the consistency conditi-
conditions and are called consistent anomalies for this reason. However, the cor-
corresponding energy-momentum tensor (T7*") does not transform covariantly
under Einstein transformations. But Bardeen and Zumino [Bardeen, Zu-
mino 1984] discovered that—as in the gauge case, which we have studied
in Chapter 10—one can also construct a covariantly transforming energy-
momentum tensor leading to a covariant gravitational anomaly.
Definition: We introduce a general variation
= ?>„„(х) A2.476)
and we regard this variation as a derivative in the space of all metrics
Sp M = {gr(x)}.
The variation 6^ is a derivative along an arbitrary (but fixed) tensor
field (p^
/c
dxip^ix)- r-r, A2.477)
°9iiv\X)
where 6 denotes the exterior derivative in Sp M- So 8^ is an operator
which gives a metric g^ an arbitrary (but fixed), symmetric increment
Einstein transformation law: In order to derive the transformation law
for the energy-momentum tensor we proceed analogously to the gauge case
of Chapter 10. We apply the following commutator to the 'quantum action'
[«f, Sv]W\g] = (<5|<5„ - MDWfol' A2.478)
where we have chosen W[g] to depend on g; on the one hand we obtain
/S с
da; <Pau{x)Sf- -f(j] - SlfiGE(vi, Г) A2.479)
°дци{х)
and on the other

Gravitational anomalies 519
Equating these results gives
^ E). A2.481)
Inserting the definition of the energy-momentum tensor A2.432)
A2.482)
provides the transformation law for that tensor.
Theorem: Einstein transformation law
Sljdxy/Ш ^„(Г"") = 26vGE(vt,T). A2.483)
Of course, we could also start by allowing <рц„ to transform covariantly as
a tensor, 69 <р^ = ?$ <рр„; then the commutator vanishes, [5|, 6^] = 0, and
the application to W[g] provides result A2.483) immediately.
What does the transformation law A2.483) mean? If there were no anom-
anomaly, the left-hand side would be Einstein transformation invariant. Since <рц„
transforms covariantly as a tensor, (Т**") must transform covariantly too.
• Hence in the presence of an anomaly—Einstein anomaly—the covari-
ant transformation law of the energy-momentum tensor is broken!
Bardeen-Zumino polynomial: Bardeen and Zumino [Bardeen, Zumino
1984] found that, analogous to the gauge case (recall Chapter 10), one can
construct a new energy-momentum tensor which transforms covariantly.
Proposition:
• There exists a local polynomial tensor—tie BZ polynomial 7"*"—
with transformation property
J dx^/Ы wP"" = -26V GE(vb Г), A2.484)
so that the covariant energy—momentum tensor
(f"") = (T"") + P"" A2.485)
transforms covariantly, which means the Einstein invariance of
«f
6\
jdxy№ w {ТП = 0. A2.486)

520 Gravitation
Definition: Then the covariant gravitational anomaly is defined by
M(f "">. A2.487)
Formulae: The explicit expression for the BZ polynomial 7"*" can be
found by solving the transformation property A2.484) for 7"*". This is pos-
possible since the consistent anomaly is known, say, by the chain term <?2п-2
(see Section 12.7). Then we can evaluate the covariant gravitational anom-
anomaly, the procedure works analogously to the YM case, which we explained
in Chapter 10. We just have to replace A —> Г, F —> R and v —> V^, and
we can read off the final result from formula A0.46). In this way we find
Bardeen-Zumino's formula in Bn — 2) dimensions:
Covariant gravitational anomaly
GE{VbY) = -n I PiV^R71'1), A2.488)
JM2n-2
where V? means
\ A2.489)
, i?™) is an invariant polynomial of V? and Rn~1 as defined in Section
7.1. We have now included the normalization of the invariant polynomial
into P and it is given by the index theorem (see Section 12.8). Note that the
leading term is a factor n larger than in the consistent case. Actually, the
covariant anomaly depends on R, but let us keep Г in the argument. As in
the gauge case the covariant anomaly does not fulfil the Stora-Zumino chain
of descent equations but is part of more complicated algebraic relations (see
e.g. [Ader, Gieres, Noirot 1991]).
12.7 BRS algebra and descent equations
We can simplify the consistency conditions described in Section 12.6.2 consi-
considerably by introducing anticommuting Grassmann variables—ghosts—and
the BRS operator, as we did in the YM gauge case in Chapter 8.
12.7.1 BRS algebra
We regard the following quantities as anticommuting Grassmann fields:
?Q(z) ... Einstein ghost
ааь(х) ... Lorentz ghost
ct(x) ... Weyl ghost.

BRS algebra and descent equations 521
Accordingly the infinitesimal transformation operators become BRS-like
«| -¦ sE
8<r —* sw
These operators act on the vielbein as the usual transformations A2.397),
A2.393), A2.416) (we consider for the moment only the trivial bundle case)
sLe\ = -aabe\
swe% = <re%. A2.490)
Defining the total BRS operator now by
s = sE + sL + sw A2.491)
we get
s e% = ? ¦ д e% + eavd^v - aabe\ +<reali. A2.492)
We require the BRS property:
s antiderivative and s2 = 0. A2.493)
This means, on any local functional F depending on e,?,a,cr, we have
s2F[e,?,a,a] =0 A2.494)
and specifically
s2 e% = 0. A2.495)
From condition A2.495) follow the BRS transformations for the ghosts
8? =
aab =
sa = Z-da. A2.496)
saab = ?-дааь- ааса°ь
(Consequently we obtain
B{ve)a0 =
= t-d(ve)af,-{ve)av(vt)v(, A2.497)
or
я«{=^-а«?-«|. A2.498)

522 Gravitation
Proposition: Total gravitational anomaly
• The BRS transformed 'quantum action' W\e] represents the total gra-
gravitational anomaly
sW[e] = С?Г(е,?,а,а) A2-499)
(s B + sL + sw)W[e] = GE(e,?)+GL(e,a)+Gw(e,a).
In terms of functional derivatives we write
GgT = s W[e) = Jdx se^^-^Wle] A2.500)
with seaц given by equations A2.490). The nilpotency A2.493) implies:
Theorem: Consistency condition for the gravitational anomaly
eG?twr(e,?,a1a)=0. A2.501)
This compact equation is equivalent to the set of conditions discussed in
Section 12.6.2. So the search for an explicit anomaly expression is reduced
to the solution of a cohomology problem. This means we are seeking local
functionals in the vielbein and in the ghosts which are linear in the ghosts
and closed under s but not exact. Hence the trivial solution is
= sG{e), A2.502)
where G[e] represents a local functional.
Lie derivative: As in the gauge case the condition A2.501) is part of a
whole chain of equations. To derive this chain we first need the appropriate
BRS algebra on the basic variables. We find it convenient to express the
BRS transformations in terms of the Lie derivative A2.152) along ?
Ct = t-d-[dt,]. A2.503)
But now ? = ?a da represents the ghost vector field of diffeomorphisms.
Rewriting the Lie derivative we find
?€ = ?€-[««,], A2.504)
where here the commutator contracts only tensor indices. The Lie deriva-
derivative operator acting on forms
C<: = цс1 - di? A2.505)

BRS algebra and descent equations 523
now contains a minus sign due to the anticommutativity of the ghost field.
We demonstrate this quickly for the example
A2.506)
(recall that ?" and dx11 anticommute). The ghost ? makes i$ act as a deri-
derivative and ?{ as an antiderivative!
BRS algebra: Now we establish the BRS algebra. Again, we neglect the
Weyl transformations for simplicity. (The Weyl anomaly was first discovered
by [Coleman, Jackiw 1971] and [Crewther 1972]. Its inclusion in the descent
equations can be found e.g. in [Ebner, Heid, Lopes Cardoso 1987].)
Let us recall next the effect of the transformation operators described in
Sections 12.1.5 and 12.2.5. Prom formulae A2.166), A2.172) and A2.309)
we can read off the following equations when considering the fields ? and a
as anticommuting ghosts
sET «-
sER(T) «-
sER(u) %
^ = —Da
¦- 6%Г = 0. A2.507)
Kor the BRS operator that we now use
s = sE + sL, A2.508)
with the transformation properties A2.496) and A2.498), we find:
URS algebra
i) for the coordinate system
sY =
sR = ?еД + [Д,«€], A2.509)
where R = df + Г3,

524 Gravitation
ii) for the tangent frame
s a = C^a — a2
s cj — C^cj - Da
s R = ??Д+[Д,а], A2.510)
where R = du + cj2.
In addition we know that for the ghost vector field ? = ?ada we have
where [?, ?] means the graded Lie bracket of the ghosts
Ы = Ы*да = 2Г9„Г9а. A2.512)
So we can rewrite the BRS transformation A2.496) for the ghost ? by
*С = \Ь* = \Ы. A2-513)
Finally, for the vielbein we find from equations A2.490)
sea =Ciea- aab eb. A2.514)
Shifted BRS algebra: As suggested by Ader, Gieres and Noirot [Ader,
Gieres, Noirot 1991] the two sets of equations A2.509) and A2.510) can be
simplified. For that we define a shifted BRS operator
st — s-Ct, sl = 0 A2.515)
(introduced by [Ne'eman, Takasugi, Thierry-Mieg 1980], [Baulieu, Bellon
1986], [Baulieu, Thierry-Mieg 1984]).
This new BRS operator s$ removes the explicit dependence on ? and
decouples the diffeomorphism from the Lorentz group. In terms of s$ we
thus get the following algebra:
Shifted BRS algebra
i) for the coordinate system
Г = -Vw$
€], A2.516)

BRS algebra and descent equations 525
ii) for the tangent frame
s^ot = -oc2
S?W = -Da
s^ R(u) — [R(u),a\. A2.517)
Additionally we have
s^ = -Ы,?] A2.518)
s?ea = -aabeb. A2.519)
12.7.2 Descent equations
The shifted BRS algebra, which we have now derived, is of the same form
as the BRS algebra in YM gauge theory. Therefore we may proceed com-
completely analogously to the gauge case (Chapter 9) in order to find the chain
of descent equations. The following is valid for gravitation with a trivial
fibre-bundle structure; we discuss the generalization to nontrivial bundles
subsequently.
First, we note the validity of the following familiar identity (due to the
shifted BRS algebra A2.516), A2.517)).
Theorem: 'Russian formula'
R(?) = ДГ + f2 = dT + Г2 = R(T)
R(Q) = AQ + Q2 = duj+uj2 = R(u), A2.520)
where
and ^
Д2 = 0 by virtue of dC^ + C^d = 0. A2.522)
Next we start from a symmetric invariant polynomial of degree n (as
defined in Section 7.1) from which we know that it is locally exact—we
have the 'transgression'
ту/jjn\ A/~\ /T1 z?"\ /10 C\O*^
r\tl ) — CLLj2n— 1\L !"•)• у1л.Ол6)
The term Qin-i represents the Chern—Simons form given by formula
G.45). This 'transgression' is also valid for the shifted fields—'shifted
transgression'
,-i(f,#) A2.524)

526 Gravitation
by virtue of the Bianchi identity
AR = AR+ [f, R] = 0. A2.525)
Then the 'Russian formula' A2.520) implies the identity
AQ2n-i(r + vz, R) = rfQ2n-i(r, R). A2.526)
Expanding the Chern-Simons form in powers of the ghost v^
2n-l
<Э2„-1(Г + vt, R)=Y, Q2n-i-kD, Г, Я); A2.527)
fc=0
comparing terms of the same order in equation A2.526) leads to:
Stora— Zumino chain of descent equations
ЛЙ„_1 = 0
"'1 = 0 A2.528)
and к = 0,1,...,2n- 2.
Identification: When we integrate the equations over a boundaryless ma-
manifold we get for к = 1
/&,-2(«е.Г,Я)=0 A2.529)
(the d-term vanishes via Stokes theorem). Since we also have (after integra-
integration)
ZeQL-г = 4dQhn-2 - diiQ\n-2 - о A2.530)
{Q\n-2 is a form of maximal degree and di^Q\n_2 is exact) we recover the
consistency condition A2.501) for the gravitational anomaly
,-2(«€>Г,Я) = 0. A2.531)
So we can identify the pure Einstein anomaly with the chain term Q\n-2-
On the other hand, the descent equations are also valid when we refer to
the tangent frame. Then we have the fields a,w,R(w) and we can identify
the Lorentz anomaly with the term Q\n~2-

BRS algebra and descent equations 527
Proposition:
• The chain term Qln-2 represents either the pure Einstein anomaly
GE(Vi,T) = - j Q\n_2{v^ Г, R) A2.532)
or the Lorentz anomaly
J1(a,u,R). A2.533)
Here the normalization is included in Qln-2-
Finally, we also want to refer to other approaches of the cohomology set-
setup for gravitational anomalies [Baulieu, Thierry-Mieg 1984], [Bonora, Pasti,
Tonin 1986], [Bandelloni 1986] and to extensions of the cohomology forma-
formalism to gravity with torsion [Moritsch, Schweda, Sorella 1994], [Moritsch,
Schweda, Sommer 1994], [Moritsch 1994].
Resume: We start with one and the same invariant polynomial P(Rn).
Since the curvature is given either by R = dT + Г2 or by R = dw + w2
we may consider the polynomial either as a function of the GLBn) curva-
curvature or of the SOBn) curvature. This leads to a first cocycle for the term
Q2n-2(viF,R) or Q2n-2(a^CJ^)- Both terms represent the anomaly—the
Einstein or the Lorentz anomaly—and satisfy the same chain. Generally, we
obtain the chain terms in version Q2n-i-fc(w?>r, R) or Q^n-i-ki01^'^-)-
We also know the explicit expressions; they are determined, for instance, by
the formulae (9.120) and (9.136) derived in Section 9.5. For example:
Chern—Simons form
/.l
Зап-1(Г,Я) = п / dtP{T,R!}-1), Rt = tR + (t2 - t)T2 A2.534)
./o
/•l
Qln^u,R) = n dt Р(ш,R?-1), Rt = tR+ (t2 - t)u2. A2.535)
./o
Anomaly term
Qan-2("€.r-R) = <n ~ 1) / *A - i)^(«€. d(r- -R?)) A2-536)
Qln-2(<*,ut,R)=n(n-l) dt(l-t)P(a,d(u,R?-2)). A2.537)
Jo
Choosing n = 2 specifically we obtain for the Chern-Simons form

528 Gravitation
Ql(T,R) ~ tr (rdT + |
B \
u}du) + -u3) A2.538)
and for the 2-dimensional anomaly term
Q\{a,u)) ~ tiadco. A2.539)
Here the invariant polynomial P(Rrl) contains a normalization constant
which we determine later on (Section 12.8).
Remark: Finally, we also observe that the polynomial vanishes
P(R2k+1) = 0 A2.540)
(k — 0,1,2,...) for odd products of R. This is due to the antisymmetry of
Rap or Rab- So n must be even, n = 2fc + 2. Consequently the gravitational
anomaly determined by <?2n-2 can occur in only 2n—2 = 4fc+2 = 2,6,10,...
dimensions!
12.7.3 BRS for nontrivial fibre bundles
The chain of descent equations that we derived previously is valid for
gravitation—considered as a SO(m) or GL(m,R) gauge theory—with a
trivial bundle structure. But as in the YM gauge case (Section 9.2) the
whole formalism can be extended to nontrivial fibre bundles. Again, this
generalization has been elaborated by Stora [Stora 1986] and his collabora-
collaborators [Langouche, Schucker, Stora 1984], a work we want to follow now.
To lift the diffeomorphism of the base into the bundle an additional
fixed background connection шо is required (see Stora's GIFT lectures [Stora
1986]). Then the BRS algebra is introduced in the following way:
BRS algebra with s2 = 0
se = —ae + ?°e
sa - -a2 + C\a - -ц
st = \[M], A2-541)

BRS algebra and descent equations 529
where the covariant Lie derivative operator acting on forms is defined
by
C\ = цО(ш0) - D(wo)i€ A2.542)
and
D(wo) = d+[wo, ]. A2.543)
The background connection is not transformed
sw0 = 0. A2.544)
Note: Clearly, for u)q — 0—the case of a trivial bundle—we recover the
BRS algebra given before (equations A2.510), A2.513), A2.514)), where
we need the identity
) - Оцш. A2.545)
As before, the gravitational anomaly GGrav is determined by the consi-
consistency condition
sGgrav(?,a) = 0. A2.546)
It is convenient to change (mix) the ghosts a —> 2, where
a := a + 1^(ш — щ). A2.547)
Then we obtain a new set of algebra which does not contain wo any more
(the BRS algebra A2.541) corresponds to a Lie algebra which seems to
depend on wo! however, one finds that actually it does not depend on wo):
BRS algebra
se — —ae + (i
sw — -Da + i
~2 1. •
sa = —a H—iti
2
si = \[U\- A2-548)
If we now perform a 'Stora shift'
U —> Ul = ш + Oi
d -> A = d + s, A2.549)
we find the following identity of the curvatures.

530 Gravitation
Theorem: 'Russian formula'
ВД = eli R{u) = A + ц + ^uh)R(u), A2.550)
where ___
R(ui) = Аш + ui2. A2.551)
To derive the chain of descent equations we have to consider the 'trans-
'transgression' for the shifted fields which is valid for nontrivial fibre bundles
(recall equations G.28), G.29) of Section 7.2).
Theorem: 'Shifted transgression'
Р(Дп(ш)) - Р(Дп(ш0)) = AQ3n-i(w,wo) A2.552)
<32n_iE,wo) = n / dtP(w - wo, ЛГ1), A2.553)
Jo
with __ ___
Rt = R{u!t) = Awt + ?t2, wt = wo + t(w-wo)- A2.554)
The homotopic curvature can be re-expressed in the following way
Rt = Rt + ti^R + -ti^R + {t2 - t)({a, и - w0] + a2), A2.555)
with
Rt = R(wt) = (kot+ ш2, ut = u0 + t{w - ш0). A2.556)
Applying the 'Russian formula' to the 'shifted transgression'
е*<Р(Д"И) - Р(ЛПМ) = AQ2n-i(?,u;o), A2.557)
expanding both sides in powers of the ghosts
i -)P(Rn(u;)) - Р(Л"(ыо))
+Q\n_2 + Q22n_3 + •••), A2.558)
and comparing terms of the same order, we obtain:
Chain of descent equations
Р(Д») - Р(Л"М) = dQln_t A2.559)
° L ")) A2.560)

Index theorem for gravitation 531
l l \u)) A2-561)
Since we have
ццР(Яп(и)) = 0 in Bn - 2) dimensions
we find that Qln-2 satisfies the consistency condition A2.546).
Proposition:
• The chain term <5гп-2 represents the gravitational anomaly
Q\n_2. A2.562)
M2n-2
The normalization is included in <?2n-2> respectively in the invariant poly-
polynomial P(BT).
Formula: The chain terms can be solved explicitly; the result is, for ex-
example [Langouche, Schiicker, Stora 1984]:
Anomaly term
QL~2 = n f dtPia,!??-1)
Jo
/•l
+ n(n-l) / dtP(w-wo,(t2-t)[a,w-wo],R?~2)
Jo
/•l
+ n(n-l)/ dtP(w-4Jo,tizR,R?-2). A2.563)
Jo
Of course, we may choose щ = 0 only for a trivial bundle and we recover
the formulae discussed before.
12.8 Index theorem for gravitation
As in the gauge case the gravitational anomalies can be completely determi-
determined by topological techniques. These rest upon the family index theorem of
Atiyah and Singer [Atiyah, Singer 1984] and, specifically for gravitation, on
the theorem of Alvarez, Singer and Zumino [Alvarez, Singer, Zumino 1984],
and, furthermore, on the local index theorems of Bismut [Bismut 1991], [Bis-
mut, FVeed 1986a,b], [Knecht, Lazzarini, Stora 1991a,b]. We, however, again

532 Gravitation
follow the procedure of Alvarez-Gaume and Ginsparg [Alvarez-Gaume, Gin-
sparg 1984] described in Section 11.5.3. The analogy to the gauge case is
very close.
12.8.1 Alvarez-Gaume and Ginsparg's index procedure
Diffeomorphisms: In gravitation we work with the topological spaces:
Sp M.... space of all metrics
Sp M. — {fif^i/} on the manifold M<m-
T>... space of all pointed diffeomorphisms
V = {x'(x)\x'(x(,) = x0, dx'(xo) = 1}.
Sp M/V... moduli space
equivalence class of all metrics up to diffeomorphism.
These spaces play the role of the spaces Sp A, G, Sp A/G in the case of
gauge theories.
Alvarez-Gaume and Ginsparg relate the anomaly in 2n dimensions to the
Atiyah-Singer index theorem in Bn+2) dimensions. The extra 2 dimensions,
parametrized by (t, в), describe 2-dimensional families of metrics on М^п—
2-dimensional, compact submanifolds in Sp M. These are projected down
to a sphere S2 in Sp M/V (the topologically nontrivial space), where the
metrics—which represent the upper and lower hemisphere in Sp M/T>—are
related to each other along the 'equator' of the sphere by a loop in the
diffeomorphism group V. The anomaly represents an obstruction to this
projection. To show the occurrence of the anomaly the determinant bundle
is considered over such a noncontractible sphere S2 in Sp M./V.
So we introduce a one-parameter family of diffeomorphisms
х->х'(в,х), A2.564)
giving rise to a loop in the diffeomorphisms. Then we get the coordinate
transformation matrix
^^, A2.565)
the exterior derivative in the additional 0-variable
S = d6-?-e, A2.566)
and the Maurer-Cartan form in group space
u?(M) = А-1(в,х)ёА(в,х). A2.567}
The Christoffel connection Г and the curvature R = dT + Г2 transform as

Index theorem for gravitation 533
Щв,х) = А-1(в,х)Щх)А(в,х). A2.568)
The anomaly occurs by considering the generating functional which re-
represents the fermion determinant.
Proposition:
• The noninvariance of the generating functional, the fermion determi-
determinant, under a general coordinate transformation occurs'in a phase
variation as we move along V. The local winding of the determinant
phase generates the anomaly!
Procedure:
i) We find a topological situation where the winding number of the phase
is nonvanishing.
ii) We construct a Bn + 2)-dimensional Dirac operator—by adding as
dimensions the variables в and t—whose index is equal to this winding
number. Then the index density supplies the gravitational anomaly.
The whole procedure is nothing but a repetition of the gauge case, des-
described extensively in Section 11.5.3, so that we may take over from formula
A1.228) the final result:
Proposition: Einstein anomaly
• The Einstein anomaly is given by
-GE(vt, Г@)) = 2тгг Index i p2n+2, A2.569)
where
г ^Jn+2 = / Index i Jp-m+i- A2.570)
Js1
index г
Index theorem: Now the index theorem enters, stating that the index
«¦tin be expressed as an integral over certain characteristic classes [Atiyah,
Singer 1968a,b,c, 1971a,b, 1984] (for further literature we refer to the books
[C.ilkey 1984), [Berline, Getzler, Vergne 1992], [Lawson, Michelsohn 1989],
JBooss, Bleecker 1985] and [Nash 1991]). In our case of the Dirac operator
A2.351) which contains the spin connection w = шаь сгаЬ/2 the characteristic
«•lass is given by the Dirac genus of the compact manifold M.

534 Gravitation
Definition: The Dirac genus is denned by
Xa/2
sinhxa/2
The quantities xa denote the skew eigenvalues of the curvature 2-form
/ 0 xi \
A2.571)
Rab
27Г
0
\
0 xn
-xn 0 /
A2.572)
which we consider as a matrix in the Lie algebra of SOBn). Each xa expres-
expresses a 2-form. The Dirac genus A(M) can be expanded to arbitrary order in
the curvature R. Rewriting the results of AGG [Alvarez-Gaume, Ginsparg
1985] we have
DтгN
Dtt)8
1
10368
(trfl2K + -±-tri?2tri?4 +
4320
5670
tr
497664
103680(
68040
+ ...
tr R2 tr Д6 +
259200
(tr Ry +
75600
tr
A2.573)
So the Dirac genus represents a sum of invariant polynomials in the cur-
curvature 2-form Л to a given finite order depending on the dimension of the
manifold.
Theorem: Atiyah—Singer index theorem
index D+= I A(M).
JM2n
A2.574)
D+ = i PP+ denotes the Weyl operator and P = Э + Ф the Dirac operator
A2.351) for gravitation. Clearly, we pick up only the n-th term of A{M)
proportional to the volume 2n-form.

Index theorem for gravitation 535
Note: Up to now we have not worried about the specific form of the ma-
manifold. However, choosing for gravitation, for instance, the sphere S2n with
radius L the Riemannian curvature (recall equation A2.275)) implies vanis-
vanishing traces .
Rab = -L ea eb => tr R2n = 0 A2.575)
and consequently
A(S2n) = 1. A2.576)
So in this case we achieve only a trivial result.
Anomaly: We apply the AS index theorem now to the Dirac operator in
Bn + 2)-dimensions chosen on a Weyl-like basis (recall equations A1.213),
A1.214)). The Bn + 2)-dimensional manifold M2n+2 constructed a la AGG
is a product manifold of a 2-dimensional sphere S2 in the variables (в, t)
with the usual compact manifold M2n of space-time
M2n+2 = S2 x M2n. A2.577)
So we can express the index A2.574) by
2тгг • index г #>2n+2 = 2ттг / A(M) = [ P(Rn+1{6)),
Js2 x м2п Js2 x м2п
A2.578)
which picks up a certain invariant polynomial P(Rn+1) to order (n + 1).
Proceeding now as usual, we rewrite the polynomial locally as the exterior
derivative (in Bn + 2) dimensions) of the Chern-Simons form
P(Rn+1F)) = (d + 6+ 6t)Q2n+1. A2.579)
We apply Stokes' theorem then we get the Chern-Simons form over the
Vquator' S1 x M2n
2ттг- index i p2n+2 = [ Q2n+i(T@)+Ч,Щ6))- A2.580)
JS1xM2n
Wo expand in powers of v$—the only quantity that carries d9—and we gain
i In; SZ chain term Q\n—the linear term in v$—as the sole contribution
2ттг • index г #2ri+2 = / <Э2п(^, Ц0), Д@)). A2.581)
JS1xM2n
('boosing finally в = 0 we find from equations A2.569), A2.570), A2.581):

536 Gravitation
Proposition: Einstein anomaly
• The pure Einstein anomaly is given by the chain term Q\n
Qln(v,T,R). A2.582)
This is in accordance with our differential-geometric chain result A2.532)
(by shifting n —» n — 1).
Now we can calculate the chain term Q\n inclusive of the normalization
from the invariant polynomial P(Rn+1) (recall equation A2.578)).
Normalization:
• The invariant polynomial—and consequently the chain term Q\n—is
determined completely by the Dirac genus
P(Rn+1) = 2ni A(M2n) • A2.583)
2n+2
The subscript 2n + 2 means that for a 2n-dimensional theory we have to
start with an invariant polynomial on a Bn + 2)-dimensional manifold. But
the indices of the curvature R^v or Rab still act only on the 2n-dimensional
space-time manifold M2n. It is the form part of R which carries the extra
dimensions.
Lorentz transformations: On the other hand, we also consider Lorentz
transformations, then we refer to the tangent frame and we work with the
topological spaces:
Sp В... space of all tangent frames
sP в = К}.
С... group of all local Lorentz transformations
Sp B/C ... moduli space.
More precisely, Sp В means the space of all sections of the tangent frame
bundle, and С means the space of all maps of the base manifold into the
Lorentz group. Actually we work on the space Sp M. x Sp В / T> x C.
We introduce the one-parameter family of local Lorentz transfor-
transformations
LF,x) A2.584)
and the MC form in group space
a@,x) = Ь~1(в,х)ёЦв,х). A2.585)
The spin connection w and the curvature R(oj) = du + ш2 transform as

Index theorem for gravitation 537
ш{в,х) = L-1(e,x)[u}{x) + d\L(e,x)
R(O,x) = Ь-1(в,х)Щх)Ь(в,х). A2.586)
To find the anomaly we proceed as described above, we just have to replace
1){-»а,Г-»ш and we obtain the final result.
Proposition: Lorentz anomaly
• The Lorentz anomaly is given by the chain term Q\n
GL(a,u>) = - f Qln(a,u>,R). A2.587)
JM2n
Again, the chain term Q\n is now correctly normalized via equation A2.583).
12.8.2 Examples in two dimensions
Let us illustrate the 2-dimensional case, n = 1, which is easy to handle. We
have to begin with the invariant polynomial (with the correct normaliza-
normalization); it is determined by equation A2.583) and expansion A2.573):
Invariant polynomial
P{R2) = 2m A(M2) = -1- tr R2. A2.588)
dim=4 9u7T
Then the chain term Q\ following from the polynomial A2.588) is given by:
Chain term
Ф(«€.Г) = 7^-*г«€Д\ Q\{ot,w) = r±- tr adv. A2.589)
Уотт Уотт
For the anomalies we find the expressions:
Pure Einstein anomaly
A2-590)

538 Gravitation
Lorentz anomaly
GL(a,w) = f d2x e aab (Tab)
tr аш. A2.591)
As discussed in Chapter 5 the factor (—г) is removed in Minkowski space.
What does this mean for the energy-momentum tensor?
Conventions: Specifically in 2 dimensions with a Minkowski signature
we use the following conventions here:
e-tensor referred to TF
e l = -e1 =1 ?oi = — ею = —1
eacecb = 6ab еаЬеаЬ = -2. A2.592)
e-tensor referred to CS , e = detea/J
W^v — — P^v F — P P
с — с *~\iv — ^ ?.y.v
go» ё10
go» = -ё10 = - ё01 = -ею = -e A2 593)
e^ eXv = S»v E»v e^ - -2
E»v = Ea>iEbveab E^
Formulae: With these notations we derive the following useful formulae
which we will need below
\ dx)idxl/ = e)iVdxodx1 = e»vd2x
ДаЬ„„ = -I eabE^1Z E^veabd^abv = П, A2.594)
where "R is the Ricci scalar A2.95)

Index theorem for gravitation 539
1l = R>iV(iV = Rabab = 2Rolol. A2.595)
Lorentz anomaly: Let us consider the case of a Lorentz anomaly
A2.591). In components we have
GL{a,u) = f d2xeaab(Tab) = ^- f d2x ааьд^ъа1/^. A2.596)
Proposition:
• The Lorentz anomaly is equivalent to the existence of an antisymme-
antisymmetric energy-momentum tensor of the following form
Уотт 1У2тг
We have shifted the anomaly into the Lorentz part which means that there is
no anomaly in the diffeomorphisms, GE{?,) = 0, no Einstein anomaly. Howe-
However, this does not mean a conservation of the energy-momentum tensor—as
discussed already in Section 12.6.1—but implies (recall formula A2.424))
V^ (Т"„) = uw (ГаЬ). A2.598)
Inserting result A2.597) we get:
Proposition:
• The Lorentz anomaly also corresponds to the following nonconserva-
tion of the energy-momentum tensor
^ пЬш*Ьг,П. A2.599)
We observe that [Т^и) does not transform in a covariant way.
Weyl anomaly: For completeness we also quote the result of the Weyl
anomaly without going into the details of the calculation. These can be
found e.g. in [Ebner, Heid, Lopes Cardoso 1987]
Gw{o) = f d2x e а (Т*M>
= -^Jd^xea^n-^e^V^abA A2.600)
and consequently we have:

540 Gravitation
Proposition:
• The Weyl anomaly expresses the following nonvanishing tensor trace
(n \ e V^b) . A2.601)
Covariant gravitational anomaly: As discussed in Section 12.6.4 we
can find a covariantly transforming energy-momentum tensor
(Т^) = (Т^)+Р^ A2.602)
by adding the local BZ polynomial 7"*". Then the covariant gravitational
anomaly is determined by (T**") and can be calculated explicitly via the
formula A2.488) of Bardeen and Zumino (n = 2). Again, the correctly nor-
normalized invariant polynomial is given by equation A2.588) and we obtain:
Covariant gravitational anomaly
= -2 I P{VbR)
In components we have (after partial integration and removing (-г) in Min-
kowski space)
^J Ra0(i^v A2.604)
so that we find for the covariant energy-momentum tensor
~ 1
** 96тг
Re-expressing in terms of the Ricci scalar 1Z (formula A2.594)) gives:
Proposition:
• The covariant gravitational anomaly represents the following noncoa-
servation of the covariant energy-momentum tensor
I. A2.606)

Index theorem for gravitation 541
Of course, the BZ polynomial has to satisfy
) A2-607)
and can be found to be of the following form:
BZ polynomial
^WcVabii + ЧршаЪа) - ё^П] . A2.608)
(Relation A2.607) can be easily checked, all we need is the identity
0.)
We notice that the trace of the polynomial is
^ A2.609)
and cancels precisely the second term in the trace A2.601) so that we get for
the trace of the covariant energy-momentum tensor—the covariant trace
anomaly
? ^ П. A2.610)
We have calculated all the results in this section within differential geo-
geometry and topology; they are in agreement with the results of [Leutwyler
1985b], [Leutwyler, Mallik 1986], [Langouche 1984], [Ebner, Heid, Lopes
Cardoso 1987] and [Alvarez-Gaume, Witten 1983] which are achieved by
quite different methods.
12.8.3 Mixed anomalies
Finally, we want to consider a fermionic theory with gravitational and Yang-
Mills gauge fields. Then we use the Dirac operator
9 = ES-fid» + AI1+ wM), ' A2.611)
with the YM field A^ = A^T* and the spin connection шц = шаъцСгаЬ/2. The
corresponding quantum field theory may generate gauge-, gravitational- and
mixed anomalies. The mixed anomalies arise in perturbation theory from a
formion loop which couples to gravitational and YM fields.
Again, these anomalies can be calculated via the index theorem [Alvarez-
Caume, Witten 1983], [Alvarez, Singer, Zumino 1984]. In this case we use
t he following index theorem (see e.g. [Gilkey 1984], [Berline, Getzler, Vergne
1992], [Lawson, Michelsohn 1989]).

542 Gravitation
Theorem: Atiyah-Singer index theorem
index D+= I ch (F)A(M), A2.612)
•>м2п
where D+ = i PP+ and lf> — ф + 4 + ф.
The expression ch (F) A(M) means the wedge product of the Chern
character A1.112) with the Dirac genus A2.571), A2.573) in a given order
corresponding to the dimension of the manifold. Wedging the expansions
A1.112) and A2.573) we gain the explicit invariant polynomials in several
dimensions
dim = 2 index D+ = ^- / tr F
JM5
dim = 4 index D+ =
dim = 6 index D± =
Me
dim = 8 index D+ = j^
Л t,. f3 + 4 tr F tr Л2
6 48
A2-613>
where г denotes the dimension of the gauge group representation.
We observe that mixed terms involving both F and R occur only for
dimensions higher than or equal to 6 and consequently there exist mixed
anomaly contributions only in these dimensions. We can calculate the an-
anomalies from the above invariant polynomials in the previously desribed way.
For further literature we refer to [Alvarez-Gaume, Witten 1983], [Alvarez-
Gaume, Ginsparg 1985], [Minkowski 1988].
12.8.4 Axial gravitational anomaly
At the end of the book we return to the ABJ anomaly of Chapter 4. The
ABJ anomaly, or axial anomaly, is described by the triangle diagram which
contains one axial current and two vector currents that couple to gauge
fields. Replacing the gauge fields by gravitational fields Delbourgo and S*-
lam [Delbourgo, Salam 1972], Eguchi and Freund [Eguchi, Freund 1976}
found that the fermion triangle with one axial current and two energy-
momentum tensors also generates an axial gravitational anomaly. We
do not consider the perturbative calculation here but instead we determine
this anomaly via the index theorem, which we are now familiar with.
As we discussed in Chapters 5 and 11 the axial anomaly—the anomalous

Index theorem for gravitation 543
divergence of the axial current—occurs in the Jacobian of the path integral
for the generating functional of the Green functions. This Jacobian contains,
after suitable regularization, an index density that leads to the following
relation (in Euclidean space, recall Theorem A1.20)).
Theorem: Anomaly—index relation
A = d* f = 2i- Index D+. A2.614)
The index density, on the other hand, is given by the AS index theorem
A2.574)
index ?>+ = / Index D+ = f A(M). A2.615)
(We do not compactify space-time to a sphere 52n, where A(S2n) = 1.)
Now we use the expansion of the Dirac genus, equation A2.573), and we
find:
Proposition: Axial gravitational anomaly
• The axial anomaly in the presence of gravitational fields is given by
= d * f = 2i A(M2n) = -i -r^ tr R2. A2.616)
n=2 УОТГ^
As usual, the factor (—i) is removed in Minkowski space. In components we
get
l ^ ^В?^. A2.617)
Its physical meaning is as in the gauge case; the anomaly signals the break-
breakdown of the external (axial) symmetry in the presence of gravitational fields.

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Index
action principle, 492
active coordinate transfor-
transformations, 470, 472-474
adjoint derivative, 51-53, 294
adjoint map, 94
adjoint operator, 412
adjoint representation, 94, 161
Adler-Bardeen theorem, 205
Aharonov-Bohm effect, 306-311
fibre bundle description, 309
phase shift, 308
quantum mechanics, 307
anomalous divergence, 188, 214,
219, 253
anomaly, 176, 188, 193-195,
212-214, 296-297, 338,
353, 354, 365, 366, 378,
410, 426
2-dimensional, 214-227,
260-261, 276, 283, 286
ABJ, 188, 197-214
Bardeen, 241, 243, 297, 351,
436, 446
consistent, 243, 272, 401
covariant, 243, 266, 270, 271,
297, 393-394, 397-400,
405
dispersion relations, 216-220
IR behaviour, 230-231
for L-currents, 190
non-Abelian, 241-244,
265-272, 297, 338, 379,
382, 435-436, 445-446
nonlocal extensions, 406-407
normalization, 369
and phase, 194-195, 440
regularization independence,
261-263
singlet, 238-241, 259, 296, 379,
381, 400, 411
UV behaviour, 231-232
anomaly cancellation, 245-247
anomaly formula, 341
anomaly shift, 517
anti-BRS transformation, 175
antiderivation rule, 43, 333
associated bundle, 103
Atiyah-Singer index theorem,
305, 319, 410-411,
426-427, 443, 534, 541
atlas, 31
axial current, 178
non-Abelian, 183 .
axial gauge, 169
background field, 371
Bardeen-Zumino counterterm,
515
Bardeen-Zumino polynomial,
392-393, 396-400,
404-405, 519, 541
base space, 95
beta function, 388
Betti numbers, 69-71
Bianchi identity, 165, 295, 354,
463, 480
first, 482
boundary, 11

558
Index
boundary operator, 62
bounded operator, 412
BRS algebra, 376, 520-524
for nontrivial fibre bundles,
528-529
shifted, 524-525
BRS operator, 172-175, 345, 350,
356, 379, 523
shifted, 524
total, 521
BRS transformation, 172-175,
345-346, 357, 358, 366,
521
Cartan structure equation,
114-116, 463, 479
Cartan's exterior algebra, 42
Cartan's homotopy formula, 334
Cauchy's integral formula, 217
chain complex, 62
chain terms, 352, 354, 369, 537
solutions, 382-383, 386, 389
chains, 61
charts, 31
Chern character, 425
Chern number, 305, 319, 327
Chern-Simons form, 296, 314,
327-328, 335-337, 354,
367, 372, 377, 379, 444,
516, 525, 527
gauge transformation, 336-337
variation, 339, 341
Chern-Simons term, 240
Chern-Simons theory, 379
chiral transformations, 181,
250-251, 256, 267
Jacobian, 280-286
chirality split, 268
Christoffel connection, 460-461
transformation, 469, 471, 474
Christoffel symbol, 460
closed set, 10
closure, 10
coboundary, 67, 364, 378
coboundary operator, 363
cochain, 363
cocycle, 67, 362, 364, 378
cocycle condition, 361, 364
coderivative, see adjoint
derivative
cohomology, 363-365, 378-379
cokernel, 413
commutation relations
functional, 391
compactification, 13, 17
compactness, 12
compatibility condition, 112, 117,
304, 317, 442, 443
conformal group, 456
conformal transformation, 456
conformally flat manifolds,
456-458
connectedness, 18
connection, 105-113, 304, 309,
317
global, 443
connection 1-form, 109-112, 442,
460
conservation laws, 177-185, 228
consistency conditions
for gravitational anoma-
anomalies, 512-515, 522, 529
consistent current, 392
gauge transformation, 392, 396,
402-403
continuous map, 13
contractible space, 329
contravariant vector, 459
coordinate transformations,
458-460, 468-474
infinitesimal, 470-471
matrix, 469
cosmological constant, 493
cotangent bundle, 104
cotangent space, 39-40

Index
559
basis, 39, 373
covariant anomaly condition, 394
covariant current, 392-393, 397,
404
gauge transformation, 393, 404
covariant derivative, 113, 162,
163, 181, 343, 376, 461,
467-468, 479, 496
covariant vector, 459
covering, 12
current 1-form, 296, 395
on Sp Л, 401
current operator, 210-214
curvature, 113-117, 304, 310,
463-465, 479, 484
transformation, 469, 471, 473
curvature 2-form, 114, 443
Cutkosky rule, 225
cycle, 63
d'Alembert operator, 420
de Rham cohomology, 67-73
group, 67
inner product, 70
de Rham complex, 68
de Rham's theorem, 69
derivative
in Sp Л, 391
descent equations, 341, 354, 369,
372, 376, 377, 385, 387,
525-527, 530
determinant
of a differential operator,
137-139
of the Dirac operator, 277-286
fermion, 439
gauge invariance, 429-430, 439
of the metric, 453
phase, 439-440
regularized, 277, 280-286
of the Weyl operator, 428
determinant bundle, 448-449
diflfeomorphism group, 469
diflfeomorphisms, 13, 532
differential forms, 40-58, 84
anticommutator, 56-58
closed, 54
coclosed, 54
coexact, 54
commutator, 56-58
exact, 54
harmonic, 54, 71-72
inner product, 51
integration of, 45-46
on Sp Л, 348
trace, 56
differential map, 73, 89
dilatation vector, 475
Dirac equation, 154, 178, 229
energy spectrum, 229
Dirac field
free, 154, 155
interacting, 155-156
Dirac genus, 534
Dirac monopole, 297-306, 380,
426
fibre bundle description, 303
field strength, 302
gauge potentials, 299, 301, 302
gauge transformation, 300-302
magnetic charge quantization,
301
winding number, 305
Dirac operator, 249, 255, 265, 497
construction on S2 x S2n,
441-443
eigenvalue equation, 409
spectrum, 431
zero modes, 409
Dirac sea, 227-233
dispersion relations, 216-220
dual p-form, 49
dual algebra *Lie G, 358
dual current form, 296, 395
duality, 68

560
Index
Einstein anomaly, 509-510, 533,
536
2-dimensional, 537
Einstein equation, 492-493
with matter, 493
without matter, 492
Einstein ghost, 521
Einstein invariance, 504-505, 519
Einstein tensor, 493
Einstein transformations, 474,
504, 518-519
generator, 511
Einstein-Hilbert action, 491
electrodynamics, 287
action, 292
current conservation, 292
field strength, 178
field strength tensor, 288
gauge potential, 156-159, 290
Lagrangian, 177
elliptic differential operators,
417-420, 422
embedding, 454
energy-momentum tensor,
493-495, 508, 519
antisymmetric, 509
covariant, 519
covariant conservation, 493,
505
symmetric, 493, 503, 504
traceless, 507
equivalence principle, 458
Euclidean space, 250
2-dimensional, 261
Euler characteristic, 69-71
exponential map, 90
extended gauge group element,
374
exterior covariant derivative, 113
exterior derivative, 42, 373, 443
in Sp A, 348, 360
in Sp M, 518
oaMxj, 374
face, 59
Faddeev-Popov determinant, 167
Faddeev-Popov ghosts, 92,
168-170, 344, 350, 356,
379
Faddeev-Popov operator
differential, 345
integrated, 345
family index theorem, 436
fermionic action, 497
fermionic Lagrangian, 500-502
Feynman gauge, 159
Feynman parameter integral, 206,
221
Feynman propagator
for fermions, 154-155
of the photon, 158-159
scalar, 137
Feynman's path integral, 124-129
formula, 126
Hamilton formalism, 126-129
fibre bundles, 95-117, 348
reconstruction, 99-101
fibres, 95, 304, 316
field strength, 116-117
flow, 78-81, 90, 93, 105
infinitesimal, 79
frame bundle, 104
Fredholm alternative, 415
Fredholm operators, 413-414, 420
direct sum, 415
product, 416
Fujikawa's method, 255-259
regularization, 257-259, 263,
268, 270, 432-435
Fujikawa's uncertainty principle,
263-265
function of an operator, 274
functional derivative, 132-135,
347
fundamental vector field, 105, 360
commutator, 401

Index
561
gamma function, 278
gauge condition, 166
gauge fixing, 158, 167, 169
gauge group, 159-161
gauge group space, 347, 374, 447
gauge operator, 343-344, 353,
356, 360
commutation relations, 343,
353
integrated, 344
gauge potential, 110-113, 162
gauge transformations, 112, 161,
270, 374
axial, 179, 181
infinitesimal, 164
left-handed, 180, 182
right-handed, 180, 182
vector, 178, 181
gauge variation, 344
Gauss theorem, 467
Gaussian cut-off, 257, 434
Gaussian integration, 137-139,
259, 435
for Grassmann variables,
151-154
Gell-Mann matrices, 160
generating functional, 140, 142,
144, 154, 155, 169,
191-197, 251, 342
graded algebra, 173
Grassmann algebra, 42, 149-154
Grassmann differentiation, 150
Grassmann integration, 150, 256,
264, 433
Grassmann measure, 256, 267
gravitational anomaly
axial, 542-543
covariant, 518-520, 540
normalization, 536
total, 522
gravitational anomaly formula,
527, 531
Green functions, 185-190
connected, 147-149
for interacting fields, 144-147
free, 139-141
Gribov ambiguity, 171-172
group parameter space, 372
harmonic oscillator, 121
heat equation, 274-275
heat kernel, 274-276, 423-425
expansion, 275
regularization, 273-276
Heisenberg picture, 118
Hodge decomposition theorem, 71
Hodge star operation, 48-51
holonomy, 109, 310-311
homeomorphism, 13, 95
homology, 58-67
group, 63, 65
homomorphism, 66
homotopic connection, 326, 336,
384, 516
homotopic covariant derivative,
385
homotopic curvature, 326, 384
homotopic equivalent spaces, 15
homotopic metric, 516
homotopy, 13-29, 329, 336, 384,
387, 439
of maps, 14-18
of operators, 416
homotopy class, 16
homotopy derivation, 333, 339
homotopy group, 18-29, 310, 449
first, 20, 22
n-th, 25, 28
non-Abelian, 23
homotopy operator, 331-335
horizontal lift, 107-109
horizontal subspace, 106-107, 109
horizontality condition, 377
hyperbolic operators, 420

562
Index
immersion, 454
index, 414-417, 423-425
density, 411
of the Weyl operator, 410, 422
index theorem, 425-427
for gravitation, 533-534
induced metric, 454-455
induced vector field, 93
inner product, 373, 452
in Sp Л, 347
instantons, 311-320, 426
BPST instanton, 313, 315
topology, 315
winding number, 318
integral curve, 78
interior product, 44
invariant polynomial, 321-325,
367
of forms, 323
invariant vector field, 89
invariant volume element, 460,
478
isometric map, 474
isometry, 455
group, 456
Jacobi identity, 91, 161, 294, 344,
379
Jacobian, 251, 257, 259, 268, 270,
275-277, 410, 433-435
Ktinneth formula, 71
kernel, 412
Killing equation, 475
Killing vector, 474-475
conformal, 475
Klein-Gordon equation, 136
Landau gauge, 159
Laplace transform, 278-279, 282
Laplacian, 53-54, 419, 421, 422
leading symbol, 419
left-action, 92
left-invariant vector field, 89
left-translation, 89
Levi-Civita connection, 462
Levi-Civita tensor, 45
Lie algebra, 56, 90-95
basis of, 90
dual, 91
Lie bracket, 43, 82-84, 89, 472
graded, 524
in Sp A, 403
Lie derivative, 81-88, 360-361,
471-472, 522
commutator, 401
covariant, 529
of p-form, 84-85
of function, 85
of tensor field, 85
of tensor-valued p-form, 86-87
of vector field, 81-82
Lie group, 88-95
limit point, 10
line bundle, 103
line element, 454
on the sphere, 485
local bundle coordinates, 96
local curvature, 116
local index theorem, 450
local trivialization, 96
canonical, 111
loops, 18
higher, 25
homotopic, 19
inverse, 21, 27
product of, 20, 26
unit element, 22, 27
Lorentz anomaly, 509-510, 537
2-dimensional, 538-539
Lorentz gauge, 291
Lorentz ghost, 521
Lorentz invariance, 503
Lorentz metric, 454

Index
563
Lorentz transformations, 477,
490, 536
generator, 511
infinitesimal, 490, 503
local, 536
spinor representation, 496
generators, 496
LSZ reduction formula, 234
Mobius strip, 101-102
magnetic flux, 307
manifolds, 29-37, 92
boundary of, 32-33
complex analytic, 32
differentiable, 29-37
dimension of, 32
orientability, 36-37
product of, 35
mass renormalization, 145-146
Maurer-Cartan form, 92, 379
on group space, 349, 350, 375,
532
Maurer-Cartan horizontality
condition, 447
Maurer-Cartan structure
equation, 91-92
in group space, 349, 375
Maxwell equations, 287-290
metric, 452-453
transformation, 473
metric space, 11
metric tensor, 453
metric topology, 11
Minkowski space
2-dimensional, 260
mixed anomalies, 541-542
moduli space, 347, 447, 532, 536
moments, 121-122
n-point function, 147, 149
neighbourhood, 10
nonlocal counterterm, 246
norm, 453
one-parameter group, 79
one-parameter subgroup, 90, 94
open set, 10
operator norm, 412
orbit, 93
orthogonal complement, 412
Palatini identity, 466
parallel transport, 107-109, 310
partition function, 123-124
passive coordinate trans-
transformations, 469
generator, 513
path integrals, 124-169
Abelian fields, 156-159
chirally transformed, 252
fermions, 154-156
measure, 251, 255, 266, 432
non-Abelian fields, 165-169
quantum mechanics, 124-129
scalars, 135-149
Pauli matrices, 160
Pauli-Villars regularization,
205-207, 263
PCAC, 235
period, 68
perturbation theory, 142-144
ф* theory, 142-149
тг° -» 77 decay, 233-238
pion amplitude, 236
pion decay rate, 237
Poincare duality, 70-71
Poincare lemma, 55, 328-331, 354
algebraic, 333, 367, 378
point splitting regularization,
210-214
Pontrjagin index, 313, 318-319,
411

564
Index
principal bundle, 102-103, 303,
309, 316, 348
projection, 95
projection operators, 179
propagators, 118-121, 124-126,
128
free, 275
proton spin crisis, 244
pseudo-Riemannian metric, 452
pseudoscalar current, 178
non-Abelian, 183
pullback, 76-78, 330, 331
pure gauge, 117, 312
quantum action, 147-149,
191-197, 351, 379, 508
invariance, 253, 254
range, 412
reference system, 478
Ricci scalar, 464
Ricci tensor, 464
Riemann tensor, 463-465, 484
Riemannian connection, 462, 481
Riemarmian geometry, 452-465
Riemannian metric, 452
right-action, 93
right-invariant vector field, 89
right-translation, 89, 90
Russian formula, 368, 371, 376,
446-447, 525, 530
scalar, 459
scalar field
free, 135-137
Schrodinger equation, 119-120
Schrodinger picture, 118
Schwinger source, 129
Schwinger term, 378, 379, 382
seagull term, 224-225
section, 102, 303, 309
Seeley coefficients, 275
signature, 454
simplex, 58
boundary of, 61-62
orientation, 60
simplicial complex, 60
space of all gauge connections,
347, 437, 447
space of diffeomorphisms, 532
space of metrics, 518, 532
space of tangent frames, 536
spin connection, 479, 483, 498
spin connection 1-form, 479, 484
Stokes' theorem, 46-48
Stora—Zumino chain of descent
equations, see descent
equations
structure constants, 56, 91,
160-161, 344
symmetric, 161
structure equations, 375-376
structure group, 95, 96, 104, 303,
309, 316, 448
submanifold, 454
superconvergence sum rule, 220
Sutherland-Veltman paradox,
235
symbol of a differential operator,
418-419
symmetries, 177-185
Abelian, 178
non-Abelian, 181
symmetrized trace, 322, 324
't Hooft-Veltman integral
formula, 207, 222
't Hooft-Veltman regularization,
207-210, 220-223
tangent bundle, 103
tangent map, 73, see differential
map
tangent space, 37-39

Index
565
basis, 38, 373
tangent vector, 38
tensor density, 459
tensor fields, 40, 85
tensor-valued forms, 86-87
tensors, 40, 459
time ordered product, 131-132
topological charge, 313, 318-319,
411
density, 313
topological current, 313
topological field theory, 379
topological invariants, 13
topological space, 10
torsion, 462, 480
torsion consistency condition, 480
total space, 95
trace anomaly, 509
covariant, 541
trace of an operator, 278-279
transformation law
for field strength, 117
for gauge potentials, 112
transgression, 367, 371, 376, 385,
387, 525
shifted, 368, 371, 376, 525, 530
transgression formula, 325-327,
335
transition amplitude, 119,
124-126, 130
transition functions, 96-102, 104,
112, 117, 304, 316, 443,
449
consistency conditions, 96
transitivity, 15
triangle graph, 197-205
triangulation, 65-66
trivial bundle, 98-99
?/A) problem, 244
vacuum functional, see generating
functional
vacuum-to-vacuum transition,
129-131
vector bundle, 103-105
vector current, 178
non-Abelian, 183
vector fields, 39, 40, 73, 81-84
on Sp A, 348
vertical subspace, 105-106
vielbein, 476-478
inverse, 477
on the sphere, 485
transformation, 478
vielbein 1-form, 476
vielbein condition, 499
inverse, 499
vielbein determinant, 477
Ward identity, 176, 185-188,
192-193, 253-254, 351
2-dimensional, 215-216,
218-219, 223
anomalous, 193, 202-204, 206,
210, 239, 253, 351, 353,
357, 361, 366, 371, 401,
436
axial, 188, 198, 206, 209-210
non-Abelian, 195-197
vector, 186, 188, 202-203, 205,
208
Ward operator, 354, 512
wedge product, 40, 91, 373
in Sp A, 347
weight, 459
Wess-Zumino consistency
condition, 243, 352-356,
358-362, 365-367, 371,
379, 401-402
Weyl anomaly, 509-510
2-dimensional, 539
Weyl ghost, 521

566 Index
Weyl group, 456
Weyl invariance, 506-507
Weyl operator, 420-422, 424, 427
Weyl transformations, 507
Wick rotation, 250
Wick theorem, 141
winding number, 17-18, 305, 318,
439-441
Yang-Mills theory, 159-165,
293-295
action, 295
covariant derivative, 162, 163,
293
current conservation, 184, 295
field equation, 164, 295
field strength, 116-117, 162,
293
gauge potential, 110-113, 162,
293
Lagrangian, 162, 169
Z-functional, see generating
functional
zeta function, 277-286
regularization, 277-286, 433
zweibein, 483