anomalies in qft
TRANSCRIPT
Anomalies in Quantum Field Theories
Andrew James Bruce
October 6, 2005
Preface
The main purpose of this thesis is to introduce the various anomalies that arise in quantumfield theories; in particular their connection with topology and geometry. A theory is said tobe anomalous when a classically conserved current is no longer conserved upon quantisationof the theory. More precisely, if there exists no regularisation procedure that preserves allthe classical symmetries then the theory is said to have an anomaly. A better name for suchanomalies would be quantum mechanical symmetry breaking as all anomalies arise as oneloop corrections to the classical conservation laws.
Anomalies shed much light on the deep nature of quantum field theories. In particular,anomalies were first calculated within perturbation theory but later shown to be related tothe global topology of the theory via index theorems.
This thesis uses the modern formulation of differential geometry and topology to describethe various anomalies that can arise in a quantum field theory. Much emphasis is placed onthe methods initiated by Stora, Zumino and Wess [74, 77] who found an algebraic formula-tion known as the descent equations. We set up the descent equations in Gauge theory viageometric and topological arguments in particular the Atiyah-Singer index theorem and theBRST transformations.
It is then explained how the descent equations are related to BRST cohomology and howone can approach the problem of finding the anomalies in this set up.
The reader is expected to have a good grasp on the basics of quantum field theory, espe-cially functional methods and perturbation theory. Knowledge of differential geometry andalgebraic topology is assumed. A familiarity with general relativity would also be useful.
The following books were found useful in preparing this thesis; [20, 34, 44, 45, 58, 53, 54,55, 60].
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Acknowledgements
I would like to thank Dr. P. M. Saffin for support and direction in the undertaking of thisthesis. I would also like to thank Drs. M. B. Hindmarsh and A. Tranberg who have bothenhanced my understanding of quantum field theory. I also wish to thank all the membersof the particle theory group at the University of Sussex all of whom have made my studiesenjoyable.
Andrew James Bruce
Sussex.
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Conventions and Notation
Throughout we use natural units where c (speed of light) = ~ (Planck’s constant) = 1, un-less otherwise explicitly stated. Einstein’s summation convention is employed: if the indexappears twice, once as a superscript and once as a subscript, then the index is summed overall possible values. For example if µ runs over 0 to m, we have,
AµBµ =m∑
µ=0
AµBµ.
The Minkowski metric is given by gµν = ηµν = diag(1,−1...,−1), while the Euclidean metricis gµν = δµν = diag(+1,+1...,+1). Lower case Greek letters will generically denote indicesrunning from 0 to m and lowercase Latin letters will denote indices running from 1 to m,where m is the dimension of the manifold
With a Euclidean metric the Dirac Matrices satisfy
㵆 = γµ,
γµ, γν = 2δµν .
Assuming the space-time to be even-dimensional (dimM = m = 2l) we also define
γm+1 = (i)lγ1...γm,
γm+1† = γm+1,
(γm+1)2 = I.
Unless otherwise stated, the metric will be taken to be Euclidean. Under some assumptionsit has been shown that within perturbation theory the results of the Euclidian theory implythe results of the corresponding Minkowskian theory. The Minkowskian theory is formallyobtained via the analytic continuation in the complex plane. This is equivalent to replacingp0 of the 4-momentum by −ip0. This is known as a Wick rotation.
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CONTENTS
1 Introduction 1
2 The Anomalies of Gauge Theory 6
2.1 The ABJ-Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The Singlet Anomaly via Fujikawa’s Method . . . . . . . . . . . . . . . . . . 122.3 The Non-Abelian Anomaly via Fujikawa’s Method . . . . . . . . . . . . . . . 182.4 The Wess-Zumino Consistency Conditions and the Descent Equations . . . . 222.5 Epitome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 BRST Cohomology and the Descent Equations 26
3.1 Local Cohomology and the Cohomology of d . . . . . . . . . . . . . . . . . . 263.2 The BRST Operator and its Cohomology . . . . . . . . . . . . . . . . . . . . 283.3 The Cohomology of s mod d . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Some Solutions to the Descent Equations . . . . . . . . . . . . . . . . . . . . 323.5 General Solutions to the Decent Equations and the Russian Formula . . . . . 343.6 The Bottom Up Approach to Solving the Descent Equations . . . . . . . . . 383.7 BRST Cohomology and Physics . . . . . . . . . . . . . . . . . . . . . . . . . 393.8 Descent Equations for Nontrivial Gauge Bundles . . . . . . . . . . . . . . . . 403.9 Epitome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 The Anomalies of Gravity 42
4.1 Chiral U(1) Gravitational Anomalies . . . . . . . . . . . . . . . . . . . . . . 434.2 The BRST Algebra for Gravitation . . . . . . . . . . . . . . . . . . . . . . . 444.3 Pure Lorentz Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 The Equivalence of Einstein and Lorentz Anomalies . . . . . . . . . . . . . . 484.5 The Descent Equations for Gravity via the Russian Formula . . . . . . . . . 494.6 Mixed Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.7 Epitome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 The Atiyah-Singer Index Theorem and Anomaly Polynomials 53
5.1 Twisted spin complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 The Rarita-Schwinger spin complex . . . . . . . . . . . . . . . . . . . . . . . 55
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5.3 The signature complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.4 The Invariant Polynomials in Various Dimensions . . . . . . . . . . . . . . . 565.5 Anomalies in Supergravity Theories . . . . . . . . . . . . . . . . . . . . . . . 575.6 Epitome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6 Conclusion 60
Appendices 61
A Non-Abelian Gauge Theory and BRST Symmetry 62
A.1 Classical Non-Abelian Gauge Theories . . . . . . . . . . . . . . . . . . . . . 62A.2 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65A.3 Faddeev-Popov Ghosts and Gauge Fixing . . . . . . . . . . . . . . . . . . . . 66A.4 BRST Quantisation of Gauge theory . . . . . . . . . . . . . . . . . . . . . . 69A.5 The Gribov Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
B Basics of Lie Algebra Cohomology 74
B.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74B.2 Lie Algebra Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75B.3 Chevalley-Eilenberg Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 76B.4 Lie Algebra Cohomology and BRST Cohomology . . . . . . . . . . . . . . . 76
C The Atiyah-Singer Index Theorem and some Characteristic Classes 78
C.1 Elliptic Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79C.2 Statement of The Atiyah-Singer Index Theorem . . . . . . . . . . . . . . . . 79C.3 Some Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Bibliography 81
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CHAPTER 1
Introduction
Symmetries and their conservation laws play a huge role in the formulation of modern physics.This is most acute in the application of quantum field theory to describe the fundamentalforces of nature. However, it is not guaranteed that a classical symmetry will be respectedin the quantum theory.
Indeed, there is little reason to expect the classical symmetry to be a symmetry of the cor-responding quantum system. Consider the fact that the classical system is on mass-shellas where the quantum system is necessarily off mass-shell. Hence it is not obvious whatproperties of the classical system are shared by the quantum system. A simple exampleis that of the vacuum energy of the harmonic oscillator in quantum mechanics. Quantummechanically the lowest energy state is greater than zero as where classical it is zero. Thissimple and well known fact cannot be seen from the classical Lagrangian and is a quantumeffect.
However, classical symmetries do play an important role in quantum field theories. The mostimportant classical symmetry in modern quantum field theory is that of gauge symmetry. Itforms the basic backbone of the standard model of particle physics. If one insists on gaugesymmetry in the quantum field theory one is lead to non-trivial constraints on the generatingfunctionals; known as the Ward-Takahashi identities [70, 67] for QED or the Slavnov-Tayloridentities [63, 68] for Yang-Mills. These identities and hence the gauge symmetry are nec-essary for the renormalisation of the theory. Without gauge symmetry, renormalisability ofthe theory is lost and the theory is inconsistent.
Not all anomalies are as disastrous for the theory as these local gauge anomalies. Generically,any local symmetry that becomes anomalous makes the theory inconsistent, as where globalsymmetries do not place any constraints on the quantum theory. In fact global anomaliescan be phenomenologically welcome.
For example, take the decay of a π meson into two photons. Early calculations by Steinberger[65] within the pion-nucleon model lead to decay rates that did not agree with experimentalobservation.
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The Feynman graph which represents the decay is the triangle graph in which two photonscouple to an internal fermion loop via two currents. We have the so called V-V-A trianglewhich is now known to lead to an anomaly. Bell and Jackiw [19] correctly calculated thedecay rate by taking into account the axial anomaly. Adler [1] discovered a similar result inQED.
The triangle diagram containing one axial and two vector currents leads to the anomalousconservation law for the axial current
∂µJ5µ(x) = A(x), (1.1)
where A is the Adler-Bell-Jackiw (ABJ) anomaly
A(x) =e2
16 π2εµναβFµνFαβ. (1.2)
A similar result is found for Yang-Mills fields; Aµ = AaµTa and Fµν = F a
µνTa. This is the so
called singlet anomaly
A(x) = ∂µj5µ =
1
16π2εµναβTrFµνFαβ. (1.3)
Bardeen [14] discovered that chiral fermions can lead to the break down of gauge symmetryfor non-Abelian gauge fields.
Ga[Aµ] = Dµ〈jµ〉a = ± 1
24π2Tr[Taε
κλµν∂(Aλ∂µAν +
1
2AλAµAν
)]. (1.4)
Where ± corresponds to right and left handed fermions.
It should be noted that the anomalies are not just artifacts of perturbation theory. It wasnoticed that the singlet anomaly can be given purely in topological terms by using theAtiyah-Singer (AS) [7, 8, 9, 10, 11] index theorem. This theorem loosely states that theindex of an operator can be expressed in terms of characteristic classes. The singlet anomalyis given by
∫dxA(x) ∝ v+ − v− = indD+. (1.5)
The AS index theorem then states that the index of the Dirac operator is given completelyin terms of the Chern character, in 4 dimensions we explicitly have
indD+ = − 1
8π2TrF ∧ F. (1.6)
Thus, the singlet anomaly is completely determined on topological grounds only.
Fujikawa [35, 36] noticed that the anomaly would appear as the Jacobian of the path inte-gral measure. This formulation is non-perturbative in nature and shows most directly theanomaly’s connection with the global properties of the theory.
Modern differential geometry and algebraic topology can also be used to describe the anoma-lies. Written as differential forms the singlet and non-abelian anomaly in 4 dimensions are
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A =1
4π2d Tr
(AdA+
2
3A3)
(1.7)
Ga[A] = ± 1
24π2TrTad
(AdA+
1
2A3). (1.8)
The non-abelian anomaly can be used to define the Wess-Zumino consistency condition.Defining
G[η,A] =∫ηaGa[A], (1.9)
allows the Wess-Zumino consistency condition to be written as
sG[η,A] = 0. (1.10)
Where we have the Faddeev-Popov ghost η = Taηa and the BRST operator s which is
nilpotent, s2 = 0. The gauge anomaly is understood as a non-trivial solution to the Wess-Zumino consistency condition. That is a solution not of the form
G[η,A] = sG[A]. (1.11)
Thus, the consistency condition defines the cohomology of s. The local form of the anomalycan be determined via local cohomology. That is the cohomology defined on the space oflocal functionals. The local variables are η, A and F . This local cohomology can be solvedvia the so called descent equations
dQ02n+2 = P2n+2
sQ02n+1 + dQ1
2n = 0
sQ12n + dQ2
2n−1 = 0...
sQ2n1 + dQ2n+1
0 = 0
sQ2n+10 = 0. (1.12)
P2n+2 is a symmetric gauge invariant polynomial in F . The lower index refers to the formdegree and the upper the power in η known as the ghost degree. Starting from a symmetricpolynomial all the terms in the descent equation can be explicitly calculated.
The term Q12n is identified as the local form of the gauge anomaly
G[η,A] = N∫
M2n
Q12n(η,A). (1.13)
The normalisation N is not fixed by the descent equations. The normalisation has to befixed via perturbation theory or careful topological analysis such as k-theory.
The important question is what symmetric invariant polynomial should be used? This de-pends on the theory and is directly related to the Atiyah-Singer index theorem for the ellipti-cal operator in question. For example, if one chooses the symmetrised trace as the invariant
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polynomial which gives the singlet anomaly, the non-abelian anomaly is the result. Thus, thesinglet anomaly in 2n+2 dimensions is related to the non-abelian anomaly in 2n dimensions.
The normalisation of the non-abelian anomaly requires the use of k-theory. Alvarez-Gaumeand Ginsparg [4, 5] demonstrated how the anomaly in 2n dimensions is related to the AStheorem in a (2n + 2) dimensional space. Their method avoids the use of k-theory, but insome sense it closely resembles it. They showed by using two-parameter families of gaugepotentials that the normalisation of the non-abelian anomaly is
N = −2πiin+1
(2π)n+1(n+ 1)!. (1.14)
The situation is very similar in gravity. Gravitation can be considered as a gauge theory ofdiffeomorphisms and Lorentz transformations. Thus, with minor modification the machinerydeveloped for gauge theory can be applied to gravitation. It turns out that the two anoma-lies, diffeomorphic and Lorentz are in fact not independent, but are really two expressionsof the same phenomenon.
Outline
The thesis is arranged as follows:
In chapter 2 we explicitly calculate the ABJ anomaly via evaluating the triangle diagramand Pauli-Villars regularization. Then we examine the singlet and non-abelian anomaly us-ing Fujikawa’s method and set up the descent equations via the Wess-Zumino consistencyconditions.
We then put the descent equations on strong mathematical ground in chapter 3. We developthe idea of local cohomology and use it to re-derive the descent equations. Solutions tothe descent equations are presented using several methods. Explicit examples are given forthe Dirac field. Finally in this chapter we set up the descent equations for non-trivial fibrebundles.
Now armed with the necessary tools to solve the descent equation in general, the questionof gravitational anomalies is tackled in chapter 4. We set up the correct BRST algebra forgravitational theories with out torsion. It is shown how this resembles the BRST algebra forgauge theory and so solutions to the descent equations are readily available. Mixed anoma-lies are briefly introduced in this chapter.
In chapter 5 we address the question of how to pick the invariant symmetric polynomials,which are also called anomaly polynomials in the literature. We construct the invariant poly-nomials for the spin-1/2 Dirac, spin-3/2 Rarita-Schwinger and (anti-)self-dual antisymmetricform fields via the AS index theorem for the relevant complexes. We introduce the necessarycomplexes and apply the AS index theorem. Using the so called anomaly polynomials weshow that 10-d supergravity theories are anomaly free.
Chapter 6 we round up the discussion and make our concluding remarks. Several appendixeshave also been included. In appendix A we quickly review Yang-Mills theory and show how
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the BRST operator arises here. Appendix B outlines the basics of Lie algebra cohomologyand shows how this is related to BRST cohomology. The Atiyah-Singer index theorem andsome characteristic classes are presented in appendix C. These results have been used ex-tensively throughout the thesis. The bibliography is extensive but by no means exhaustive.The study of anomalies is huge and as such it would be impossible to list all references onthe subject.
Omissions
Due to time and length restrictions several important topics have been omitted from thisthesis. These include;
1. any detailed discussion about k-theory and the normalisation of anomalies. See [54]for an accessible account. We do not discuss in any detail the Alvarez-Gaume andGinsparg index procedure. For a nice account see [20].
2. anomalies in the Batalin-Vilkovisky (BV) formulism [16, 17]. See [13, 37].
3. gravity theories with torsion. See [30, 51, 52, 75].
4. anomalies in non-commutative gauge theories. See [25, 47, 50] for a discussion ofanomalies. For an introduction to non-commutative geometry see [27, 48]. For a shortintroduction to non-commutative field theory see [12].
5. branes, M-theory and anomaly inflow. See [21, 22, 26, 33].
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CHAPTER 2
The Anomalies of Gauge Theory
Here we consider the anomalies that can arise in gauge theory. There are two kinds ofanomaly; global chiral and local gauge. The global chiral anomaly is the quantum break-down of the chiral symmetry (or a modification to it in the massive case). As it is a globalsymmetry is does not affect the renormalisation and consistency of the field theory. It isindeed welcomed phenomenologically.
When we have a chiral theory, that is the left and right handed fermions do not couple tothe gauge fields in the same way we have the possibility of a gauge symmetry being brokenquantum mechanically. Thus, the local gauge anomaly is catastrophic for a consistent quan-tum field theory and must be cancelled at all costs.
In this chapter we explicitly calculate the so called ABJ chiral anomaly in QED [1, 19] viaexamining the triangle graph and use Fujikawa’s method [36, 35] to determine the generali-sation of this to Yang-Mills theory and to calculate the non-abelian anomaly in chiral gaugetheory.
Fujikawa’s method of determining the anomalies is the most direct way to see their geomet-ric and topological nature. The anomalies can be seen as the failure of the regularisationscheme to preserve the classical symmetry. Fujikawa’s method explicitly shows that thisfailure occurs in the path integral measure. This method also directly shows the topologicalnature of anomalies.
The non-abelian anomaly is also explained as the lack of BRST invariance of the effectiveaction. This leads to the so called Wess-Zumino consistency condition. Solutions to thiscondition can be found via the descent equations.
2.1 The ABJ-Anomaly
Here we examine the Adler-Bell-Jackiw (ABJ) anomaly in QED via direct computation ofthe so called V-V-A triangle graph (see figure 1.1). We employ the Pauli-Villars regulisation[57], which as we explicitly add a massive fermion to the theory directly breaks chiral sym-
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metry. It should be noted that the anomaly exists for all other regulisation schemes. Forexample, when one uses dimensional regulisation we extend the theory to a higher dimen-sional complex space. As there is no generalistion of γ5 in these spaces there is no notion ofchirality and so chiral symmetry is spoiled.
We explicitly calculate the chiral anomaly in the simplest gauge theory, QED. Here weassume a Minkowski metric of signature (1, 1, 1,−1). Consider the fermionic part of theLagrangian for QED.
L = ψ(i/∂ −m+ e /A)ψ. (2.1)
Where we have used the “slashed” notation for /∂ = γµ∂µ and /A = γµAµ, with Aµ being theU(1) gauge field.
We also define three bilinear currents
jµ(x) = ψ(x)γµψ(x)− vector
j5µ(x) = ψ(x)γµγ5ψ(x)− axial
P (x) = ψ(x)γ5ψ(x)− pseudoscalar. (2.2)
Using the Dirac equation we can derive conservation laws
∂µjµ = ψ←−/∂ ψ + ψ/∂ψ
= iψ(m− e /A)ψ + iψ(−m+ e /A)ψ
= 0 (2.3)
∂µj5µ = ψ
←−/∂ γ5ψ + ψ/∂γ5ψ
= ψ←−/∂ γ5ψ − ψγ5/∂γ5ψ
= iψ(m− e /A)γ5ψ − iψγ5(−m+ e /A)ψ
= 2imP. (2.4)
Equation (2.4) vanishes in the massless limit and then we classically have both vector cur-rent and axial current symmetry. This is not the case when we quantise the theory as thedivergence of the axial current is non-zero even in the massless case. One needs to examinethe V-V-A triangle graph which breaks the axial symmetry.
For this we consider a Lagrangian with both vector and axial gauge fields Vµ and Aµ respec-tively.
L = ψ(i/∂ + /V + /Aγ5 −m)ψ. (2.5)
The lagrangian (2.5) is invariant under local vector gauge transformations
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ψ → eiθ(x)ψ
ψ → ψe−iθ(x)
Vµ → Vµ + ∂µθ(x). (2.6)
There is also a local axial gauge transformation which leaves (2.5) invariant
ψ → eiα(x)γ5ψ
ψ → ψe−iα(x)γ5
Aµ → Aµ + ∂µα(x). (2.7)
All together the Lagrangian (2.5) is said to have a local UV (1)× UA(1) symmetry.
q
pp− q
k1k2k1k2
p− q p
q
p− k1p− k1
ν µ ν µ
λ
FIGURE 1.1: The V-V-A and V-V-P triangle graphs that lead to the anomaly.
Now, examining the V-V-A and V-V-P triangle graphs we see that the corresponding matrixelements are
Tµνρ(k1, k2, q) =∫d4xe−ik1x1−ik2x2+iqx3〈0|T [jµ(x1)jν(x2)j
5ρ(x3)]|0〉. (2.8)
Tµν(k1, k2) =∫d4xe−ik1x1−ik2x2+iqx3〈0|T [jµ(x1)jν(x2)P (x3)]|0〉. (2.9)
Where we have defined d4x = d4x1d4x2d
4x3. Equations (2.8) and (2.9) are essentially multi-dimensional Fourier transformations. By taking derivatives we derive the important relations
qρTµνρ = ∂ρx3
[∫d4x e−ik1x1−ik2x2+iqx3
]〈0|T [jµ(x1)jν(x2)j
5ρ(x3)]|0〉
=∫d4x 〈0|T [jµ(x1)jν(x2)∂
ρx3j5ρ(x3)]|0〉e−ik1x1−ik2x2+iqx3
=∫d4x [−2m〈0|T [jµ(x1)jν(x2)P (x3)]|0〉] e−ik1x1−ik2x2+iqx3
= 2mTµν . (2.10)
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In order to preserve gauge symmetry we insist that ∂µjµ = 0 which means that
kµTµνρ = kνTµνρ = 0. (2.11)
Equation (2.10) is known as the axial Ward identity and (2.11) is the vector Ward identity.Equation (2.11) can be relaxed, leading to an anomaly in the vector current. However, byinsisting that we have gauge symmetry forces the anomaly into the axial current.
Calculating the amplitudes Tµνρ and Tµν directly using the Feynman rules one sees that thisdoes not agree with equation (2.10) and (2.11). The amplitudes are
Tµνλ = −i∫ d4p
(2π)4tr
i
/p−mγλγ5
i
/p− /q −mγν
i
/p− /k1 −mγµ
+
(k1 → k2
µ→ ν
)
. (2.12)
Tµν = −i∫ d4p
(2π)4tr
i
/p−mγ5
i
/p− /q −mγν
i
/p− /k1 −mγµ
+
(k1 → k2
µ→ ν
)
. (2.13)
With q = k1 + k2.
The Pauli-Villars regulated amplitude is defined as the difference between the unregulatedamplitude and the amplitude with mass M .
T regµνλ = Tµνλ(m)− Tµνλ(M). (2.14)
The physical amplitude T physµνλ is then understood as the limit of the regulated amplitude asthe regulator mass tends to infinity, that is
T physµνλ = limM→∞
T regµνλ. (2.15)
The same applies to T physµν , but since Tµν(M) ∼ 1/M we see that Tµν is convergent and doesnot need regularisation
T physµν = limM→∞
[Tµν(m)− Tµν(M)] = Tµν(m). (2.16)
Then by using equation (2.10) we see that the axial conservation law becomes
qλT physµνλ = 2mTµν(m)− limM→∞
2MTµν(M). (2.17)
Then the AJB anomaly is completely described by the second term in equation (2.4).
Aµν = − limM→∞
2MTµν(M). (2.18)
In order to evaluate equation(2.18) we need to write equation (2.9) in a more useful form.First we need to introduce the Feynman parameter integral
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1
abc= 2
∫ 1
0dx1
∫ 1−x1
0dx2
1
[ax1 + b(1− x1 − x2) + cx1]3. (2.19)
Then we can write
Tµν = −i∫ d4p
(2π)4tr
[i(/p+m)
p2 −m2γ5
i(/p− /q +m
(p− q)2 −m2γµ
i(/p− /k1 +m)
(p− k1)2 −m2γν
]
= −2i∫ d4p
(2π)4
∫ 1
0dx1
∫ 1−x1
0dx2
× tr[i(/p+m)γ5i(/p− /q +m)γµi(/p− /k1 +m)γν ]
[(p2 −m2)x2 + ((p− q)2 −m2)(1− x1 − x2) + ((p− k1)2 −m2)x3]3
+
(k1 → k2
µ→ ν
)
. (2.20)
Now we need to explicitly evaluate the trace. Expanding the trace term it is easy to see thatthis term reduces to
tr[i(/p+m)γ5i(/p− /q +m)γµi(/p− /k1 +m)γν ] = −imtr[γ5/qγν/k1γµ]. (2.21)
Then we have
Tµν = 2i∫ d4p
(2π)4
∫ 1
0dx1
∫ 1−x1
0
× im tr[γ5/qγν/k1γµ]
[(p2 −m2)x2 + ((p− q)2 −m2)(1− x1 − x2) + ((p− k1)2 −m2)x3]3
+
(k1 → k2
µ→ ν
)
. (2.22)
Now we explicitly evaluate the trace.
tr[γ5/qγµ/k1γν ] = tr[γ5(/k1 + /k2)γµ/k1γν ]
= tr[γ5γαkα1 γµγβk
β1 γν ]
+tr[γ5γαkα2 γµγβk
β2 γν ]
= tr[γ5γαγµγβγν ]kα1 k
β1
+tr[γ5γαγµγβγν ]kα2 k
β1
= 4iεαµβνkα2 k
β1 . (2.23)
Then we have
Tνµ(m) = 2i∫ d4p
(2π)4
∫ 1
0dx1
∫ 1−x1
0dx22
im4iεαµβνkα2 k
β1
[p3 − 2pk −m 2]3. (2.24)
In the above we have included a factor of 2 which arises from k1 → k2 and µ→ ν. We havealso defined
10
k = q(1− x1 − x2) + k1x1, (2.25)
m 2 = m2 − q2(1− x1 − x2). (2.26)
We now use the ’t Hooft-Veltman integral formula [66]
∫ dnp
(p2 − 2pk −m 2)α= i1−2απ
n2
Γ(α− n2)
Γ(α)
1
(k2 +m 2)α−n2
. (2.27)
Here we have α = 3 and n = 4. Thus,
i1−6π2 Γ(3− 4/2)
Γ(3)
1
(k2 +m 2)3−2= i−5π2 1
(k2 +m 2)
Γ(1)
Γ(3)
= −iπ2
2
1
(k2 +m 2). (2.28)
For large M the integral clearly behaves as 1/M2. Therefore,
limM→∞
2MTµν(M) = limM→∞
22i
(2π)4
π2
2i2iM2
M24iεµναβk
α1 k
β2
=23π2
(2π)44εµναβk
α1 k
β2
=1
2π2εµναβk
α1 k
β2 . (2.29)
Which is the momentum space expression for the anomaly. Hence even in the massless case(m = 0), a local chiral rotation is not a symmetry of the quantum action.
qλT physµνλ = 2mTµν −1
2π2εµναβk
α1 k
β2 . (2.30)
Then in x space the anomalous conservation law can be written as
∂µj5µ = +2miP +
1
16π2εµναβF
αβF µν . (2.31)
In the massless case we have
A(x) = ∂µj5µ =
1
16π2εµναβF
αβF µν . (2.32)
We call the term A(x) the singlet (or abelian)anomaly.
Expressions (2.30) and (2.31) are equivalent, this can be shown by considering the expecta-tion value with two external photons with momentum k1, k2 and polarisation vectors ε1, ε2.See [2] for details.
So far, we have calculated the singlet anomaly in QED via a one-loop perturbation calcu-lation. It is important to realise that higher order contributions only renormalise the fields
11
and their couplings, they do not effect the anomaly. Thus all the structure of the singletanomaly is contained within the triangle graph. This is known as the Adler-Bardeen theorem[2]. This theorem has been proved for QED and QCD. However, it is not at all clear if thistheorem holds true in general gauge theories.
The singlet anomaly (2.32) is independent of the regularisation employed. The ’t Hooft-Veltman dimensional regularisation will yield the same result.
The anomaly also appears in x-space (rather that momentum space) when one considerscurrent operators. As the product of quantum fields at at the same point in space is infinite,the axial current
j5µ(x) = ψ(x)γµγ5ψ(x), (2.33)
is formally infinite. This current requires regularisation which induces the singlet anomaly.See [42] for a further discussion.
2.2 The Singlet Anomaly via Fujikawa’s Method
Here we work on a 4-dimensional manifold (M) with Euclidean signature metric. As weare dealing with spin bundles we need the manifold to have trivial first ω1(M) and secondω2(M) Stiefel-Whitney classes. Such a manifold will be called a spin manifold. This is nota severe restriction and is satisfied for example by spheres.
The first Stiefel-Whitney class ω1(M) is an obstruction to the orientability of a manifold.Thus, to be orientable ω1(M) must be trivial.
The second Stiefel-Whitney class ω2(M) can be thought of as an obstruction to the existenceof a spin bundle. Spin bundles over orientiable manifolds only exist if ω2(M) is trivial.In this section we have also restricted our attention to manifolds that have trivial A-genus.
Let ψ be a massless Dirac Field interacting with a non-abelian gauge field Aµ = AαµTα.Where Tα are the generators of some non-abelian group G. The Lagrangian is given by
L = iψγµ(∂µ − Aµ)ψ. (2.34)
This Lagrangian is invariant under a local gauge transformation
ψ(x)→ g−1ψ(x), Aµ(x)→ g−1[A(x) + ∂µ]g. (2.35)
There is also a global symmetry
ψ(x)→ eiγ5αψ(x), ψ(x)→ ψ(x)eiγ5α. (2.36)
This is the global chiral symmetry which generates the Noether chiral current jµ5 .
12
jµ5 (x) = ψ(x)γµγ5ψ(x). (2.37)
Classically we have ∂µjµ5 = 0, however when one considers the quantum effective action the
above conservation law does not hold. This is seen as a non-trivial transformation of thepath integral measure under a chiral rotation. The effective action is defined as
e−W [A] =∫DψDψe−
∫dxψi /Dψ. (2.38)
Where we have used the Feynman slash notation to define i /D = iγµ(∂µ + Aµ).
The chiral current jµ5 can be calculated by considering an infinitesimal chiral transformation.Now we perform a local chiral rotation of the spinor fields
ψ(x)→ eiγ5α(x)ψ(x) ≈ ψ + iα(x)γ5ψ (2.39)
ψ(x)→ ψ(x)eiγ5α(x) ≈ ψ + iψα(x)γ5.
Thinking of the local chiral rotations as a coordinate transformation, it is clear that thesetransformations will be symmetries of the path integral. This is in analogy to standardintegrals. If we insist that
∫dnx f(x) =
∫dnx′ f(x′), (2.40)
then we need
dnx′ =
∣∣∣∣∣∂x′
∂x
∣∣∣∣∣ dnx. (2.41)
Thus in changing variables, the Jacobian factor is important in the evaluation of integrals.
Under a local chiral rotation, the classical action transforms under this rotation as
∫dxψi /Dψ →
∫dx(ψ + iψαγ5)i /D(ψ + iαγ5ψ)
=∫dxψi /Dψ + i
∫dx[αψγ5i /Dψ + ψi /D(αγ5ψ)]
=∫dxψi /Dψ −
∫dx[αψγ5γµ(∂µ + Aµ)ψ
+ψγµ(∂µ + Aµ)(αγ5ψ)]
=∫dxψi /Dψ +
∫dxα(x)∂µj
µ5 . (2.42)
However, as stated earlier not only does the classical action change under the chiral ro-tations also the quantum measure changes. To see this define the chiral (infinitesimally)rotated fields as
ψ′ = ψ + iαγ5ψ =∑
a′iψ (2.43)
ψ′= ψ + iψαγ5 =
∑b′
iψ†i . (2.44)
13
Where a,b are anti-commuting Grassmann variables and ψi are eigenspinors of the Diracoperator
i /Dψi = λiψi. (2.45)
The eigenspinors are normalised such that
〈ψi|ψj〉 =∫dxψ†
i (x)ψj(x) = δij, (2.46)
which can be achieved provided M is compact, which we assume. This allows us to writethe fermionic quantum measure in terms of the Grassmann parameters a and b. We definethe change in this measure under a chiral rotation as
∫ ∏
i
daidbi →∫ ∏
i
da′idb′
i. (2.47)
We now want an expression for a′i and b′i.
a′i = 〈ψi|ψ′〉= 〈ψi|(1 + iαγ5)|ψ〉=
∑
j
〈ψi|(1 + iαγ5)aj|ψj〉
=∑
j
〈ψi|ψj〉+∑
j
iαaj〈ψi|γ5|ψj〉
=∑
j
δij +∑
j
iαaj〈ψi|γ5|ψj〉
=∑
j
Cijaj (2.48)
Then the Jacobian is given by
∏da′j = (det[Cij])
−1∏dai
= exp[−tr lnCij]∏dai
= exp[−tr ln(1 + iα〈ψi|γ5|ψj〉)]∏dai
≈ exp[−tr iα〈ψi|γ5|ψj〉]∏dai
= exp
[
−iα∑
i
〈ψi|γ5|ψi〉]∏dai. (2.49)
It is the inverse of the determinant that is used in the above as we are dealing with Grassmannnumbers. The contribution for
∏b′
i is identical to∏a′i. Hence the measure changes overall
as
∏
i
daidbi →∏
i
da′idbi′ exp
[
−2α∑
n
〈ψn|γ5|ψn〉]
=∏
i
da′idbi′ exp
[
−2i∫dxα(x)
∑
n
ψ†n(x)γ5ψn(x)
]
. (2.50)
14
Then we have two expressions for the effective action
e−W [A] =∫ ∏
i
daidbi exp[−∫ψi /Dψdx]
=∫ ∏
i
da′idbi′ exp[−
∫dx(ψi /Dψ − α(x)∂µj
µ5
−2iα(x)∑
n
ψ†nγ5ψn)]. (2.51)
Since α(x) is arbitrary we require the chiral conservation law
∂µjµ5 = −2i
∑
n
ψ†nγ5ψn. (2.52)
The term on the right up to factor of −2i is known as the Abelian anomaly
A(x) =∑
n
ψ†nγ5ψn. (2.53)
The above expression for the anomaly is not well defined and requires regularisation. Herewe use the heat kernel method.
A(x) = limM→∞
∑
n
ψ†n(x)γ5 exp[(i /D/M)2]ψn(x). (2.54)
We change basis to momentum space as
ψn(x) =∫ d4k
(2π)2eik.xψn(k). (2.55)
Using this the trace of any matrix becomes
trM(x) =∫ d4k
(2π)4eik.xM(x)e−ik.x. (2.56)
Then the anomaly can be expressed as
A(x) = limM→∞
tr∫ d4k
(2π)4eik.xγ5e
−(i /D/M)2e−ik.x. (2.57)
Now we expand the square of the Dirac operator as
(i /D)2 = −DµDµ − 1
4[γµ, γν ]Fµν . (2.58)
Where we have made use of the definition Fµν = [Dµ, Dν ]. Then we have
A(x) = limM→∞
tr∫ d4k
(2π)4eik.xγ5 exp
[∇µ∇ν + 1
4[γµ, γν ]FµνM2
]
e−ik.x
= limM→∞
tr∫ d4k
(2π)4eik.xγ5 exp
[−k2 + 1
4[γµ, γν ]FµνM2
]
e−ik.x
= limM→∞
tr(γ5 exp
[1
4M2[γµ, γν ]Fµν
]) ∫ d4k
(2π)4e−
k2
M2 . (2.59)
15
Expanding the first exponential
exp[
1
4M2[γµ, γν ]Fµν
]≈ 1 +
1
4M2[γµ, γν ]Fµν +
1
2
1
(4M2)2([γµ, γν ]Fµν)
2 . (2.60)
Then using
trγ5 = trγ5γµγν = 0, (2.61)
we arrive at
A(x) = limM→∞
1
2tr
[
γ51
(4M2)2([γµ, γν ]Fµν)
2
] ∫ d4k
(2π)4e−
k2
M2
= limM→∞
1
2tr
[
γ51
(4M2)2([γµ, γν ]Fµν)
2
]M4π2
(2π)4
=1
2
1
162
1
π2tr[γ5([γ
µ, γν ]Fµν)2]
=4
2 162π2tr[γ5γ
κγλγµγνFκλFµν]. (2.62)
Now use
tr [γ5γµγνγργσ] = −4εµνρσ. (2.63)
Then we obtain
A(x) = − 1
32π2trεκλµνFκλFµν . (2.64)
Then the chiral conservation law becomes
∂µjµ5 =
1
16π2trεκλµνFκλFµν
=1
4π2tr[εκλµν∂κ(Aλ∂µAν +
2
3AλAµAν
]. (2.65)
Which can be regarded as the local version of the Atiyah-Singer index theorem. Notice theChern-Simons form that appears in the local form of the anomaly. By local, we mean thatwe are not integrating over the manifold as one does in the standard Atiyah-Singer indextheorem.
If one integrates over the manifold we make direct contact with the Atiyah-Singer indextheorem.
∫dxA(x) =
∫dx∑
n
ψ†nγ5ψn exp
[−λ2
n/M2]|M→∞,
=∑
n
〈ψn|γ5 exp[−(i /D/M)2
]|ψn〉|M→∞. (2.66)
16
Now, suppose we have i /D|ψn〉 = λn|ψn〉 with λn non-vanishing. Then we have another spinor|φn〉 ≡ γ5|ψn〉 such that i /D|φn〉 = −λn|φn〉. To see this we note that
i /D|φn〉 = i /Dγ5|ψn〉= −λnγ5|ψn〉= −λn|φn〉. (2.67)
Then using the above it is clear that we must have
〈ψn|φn〉 = 〈ψn|γ5|ψn〉 = 0, (2.68)
due to the orthogonality of the eigenspinors. Then contributions to the anomaly by thespinors with non-vanishing eigenvalues is
〈ψn|γ5 exp[−(i /D/M
)2]|ψn〉 = 〈ψn|γ5|ψn〉 exp
[− (λn/M)2
]= 0. (2.69)
Then the only possible contribution to the anomaly must be from the zero modes. Thisobservation directly links the anomaly to an index theorem.
Let |0, i〉 be zero modes of the Dirac operator, remembering that the Dirac operator is ellipticon a compact manifold and hence has a finite dimensional kernel and co-kernel. These canbe classified by their eigenvalue of γ5 as they are not an irreducible representation of thespin algebra. Then,
γ5|0, i〉± = ±|0, i〉±. (2.70)
Then we have,
∫dxA(x) =
∑〈ψi|γ5 exp[−(i /D/M)2]|ψi〉|M→∞
=∑
i+
〈0, i|0, i〉+ −∑
i−
〈0, i|0, i〉−
= υ+ − υ− = indi /D+. (2.71)
Where ν± is the number of positive and negative chirality zero modes. This is precisely theAtiyah- Singer index theorem.
Writing F = 12Fµνdx
µ ∧ dxν , we see that
ν+ − ν− =∫
Mdx∂µj
µ5 =
∫
Mch2(F ). (2.72)
Which is the AS index theorem for a twisted Dirac complex on a compact 4-manifold withtrivial A-roof genus.
In the language of differential forms the anomalous conservation law can be written as
d ∗ J = − 1
4π2tr(F ∧ F ) (2.73)
17
Where the current one-form is defined as J = j5µdx
µ, d is the exterior derivative and ∗ is theHodge dual.
This result can be generalised to any compact spin-manifold with trivial A-roof genus ofarbitrary (but even) dimension Dim M = 2l as
ν+ − ν− =∫
Mdx∂µj
µ5 =
∫
Mchl(F ), (2.74)
which is the AS-index theorem for twisted spin complexes on compact spin-manifolds withtrivial A-roof genus without boundary.
It should be noted that the anomaly is independent of the regularisation used. Instead ofusing the exponential function, we could have used any function f which is smooth anddecreases rapidly enough at infinity
f
(λnM2
)
, (2.75)
with
f(∞) = f ′(∞) = f ′′(∞) = . . . = 0
f(0) = 1. (2.76)
For example, we could use
f
(λnM2
)
=1
1 + λ2n/M
2(2.77)
as the regularisation function, which corresponds to Pauli-Villars regularisation.
2.3 The Non-Abelian Anomaly via Fujikawa’s Method
Here we examine the case where we have a chiral theory and so the left and right-handedWeyl fermions ψ do not couple equally to the gauge field Aµ. The result is a theory in whichthe effective action is not gauge (or more precisely BRST) invariant, whereas the classicalaction is. In what follows we ignore all possible gauge fixing and FP ghost terms. This is ajustified restriction as the anomaly only arises in the fermionic sector of the theory. Again,we assume the manifold is a compact spin-manifold with trivial geometry and Euclideanmetric. We use the techniques developed for the abelian anomaly, but pointing out the maindifferences and complications as they arise.
We construct an effective action in which the gauge field A couples only to the left-handedWeyl fermion. ψ also transforms under a complex representation of the gauge group G. Theeffective action is
e−W [A] =∫DψDψ exp
[−∫dxψi /D+ψ
]. (2.78)
18
Where we have defined
i /D+ = i /DP+ P± =1
2(1± γ5). (2.79)
The gauge current is given by
jµα = iψγµTαP+ψ. (2.80)
Now we consider an infinitesimal gauge transformation g = 1− v, with v = vαTα. Then wehave
Aµ → (1 + v)(Aµ + d)(1− v) = Aµ −Dµv, (2.81)
where the covariant derivative is defined as Dµv ≡ ∂µ + [Aµ, v]. The effective action thentransforms as
W [A] → W [A−Dv]
= W [A]−∫dx tr
(
Dv δ
δAW [A]
)
= W [A]−∫dx tr
(∂µv
α + fαβγAβµv
γ) δ
δAαµW [A]
= W [A] +∫dx tr
(
vαD δ
δAW [A]α
)
. (2.82)
We can then write
W [A−Dv]−W [A] =∫dx tr(vαDµ〈jµ〉α). (2.83)
Where we have used
δ
δAαµW [A] = 〈iψγµTα
1
2(1 + γ5)ψ〉A = 〈jµα〉. (2.84)
One would now like to preceded by writing the effective action as the determinant of theDirac operator det(i /D) =
∏λi, with λ being eigenvalues of i /D. However, this construction
has no meaning as is not a well posed eigenvalue problem. The difficulty arises due to thefact that i /D+ maps sections of S+⊗E to S−⊗E. Where S± are the spin bundles of positiveand negative chirality and E is the associated vector bundle to the principle G-bundle.
We formally overcome this complication by introducing a Dirac spinor ψ and defining theeffective action as
e−W [A] =∫DψDψ exp
(∫dxψiDψ
). (2.85)
Where the operator iD is defined as
iD ≡ iγµ(∂µ + iAµP+) =
(0 i/∂
i /D+ 0
)
. (2.86)
19
In this construction it is clear that the gauge field only couples to the Dirac spinor of positivechirality. The eigenvalue problem iDψi = λiψi is well posed. However, one must note thatiD is not Hermitian and as such λi will be a complex number in general. As such, one needsto introduce right and left eigenfunctions.
iDψi = λi (2.87)
χ†i i←−D = λiχ
†i , (iD)†χi = λiχi. (2.88)
As these define a complete set of eigenvectors we can use them to define an orthonormalbasis,
∫χ†iψj = δij. (2.89)
However, the eigenvalues λi are not gauge invariant. This is clearly seen as the operatoriD is in fact not gauge invariant. This is not really a problem as we are interested in theproduct of all the eigenvalues and this is gauge invariant. We see this using
det(iD) det((iD)†) = det(iD(iD)†)
= det
((i/∂−)(i/∂+) 0
0 (i /D−)(i /D+)
)
= det(i/∂−i/∂+) det(i /D−i /D+). (2.90)
With i/∂+ = (i/∂−)† and i /D− = (i /D+)†. This is up to a numerical factor just the Diracdeterminant which is gauge invariant. In the above we have implicitly used a gauge invariantregularisation scheme such as the Pauli-Villars. Explicitly we have
(det(i /D))2 =
((i /D−)(i /D+) 0
0 (i /D+)(i /D−)
)
= (det(i /D+i /D−))2. (2.91)
As the Dirac determinant is gauge invariant so is | det(iD)|. Now if we examine the real partof the effective action Re W [A] we see that it is gauge invariant
exp(−W [A]) exp(−W [A]) = det(iD) det((iD)†) ∝ det(i /D+i /D−). (2.92)
This is a crucial observation, only the imaginary part of the effective action, that is the phaseof det(iD), may be anomalous. This means that in order to have gauge anomalies the gaugegroup must have a complex representation.
We can now evaluate the anomaly by examining the Jacobian of the fermionic measure, aswe did for the abelian anomaly.
We write Weyl fermions in terms of the Dirac spinors which are eigenspinors of D.
ψ =∑
i
aiψi (2.93)
ψ =∑
i
biχi (2.94)
20
Then consider infinitesimal gauge transformations,
A→ A−Dv ψ → ψ + vψ+ ψ → ψ − ψ−v. (2.95)
We can thus define the gauge transformed fields as
ψ → ψ + v(x)P+ψ =∑
i
a′iψi
ψ → ψ − v(x)P−ψ =∑
i
b ′iχi. (2.96)
Then we can form the Jacobian in the same way as for the abelian anomaly. We first haveto calculate the transformation properties of the ai and bi.
a′i = 〈ψi|ψ′〉 = 〈ψi|1 + v(x)P+|ψ〉 =∑
j
〈ψi|1 + v(x)P+|ψj〉aj. (2.97)
Then using the same arguments as before the Jacobian for dai is given by
∏da′j = exp[−tr ln(1 + v(x)〈ψi|P+|ψj〉)]
∏dai
≈ exp[−tr v(x)〈ψi|P+|ψi〉)]∏daj. (2.98)
In a similar way we calculated the Jacobian for dbi.
b ′i = 〈ψ′|χi〉 = 〈ψ|1− P−v(x)|χi〉 =
∑
j
bj〈ψj|1− P−v(x)|χi〉. (2.99)
Then we have
∏db ′
j ≈ exp[tr v(x)〈ψi|P−|χi〉]∏dbj. (2.100)
Then the Jacobian factor in the effective action is given by∫dxtr v(x)
∑
n
〈χn|γ5|ψn〉 (2.101)
We then pick a representation of the spinors such that 〈x|n〉 = ψn(x) and (n|x〉 = χ†n(x).
Notice that (n| is not the Hermitian conjugate of |n〉. This integral is not well defined andrequires regulisation. Again, we use the heat kernel regulisation,
∫dx lim
M→∞tr v(x)
∑
n
(x|y〉γ5〈x|e−(iD)/M2|n〉
=∫dx lim
M→∞tr v(x)γ5e−(iD)/M2
δ(x− y). (2.102)
Where we have used the completeness relation
∑
n
|n〉(n| = I. (2.103)
Then asW [A] is gauge invariant, its gauge variation (2.83) must be cancelled by the Jacobianfactor. Thus, we are able to write
21
∫dx vα(x)Dµ
(δ
δAαµW [A]
)
=∫dx lim
M→∞tr[ v(x)γ5e−(iDx)/M2
δ(x− y)]. (2.104)
We now “split” the trace into two pieces depending on their chirality.
tr[v(x)γ5e−(iDx)2/M2
] = tr[v(x)(P+ − P−)e−(i/∂−i /D+)−(i /D
−i/∂+)/M2
(2.105)
= tr[v(x)P+e(i/∂i /D)/M2
]− tr[v(x)P−e(i /Di/∂)/M2
].
This can then be evaluated in the plane wave basis, see [40] for details.
W [A−Dv]−W [A] =1
24π2
∫dx tr
(vαTαε
κλµν∂κ
[Aλ∂µAν +
1
2AλAµAν
])
=1
24π2
∫tr(v d
[AdA+
1
2A3]). (2.106)
Then the anomalous conservation law can be written as
Gα[A] = Dµ〈jµ〉α =1
24π2tr[Tαε
κλµν∂κ
(Aλ∂µAν +
1
2AλAµAν
)]. (2.107)
Notice the difference in the factors in front of the A3 between the abelian and non-abeliananomaly. This difference has a deep topological origin which will be explained in the nextsection and proceeding chapter.
We have calculated the non-abelian anomaly for left chiral spin- 1/2 fermions coupled toan external Yang-Mills field. The whole calculation could be carried out for right handedfermions. The final result is that the non-abelian anomaly for the right handed fermions isidentical to the left handed fermions apart from an overall minus sign.
Grightα [A] = −Gleft
α [A] = − 1
24π2tr[Tαε
κλµν∂κ
(Aλ∂µAν +
1
2AλAµAν
)]. (2.108)
Thus, in a non-chiral theory the two sectors cancel resulting in an anomaly free theory.
2.4 The Wess-Zumino Consistency Conditions and the
Descent Equations
As shown earlier the effective action W [A] of a Weyl fermion in a complex representation ofa gauge group G transforms under an infinitesimal gauge transformation as
δνW [A] = −∫
(Dµν)αδ
δAαµW [A] =
∫ναDµ〈jν〉α. (2.109)
Where δνA = −Dν. However, the correct statement in quantum field theory is that thegauge fixed effective action should be BRST invariant (see Appendix A). That is the BRSToperator should annihilate the effective action. The presence of the gauge anomaly transpires
22
to be non-invariance under the BRST transformations.
Define Gα[A] as the local form of the non-abelian anomaly
Gα = Dµ〈jµ〉α, (2.110)
Then we define the global form of the non-abelian anomaly in terms of the BRST transfor-mations and ghost fields
sW [A] = G[η,A] =∫dxηαGα[A] 6= 0, (2.111)
if the effective action is no longer BRST invariant. As s is a nilpotent we have
sG[η,A] = s2W [A] = 0. (2.112)
Equation (2.112) is known as the Wess-Zumino consistency condition and is strong enoughto determine the gauge anomaly up to normalisation.
Now in order to solve the consistency equation one defines a new operator
d = d+ s (2.113)
which itself is also a nilpotent and anticommutes with the exterior derivative.
d2 = d2 + ds+ sd+ s2 = 0. (2.114)
We also define
A ≡ g−1(A+ d)g. (2.115)
From this we define the field strength
F ≡ dA + A2 = g−1Fg. (2.116)
The Faddeev-Popov ghost is identified with
η ≡ g−1sg. (2.117)
We also define
A ≡ g−1(A+ d)g = A + η, (2.118)
F ≡ dA + A2 = F. (2.119)
Stora and Zumino constructed solutions to the Wess-Zumino consistency conditions via theso called decent equations. The Abelian anomaly in (2l + 1) dimensions is given by thefollowing Chern-Simons form as we have seen
chl+1(F) =1
(l + 1)!tr
(iF
2π
)l+1
. (2.120)
Let Q2l+1(A,F) be the Chern-Simons form
23
chl+1(F) = dQ2l+1(A,F). (2.121)
As the algebraic structure of (d, A, F) is identical of that of (d,A,F) we have
chl+1(F) = dQ2l+1(A, F) = dQ2l+1(A + η,F). (2.122)
Expanding the above in powers of η
Q2l+1(A, F) = Q02l+1(A,F) +Q1
2l(η,A,F) +Q22l−1(η,A,F)
+ ...+Q2l+10 (η,A,F). (2.123)
Where Qsr is s-th order in η and r + s = 2l + 1.
We see that
Q2l+1(A, F) = dQ2l+1(A,F), (2.124)
as we have chl+1(F) = chl+1(F). Then we have
(d+ s)[Q02l+1(A,F) +Q1
2l(η,A,F)
+...+Q2l+10 (η,A,F)] = dQ0
2l+1(A,F). (2.125)
Then collecting terms order by order in η we arrive at the decent equations
dQ02n+1 = chl+1(F)
sQ02l+1(A,F) + dQ1
2l(η,A,F) = 0
sQ12l(ηA,F) + dQ2
2l−1(η,A,F) = 0...
sQ2l1 (η,A,F) + dQ2l+1
0 (η,A,F) = 0
sQ2l+10 (η,A,F) = 0. (2.126)
Notice that if we take the 2l-form Q12l and define
G[η,A,F] =∫
MQ1
2l(η,A,F), (2.127)
then G[η,A,F] automatically satisfies the Wess-Zumino consistency condition.
sG[η,A,F] = s∫
MQ1
2l(η,A,F) = −∫
MdQ2
2l−1(η,A,F)
= −∫
∂MQ2
2l−1(η,A,F) = 0. (2.128)
Where we have explicitly assumed that the manifold has no boundary. Hence, finding theanomaly becomes a case of finding Q1
2l. This is further discussed in the next chapter. How-ever, it should be noted that that this approach to calculating the gauge anomaly has someunanswered questions: i)the consistency condition does not fix the normalisation of theanomaly. Hence, the anomaly is not unique. ii) it is not clear why we start form the abeliananomaly in (m+ 2) dimensions.
24
2.5 Epitome
In gauge theory there are two types of anomaly (in 4-dimensions)
1. Singlet anomaly
A(x) =1
4π2Trd
[AdA+
3
2A3]
(2.129)
2. Non-abelian anomlay
Gα[A] =1
24π2Trd
[AdA+
1
2A3]. (2.130)
These were calculated via Fujikawa’s method in which these anomalies show up as Jacobiansfor the path integral measure under a chiral rotation and a gauge transformation for thesinglet and non-abelian anomaly respectively.
In arbitrary, but even dimensions dimM = 2l the singlet anomaly can be calculated via theAtiyah-Singer index theorem. Considering only manifolds with trivial A-genus we have
∫dxA(x) = ind i /D+ =
∫
Mchl(F ). (2.131)
The non-abelian anomaly is completely characterised by the Wess-Zumino consistency con-ditions
sG[η,A] = 0, (2.132)
which directly lead to the descent equations (2.126). The local form non-abelian anomalyis given by the Q1
2l term in the descent equations, at least up to normalisation. Thus thenon-abelian anomaly becomes a case of solving the descent equations.
25
CHAPTER 3
BRST Cohomology and the Descent Equations
As shown in Chapter 3 the Wess-Zumino consistency conditions lead directly to a BRSTcohomology problem. The consistency equations take a very simple form
sG[η,A] = 0, G[η,A] 6= sF [A], s2 = 0, (3.1)
where the anomaly G is a functional
G[η,A] =∫ηαGα[A]. (3.2)
The first equation defines the BRST cohomology in the space of local functions and wewill denote this H(s). The consistency condition may also be written in terms of the localfunctionals,
sQ12l + dQ2
2l−1 = 0, Q12l 6= sQ0
2l + dQ12l−1. (3.3)
Thus the relevant cohomology is the BRST cohomology in the space of local functionalswhich we denote as H(s|d). The consistency condition may also be also be studied for ghostnumber other than one; which corresponds to the gauge anomalies. For example for zeroghost number the cohomology controls the possible counterterms in the renormalistion ofYang-Mills.
In this section we are using differential forms and as such the wedge product is understoodwhen we multiply forms.
3.1 Local Cohomology and the Cohomology of d
Local forms are defined as the differential forms Q whose coefficients Qµ1...µp(x) are local
functionals. That is the coefficients are polynomials in the fields and their derivatives takenat the same point x. The integral of a local form is also called a local functional.
Let V(A, dA, η, ζ = dη) be the space of polynomials of differential forms. We define theone-form gauge connection as A = TaA
aµdx
µ and the zero-form ghost as η = Taηa, which
26
can be identified as the Maurer-Cartan form of the gauge group, see [61, 62]. We wish tocompute the various local cohomologies on V(A, dA, η, ζ), first we consider the cohomologyof the exterior derivative.
Local cohomolgies are in general difficult to compute. Introducing a filtration operator Nand a simpler exterior derivative operator d0 will prove to be useful in calculating the localcohomology. The filtration operator N : V → V is defined as
N = ζ∂
∂ζ+ η
∂
∂η+ A
∂
∂A+ F
∂
∂F. (3.4)
The integral of which counts the number of fields in a given monomial. Direct computationshows that
NA = A , NF = F
Nη = η , Nζ = ζ. (3.5)
Acting on V the exterior derivative acts as an ordinary differential operator
d = dA∂
∂A+ dη
∂
∂η+ dF
∂
∂F
= (F − A2)∂
∂A+ ζ
∂
∂η+ [F,A]
∂
∂F. (3.6)
Where we have used the Bianchi identity (A.16) and the definition of the Yang-Mills curva-ture (A.14).
Now we suppose that the forms Qqp can be expanded in terms of eigenvalues of the filtration
operator
Qqp = Qq
p(0) +Qqp(1) , NQq
p(n) = nQqp(n). (3.7)
where n = 0, 1.
From (3.6) it is clear that the exterior derivative decomposes as
d = d0 + d1 , [N, dn] = ndn, (3.8)
Where
d0 = F∂
∂A+ ζ
∂
∂η. (3.9)
We have that
d0F = 0,
d0ζ = 0. (3.10)
27
Also it is easy to see that
d0A = F,
d0η = ζ. (3.11)
We now use the following general results from homological algebra, see for example [41, 59];
1. d0 is a coboundary operator.
2. The cohomology of d is isomorphic to a subspace of the cohomology of d0.
Then from (3.10) and (3.11) it is clear that the cohomology of d0 is trivial. As such, thecohomology of d on V(A, dA, η, ζ) must be also trivial. This result will be important laterin deriving the descent equations.
3.2 The BRST Operator and its Cohomology
The classical BRST transformations (A.52) on a gauge field Aµ, a ghost field η = Taηa and
the field strength Fµν are given by
sAµ = Dµη,
sη = −1
2[η, η] = −η2
sFµν = [Fµν , η]. (3.12)
The exterior algebra and the BRST algebra (3.12) can be combined into a graded algebra.This algebra is graded by the total degree Q which is defined as the sum of the form degreep and the ghost degree q. On this graded algebra both d and s act as antiderivatives andanticommute
s2 = d2 = s, d = 0. (3.13)
These two coboundary operators act on the differential forms Ωqp with their degrees being
(q, p). The action of s increases q by one unit and the action d increases p by one. Thedifferential forms are given by
Ωqp =
1
p!Ωqµ1...µp
dxµ1 ...dxµp . (3.14)
These forms are partitioned into even and odd forms according to the parity of the totaldegree (q+p). The coefficients Ωq
µ1...µp(x) are local polynomials in the fields and their deriva-
tives.
The definition of the exterior derivative is chosen to be
dΩqp =
1
p!dxν∂νΩ
qµ1...µp
dxµ1 ...dxµp . (3.15)
28
......
...
· · · Ωnm
d- Ωn
m+1
d- Ωn
m+2 · · ·
· · · Ωn+1m
s
? d- Ωn+1
m+1
s
? d- Ωn+1
m+2
s
?
· · ·
......
...
Figure 3.1: The BRST bicomplex whose local cohomology is important in the calculation of
possible anomalies in gauge theories.
Then the BRST transformations (3.12) are written in terms of the differential forms A andF
sA = −(dη + [A, η])
sη = −1
2[η, η]
sF = [F, η]. (3.16)
In the BRST transformations (3.16), [, ] is the graded commutator,
[Ω,Λ] = ΩΛ− (−1)degΩ.degΛΛΩ. (3.17)
In particular we have
[A, η] = Aη + ηA
[F, η] = Fη − ηF[η, η] = ηη + ηη = 2η2. (3.18)
The BRST algebra (3.16) is independent of any Lagrangian. All that is needed is a principlegauge bundle over a manifold. The BRST algebra is a fundamental symmetry that any
gauge theory must obey. Thus, in this section and following sections the physical theory isnot fixed by a particular Lagrangian. Only BRST symmetry need be assumed.
We briefly comment that an anti-BRST operator s can also be constructed in a similar wayto the BRST operator s. Here s decreases the the ghost number by one. The anit-ghost ηhad ghost number −1. The graded anti-BRST algebra is defined as
sA = −DηsF = [F, η]
29
sη = −1
2[η, η]
s2 = sd+ ds = 0. (3.19)
The anti-BRST transformations play no role in the following discussions on the anomaliesand as such will not feature in subsequent sections of this thesis.
As the BRST operator is a nilpotent it can be used to define a cohomology. The BRSTcocycles A are forms that are BRST-closed and hence in the kernel of s,
sΩ = 0. (3.20)
The BRST coboundaries are defined as forms that are BRST-exact (in the image of s) andas such are also BRST-closed
Ω = sΩ. (3.21)
The BRST-cohomology is defined as the quotient space Kers/Ims,
H(s) =Kers
Ims. (3.22)
An element of H(s) defines an equivalence class of BRST cocycles, where the cocycles areequivalent if they differ by a BRST coboundary. Note the similarity here with de Rhamcohomology.
The cohomology of s on V(A, dA, η, ζ) [24, 64] is spanned by polynomials in F and η,generated by
(
Trη2m+1
(2m+ 1)!
)
P2n+2(F ), (3.23)
where n,m = 1, 2... and P2n+1(F ) is an invariant polynomial of degree 2n + 1. One suchexample is
P2n+2(F ) = Tr(F n+1), (3.24)
which will turn out to be the relevant monomial for evaluating the non-abelian anomaly fora spin-1/2 fermion.
The invariant polynomial (3.24) is certainly not the most general expression possible. Theexact form of the invariant polynomial depends on the physical theory in question. Generi-cally the invariant polynomial will be the sum of products of traces.
3.3 The Cohomology of s mod d
The cohomology of the exterior derivative on V(A, dA, η, dη) is vanishing, [15, 24].
From the Bianchi identity it is clear that P2n+2(F ) is d-closed,
dP2n+2(F ) = 0. (3.25)
30
As the cohomology is trivial we see that
P2n+2 = dQ02n+1. (3.26)
Using equation (3.14) the descent equations (2.126) can easily be derived. Acting on (3.26)with s gives
sdQ02n+1 = sP2n+1 = 0. (3.27)
Where we have used the fact that P2n+1 is gauge invariant and hence it must also be BRSTinvariant. Using (3.14) we see that
d sQ02n+1 = 0. (3.28)
As the cohomology of d on V(A, dA, η, ζ) is trivial, every closed form is exact. This allowsus to write
sQ02n+1 + dQ1
2n = 0, (3.29)
which we recognise as the first line in the descent equations. Now apply s to (3.29) we get
sdQ12n = 0. (3.30)
Then by interchanging s and d, and using the triviality of the cohomology of d as before wearrive at
sQ12n + dQ2
2n−1 = 0. (3.31)
Continuing this iteratively until one reaches a differential form with negative degree. Thedescent equations must end here and the final line must be given by
sQ2l+10 = 0. (3.32)
The descent equations are thus given by
dQ02n+2 = P2n+1
sQ02n+1 + dQ1
2n = 0
sQ12n + dQ2
2n−1 = 0...
sQ2n1 + dQ2n+1
0 = 0
sQ2n+10 = 0. (3.33)
We define a s-ladder Q as the set of differential forms that satisfy the descent equations.
Thus the problem of solving the descent equations (3.33) is the problem of solving thecohomology of s mod d. The elements of this cohomology are the solutions of the descentequations which are not trivial, that is not of the form
31
Qqp = sQq−1
p + dQqp−1, for p 6= 0,
Q2n+10 = sQ2n
0 . (3.34)
Where Q is some ladder.
Two ladders, Q and Q′ are said to be equivalent if their difference is a trivial latter (3.34).Elements of the cohomology of s mod d define the corresponding equivalence classes.
3.4 Some Solutions to the Descent Equations
We present solutions to the Descent equations as first solved by Stora and Zumino. Thesolution relies on the triviality of the cohomology of d on V(A, dA, η, ζ) which means thatthe invariant polynomials are always exact.
P2n+2(F ) = dQ2n+1(A,F ), (3.35)
where Q2n+1(A,F ) is a Lie algebra valued 2n + 1 form known as the Chern-Simons form.This form may be written as
Q2n+1(A,F ) =1
(n)!
(i
2π
)n+1 ∫ 1
0dt str(A,F n
t ). (3.36)
Where we have
Ft = tF + (t2 − t)A2. (3.37)
Some examples that will come in useful later are;
Q1(A,F ) =i
2π
∫ 1
0dt A =
i
2πTr(A). (3.38)
Q3(A,F ) =(i
2π
)2 ∫ 1
0dt str(A, tdA+ t2A2)
=1
2
(i
2π
)2
Tr(AdA+
2
3A3). (3.39)
Q5(A,F ) =1
2
(i
2π
)3 ∫ 1
0dt str(A, (tdA+ t2A2)2)
=1
6
(i
2π
)3
Tr(A(dA)2 +
3
2A3dA+
3
5A5). (3.40)
The method is then to “feed in” the solution for Q02n+1(A,F ) and then “descend” the equa-
tions.
Lets consider the descent equations for n = 1 as an example. Explicitly we have
32
sQ03 + dQ1
2 = 0 (3.41)
sQ12 + dQ2
1 = 0 (3.42)
sQ21 + dQ3
0 = 0 (3.43)
sQ30 = 0. (3.44)
Also we have
Q03 = Tr
(AdA+
2
3A3)
= Tr(AF − 1
3A3). (3.45)
Here we have omitted the normalisation of the Chern-Simons form as it is arbitrary as thedescent equations are concerned. Then it is straight forward to verify that
sQ03 = Tr (sAF − AsF − sAA2)
= −Tr (d(ηF )− d(ηA2))
= −dTr η(F − A2). (3.46)
Then consulting the descent equations we see
sQ03 = −dQ1
2, (3.47)
gives
Q12 = Tr η(F − A2) = Tr ηdA. (3.48)
Notice that up to a factor this is the non-abelian anomaly in 2 dimensions.Now we apply s to Q1
2 to give the next term in the chain.
s Q12 = d Tr η2A. (3.49)
Then
Q21 = −Tr η2 A. (3.50)
Again looking at the chain we see that
Q30 = −Tr
1
3η3. (3.51)
So the full s-ladder is
Q03 = Tr
(AdA+
2
3A3)
= Tr(AF − 1
3A3),
Q12 = Tr η(F − A2) = Tr ηdA,
Q21 = −Tr η2 A,
Q30 = −Tr
1
3η3. (3.52)
33
Now we consider the n = 2 case which produces the anomaly in 4 dimensions. Againstarting from the Chern-Simons form in 5 dimensions for Q0
5, direct computation of thedecent equations gives the ladder
Q05 = Tr
(AF 2 − 1
2A3F +
1
10A5),
Q14 = Tr η d
(AdA+
1
2A3),
Q23 = −1
2Tr((η2A+ ηAη + Aη2)dA+ η2A3
),
Q32 =
1
2Tr(−η3dA+ AηAη2
),
Q41 =
1
2Tr η4A,
Q50 =
1
10Tr η5. (3.53)
Notice that dQ05 is the U(1) anomaly in 6 dimensions and that Q1
4 is the non-Abelian anomalyin 4 dimensions, up to normalisation.
The non-abelian anomaly is given by
G[η,A] = (±)(−2πi)in+1
(2π)n+1(n+ 1)!
∫
MQ1
2n(η,A). (3.54)
The normalisation can be calculated via k-theory or perturbation theory.
This method can be applied to any value of n. However, it quickly becomes lengthy andcomplicated. A general expression for arbitrary n is developed in the next section.
3.5 General Solutions to the Decent Equations and the
Russian Formula
A general solution (up to elements in the local cohomology of s) can be obtained for thedescent equations using a generalised transgression formula known as the Russian formula.Again we define a generalised exterior derivative
d = d+ s, (3.55)
which is nilpotent. We also introduce a family of gauge one-forms and there curvatures
At = tA+ η, t ∈ [0, 1], (3.56)
where
A1 = A+ η, A0 = η. (3.57)
34
We define the associated curvature as
Ft = dAt + A2t
= Ft + (1− t)dη. (3.58)
Where we have defined Ft = tF + (t2 − t)A2.
The BRST transformations are equivalent to the horizontality condition
F1(A1) = dA1 + A21 = dA+ A2 = F (A), (3.59)
which is known as the “Russian Formula”. We also have that
F0 = dη. (3.60)
Given a G-invariant symmetric polynomial P we consider a generalised transgression formulawith l = n+ 1
P(F1l)− P((dη)l) = dQ2l−1, (3.61)
where
Q2l−1 = l∫ 1
0dt P(A, (Ft)
l−1). (3.62)
From (3.62) the first part of the descent equations can be derived by expanding Q2l−1 inpowers of dη and using the transgression formula (3.59).
The symmetric polynomials can be expanded as
P(A, (Ft + (1− t)dη)l−1) =l−1∑
k=0
(l − 1)!
k!(l − k − 1)!(1− t)kP((dη)k, A, (Ft)
l−1−k), (3.63)
and then the writing
Q2l−1 = Q02l−1 +Q1
2l−1 + ...+Ql−1l , (3.64)
the first part of the descent equations are obtained
P(F l)− dQ02l−1 = 0
sQ02l−1 + dQ1
2l−2 = 0...
sQl−1l + P((dη)l) = 0. (3.65)
Then we can write
Qk2l−1−k =
(l − 1)!
k!(l − k − 1)!
∫ 1
0dt (1− t)kP((dη)k, A, (Ft)
l−1−k). (3.66)
35
With 0 ≤ k ≤ l − 1. This formula gives the terms in the descent equations up to k ≤ l − 1in the power of dη.
For k = 0 we have the definition of the Chern-Simons form Q02l−1. For k = 1 we have an
explicit solution for the anomaly
Q12l−2 = l(l − 1)
∫ 1
0dt(1− t)P(dη,A, F l−2
t ). (3.67)
To express the anomaly in terms of η rather than dη we can shift the derivative from η to Aand Ft. This picks up a d-exact term which is irrelevant. Thus the anomaly can be expressedas
Q12l−2 = (l − 1)
∫ 1
0dt(1− t)P(η, dA, d(Ft)
l−2). (3.68)
For the higher terms one needs to consider fields of a different homotopy type.
At = tη. (3.69)
In this case we have
A1 = η, A0 = 0. (3.70)
Then the curvature is
Ft = dAt + A2t = t(d+ s)η + t2η2 = tdη + (t2 − t)η2, (3.71)
thus
F1 = dη, F0 = 0. (3.72)
Now we apply the transgression formula
P((F1)l)− P((F0)
l) = dQl−1, (3.73)
with
Ql−1 = n∫ 1
0dt P(η, (tdη + (t2 − t)η2)l−1). (3.74)
The invariant polynomials in this case can be expanded as
P(η, F l−1t ) =
l−1∑
k=0
(l − 1)!
k!(n− k − 1)!tl−1−k(t2 − t)kP((dη)l−1−k, η, (η2)k). (3.75)
Now we expand Ql−1 as
Ql−1 = Qll−1 +Ql+1
l−2 + · · ·+Q2l−10 . (3.76)
Then (3.73) can be expanded as
P((dη)l) = (s+ d)Qll−1 + (s+ d)Ql+1
l−2 + · · ·+ (s+ d)Q2l−10 . (3.77)
36
Collecting terms in the same order of dη produces the second part of the descent equations
P((dη)l)− dQll−1 = 0
sQll−1 + dQl+1
l−2 = 0
...
sQ2l−21 + dQ2l−1
0 = 0
sQ2l−10 = 0. (3.78)
From (3.75), after evaluating the integrals in terms of beta functions the terms in chain(3.78) can be written as
Ql+kl−1−k = (−1)k
l!(l − 1)!
(l − 1− k)!(l + k)!P((dη)l−1−k, η, (η2)k), (3.79)
with 0 ≤ k ≤ l − 1.
Notice how these terms do not depend on A or F , but just the ghost form η. Although thechain terms are of different homotopy they do indeed belong to the same chain.
Using equation (3.79) the last term in the chain can be calculated,
Q2l−10 (η) = (−1)l−1 l!(l − 1)!
(2l − 1)!P(η2l−1). (3.80)
In order to compare these results to Section 3.4 the symmetric polynomial P is taken to bethe symmetrised trace. We calculate the ladder for l = 2 explicitly and then compare themwith (3.52). For k = 0 and using (3.66) we immediately get the Chern-Simons form
Q03 = 2!
∫ 1
0dtStr(A,Ft)
=∫ 1
0dtTr(A, tF + (t2 − t)A2)
=∫ 1
0dtTr
(tAF + (t2 − t)A3
)
=1
2Tr(AF − 1
3A3)
=1
2Tr(AdA+
2
3A3), (3.81)
which up to normalisation agrees with (3.52). Using (3.68) it is clear that the 2-dimensionalanomaly is
Q12 =
1
2
∫ 1
0(1− t)Tr(ηdA)
=1
4Tr(ηdA). (3.82)
37
Which clearly agrees with the previous analysis again up to normalisation. Now we need touse formula (3.79) to calculate terms lower in the ladder. It is easy to see that using k = 0and k = 1
Q21 = Tr dηη, (3.83)
and
Q30 = −1
3Trη3. (3.84)
Notice that the Schwinger term (3.83) does not not appear to be the same as (3.50). However,it is straightforward to see that the difference between the two expressions is BRST-exact,hence they are BRST cohomologous.
(3.83)− (3.50) = Tr(dηη + η2A)
= Trη(dη + Aη − ηA+ ηA)
= Trη(dη + [A, η] + ηA)
= −Trη(sA− Aη)= Trs(ηA). (3.85)
3.6 The Bottom Up Approach to Solving the Descent
Equations
So far we have presented the “top-down” approach to solving the descent equations usinghomotopic arguments such as the Russian formula. There also exists a “bottom-up” approachwhich starts by solving the bottom descent equation. We follow [58].
sQ2n+10 = 0. (3.86)
The solution to the above is clearly
Q2n+10 = A2n+1
0 + sQ2n0 . (3.87)
Where A2n+10 is a representative of the cohomology of s. Now inserting (3.87) into the next
line of the descent equations, i.e. p = 1 gives
s(Q2n1 − dQ2n
0 ) + dA2n+10 = 0 (3.88)
To solve (3.88)one assumes that a special solution A2n1 to the descent equations exists
sA2n1 + dA2n+1
0 = 0. (3.89)
Then we can write
s(Q2n
1 − dQ2n0 − A2n
0
)= 0. (3.90)
Thus we are solving for the cohomology of s again. Hence we have,
38
Q2n1 = A2n
1 + A2n1 + dQ2n
0 + sQ2n−11 , (3.91)
where A2n1 is another representative of the cohomology of s. We can continue this procedure
in an iterative manner
Q2n+1−pp = A2n+1−p
p + A2n+1−pp + dQ2n+1−p
p−1 + sQ2n+1−(p+1)p , (3.92)
for 1 ≤ p ≤ D− 1, where D is the space-time dimensions. A2n+1−pp is a representative of the
cohomlogy of s in the sector of forms of degrees (2n+ 1− p, p) and A2n+1−pp depends on the
cohomology of s in the sectors of degree less that p−1. Then the next line up in the descentequations is
sQ2n+1−(p+1) + d(A2n+1−pp + A2n+1−p
p
)− sdQ2n+1−(p+1)
p = 0. (3.93)
Now assume as before the existence of a special solution A2n+1−(p+1)p+1
sA2n+1−(p+1)p+1 + d
(A2n+1−pp + A2n+1−p
p
)= 0. (3.94)
Then applying (3.92) with p→ p+ 1
Q2n+1−(p+1)p+1 = A
2n+1−(p+1)p+1 + A
2n+1−(p+1)p+1 + dQ2n+1−p+1
p + sQ2n+1−(p+2)p+1 , (3.95)
here A2n+1−(p+1) is a representative of the chomology of s in the sector of forms with de-grees (2n+1−(p+1), p+1). Equation (3.95) is the general solution to the descent equations.
It is worth noting that the condition (3.94) may not be satisfied for all p.
3.7 BRST Cohomology and Physics
So far we have concentrated on the singlet and gauge anomalies that arise in gauge theories.Other elements in the chain have a physical interpretation.
1. Abelian Anomaly - TrF n+1
The trace of the curvatures gives the abelian anomaly in in (2n+ 1) dimensions. Thecorrect normalisation needs to be calculated via perturbation theory or by the indextheorem. See chapter 2.
2. Topological Field Theory - Q02n+1
The Chern-Simons form is the starting point of all topological field theories.
3. Gauge Anomalies - Q12n
This term is the non-abelian anomaly in 2n dimensions. Again the correct normal-isation is calculated via pertubation theory or via a generalised index theorem. Seechapter 2.
4. Schwinger terms - Q22n−1
This term to order η2 has been identified with the Schwinger term in an equal timecommutatior of Gauss-law operators for Yang-Mills theory [32].
39
5. Magnetic monopoles and the Jacobi identity - Q32n−2
This term is related to the breakdown of the Jacobi identity for velocity operators inthe presence of a magnetic monopole [43].
However, higher order terms in the descent equations have so far lacked any physical inter-pretation.
3.8 Descent Equations for Nontrivial Gauge Bundles
So far in deriving the descent equations the gauge theories considered had trivial fibre-bundlestructure. That is the principle bundles P (M,G) with group structure G over manifold Mwere globally M ×G. The structure of the anomaly and the descent equations can be gen-eralised to nontrivial gauge bundles via the introduction of reference connection A0. Thisconstruction was first proposed by Manes, Stora and Zumino [49]
The reference potential does not transform under a BRST transformation
s A0 = 0 (3.96)
The anomalous Ward identity becomes
sW [A,A0] = G[η,A,A0] =∫
MηαGα[A,A0]. (3.97)
The nilpotency of s produces the Wess-Zumino consistency condition
sG[η,A,A0] = 0. (3.98)
Following [49] we a transgression formula for non-trivial bundles
P(F l(A))− P(F l(A0)) = dQ2l−1(A,A0) (3.99)
= ld∫ 1
0dtP(A− A0, F
l−1t ), (3.100)
where we have defined
Ft = dAt + A2t , At = tA+ (1− t)A0. (3.101)
Now we shift the gauge field A and define a generalised exterior derivative as before
A = A+ η
d = d+ s. (3.102)
Applying the “Russian Formula” (3.59) to the “shifted transgression” formula
dQ2l−1(A,A0) = dQ2l−1(A+ η,A0)
= ld∫ 1
0dtP(A+ η − A0, F
l−1t ), (3.103)
40
where we have the homotopies
Ft = dAt + A2t ,
At = t(A+ η) + (1− t)A0. (3.104)
Expanding the Chern-Simons form in powers of η as before
Q2l−1(A+ η,A0) = Q02l−1(A, η,A0) +Q1
2l−2(A, η,A0) + · · ·+Q2l−10 (η), (3.105)
and then collecting terms of the same order produces the descent equations for non-trivialbundles
P (F n(A))− P (F n(A0))− dQ02n−1(A,A0) = 0
sQp2n−1−p(η,A,A0) + dQp+1
2n−2−p(η,A,A0) = 0
sQ2n−10 (η) = 0 (3.106)
with p = 0, 1, ..., 2n− 2.
Such a generalisation to non-trivial bundles would be needed if magnetic monopoles arepresent. However, all fibre bundles are locally trivial and hence our discussion so far islocally valid provided we are not actually at a monopole.
3.9 Epitome
The local cohomology of the exterior derivative d on the space of form polynomials V(A,F, η, ζ)is trivial. This allows the descent equations to be derived in a mathematically sound manner.The descent equations in fact define the local cohomology of s mod d.
Solutions to the descent equations can be obtained by picking an invariant polynomial P2n+2,applying the graded BRST algebra and consulting the descent equations. A “bottom up”approach also exists.
Calculating solutions in arbitrary dimensions becomes difficult. General solutions can beobtained via transgression formula and the “Russian Formula”. The non-abelian anomalyis given by
Q12n = n
∫ 1
0dt(1− t)P(η, dA, d(Ft)
n−1), (3.107)
where Ft = tF + (t2 − t)A2.
The remaining question is how to pick the symmetric polynomial P2n+1? If the symmetrisedtrace is used then one recovers the non-abelian anomaly in 2n dimensions starting from thesinglet anomaly in (2n+ 2) dimensions.
41
CHAPTER 4
The Anomalies of Gravity
Here we investigate the anomalies that are present when matter fields are coupled to gravityas calculated by Alvarez-Gaume and Witten [3, 6].//We show that there is a generalisation of the global U(1) chiral anomaly for standard gaugetheory. This is shown to be related to the Dirac genus of the underlying manifold, whichmust be of even dimensions. Fujikawa’s method is used to explicitly calculate this anomaly.
There are three kinds of gravitational anomaly each related to the three symmetries ofgravity; Einstein (diffeomorphisms), Lorentz and Weyl. These anomalies destroy the usualproperties of the energy-momentum tensor.
1. The anomaly in local Lorentz transformations is equivalent to the existence of anantisymmetric part of the energy momentum tensor
〈T ab〉 − 〈T ba〉 6= 0. (4.1)
2. The Einstein anomaly is understood as the nonconservation of the energy-momentumtensor
∇µ〈T µν〉 6= 0. (4.2)
3. The Weyl anomaly is equivelent the nonvanishing of the trace of the energy-momentumtensor and is thus also know as the trace anomaly
〈T µµ〉 6= 0. (4.3)
We consider only the local Lorentz and Einstein anomalies here. We set up the correct ghoststructure for gravitational theories and construct the correct BRST algebra. Then using thesame arguments as for the gauge theory case the Descent equations are presented and solved.Using this construction the equivalence of the Lorentz and Einstein anomalies is shown. Weassume that the connection is metric compatible and torsionless.
42
4.1 Chiral U(1) Gravitational Anomalies
We consider a Dirac field coupled to gravity via the Lagrangian
L =√g ψ
(iγαeµα(∂µ −
i
4ωabµ σab)
)ψ. (4.4)
The Dirac operator, which is hermitian is defined as
i /D = iγαeµα(∂µ −i
4ωabµ σab), (4.5)
where have the spin connection ωabµ and vielbein eµα ∈ GL(m,R). We have also defined
σa,b =i
2[γa, γb], (4.6)
A complete orthonormal set of eigenspinors is defined as
/D = λnψn,∫dmx√gψ†
n(x)ψ(x) = δn,m. (4.7)
In exactly the same way as for the situation in gauge theory, we expand the path integralmeasure in terms of these eigenspinors we obtain the following relation
∫DψDψ exp[S] (4.8)
=∫ ∑
i
da′idb′
i exp[S +∫dmx√gα(x)
(
DµJµ5 (x)− 2i lim
N→∞
∞∑
l=1
ψ†l (x)γm+1ψl(x)
)
].
Thus we identify the anomalous conservation law as
DµJµ5 = 2iA5(x). (4.9)
With Jµ5 = ψ(x)eµαγαγm+1ψ(x) and
A5(x) = limN→∞
∞∑
l=1
ψ†l (x)γm+1ψl(x). (4.10)
The above expression is not well defined and needs to be regulated. As before, we pick theheat kernel method.
∫dmx√gA5(x) = lim
N→∞
N∑
l=1
ψ†l (x)γm+1ψ(x)
= limM→∞
∫dmx√g
∞∑
l=1
ψ†l (x)γm+1 exp
[−λ2
l /M2]ψ(x)
= limM→∞
∫dmx√g
∞∑
l=1
ψ†l (x)γm+1 exp
[− /D2
/M2]ψ(x)
= Tr〈ψl|γm+1 exp[− /D2
/M2]|ψl〉|M→∞. (4.11)
Using the othogonality condition of the eigenspinors the only contribution to the anomalycomes from the zero modes and hence
43
∫dmxA5(x) = Tr〈ψl|γm+1 exp
[− /D2
/M2]|ψl〉|M→∞
= v+ − v−.
Where v± are the number of normalisable zero modes of the Dirac operator with positiveand negative chirality.
Now we apply the AS index theorem for a spin complex,
v+ − v− =∫
mA(TM)|vol. (4.12)
Where A(TM) is the Dirac or A-roof genus which is defined as
A(TM) =k∏
j=1
xj/2
sinh(xj/2). (4.13)
The above can be expressed in terms of the Pontrjagin classes and hence in terms of thecurvature. For example in 4 dimensions the local form of the anomaly is
A5 = DµJµ5 (x) = − i
348 π2εαβλρRµν
αβRλρµν . (4.14)
4.2 The BRST Algebra for Gravitation
In analogy to gauge theory ghosts are introduced for gravity [56, 71]. We have the followingghost structure (we consider connections with zero torsion and only trivial bundles, alsoignoring any complications due to the Gribov ambiguity). We follow [73]. Generalisations togravity theories including torsion have been constructed, see [52]. For an overview of gravitytheories with torsion and Poincare gravity see [23].
ξα(x) − Einstein Ghost
αab(x) − Lorentz Ghost
σ(x) − Weyl Ghost. (4.15)
There are three BRST operators associated with GR each corresponding to a classical sym-metry,
sE − Einstein general coordinate transformation
sL − Local Lorentz transformation
sW − Weyl transformation. (4.16)
Consider just the Lorentz BRST transformation. We define the spin connection one form ωand the zero form Lorentz ghost α
44
ω =1
2Labωab, (4.17)
α =1
2Labαab. (4.18)
Both ωab and αab are antisymmetric in their indices. Lab are the hermitian generators ofSO(k) in some representation, k is the dimension of the Euclidean space.
Let V(ω, dω, α, ρ = dα) be the space of form polynomials. The local space V has a naturalgrading given by the sum of the form degree and the ghost number. ω has ghost numberzero where α has ghost number one. In general any p-form with ghost number q is denotedΩqp.
The (Lorentz) BRST transformations on the spin connection and Lorentz ghost are
sL ω = −(dα+ [ω, α]),
sL α = −α2. (4.19)
The BRST operator is a nilpotent, s2L = 0. It is worth comparing these transformations with
the gauge theory case. They are almost identical, which is what we would expect as we aretreating gravity as a gauge theory of the Lorentz group.
The Riemann curvature two-form is defined as
R(ω) = dω + ω2. (4.20)
Under the Lorentz BRST the curvature two-form transforms as
sL R(ω) = [R(ω), α]. (4.21)
Also the Bianchi identity holds
dR = [R,ω]. (4.22)
Now consider the Einstein BRST transformations. We define F(Γ, dΓξ, v = dξ) as the spaceof polynomials in (Γ, dΓ, ξ, v). Here we have defined
Γ = e−1de+ e−1ωe, (4.23)
to be the Christoffel connection one-from, and we have defined the vielbein e = eµdxµ with
eµ being a coordinate basis for the tangent bundle.
The Einstein BRST transformations act on the connection and ghost as
sEΓ = LξΓ−∇v
sEξ =1
2[ξ, ξ]. (4.24)
45
Here ∇ = d + [Γ, ] and [ξ, ξ] is the graded Lie bracket. Lξ is the Lie derivative taken withrespect to the ghost parameter ξ.
It is convenient to shift this BRST operator by the Lie derivative and define a new BRSToperator
sξ = sE − Lξ. (4.25)
It is then also useful to redefine the basic variables as F(Γ, R(Γ), ξ, v), where R(Γ) = dΓ +Γ2. Under the new shifted transformations we obtain the “Yang-Mills like” Einstein BRSTtransformations
sξv = −v2
sξΓ = −∇vsξR(Γ) = [R(Γ), v]. (4.26)
Notice that both the Lorentz and Einstein BRST transformations are now cast in the sameform as the BRST transformations for gauge theory. This means that we can apply thepowerful machinery developed in section 3 to the gravitational case.
4.3 Pure Lorentz Anomalies
We define a local Lorentz anomaly to be the non-vanishing action of sL on the quantumaction
sLW [ω] = GL[ω, α]. (4.27)
This leads to the Lorentz anomaly consistency condition
sLGL[ω, α] = 0. (4.28)
As discussed in chapter 3, we seek local functionals that are linear in the ghost fields whichare s-closed but not s-exact. The solution can be calculated via the descent equations.
The cohomology of d on V(ω, dω, α, ρ) is important in deriving and solving the descent equa-tions. As for gauge theory, we use a filtration operator to reduce the calculation to that of asimpler exterior derivative d0. It will be continent to change the local variables to (ω,R, α, ρ)
In analogy to the gauge theory case, we define the exterior derivative and a filtration operatoron V(ω,R, α, ρ)
d = dω∂
∂ω+ dα
∂
∂α+ dR
∂
∂R
N = ρ∂
∂ρ+ α
∂
∂α+ ω
∂
∂ω+R
∂
∂R. (4.29)
The exterior derivative decomposes as
46
d = d0 + d1, (4.30)
with
[N, dn] = ndn n = 0, 1. (4.31)
d0d0 = 0. (4.32)
It is easy to see that
d0ω = R; d0α = ρ
d0R = 0; d0ρ = 0. (4.33)
Thus (4.33) shows that d0 has vanishing cohomology and thus d must also have vanishingcohomology as the cohomology of d is isomorphic to a subspace of the cohomology of d0.
We start from with an invariant polynomial of degree (2p + 2). All such polynomials areexpressed in terms of sums and products of traces in the curvature R(ω)
P2p+2(R) ∼ TrRp+1. (4.34)
However, such invariant monomials vanish unless p = (2n − 1). Thus as we shall see, pureLorentz anomalies can only occur in 4n− 2 space-time dimensions.
Via the Bianchi identity P4n is d-closed,
dP4n = 0. (4.35)
As the cohomology of d is trivial, P4n must also be d-exact
P4n(R) = dQ04n−1. (4.36)
Then using the fact that the cohomology of d is trival and the anticommuting properties ofd and s, the gravitational descent equations can be derived
sLQ04n−1 + dQ1
4n−2 = 0
sLQ14n−2 + dQ2
4n−3 = 0...
sLQ4n−21 + dQ4n−1 = 0
sLQ4n−10 = 0. (4.37)
We recognise Q14n−2 as the Lorentz SO(4n− 2) anomaly.
As an explicit example, lets consider the n = 1 case. The descent equations are
47
sLQ03 + dQ1
2 = 0 (4.38)
sLQ12 + dQ2
1 = 0 (4.39)
sLQ21 + dQ3
0 = 0 (4.40)
sLQ30 = 0. (4.41)
Lets consider the case for the Dirac field. In this instance we have that
P4n = A|vol. (4.42)
In two dimensions A|vol is given by the first Pontrjagin class p1 ∝ TrR2. Hence, we startwith the the SO(3) Chern-Simons form
Q03 = Tr
(ωR− 1
3ω3), (4.43)
the SO(2) Lorentz anomaly is easily calculated via (4.38). We see that
sLQ03 = −dQ1
2 (4.44)
Acting of the Chern-Simons form with sL gives
sLQ03 = −dTrα(R− ω2) = −dTrαdω. (4.45)
Then the Lorentz anomaly (up to normalisation) is given by
Q12 = Trαdω (4.46)
Other terms in the ladder can be calculated in this way. For completeness we quote themhere. Notice that the calculation is identical to the gauge theory example in Chapter 3.
Q03 = Tr
(ωdω +
2
3ω3)
= Tr(ωR− 1
3ω3),
Q12 = Tr α(R− ω2) = Tr αdω,
Q21 = −Tr α2 ω,
Q30 = −Tr
1
3α3. (4.47)
Higher dimensional examples can be calculated in this way, however it clearly becomesclumsy and long winded. A general solution can be obtained via the “Russian Formula”which is presented in a later section.
Again, we stress that the normalisation of the anomaly is not fixed by the descent equations.
4.4 The Equivalence of Einstein and Lorentz Anoma-
lies
Einstein anomalies are defined in a similar fashion to the Lorentz anomalies,
48
sEW [e] = GE[ξ,Γ]. (4.48)
This leads to the Einstein consistency condition
sEGE[ξ,Γ] = 0. (4.49)
As for the Lorentz anomaly we want to set up the descent equations and provide a systematicmethod of solving them. Notice that the shifted Einstein BRST algebra (4.26) is identical tothe Lorentz BRST algebra (4.19),(4.21) with Γ and v playing the role of the spin-connectionω and the Lorentz ghost α. Thus the cohomological arguments in section 4.3 also hold onF(Γ, R(Γ), ξ, v). Thus the structure of the descent equations is identical to (4.37), exceptnow the invariant polynomials are given in terms of the Gl(2n) curvature.
So, the term in the descent equations relevant for the Einstein anomaly is
sξQ14n−2 + dQ2
4n−3 = 0. (4.50)
To show that (4.50) does indeed lead back to the consistency condition (4.49) we first assumethat the manifold has no boundary. Thus via Stokes theorem
sξ
∫Q1
4n−2(v,Γ) = 0. (4.51)
The Lie derivative Lξ acting on a form of maximal space-time degree gives a d-exact term
LξQqmax = dQq+1
max−1, (4.52)
thus we recover the consistency condition for the Einstein anomaly.
sE
∫Q1
2n−2(v,Γ) = 0. (4.53)
In order to calculate the Einstein anomaly it is sufficient to calculate the Lorentz anomalyand then replace the spin-connection ω with the Christoffel connection Γ and the Lorentzghost α with the ghost variable v. As an explicit example, the two dimensional Einsteinanomaly for the Dirac field viz (4.47) is given by
Q12(v,Γ) = TrvdΓ. (4.54)
4.5 The Descent Equations for Gravity via the Russian
Formula
We consider only trivial fibre-bundles in this section. The “Russian Formula” equally holdsfor both the Lorentz and Einstein sectors
R(Γ) = ∆Γ + Γ2 = dΓ + Γ2 = R(Γ)
R(ω) = ∆ω + ω2 = dω + ω2 = R(ω). (4.55)
Where we have defined
49
Γ = Γ + v, ω = ω + α, d = d+ sξ. (4.56)
We also notice that d is a nilpotent
d2 = 0. (4.57)
Again we start with a symmetric invariant polynomial and use the transgression formula.Consider for the moment just the Einstein sector.
P(Rl) = dQ2l−1(Γ, R). (4.58)
Here Q2l−1 is the GL(2l) Chern-Simons form. The transgression formula also holds for theshifted fields
P(Rl) = dQ2l−1(Γ, R). (4.59)
The “Russian Formula” then implies
dQ2l−1(Γ + v,R) = dQ2l−1(Γ, R). (4.60)
Then expanding the Chern-Simons form in powers of the ghost v we obtain the gravitationaldescent equations
P(Rl)− dQ02l−1 = 0
sξQ02l−1 + dQ1
2l−2 = 0...
sξQ2l−10 = 0. (4.61)
The anomaly is given by
Q12l−1(v,Γ) = l(l − 1)
∫ 1
0dt(1− t)P(v, dΓ, dRl−2
t ) (4.62)
Rt = tR(Γ) + (t2 − t)Γ
Q12l−1(α, ω) = l(l − 1)
∫ 1
0dt(1− t)P(α, dω, dRl−2
t ) (4.63)
Rt = tR(ω) + (t2 − t)ω.
4.6 Mixed Anomalies
Consider a theory in which the Dirac field is coupled to the gravitational field and a Yang-Mills field. The corresponding Dirac operator is
/D = γaeµa(∂µ + Aµ + ωµ). (4.64)
50
The above operator will in general generate gauge, gravitational and mixed anomalies. Theseanomalies are best tackled via the index theorem, which is used to generate the necessaryinvariant polynomials needed at the top of the descent equations.
ind /D+ =∫
Mch(F ) ∧ A(TM)|vol. (4.65)
Where D+ = i /DP+.
The symmetric polynomials for the first few dimensions are
2− dim ind D+ =i
2π
∫
M2
TrF
4− dim ind D+ =1
(2π)2
∫
M4
[−1
2TrF 2 +
r
48TrR2
](4.66)
6− dim ind D+ =1
(2π)3
∫
M6
[− i
6TrF 3 +
i
48TrFTrR2
].
Where r is the dimensions of the gauge group representations. The various anomalies canthen be calculated from these invariant polynomials using the methods described earlier.
For other fields coupled to gravity and gauge fields, the correct invariant polynomials canbe constructed (including the correct normalisation) from the Atiyah-Singer index theoremfor the relevant elliptical operator. This is described in the next chapter.
4.7 Epitome
The Dirac field coupled to gravity has a chiral anomaly which is the analogue of the singletanomaly in gauge theory. Via Fujikawa’s method and appealing to the AS index theoremthis anomaly is
∫dxA5 =
∫
mA(TM)|vol. (4.67)
The BRST algebra for both the Lorentz transformations and Einstein differomorpisms havethe same structure as the BRST algebra for gauge theory. This enables us to apply themachinery of chapter 3 to gravity. As the theory is assumed to be torsion free the curvaturesdefined by either the Christoffel connection one-form or the spin-connection are identical.This allows the Lorentz and Einstein anomalies to be identified.
The “Russian Formula” and transgression formula give an explicit form for the Lorentzanomaly
Q12n(α, ω) = n
∫ 1
0dt(1− t)P(α, dω, d(Rt)
n−1) (4.68)
The Einstein anomaly is given by the replacement of ω with Γ and α with v.
If the invariant polynomial P4n is given by the A-genus then the in complete analogy to thegauge theory case we recover the gravitational anomaly in terms of the chiral U(1) anomaly
51
for the Dirac field coupled to gravity.
52
CHAPTER 5
The Atiyah-Singer Index Theorem andAnomaly Polynomials
In the previous chapters the invariant polynomial P2n+2 have been left undetermined exceptwhen explicit calculations were preformed for the Dirac field. In this case we used the chi-ral anomaly in (2n + 2) as calculated by the Atiyah-Singer index theorem as the invariantpolynomial (ignoring at that stage normalisation).
It is well-known that the only possible fields that can give rise to anomalies are the spin-1/2Dirac field, the spin-3/2 Rarita-Schwinger field and the (anti-)self-dual antisymmetric tensorfield. The question is what invariant polynomial should be used?
Although the anomaly is not uniquely determined by the descent equations (due to topologi-cally trivial terms), we propose that the invariant polynomials are fixed including the correctnormalistion if one appeals to the Atiyah-Singer index theorem for the relevant operator inquestion. Thus, the anomaly including normalisation is determined.
The mathematical proof of the above statement requires k-theory. For a very accessibleintroduction to k-theory (both differential and algebraic) see [72]. For the application ofk-theory to anomalies see [54]. Alvarez-Gaume and Ginsparg [4, 5] demonstrated how theanomaly in 2n dimensions is related to the AS theorem in a (2n+2) dimensional space. Theirmethod avoids the use of k-theory, but in some sense it closely resembles it. See [20, 53] fora modern account.In this chapter we present the necessary complexes associated with the Dirac, Rarita-Schwinger and self-dual tensor fields. We then apply the AS-theorem to generate the invari-ant polynomials.
We then present the symmetric polynomials for the spin-1/2, spin-3/2 and (anti-)self-dualtensor field in four, six and ten dimensions. Then the anomaly content of effective super-garvity theories in ten dimensions are examined. These supergravity theories are of partic-ular interest as they are the low energy effective theories of the known five consistent stringtheories.
53
5.1 Twisted spin complexes
Consider a spin bundle S(M) over an m-dimensional orientable manifold M . The sections ofthis bundle are denoted ∆(M) = Γ(M,S(M)). A Dirac spinor ψ ∈ ∆(M) is an irreduciblerepresentation of the Clifford algebra, but is not an irreducible representation of the SPIN(m)group generated by the Dirac matrices. ∆(M) can be decomposed in to eigenvectors of γm+1,with the eigenvalues ±1 being the chirality.
∆(M) = ∆+(M)⊕
∆−(M), (5.1)
The irreducible representations of SPIN(m) are the spinors in ∆±, with
ψ+ ∈ ∆+(M), ψ− ∈ ∆−(M). (5.2)
Where we have γm+1ψ± = ±ψ±. The Dirac operator is defined as
i /D = iγae µa (∂µ + ωµ). (5.3)
ωµ is the spin connection; ωµ = −i/4 ωabmuσab. Defining the projection operators as
P+ =1
2(I + γm+1), P− =
1
2(I − γm+1), (5.4)
allows us to define
D = i /DP+ : ∆+(M)→ ∆−(M)
D† = i /DP− : ∆−(M)→ ∆+(M). (5.5)
Hence we have the two term complex
∆+(M)D
-
D†∆−(M) (5.6)
which is known as the spin complex. The analytical index of this complex gives the differencein the number of zero-modes with different chirality
v+ − v− = indD = dim kerD − dim kerD†. (5.7)
Applying the AS-index theorem to this spin complex gives
v+ − v− =∫
Mch(∆+(M)−∆−(M))
Td(TMC)
e(TM)|vol
=∫
MA(TM)|vol. (5.8)
Here A is the Dirac or A-roof genus.
Next we consider the twisted spin complex. A spinor that belongs to a representation ofa Lie group G is a section of the bundle S(M)
⊗E, where is the associated vector bundle
P (M,G) in a given representation. The Dirac operator is defined as
54
DE = iγαe µα (∂µ + ωµ + Aµ)P
+. (5.9)
Where Aµ is the gauge field on E. The AS index theorem for the twisted spin complex is
v+ − v− =∫
MA(TM)ch(E)|vol. (5.10)
5.2 The Rarita-Schwinger spin complex
The full construction of the Rarita-Schwinger spin complex requires the use of K-theory.The usual method is to construct Rarita-Schwinger chiral bundles ∆±
3/2(M) with local coor-dinated (xµ, ψν) as virtual bundles.
The twisted Rarita-Schwinger complex is defined in analogy to the spin complex. The spinorsare sections of the bundle ∆±
3/2
⊗E. The AS index theorem for the twisted Rarita-Schwinger
complex gives the difference of normalisable zero-modes of the twisted Rarita-Schwingeroperator DRS
v3/2+ − v3/2
− = indDRS =∫
M[A(TM)(Tr exp(iR/2π)− 1)ch(E)]|vol. (5.11)
5.3 The signature complex
The antisymmetric tensor fields with self-dual (+) or anti-self-dual field strengths form bun-dles, Λ±(M) over the manifold M . The signature complex is the correct mathematicaldescription of these fields.
In order to define the signature complex we need to introduce an operator that maps Ωp(M)into Ωn−p;τ . Here Ωp denotes the space of complex p-forms over M and n is the dimensionof the manifold and is taken to be even. τ is in fact related to the Hodge star ∗,
τ = ip(p−1)+n/2 ∗ . (5.12)
From the definition of τ and using the fact that ∗2 = (−1)p implies that τ 2 = 1 and haseigenvalues ∓1. The exterior algebra can thus be decomposed into the eigenspaces of τ
Ω∗(M) =⊕
p
Ωp(M) = Ω+ ⊕ Ω−. (5.13)
Notice that d + d∗ maps Ω+ into Ω− and vice versa. We define the operator d+ as therestriction of d+ d∗ to Ω+.
d+ : Ω+ → Ω−. (5.14)
It is easy to see that d− is indeed the adjoint of d+ and is the restriction of d + d∗ to Ω−.We can thus define the two term signature complex
Ω+(M)d+
-
d−Ω−(M) (5.15)
55
The index of d+ is known as the sign of M , Sign(M). See [54] for a deeper discussion on thesign of M .
Applying the AS index theorem to the signature complex gives
ind d+ =∫
M
n∏
i=1
xitanhxi
|vol. (5.16)
It is customary to define the Hirzebruch L-polynomial,
L(TM) =n∏
i=1
xitanhxi
, (5.17)
and write (5.16) as
ind d+ =∫
ML(TM)|vol. (5.18)
Now the signature complex can be twisted by taking the coefficients to belong to an arbitrarybundle. The AS index theorem for the twisted signature complex is
ind dE+ =∫
Mch(E) ∧ L(TM). (5.19)
5.4 The Invariant Polynomials in Various Dimensions
The so called “master” equation for the anomalies is the BRST variation of the quantumaction
sW = −2πi∫
MG1
2n (5.20)
The descent equations then allow the anomaly G12n to an invariant polynomial P2n+1.
dQ02n+1 = P2n+2
sQ02n+1 + dQ1
2n = 0. (5.21)
For the three kinds of potentially anomalous matter we have the following symmetric poly-nomials 1
P(1/2)2n+2 =
[A(TM)ch(E)
]∣∣∣vol
(5.22)
P3/22n+2 =
[A(TM) (Tr exp(i/2π R)− 1) ch(E)
]∣∣∣vol
(5.23)
PA2n+1 =[−1
2
1
4L(TM)
]∣∣∣∣vol, (5.24)
which correspond to the Dirac field, Rarita-Schwinger and self-dual form field. In the lattera factor of −1/2 is needed from the duality conditions.
The invariant polynomials can be expanded in terms of Pontrjagin classes. We have in two,six and ten dimensions the following polynomials
1String theorists use I to denote the symmetric polynomials.
56
• Anomaly Polynomials in four dimensionsOnly the chiral spin-1/2 fermion can contribute to the anomaly
P1/2 =1
(2π)3
[− i
6TrF 3 +
i
48TrFTrR2
]. (5.25)
• Anomaly Polynomials in six dimensionsAll three kinds of fields generate potential anomalies.
P1/2 =1
4!(2π)4
[TrF 4 + r
(1
246TrR4 +
1
192(TrR2)2
)− 1
4TrR2TrF 2
]
P3/2 =1
4!(2π)4
[49
48TrR4 − 43
192(TrR2)2
]
PA =1
4!(2π)4
[7
60TrR4 − 1
24(TrR2)2
]. (5.26)
• Anomaly Polynomials in ten dimensionsAgain all three fields contribute to potential anomalies.
P1/2 =1
6!(2π)6
[−TrF 6 + r
(1
504TrR6 +
1
384TrR4TrR2 +
5
4608(TrR2)3
)
− 1
16TrR4TrF 2 +
5
64(TrR2)2TrF 2 − 5
8TrR2TrF 4
]
P3/2 =1
6!(2π)6
[−55
56TrR6 +
75
128TrR4TrR3 − 35
512(TrR2)3
]
PA =1
6!(2π)6
[496
504TrR6 − 7
12TrR4TrR2 +
5
72(TrR2)3
]. (5.27)
5.5 Anomalies in Supergravity Theories
Supergravity theories are known to be the low energy limit of of the five known stringtheories. It is interesting to note that they are in fact anomaly free. We can use theinvariant polynomials as defined in the previous section to prove this statement. We discussthe various supergravity theories individually
• Type IIA SupergravityThis theory is non-chiral and thus is free of (local) anomalies.
• Type IIB SupergravityIn 10 dimensions type IIB supergravity has a self-dual five-form field, a pair of chiralspin-3/2 Majorana-Weyl gravitinos and a pair of anti-chiral Majorana-Weyl spin-1/2fermions. The total anomaly polynomial is thus
P(R) = PA(R) + 21
2P1/2(R)− 1
2P3/2(R). (5.28)
57
Where we have used just the gravitational part of the anomaly for the spin-1/2 fields.The factor of 1/2 is included as we have Majorana-Weyl spinors and not Dirac spinors.Thus, P = 0 when we add up all the terms. Type IIB supergravity has no anomalies.
• Type I Supergravity coupled to super Yang-MillsType I supergravity theory is chiral and in general will posses anomalies. These anoma-lies vanish if type I supergravity is coupled to E8 × E8 or SO(32) super-Yang-Mills,this is the well known Green-Schwartz mechanism [38].
Type I supergravity in ten dimensions has a chiral Majorana-Weyl spin-3/2 gravitinoand a anti-chiral Majorana-Weyl spin-1/2 dilatino. The super-Yang-Mills theory con-tains a chiral Majorana-Weyl spin-1/2 gauginos transforming under the adjoint rep-resentation of some as of yet unspecified Lie group G. The anomaly polynomial isthus
P(R,F ) =1
2(P(R)3/2 − P(R)1/2 + P(R,F )1/2). (5.29)
Here I1/2(R) is the term from the pure gravitational anomaly of the gravitino andI1/2(R,F ) is the anomaly polynomial for the super-Yang-Mills gaugino which containsa gravitational, gauge and mixed part. Using the explicit formula (5.27) we have
P(R,F ) =−1
2(2π)66!
(496− r
504TrR6 − 224 + n
384TrR4TrR2 +
5
4608(64− r)(TrR2)3
+1
16TrR4TrF 2 +
5
64(TrR2)2TrF 2 − 5
8TrR2TrF 4 + TrF 6
)(5.30)
In order for the anomaly to be cancelled by a local counter term the above anomalypolynomial has to factorise into a four-form and an eight-form. The term TrR6 doesnot allow this factorisation and so it must vanish. Thus we are led to the first conditionon the gauge group
r = 496. (5.31)
To factorise the remaining anomaly polynomial we require that
TrF 6 =1
48TrF 4TrF 2 − 1
14400(TrF 2)3. (5.32)
The 496-dimensional Lie groups SO(32) and E8 × E8 have the above property. Theyalso hold for E8×U(1)248 and U(1)496, however no such string theories are known. Theanomaly polynomial reads
P = −1
2
1
(2π)22!
(1
30TrF 2 − TrR2
)X, (5.33)
where we have defined
58
X =1
(2π)44!
(1
8TrR4 +
1
32(TrR2)2 +
1
240TrR2TrF 2 1
24TrF 4 − 1
7200(TrF 2)2
). (5.34)
5.6 Epitome
The non-abelian anomaly in 2n dimensions G[η,A], is given via the descent equations interms of a symmetric invariant polynomial in 2n + 2 dimensions P2n+2. The exact formincluding the normalisation of the invariant polynomial is given by the Atiyah-Singer indextheorem in 2n + 2 dimensions. Thus, classical index theorems completely determine thequantum anomalies in chiral theories.
The invariant polynomials are calculated by applying the Atiyah-Singer index theorem tothe relevant complexes. We have for spin-1/2 Dirac fermions, spin-1/2 Rarita-Schwingerfermions and the (anti-)self-dual form field respectively
P(1/2)2n+2 =
[A(TM)ch(E)
]∣∣∣vol
(5.35)
P3/22n+2 =
[A(TM) (Tr exp(i/2π R)− 1) ch(E)
]∣∣∣vol
(5.36)
PA2n+2 =[−1
2
1
4L(TM)
]∣∣∣∣vol. (5.37)
Using these invariant polynomials one can examine the anomalies in various theories withoutknowing the details of the theory. The matter content is all that is needed. For example, itwas shown that the supergravity theories that are the low energy limit of the five consistentsuperstring theories are anomaly free after applying the Green-Schwartz mechanism.
59
CHAPTER 6
Conclusion
This thesis addressed the calculation of potential anomalies in quantum field theories. Themain emphasis was on the calculation of consistent anomalies via the Wess-Zumino consis-tency conditions and the descent equations. This approach was adopted as the graded BRSTalgebra and the descent equations do not depend explicitly on a specific Lagrangian, butare a generic feature of gauge (gravity) theories. The only drawback to this approach is thenormalisation is not fixed. This requires topological analysis or perturbation theory.
This approach is powerful enough to calculate the anomalies that arise when Dirac fermions,Rarita-Schwinger fermions or (anti-)self-dual form fields are coupled to gauge and/or gravityfields. Only the symmetric polynomial at the “top” of the descent equations fixes the rest ofthe s-ladder. The symmetric polynomial is selected via the Atiyah-Singer index theorem forthe relevant elliptical operator. Amazingly, using the index theorem in (2n+ 2) dimensionsgives the correct normalisation of the anomaly in 2n dimensions. Thus, the anomalies canbe calculated with relative ease.
The Alvarez-Gaume and Ginsparg index procedure relates the singlet anomaly in (2n + 2)dimensions to the non-abelian anomaly in 2n dimensions, it also gives the correct normalisa-tion. k-theory is needed to show that the Atiyah-Singer index theorem in 2n+ 2 dimensionsgives the correct anomaly in 2n dimensions in general. Due to time and space restrictionsthis was not elaborated on in this thesis.
As the anomalies are calculated independently of any lagrangian, only the field content isneed in order to examine the anomalies. This leads means that the potential anomalies instring theories, supergravity theories etc. can be examined without detailed analysis of thetheory.
60
Appendices
61
APPENDIX A
Non-Abelian Gauge Theory and BRSTSymmetry
A.1 Classical Non-Abelian Gauge Theories
Yang and Mills [76] generalised electromagnetism to include non-abelian gauge groups. Herewe briefly review the classical (pure) Yang-Mills theory needed for later sections. Yang-Millstheory is a generalisation for Maxwell’s electromagnetism from an abelian gauge group U(1)to a nonabelian group, G. The essence of gauge theory is to extend the notion of a globalsymmetry to one that is now local, i.e. it varies with space-time. This seeming simpleextension leads to highly nontrivial and nonlinear constraints on the quantum field theory.First some basic facts about Lie groups and Lie algebras are needed. A representation of aLie algebra is a set of N anti-Hermitian matrices T a, a = 1, 2...N obeying
[T a, T b] = fabcT c. (A.1)
Where the f’s are the structure constants of the Lie group G.
For SU(2), the fabc will be equal to εabc. Thus in the isospin representation we have
T a = −iσa
2. (A.2)
Where σa are the Pauli spin matrices.
The gauge potentials are vector fields Aaµ(x). It is often convenient to define matrix valuedvector field Aµ known as the Yang-Mills connection.
Aµ = AaµTa. (A.3)
Where g is the gauge coupling constant.
Let the field φi (which could be fermionic or bosonic) transform under some representationof G.
62
φi(x)→ Λij(x)φj(x). (A.4)
Where Λij ∈ G. We do not restrict ourselves to the fundamental representation. The groupelement, which is a function of space-time can be parameterised as
Λij(x) = (exp[−iθa(x)T a])ij. (A.5)
Where the parameters θa(x) are local variables and T a is defined in the representation underconsideration.
Using the gauge potential it is now possible to construct the covariant derivative Dµ.
Dµ = ∂µ − igAµ. (A.6)
The important point is that the covariant derivative of a field φ is gauge covariant. That is
(Dµφ)′ = ∂φ′ − igA′µφ
′
= Λ∂µφ+ (∂µΛ)φ− igA′µΛφ
= ΛDµφ. (A.7)
This can be achieved if the term ∂µΛ is cancelled by the variation of Aµ
A′µ = − i
g[∂µΛ(x)]Λ−1(x) + Λ(x)Aµ(x)Λ
−1(x). (A.8)
Infinitesimal variations are
δAaµ = −1
g∂µθ
a + fabcθbAcµ
δφ = −igθaT aφ. (A.9)
From the covariant derivative the Yang-Mills field strength tensor is constructed as thecurvature with respect to the Yang-Mills connection. The Yang-Mills field strength tensorFµν is defined as
Fµν =i
g[Dµ, Dν ]
= ∂µAν − ∂νAµ + g[Aµ, Aν ]
= (∂µAaν − ∂νAaµ + gfabcAbµA
cν)T
a
= F aνµT
a. (A.10)
We wish to construct an invariant action out of the field strength tensor. As Dµ is covariantwe have
Fµν = ΛFµνΛ−1. (A.11)
Where Λ ∈ G.
63
The unique action with only two derivatives is constructed from the trace. However, itpossible to define an action using other functionals. The problem with these are that theywill contain higher derivatives and hence ghosts which spoil the unitary properties neededto be physical. So we use the functional
tr(ΛFµνΛ−1ΛF µνΛ−1) = tr(FµνF
µν), (A.12)
which is clearly gauge invariant and it is also Lorentz invariant. Then we construct theYang-Mills action as
S =1
4
∫tr(FµνF
µν)dx4. (A.13)
All of the above can be set in the language of differential forms by defining
F =1
2F aµνTadx
µ ∧ dxν
=1
2Fµνdx
µ ∧ dxν
= dA+ A ∧ A. (A.14)
Where we have defined
A = AaµTadxµ
= Aµdxµ. (A.15)
The Bianchi identity expressed in terms of the exterior derivative d, becomes
DF = dF + [A,F ] = 0. (A.16)
Then in this notation the Yang-Mills action becomes
S = −1
2
∫
mTr(F ∧ ∗F ) (A.17)
Where ∗ is the Hodge dual, which on a Euclidean manifold in local coordinates can bedefined as
∗Fµν =1
2εµναβF
αβ (A.18)
Variation of the action with respect to Aµ produced the field equation
DµFµν = 0. (A.19)
Or in form notation
D ∗ F = 0. (A.20)
64
A.2 Instantons
Instantons are self-dual solutions of Yang-Mills theory in a Euclidean space, for which thefield strength tensor is maximal. By dual we mean ∗F = ±F , self-dual is + and anti-self-dualis −. As instantons mediate between different vacua they are very important in understand-ing the vacua in gauge theories.
In order to solve the self or anti-self dual equations, one needs to consider the homotopy ofthe gauge group G. The boundary conditions are asymptotic conditions which describe howA behaves at infinity in <4. The requirement that the action must be finite means that Fµνmust be square integrable.
Fµν(x) → 0,
|x| → ∞. (A.21)
Which also imply
Aµ(x) →i
g∂µΛ(x)Λ−1(x),
|x| → ∞. (A.22)
Λ(x) can be thought of as being defined by a sphere at infinity in <4, i.e. on S3. ChooseG = SU(2), then for each x, Λ(x) ∈ SU(2) gives a continuous map
Λ : S3 → SU(2). (A.23)
Such maps fall into homotopy classes and are elements of π3(SU(2)). But SU(2) is topolog-ically S3
π3(SU(2)) = π3(S3) = Z. (A.24)
The mappings λ : S3 → S3 are characterised by integers, that is the points of one S3 canbe mapped smoothly to another S3 such that the winding around S3 an integer number oftimes k.
Using the language and formalism of principle fiber bundles with group SU(N), it can beshown that the instanton winding number is equal to the 2nd Chern class. As k must be aninteger we have 3 possibilities
1. If k = 0 then S = 0, so we have a flat connection Aν(x),
2. If k > 0 then F = ∗F , we have self dual solutions and S = 8π2n,
3. If k < 0 then F = − ∗ F and we have anti-self dual solutions.
Remembering that G = SU(N), the Chern characters are
1. ch1 = 0 = c1,
65
2. ch2 = − 18π2Tr(F ∧ F ).
Then the winding number can be expressed in terms of the Chern classes, c2 = 18π2
∫Tr(F ∧
F ) = −n, therefore
n = − 1
8π2
∫Tr(F ∧ F ) (A.25)
Using the definitions of the forms in local coordinates, it is straight forward to show
n = − 1
16π2
∫Tr(Fνµ ∗ Fνµ) (A.26)
A.3 Faddeev-Popov Ghosts and Gauge Fixing
Now we consider how to quantise pure Yang-Mills theory via path integrals. There are sev-eral complications that do not arise in abelian gauge theory. These originate primarily fromthe gauge fixing, which effects the path integration measure in a non-trivial way. We brieflyreview the issues of gauge fixing which leads to the Faddeev-Popov determinant [31], whichproduces the correct measure. This determinant can be reexpressed in terms of the nowfamous Faddeev-Popov ghost fields.
It was noticed that the gauge fixed Lagrangian possess a new global symmetry that rotatesthe gauge fields into ghosts. This is known as the BRST transformation [15],[69]. This sym-metry can be used in a similar way to the Gupta-Bleuler constraint to remove unphysicalstates from the Fock space. However, there is one potential source of difficulties. We demon-strate the Gribov ambiguity [39] in the Coulomb gauge means that much of the formalismdeveloped may not exist outside of perturbation theory.
Feynman in 1962 showed that using standard quantisation methods available at the time,Yang-Mills theory was not unitary. Feynman also showed that counter terms could be addedthat removed the nonunitary parts. These terms are now known as Faddeev-Popov Ghosts.
In gauge theories like Maxwell’s theory or Yang-Mills the path integral is ill-defined due tothe gauge degree of freedom. In these gauge theories both the path integral measure DAand the action S are gauge invariant and so when we functionally integrate over DA we overcount the degrees of freedom. To see this consider QED. Maxwell’s theory is invariant undergauge transformations of the form
AΛµ = Aµ + ∂µΛ, (A.27)
so that when one functionally integrates over both AΛµ and Aµ the integrand is infinity over
counted.∫DA eiS =∞. (A.28)
The problem is that when one starts with a particular configuration Aµ and then considerall possible gauge equivalent configurations AΛ
µ we over count the degrees of freedom. Thuswe are sweeping out an orbit in the space of all gauge connections. As Λ changes along the
66
orbits we inevitably over count the path integral an infinite number of times.
The solution to this is to fix the gauge. Traditionally one adds the term
LGF = − 1
2α(∂µA
µ)2, (A.29)
to the action. However for the following argument we will fix the gauge by inserting anarbitrary function of the gauge fields into the integration
δ(F (Aµ)) (A.30)
which forces the gauge condition to be F (Aµ) = 0.
However, by inserting a delta function into the path integral we change the functional inte-gration measure DA. Inserting delta functions always produces an ambiguity in the measure.To remove this ambiguity we insert 1 into the integral.
1 = ∆FP
∫DΛ δ(F (AΛ
µ)), (A.31)
here ∆FP is the Faddeev-Popov determinant which gives the correct measure. DΛ =∏x dΛ(x) is the invariant group measure. Inserting this into the functional integration we
obtain
∫DA
(∆FP
∫DΛ δ(F (AΛ
µ))
exp(i∫d4x L(A)
). (A.32)
It is important to notice that the Faddeev-Popov determinant is independent of the gauge.This is clear as by construction we integrate over all gauge factors Λ.
∆FP (Aµ) = ∆FP (AΛµ). (A.33)
Now we make a gauge the transformation Aµ → AΛ−1
µ . The measure DA, the Faddeev-Popov determinant ∆FP and the action S are all gauge invariant. The only part that is notis F (AΛ
µ). The path integral becomes
∫DΛ
∫DA∆FP δ(F (Aµ)) exp
(i∫d4x L(A)
). (A.34)
Now the infinite part of the integration has been isolated,∫DΛ = ∞ which is a volume
factor. Hence, simply dividing by this factor renders the integral finite. The gauge conditionis enforced by the delta function δ(F (Aµ)). However, the most important result is thatthe Faddeev-Popov determinant ∆FP gives the correct integration measure. The finite pathintegral is thus
∫DA∆FP δ(F (Aµ)) exp
(i∫d4x L(A)
). (A.35)
The problem of gauge fixing then becomes one of finding an explicit form of the determi-nant.The method is to use equation (A.31) and to change variables in the integration fromΛ to F . Thus, the Jacobian is calculated
DF = det
[δF
δΛ
]
DΛ. (A.36)
67
Then the Faddeev-Popov determinant can be written as
∆−1FP =
∫DF det
[δΛ
δF
]
δF
= det
[δΛ
δF
]
F=0
. (A.37)
Where hence we can write
∆FP = det
[δF
δΛ
]
F=0
. (A.38)
In order to do calculations it will be convenient to write the determinant in such a way asnew Feynman rules can be derived. We power expand the function F (AΛ
µ) for small groupparameter θ.
F (AΛµ(x)) = F (Aµ(x)) +
∫d4yM(x, y)θ(y) +O(θ2). (A.39)
When one takes the derivative only the matrix M survives. Then the Faddeev-Popov deter-minant can be written as the determinant of the matrix M . The matrix determinant canthen be converted to a Gaussian integral over Grassmann fields η and eta†.
∆FP = detM =∫Dη Dη† exp
[i∫d4xd4y η†(x)M(x, y)η(y)
]. (A.40)
The Grassmann fields η and η† are ghost fields. That is they do not obey the spin-statisticstheorem, in fact they are scalar fields that obey Fermi-Dirac statistics. These fields areknown as the Faddeev-Popov ghosts.
Now we explicitly calculate the Fadeev-Popov ghosts for QED. We choose the Lorentz gaugecondition
F (Aµ) = ∂µAµ = 0. (A.41)
Clearly under a infinitesimal gauge transformation we have
F (AΛµ) = ∂µAµ + ∂µ∂µθ. (A.42)
Then the matrix M is given by
M(x, y) = [∂µ∂µ]x,y. (A.43)
Writing the determinant in terms of the ghost fields means we must include the followingterm in the action
∫d4xd4y η†(x)∂µ∂µη(y). (A.44)
This determinant does not couple to any of the gauge fields or fermions of QED and hencedecouples from the theory giving only a multiplicative factor which can be removed at will.This is why in QED the gauge fixing can be done without any worry about the effect on the
68
measure, the Faddeev-Popov determinant is trivial.
The same cannot be said of Yang-Mills theory. Here the Faddeev-Popov ghosts give non-trivial corrections. This is partly why Yang-Mills theory took so long to quantise. Again wepick the Lorentz condition
∂µAaµ = 0. (A.45)
Then when placing this condition into the integration we must also include the term
∆FP = det(M)x,y;a,b. (A.46)
Here x and y are discretised space-time variables and a and b are gauge indices. The matrixM is easily calculated
Ma,b(x, y) =δ
δθa(x)
(
∂µ−1
g(∂µθ
b(y) + gf bcdAcµ(y)θd(y))
)
=1
g
(−δab∂µ∂µδ4(x− y) + gf bcd∂µAcµδ
adδ4(x− y))
=1
g
(−δab∂µ∂µ + gfabc∂µAcµ
)
x,yδ4(x− y). (A.47)
After rescaling and preforming the y integrations we see that the term added to the actionis
∫d4x η† a(x)
(δab∂µ∂µ − gfabc∂µAcµ(x)
)ηb(x). (A.48)
The ghost fields have some unusual properties:
1. They violate the spin-statistics theorem. They transform as scalars under the Lorentzgroup, but have fermionic statistics.
2. They couple only to the gauge field for Yang-Mills theory. They do not appear asexternal states and as such cannot be present in tree diagrams. They only appear asin loop diagrams.
3. They decouple from the physical states and should be viewed as a mathematical trickof quantisation.
A.4 BRST Quantisation of Gauge theory
Once the gauge has been fixed the theory is no longer gauge invariant. The gauge fixing termremoves the local degrees of freedom (up to Gribov ambiguity discussed later). However,the theory now has a new global symmetry involving the rotation of gauge fields into ghostsand anti-ghost fields. This symmetry is named after it’s discoverers Becchi, Rouet, Stora[18] and independently Tyupin [69], thus BRST symmetry. As this is a global symmetry nonew degrees of freedom can be eliminated.
69
We write the gauge fixed action including the Ghost term as
L = −1
2tr (FµνF
µν)− 1
2α(∂.A)2 − η a∂µDµη
a. (A.49)
Before gauge fixing the action was invariant under gauge transformations of the form
δAaµ =1
g∂µΘ
a + fabcAbµΘc. (A.50)
Redefining the gauge parameter as
Θa = −ηaλ, (A.51)
here ηa and λ are Grassmann variables and λ is constant. The gauge fixed action equation(A.49) is invariant under the global symmetry
δAaµ = −1
g(Dµη
a)λ
δηa = −1
2fabcηbηcλ
δη a = − 1
αg(∂µAaµ)λ. (A.52)
It is straight forward to see that the Lagrangian (A.49) is indeed BRST invariant. Thetr(FµνF
µν) term is clearly BRST invariant as it is just a reparametrisation of the gaugetransformation. Next consider the ghost term
δLFP = −(δη a)∂µDµηa − η a∂µδ(Dµη
a). (A.53)
Now consider the gauge fixed part
δLGF = − 1
α(∂µAaµ)∂
νδAaν
=1
αg(∂µAaµ)(∂
νDνηa)λ. (A.54)
Adding these two contributions together
δL =1
αg(∂µAaµ)(∂
νDνηa)λ− (δη a)∂µDµη
a − η a∂µδ(Dµηa)
=1
αg(∂µAaµ)(∂
νDνηa)λ+
1
αg(∂µAaµ)λ(∂νDνη
a)
−η a∂µδ(Dµηa)
= −η a∂µδ(Dµηa). (A.55)
Remembering that λ and η being Grassmann variables anti-commute. Provided δ(Dµηa) = 0
then the theory will be BRST invariant. We now proceed to prove this
70
δ(Dµηa) = δ[∂µη
a + g fabcAµηc]
= ∂µδηa + g fabc(δAbµ)η
c + g fabcAbµ(δηc)
= −1
2fabc∂µ(η
bηc)λ− fabc(Dµηb)ληc
−g2fabcf cmnAbµη
mηnλ. (A.56)
Now consider the derivative in the first term. It is easy to show that
fabc∂µ(ηbηc) = 2fabc(∂µη
b)ηc. (A.57)
Then we obtain
δ(Dµηa) = −fabc(∂µηb)ηcλ− fabc(∂µηb)ληc
+gfabcf bmnAmµ ηnηcλ− g
2fabcf cnmAbµη
mηnλ
= g(fabcf bmnAmµ η
nηc − 1
2fabcf cnmAbµη
mηn)λ. (A.58)
Then using the Jacobi identity
fabcf cnm + famcf cnb + fancf cbm = 0. (A.59)
We obtain
δL = g(fabcf bmnAmµ ηnηc
1
2famcf cnbAbµη
mηn +1
2fancf cbmAbµη
mηn)λ. (A.60)
Interchanging n and m in the last term shows that this term is identical to the middle term.Then we obtain
δL = g(fabcf bmn + fanbf bcm)Amµ ηnηcλ
= g(−fabnf bmc + fanbf bcm)Amµ ηnηcλ
= 0. (A.61)
Hence we have proved that gauge fixed Yang-Mills is indeed invariant under the BRST trans-formations.
It is straight forward to see that the BRST variational is a nilpotent
δ2BRST = 0. (A.62)
From this symmetry a Noether current can be constructed
Jµ =∑
i
δLδ∂µφi
δBRSTφiδλ
=(−F a
µνDνηa − g
2∂µη
afabcηbηc). (A.63)
71
Then in the usual way the BRST charge is defined
QBRST =∫d3xJ0, (A.64)
which itself is a nilpotent
Q2BRST = 0. (A.65)
The Fock space of the theory is greatly increased due to the presence of the ghost and anti-ghost fields. The ghost and anti-ghost stated are unphysical and have to be removed fromthe theory. This can be done in a similar way to the Gupta-Bleuler condition in QED by
QBRST |Ψ〉 = 0. (A.66)
States that satisfy the above are the physical states.
A.5 The Gribov Ambiguity
One has to be very careful if choosing a gauge condition completely removes the gauge de-grees of freedom. In fact for the Coulomb gauge condition ∇iA
ai = 0 does not remove the
gauge degree of freedom. This leads one to question the existence of the Yang-Mills propa-gator ∆ab(x− y;A) and the over counting problem stated earlier.
Given ∇iAai = 0 it is always possible to find a A′
i which is gauge equivalent to Ai (here wesuppress gauge indices) that also satisfies
∇iA′i = 0. (A.67)
This can easily be seen if one explicitly preforms the gauge transformation
A′i = ΛAiΛ
−1 − i
g(∂iΛ)Λ−1. (A.68)
Notice that the second term is proportional to g−1. This means that perturbation theorywill never pick up this factor. That is, provided we stay within perturbation theory theCoulomb gauge will in effect remove the gauge degrees of freedom. Picking SU(2) as aconcrete example, and take Ai to be gauge equivalent to 0.
Ai = − ig(∂iΛ)Λ−1, (A.69)
where we have chosen the field such that ∇iAi = 0. We parameterise the gauge using radialcoordinates
Λ = cos(ω(r)/2) + iσini sin(ω(r)/2). (A.70)
Where nini = 1 and ni = xi/r. Substituting this into the Coulomb condition we obtain
d2ω
dt2+dω
dt− sin(2ω) = 0. (A.71)
72
Where t = log(r) This is simply the equation for a damped harmonic oscillator in a constantgravitational field. The solution ω = 0 leads to Ai = 0, which is the solution stated earlier.However, it is clear that there are many other solutions to the equation with ω 6= 0 andhence the Coulomb gauge condition is not unique. For non singular solutions all we need isω = 0, 2π, 4π, ... when t = −∞. For non-trivial solutions at t = ∞ we have the asymptoticcondition
Λ→ ±iσixi
r. (A.72)
As the Coulomb gauge does not fix the gauge completely there is a infinite sequence of fieldswhich are gauge equivalent that satisfy the Coulomb condition. These copies are known asGribov copies [39].
This ambiguity can also be explained by the fact that the matrix Mab(x, y;A) as defined inequation (??) possess zero modes. Suppose we have non-zero eigenvalues λn
Mabψbn(x) = λnψan(x). (A.73)
However as stated earlier in general we also have the zero modes
Mabϕbm(x) = 0. (A.74)
Now consider the Faddeev-Popov determinant.
detMab =∏
n
λn∏
m
0 = 0. (A.75)
As this determinant vanishes the matrix is non-invertible, thus the formulas stated earlier areincorrect for the Coulomb gauge. As such the zero modes spoil the possibility of quantisingthe system canonically, at least technically. However as long as one stays within perturbationtheory these zero modes will not make any contribution to the quantisation procedure. Thisis due to the fact that the zero modes do not couple to the physical states.
Feynman rules are by their very nature pertubative. Perturbation theory is really just theanalytic part of the generating functional as g → 0, thus it is restricted to the first Gribovcopy as all other copies have a factor of 1/g associated with them.
73
APPENDIX B
Basics of Lie Algebra Cohomology
Here we present the basics of Lie algebra cohomology needed to understand the constructionof the BRST operator and it’s cohomology. We follow the conventions of [29], [28] and alsorecommend [46] for a mathematical review of Lie Algebra cohomology. For a overview of Liealgebras from a geometry point of view see [53].
B.1 Lie Algebras
Let a and g be elements of a Lie group G. The left-translation La : G → G of g by a aredefined as
Lag = ag. (B.1)
The left-translation La is defined in a similar way. As La is a diffeomorphism from G to G.This induces a diffeomorphism La? : TgG→ TagG.
On a Lie group there exists a special class of vector fields defined by their invariance undergroup translations. Let X be a vector field a vector field on a Lie group G. X is a left-invariant vector field if La?X|ag = X|ag.
A vector V ∈ TeG defines a unique left-invariant vector field XV everywhere defined on Gvia
XV |g = Lg? = Lg?V, (B.2)
with g ∈ G. Furthermore, a left-invariant vector field X defines a unique vector V = X|e ∈TeG. The set of left-invariant vector fields on G is denoted by g.
Let X = Xµ∂/∂xµ and Y = Y µ∂/∂xµ be vector fields on G. The Lie bracket [X,Y ] isdefined as
[X,Y ]f = X[Y [f ]]− Y [X[f ]], (B.3)
74
where f ∈ F (G) and F (G) is the space of smooth functions on G. It is straight forward tosee that [X,Y ] is another vector field. The Lie bracket satisfies the following
1. Bilinearity[X, c1Y1 + c2Y2] = c1[X,Y1] + c2[X,Y2] (B.4)
[c1X1 + c2X2, Y ] = c1[X1, Y ] + c2[X2, Y ]. (B.5)
For any constants c1 and c2.
2. Skew-symmetry
[X,Y ] = −[Y,X]. (B.6)
3. The Jacobi identity
[[X,Y ], Z] + [[Z,X], Y ] + [[Y, Z], X] = 0. (B.7)
If one takes two points g and ag = Lag in G and apply La? to the Lie bracket [X,Y ] ofX,Y ∈ g, we see that
La?[X,Y ]|g = [La?X|g, La?Y |g] = [X,Y ]|ag, (B.8)
and hence [X,Y ] ∈ g and thus g is closed under the Lie bracket.
The set of left-invariant vector fields g with the Lie bracket [, ] : g×g→ g is the Lie algebraof the Lie group G.
B.2 Lie Algebra Cohomology
Let V be a vector space. A V -valued n-dimensional cochain Ωn on g is a skew-symmetricn-linear mapping
Ωn : g ∧ ..n. ∧ g→ V, (B.9)
ΩAn =
1
n!ΩAi1...in
ωi1 ∧ ... ∧ ωin , (B.10)
ω(i) is a basis of g and the upper index A labels the components in V . We denote theabelian group of all n-cochains as Cn(g, V ).
Thinking of V as a left ρ(g)-module, where ρ is a representation of the Lie algebra g,ρ(Xi)
AC ρ(Xj)
CB − ρ(Xj)
AC ρ(Xi)
CB = ρ([Xi, Xj])
AB. The coboundary operator δ : Cn(g, V ) →
Cn+1(g, V ) is defined as
75
(δΩn)A(X1, ..., Xn+1) =
n+1∑
i=1
(−1)i+1ρ(X1)AB(ΩB
n (X1, ..., Xi, ..., Xn+1)) (B.11)
+n+1∑
j,k=1 j<k
(−1)j+kΩAn ([Xj, Xk], X1, ..., Xj, ..., Xk, ...Xn+1).
From the definition of δ it is straight forward to see that it is indeed a nilpotent, δ2 = 0.
A n-cochain is a cocycle when δΩn = 0. The group of all n-cocycles is denoted Znρ (g, V ).
If a cocycle can be written as Ωn = δΩ′n−1 then Ωn is a coboundary. The group of all
n-coboundaries is denoted Bnρ (g, V ). The n-th Lie algebra cohomology group is defined as
Hnρ (g, V ) = Zn
ρ (g, V )/Bnρ (g, V ). (B.12)
B.3 Chevalley-Eilenberg Cohomology
Let V be R and the representation ρ be trivial. In equation (B.11)the first term is notpresent and hence on left-invariant one-forms δ and d (the exterior derivative) act the same.As there is a one-to-one mapping between n-antisymmetric maps on g and left-invariantn-forms on G a cochain Ωn ∈ Cn(g, R) can be written as a left-invariant n-form on G
Ωn(g) =1
n!Ωi1...inω
(i1)(g) ∧ ... ∧ ωin(g), (B.13)
where g ∈ G and the cobounary operator is now the exterior derivative d.
However, it should be noticed that the Chevalley-Eilenberg Lie algebra cohomology is ingeneral different to the de Rham cohomology. For instance, a differential form α on G maybe exact, α = dβ, but β may not be a left-invariant form and hence not a cochain.
B.4 Lie Algebra Cohomology and BRST Cohomology
Let ηi be anticommuting fields, known in physics a ghosts:
ηi, ηj = 0, (B.14)
where i, j = 1, ..., dim g. The BRST operator s is defined by
s = ηiρ(Xi) +1
2f iijη
iηj∂
∂ηk. (B.15)
From this definition it is straight forward to show that it is indeed a nilpotent.
In the trivial representation the BRST operator becomes
s =1
2fkijη
iηj∂
∂ηk. (B.16)
76
Here the BRST operator s acts on the ghosts as the exterior derivative d acts on the leftinvariant one-forms.
The BRST cochains are given by
ΩAn =
1
n!ΩAi1...in
ηi1ηin . (B.17)
Notice that the action of s (in a given representation ρ) is the same as δ and hence the BRSTcohomology is identical the the standard Lie algebra cohomology.
77
APPENDIX C
The Atiyah-Singer Index Theorem and someCharacteristic Classes
In this appendix we outline the Atiyah-Singer index theorem for elliptical complexes on com-pact manifolds without boundary. We follow [53] and also recommend [55].
A differential operator, D on a manifold M is a map between sections of vector bundles overthe manifold
D : Γ(M,E)→ Γ(M,F ), (C.1)
where E and F are some vector bundles. If inner products are defined on E and F , as isusually the case in physics the adjoint of D can be defined,
D† : Γ(M,F )→ Γ(M,E). (C.2)
The kernel of D and D† are defined as
kerD ≡ s ∈ Γ(M,E)|Ds = 0kerD ≡ s ∈ Γ(M,F )|D†s = 0. (C.3)
The analytical index is defined as
indD = dim kerD − dim kerD†. (C.4)
The Atiyah-Singer index theorem states that the analytical index is a topological invariantand can be expressed in terms of integrals over characteristic classes. Thus the index of anoperator completely determined by the topology of the manifold.
78
C.1 Elliptic Complexes
Consider a sequence of elliptic operators,
· · · - Γ(M,Ei−1)Di−1
- Γ(M,Ei)Di- Γ(M,Ei+1)
Di+1- · · · (C.5)
Here Ei is a sequence of vector bundles over M . The sequence (Ei,Di) is called an ellipticcomplex if Di is nilpotent.
C.2 Statement of The Atiyah-Singer Index Theorem
Let (E,D) be an elliptic complex over an m-dimensional compact manifold without bound-ary. The topological index of this complex is given by
ind(E,D) = (−1)m(m+1)/2∫
Mch (⊕r(−1)rEr)
Td(TMC)
e(TM)
∣∣∣∣∣vol
(C.6)
The Index theorem states that the analytical and topological index are the same.
If we have a two term complex then index theorem simplifies slightly. Let
Γ(M,E)D- Γ(M,F ) (C.7)
be a two-term elliptic complex. The index of D is given by
indD = (−1)m(m+1)/2∫
M(chE − chF )
Td(TMC)
e(TM)
∣∣∣∣∣vol
. (C.8)
For the exact definitions of the various characteristic classes stated in (C.6) and (C.8) see[53]
C.3 Some Characteristic Classes
In applying the Atiyah-Singer index theorem to the spin, Rarita-Schwinger complexes weencountered the A-genus, the total Chern character and the Hirzebruch L-polynomial. Herewe present some basic facts about these topological objects.
Consider a principle fibre bundle P (M,G) over an even dimensional compact manifold Mof dimension 2n without boundary. The group G is taken to be a Lie group. Let E be theassociated vector bundle with dimE = k. Let the curvature be denoted by F .The total Chern character is defined by
ch(F ) ≡ Tr exp(iF
2π
)=∑
j=1
1
j!Tr(iF
2π
)j. (C.9)
The jth Chern character chj(F ) is
chj(F ) =1
j!Tr(iF
2π
)j. (C.10)
79
The A-genus is defined as
A(R) =n∏
j=1
xj/2
sinh(xj/2). (C.11)
Where xj is defined via the Pontrjagin class
p(F ) = det(I +F
2π) =
[k/2]∑
i=1
(1 + x2i ) = 1 + p1(F ) + p2(F ) + · · · , (C.12)
Here
[k/2] =
k/2 if k is even;(k − 1)/2 if k is odd.
(C.13)
The xj are the eigenvalues of F/2π once it has been diagonalised.
The A(R)-genus and the Chern character ch(F ) can be expanded in term of traces of cur-vatures as
A(R) = 1 +1
(4π)2
1
12TrR2 +
1
(4π)4
[1
288(TrR2)2 +
1
360TrR4
]
+1
(4π)6
[1
10368(TrR2)3 +
1
4320TrR2R4 +
1
5670TrR6
]+ · · · (C.14)
ch(F ) = k +i
2πTrF +
1
2
(i
2π
)2
+ · · · . (C.15)
The Hirzebruch L-polynomial is defined as
L(M) =n∏
j=1
xj/2
tanh(xj/2). (C.16)
This can also be expanded in terms of traces
L(M) = 1− 1
(2π)2
1
6TrR2 +
1
(2π)4
[1
72(TrR2)2 − 7
180TrR4
]
+1
(2π)6
[− 1
1296(TrR2)3 +
7
1080TrR2TrR4 − 31
2835TrR6
]+ · · · . (C.17)
80
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