Anomalies in Quantum Field Theory

Marco Serone

SISSA, via Bonomea 265, I-34136 Trieste, Italy

Lectures given at the “37th Heidelberg Physics Graduate Days”, 10-14 October 2016

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Contents

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Lecture I 5

1.1 Basics of Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Vierbeins and Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Lecture II 18

2.1 Supersymmetric Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Adding Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Symmetries in QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 WT Identities in QED . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Lecture III 30

3.1 The Chiral Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Consistency Conditions for Gauge and Gravitational Anomalies . . . . . . . 36

4 Lecture IV 39

4.1 The Stora–Zumino Descent Relations . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Path Integral for Gauge and Gravitational Anomalies . . . . . . . . . . . . 42

4.3 Explicit form of Gauge Anomaly . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Lecture V 48

5.1 Chiral and Gauge Anomalies from One-Loop Graphs . . . . . . . . . . . . . 48

5.2 A Relevant Example: Cancellation of Gauge Anomalies in the SM . . . . . 51

5.3 ’t Hooft Anomaly Matching and the Wess-Zumino-Witten Term . . . . . . 52

5.4 Gravitational Anomalies for Spin 3/2 and Self-Dual Tensors . . . . . . . . . 54

2

0.1 Introduction

There are different ways of regularizing a Quantum Field Theory (QFT). The best choice

of regulator is the one which keeps the maximum number of symmetries of the classical

action unbroken. Cut-off regularization, for instance, breaks gauge invariance and that’s

why we prefer to work in the somewhat more exotic Dimensional Regularization (DR),

where instead gauge invariance is manifestly unbroken. It might happen, however, that

there exists no regulator that preserves a given classical symmetry. When this happens,

we say that the symmetry is anomalous, namely the quantum theory necessarily breaks

it, independently of the choice of regulator.

Roughly speaking, anomalies can affect global or local symmetries. The latter case

is particularly important, because local symmetries are needed to decouple unphysical

fields. Indeed, anomalies in linearly realized local gauge symmetries lead to inconsistent

theories. Theories with anomalous global symmetries are instead consistent, yet the effect

of the anomaly can have important effects on the theories. Historically, the first anomaly,

discovered by Adler, Bell and Jackiw [1], was associated to the non-conservation of the

axial current in QCD. Among other things, the axial anomaly resolved a puzzle related

to the π0 → 2γ decay rate, predicted by effective Lagrangian considerations to be about

three orders of magnitude smaller than the observed one.

In these lectures we will study in some detail a particularly relevant class of anomalies,

those associated to chiral currents (so called chiral anomaly) and their related anoma-

lies in local symmetries. Particular emphasis will be given to some mathematical aspects

and to the close connection between anomalies and the so called index theorems. The

computation of anomalies will be mapped to the evaluation of the partition function of

a certain (supersymmetric) quantum mechanical model. Although this might sound un-

usual, if compared to the more standard treatments using one-loop Feynman diagrams,

and requires a bit of background material to be attached, the ending result will be very re-

warding. This computation will allow us to get the anomalies associated to gauge currents

(so called gauge anomalies) and stress-energy tensor (so called gravitational anomalies) in

any number of space-time dimensions! Moreover, it is the best way to reveal the above

mentioned connection with index theorems. Since the notion of chirality is restricted to

spaces with an even number of space-time dimensions, and the connection with index

theorems is best seen for euclidean spaces, we will consider in the following QFT on even

dimensional euclidean spaces.

The plan of the lectures is as follows. Lecture I is devoted to review minimum basic

notions of differential geometry. In lecture II we will first construct step by step the su-

3

persymmetric quantum mechanical model of interest, related to anomalies. Afterwards,

we will review how symmetries are implemented in QFT by means of identities between

correlation functions. Equipped with these notions, we finally attack anomalies in lecture

3, where the contribution to the chiral anomaly of a spinor coupled to gravity and gauge

fields is computed in any even number 2n of dimensions. In lecture 4 we turn to gauge

and gravitational anomalies and explain how in 2n dimensions these can be consistently

extracted from the form of the chiral anomaly in n+2 dimensions. Lecture 5 is devoted to

some more physical applications of anomalies: we will discuss the ’t Hooft anomaly match-

ing condition, anomalies in QCD and their matching to the low energy pion Lagrangian,

cancellation of anomalies in the Standard Model. We will also briefly mention to gravita-

tional anomalies induced by other fields and how these cancel in a certain 10-dimensional

QFT.

These lecture notes do assume that the reader has a basic knowledge of QFT. Quan-

tizations of spin 0, spin 1/2 and spin 1 fields are assumed, as well as basic notions of the

path integral formulation of QFT (including Berezin integration for fermions), Feynman

rules, basic knowledge of renormalization and the notion of functional generators of Green

functions. Although some basics of differential geometry will be given in Lecture I, some

familiarity with fundamental notions such as metric, Levi-Civita connection, curvature

tensors and general relativity will be assumed.

Some exercises are given during the lectures (among rows and in bold face in the text).

They are simple, yet useful, and the reader should be able to solve them without any

problem.

I have essentially followed the review paper [2], which in turn is mostly based on the

seminal paper [3] by Alvarez-Gaume and Witten. Some other parts are taken from the

QFT course given by the author at SISSA [4]. Basic, far from being exhaustive, references

to some of the main original papers are given in the text.

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Chapter 1

Lecture I

1.1 Basics of Differential Geometry

We remind here some very basic knowledge of differential geometry that will be useful

in understanding anomalies in QFT. We warn the reader that in no way this should be

considered a comprehensive review. We will mention the basic notions needed for our

purposes, without giving precise mathematical definitions, presented in a “physical” way

with no details. There are several excellent introductory books in differential geometry.

See e.g. ref.[5] for a perspective close to physics.

Given a manifold M of dimension d, a tensor of type (p, q) is given by

T = Tµ1...µp

ν1...νq∂

∂xµ1. . .

∂

∂xµpdxν1 . . . dxνq , (1.1.1)

where µi, νi = 1, . . . , d. A manifold is called Riemannian if it admits a positive definite

symmetric (0,2) tensor. We call such tensor the metric g = gµνdxµdxν . On a general

non-trivial manifold, a vector acting on a (p, q) tensor does not give rise to a well-defined

tensor, e.g. if Vν are the components of a (0, 1) tensor, ∂µVν are not the components of a

(0, 2) tensor. This problem is solved by introducing the notion of connection and covariant

derivative. If M has no torsion (as will be assumed from now on), then ∂µVν → ∇µVν ,

where

∇µVν = ∂µVν − ΓρµνVρ , Γρµν =1

2gρσ(∂µgνσ + ∂νgµσ − ∂σgµν) . (1.1.2)

The (not tensor) field Γρµν is called the Levi-Civita connection. In terms of it, we can

construct the fundamental curvature (or Riemann) tensor

Rµνρσ = ∂ρΓµνσ − ∂σΓ

µνρ + ΓασνΓ

µαρ − ΓαρνΓ

µασ . (1.1.3)

5

Other relevant tensors constructed from the Riemann tensor are the Ricci tensor Rνσ =

Rµνµσ and the scalar curvature R = gνσRνσ.

Particular interesting for our purposes will be the completely antisymmetric tensors

of type (0, p), also denoted differential forms. A differential form ωp of type (0, p) is

commonly denoted a p-form and is represented as

ωp =1

p!ωµ1...µpdx

µ1 ∧ . . . ∧ dxµp . (1.1.4)

The wedge product ∧ in eq.(1.1.4) represents the completely antisymmetric product of the

basic 1-forms dxµi :

dxµ1 ∧ . . . ∧ dxµp =∑

P∈Sp

sign (P )dxµP (1) . . . dxµP (p) , (1.1.5)

where P denotes a permutation of the indices and Sp is the group of permutations of p

objects. For each point in a manifold of dimension m, the dimension of the vector space

spanned by ωp equals (d

p

)=

d!

(d− p)!p!. (1.1.6)

The total vector space spanned by all p-forms, p = 0, . . . , d, equals

d∑

p=0

(d

p

)= 2d . (1.1.7)

Given a p-form ωp and a q-form χq, their wedge product ωp ∧ χq is a (p + q) form. From

eq.(1.1.5) one has

ωp ∧ χq = (−)pqχq ∧ ωp(ωp ∧ χq) ∧ ηr = ωp ∧ (χq ∧ ηr) .

(1.1.8)

One has clearly ωp ∧ χq = 0 if p+ q > d and ωp ∧ ωp = 0 if p is odd. Given a p-form ωp,

we define the exterior derivative dp as the (p+ 1)-form given by

dpωp =1

p!∂µp+1ωµ1...µpdx

µp+1 ∧ dxµ1 ∧ . . . ∧ dxµp . (1.1.9)

Notice that in eq.(1.1.9) we have the ordinary, rather than the covariant derivative, because

upon antisymmetrization their action is the same:

∇[µp+1ωµ1...µp] = ∂[µp+1

ωµ1...µp] − Γα[µp+1µ1ωα|µ2...µp] + . . . = ∂[µp+1

ωµ1...µp] , (1.1.10)

the Levi-Civita connection being symmetric in its two lower indices. Given eq.(1.1.8), we

have

dp+q(ωp ∧ χq) = dpωp ∧ χq + (−)pωp ∧ dqχq . (1.1.11)

6

From the definition (1.1.9) it is clear that dp+1dp = 0 on any p-form.

A p-form is called closed if dpωp = 0 and exact if ωp = dp−1χp−1 for some (p − 1)-

form χp−1. Clearly, if ωp = dp−1χp−1, then dpωp = 0 (exact → closed) but not viceversa.

Correspondingly, the image of dp−1, the space of p-forms such that ωp = dp−1χp−1, is

included in the kernel of dp, the space of p-forms such that dpωp = 0: Im dp−1 ⊂ Ker dp.

The coset space Ker dp/Im dp−1 is called the de Rham cohomology group on the manifold

M. By Stokes theorem, the integral of an exact p-form on a p-dimensional cycle Cp of Mwith no boundaries necessarily vanish:

∫

Cp

ωp =

∫

Cp

dpχp−1 =

∫

∂Cp

χp−1 = 0 . (1.1.12)

Since the action of dp on a p-form is clear from the context, we will drop from now on the

subscript in the exterior derivative d.

A U(1) gauge theory is simply written in terms of forms. The gauge connection is a

one-form A = Aµdxµ, with

F = dA =1

2(∂µAν − ∂νAµ)dx

µ ∧ dxν =1

2Fµνdx

µ ∧ dxν (1.1.13)

being the field strength two-form. In order to generalize this geometric picture of gauge

theories to the non-abelian case, we have to introduce the notion of Lie-valued differential

forms. We then define

A = AaT a, Aa = Aaµdxµ , (1.1.14)

where T a are the generators of the corresponding Lie algebra. We take them to be anti-

hermitian: (T a)† = −T a. This unconventional (in physics) choice allows us to get rid of

some unwanted factors of i’s in the formulas that will follow. Given two Lie-valued form

ωp and χq, we define their commutator as

[ωp, χq] ≡ ωap ∧ χbq[T a, T b] = ωap ∧ χbqT aT b − (−)pqχbq ∧ ωap T bT a . (1.1.15)

Notice that the square of a Lie valued p-form does not necessarily vanish when p is odd,

like ordinary p-forms. One has now

ωp ∧ ωp =1

2[T a, T b]ωaµ1...µpω

bµp+1...µ2pdx

µ1 ∧ . . . ∧ dxµ2p =1

2[ωp, ωp] . (1.1.16)

The action of the exterior derivative d on a Lie-valued p-form does not give rise to a well-

defined covariant p+1-form. Under a gauge transformation g, the connection A transforms

as

A→ g−1Ag + g−1dg , (1.1.17)

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and clearly F = dA does not transform properly. The correct exterior derivative in this

case is the covariant exterior derivative

F = DA = dA+A ∧A . (1.1.18)

Exercise 1: i) Show that F → g−1Fg under a gauge transformation. ii) Show

that dF = −[A,F ], so that DF = dF + [A,F ]=0 (Bianchi identity).

Mathematically speaking, a gauge field is a connection on a fibre bundle F , namely

a vector space that can be associated to each point of the manifold (called also the base

space in this context). The total space given by the base space and the fibre can always

be locally written as a direct product M × F . Globally, only trivial bundles admit such

decomposition, very much like the choice of local coordinates in a non-trivial manifold. In

order to not go too much into mathematics (and too less in physics afterwards ...) I will

skip altogether the reformulation of non-abelian gauge theories in terms of fibre bundles.

1.2 Vierbeins and Spinors

The transformation properties under change of coordinates of (p, q) tensors is easily ob-

tained by looking at eq.(1.1.1). For instance, under a diffeomorphism xµ → yµ(x), a vector

transforms as

V µ(x) → ∂xµ

∂yνV ν(y) . (1.2.1)

Contrary to tensors, spinors do not have a well defined transformation properties under

diffeomorphisms. We have instead to equivalently consider local Lorentz transformations,

since the Lorentz properties of spinors are determined. At each point of the manifold, we

can write the metric as

gµν(x) = eaµ(x)ebν(x)δab , (1.2.2)

where a, b = 1, . . . , d.1 The fields eaµ(x) are called vierbeins. They are not uniquely defined

by eq.(1.2.2). If we perform the local Lorentz transformation

eaµ(x) → (Λ−1)ab(x)ebµ(x) , (1.2.3)

1In a manifold with Lorentzian signature, δab would be replaced by the Minkowski metric ηab.

8

the metric is left invariant. It is important to distinguish the index µ form the index a.

The first transforms under diffeomorphisms and is invariant under local Lorentz transfor-

mations, viceversa for the second: invariant under diffeomorphisms and transforms under

local Lorentz transformations. They are sometimes denoted curved and flat indices, re-

spectively. We will denote a curved index with a greek letter, and a flat index with a

roman letter. From eq.(1.2.2) we have g ≡ det gµν = (det eaµ)2. Since the metric is positive

definite, this implies that eaµ is a non-singular m×m matrix at any point. We denote its

inverse by eµa : eµaebµ = δab, eµaeaν = δµν . It is easy to see that eµa is obtained from eaµ by using

the inverse metric gµν : eµa = gµνeaν . Curved indices are lowered and raised by the metric

gµν and its inverse gµν . In euclidean space the position of flat indices does not matter,

being raised and lowered by the identity matrix.2 The vierbeins eµa can be seen as a set of

orthonormal (1,0) vectors in the tangent space. As such, we can decompose any vector V µ

in their basis: V µ = eµaV a. Similarly for the (0,1) vectors in cotangent space: Vµ = eaµVa.

So any tensor with curved indices can be converted to a tensor with flat indices. In other

words, we can project the (p,q) tensor (1.1.1) in the basis

T = Ta1...ap

b1...bqEa1 . . . Eape

b1 . . . ebq , (1.2.4)

where

Ea = eµa∂

∂xµ, ea = eaµdx

µ . (1.2.5)

We can then trade, as already mentioned, diffeomorphisms for local Lorentz transforma-

tions. The vierbeins eaµ can be seen as the components of a Lie-valued one-form with

group G = SO(d), where the latter is generically the tangent space group of a manifold

of dimension d. The vierbein components eaµ transform in the fundamental representation

of SO(d). As we have seen, Lie-valued one-forms require the introduction of a connection

in order to give a well-defined notion of (covariant) exterior derivative. We denote by

ωabµ the components of the connection. They transform in the adjoint representation of

SO(d), which can be represented as an antisymmetric matrix ωabµ = −ωbaµ , a, b = 1, . . . d.

It is called the spin connection and is the analogue of the Levi-Civita connection Γρµν for

diffeomorphisms. The Levi-Civita connection is a function of the metric, and thus we ex-

pect that ωabµ cannot be an independent field. If this were the case, the degrees of freedom

would not match. Indeed, the spin connection is determined as a function of the vierbeins

by the condition

Dea = dea + ωab ∧ eb = 0 . (1.2.6)

2Do not forget however that this is no longer true in Lorentzian signature, where the matrix in question

is ηab and its inverse.

9

The above expression is called the torsion two-form and demanding it to vanish goes under

the name of torsion-free condition. We will not need the explicit expression of ωab = ωab(e)

obtained by solving eq.(1.2.6). Derivative of tensors with flat indices are then promoted

to covariant derivatives:

DµTa = ∂µT

a + ωaµ,bTb . (1.2.7)

If we denote by ∇µ the covariant derivative ∇µTν = ∂µTν − ΓρµνTρ, the property below

follows.

Exercise 2: Using the property that ∇µeaν + ωaµ,be

bν = 0, prove that

∇µTν = eaνDµTa . (1.2.8)

In complete analogy to eq.(1.1.18), we can define the curvature two-form

Rab = Dωab = dωab + ωac ∧ ωcb . (1.2.9)

By using eq.(1.2.8) and the properties [∇µ,∇ν ]Vρ = RρµνσV σ, [Dµ,Dν ]V

a = Raµν,bTb, we

have

Raµν,b = eaσeρbR

σµνρ . (1.2.10)

A differential form notation can also be introduced to describe the Levi-Civita connection

(1.1.2) and the Riemann tensor (1.1.3). One defines the matrix of one-forms Γµν = Γµρνdxρ

and the curvature two-form

Rµν = dΓµν + Γµσ ∧ Γσν =1

2Rµνρσdx

ρ ∧ dxσ . (1.2.11)

We can finally come back to spinors. First of all, let us recall that the definition of gamma

matrices in d euclidean dimensions is a straightforward generalization of the usual one

known in four dimensions:

γa, γb = 2δab . (1.2.12)

Notice that we have written γa and not γµ because in a curved space γµ(x) = eµa(x)γa 6=γa. The algebra defined by eq.(1.2.12) is called Clifford algebra. For d is even, the

case relevant in these lectures, the lowest dimensional representation in d dimensions

require 2d/2 × 2d/2 matrices. Hence a (Dirac) spinor in d dimensions has 2d/2 components.

A linearly independent basis of matrices is given by antisymmetric products of gamma

matrices:

I , γa , γa1a2 , . . . γ1...d , (1.2.13)

10

where

γa1...ap =1

p!

(γa1 . . . γap ± (p− 1)! perms

)(1.2.14)

is completely antisymmetrized in the indices a1 . . . ap. The total number of elements of

the base isd∑

p=0

(d

p

)= 2d = 2d/2 × 2d/2 . (1.2.15)

which is then complete. Notice the close analogy between gamma matrices and differential

forms, analogy that we will use in a crucial way in the next lectures. In d euclidean

dimensions, the gamma matrices can be taken all hermitian: (γa)† = γa. We can also

define a matrix γd+1, commuting with all other γa’s:

γd+1 = ind∏

a=1

γa . (1.2.16)

The factor in ensures that γ2d+1 = 1 for any d. Needless to say, γd+1 is te generalization

of the usual chiral gamma matrix γ5 in 4d QFT. A spinor ψ can be decomposed in two

irreducible components: ψ = ψ++ψ−, with γd+1ψ± = ±ψ±. Weyl spinors in d dimensions

have then 2d/2−1 components. Using eq.(1.2.12), the matrices

Jab =1

4[γa, γb] =

1

2γab (1.2.17)

are shown to be (anti-hermitian) generators of SO(d). Under a global Lorentz transfor-

mation, a spinor ψ transforms as follows:

ψ → e12λabJ

ab

ψ , (1.2.18)

where λab are the d(d − 1)/2 parameters of the transformation. The covariant derivative

of a fermion in curved space is obtained by replacing

∂µ → Dµ = ∂µ +1

2ωabµ Jab . (1.2.19)

The Lagrangian density of a massive Dirac fermion coupled to gravity is finally given by

Lψ = e ψ(ieµaγ

aDµ −m)ψ , (1.2.20)

where e ≡ det eaµ.

11

1.3 Characteristic Classes

Characteristic classes is the name for certain cohomology classes that measure the non-

triviality of a manifold or of its gauge bundle. They are essentially given by integrals of

gauge invariant combinations of curvature two-forms, like (omitting flat SO(d) indices)

tr R∧ . . .∧R or tr F ∧ . . .∧F . For simplicity of notation we will omit from now the wedge

product among forms. Let us consider a G-valued two form field-strength F and define

the gauge-invariant 2n-form Q2n(F ) ≡ trFn. Using the results of Exercise 1, it is easy to

show that Q2n(F ) is closed:

dQ2n(F ) = −n−1∑

i=0

trF i[A,F ]Fn−i−1 = −tr [A,Fn] = 0 . (1.3.1)

Given two field strengths F an F ′, the difference Q2n(F ) − Q2n(F′) = dχ2n−1 for some

2n−1-form χ2n−1. The integral of Q2n(F ) over a 2n-dimensional sub-manifold C2n without

boundary of the manifold M (or over the entire M) does not depend on F :∫

C2n

Q2n(F ) =

∫

C2n

Q2n(F′) , (1.3.2)

and defines a cohomology class, called the characteristic class of the polynomial Q2n(F ).

Let us prove thatQ2n(F )−Q2n(F′) is an exact form. Although the proof is a bit involved, it

will turn out to be very useful when we will discuss the so called Wess-Zumino consistency

conditions for anomalies. Let A and A′ be one-form connections associated to F and F ′.

Let us then define an interpolating connection At defined as

At = A+ t(A′ −A) , t ∈ [0, 1] . (1.3.3)

We clearly have A0 = A, A1 = A′. Let us also denote by θ = A′ −A the difference of the

two connections. We have (recall eq.(1.1.15))

Ft = dAt +A2t = dA+ tdθ + (A+ tθ)2 = F + t(dθ + [A, θ]) + t2θ2 . (1.3.4)

We can also write

Q2n(F′)−Q2(F ) =

∫ 1

0dt

d

dtQ2n(Ft) = n

∫ 1

0dt tr

(dFtdtFn−1t

)

= n

∫ 1

0dt(tr DθFn−1

t + 2t tr θ2Fn−1t

).

(1.3.5)

On the other hand, we have

dtr θFn−1t = tr dθFn−1

t −n−2∑

i=0

tr θF it dFtFn−i−2t (1.3.6)

= tr DθFn−1t − tr [A, θ]Fn−1

t −n−2∑

i=0

(tr θF itDFtF

n−i−2t − tr θF it [A,Ft]F

n−i−2t

).

12

The second and fourth terms in eq.(1.3.6) combine into a total commutator that vanishes

inside a trace: tr [A, θFn−1t ] = 0. The derivative DFt equals

DFt = dFt + [A,Ft] = dFt + [At −At +A,Ft] = DtFt − t[θ, Ft] = −t[θ, Ft] , (1.3.7)

given that DtFt = 0. We also have

0 = tr [θ, θFn−1t ] = tr 2θ2Fn−1

t −n−2∑

i=0

tr θF it [θ, Ft]Fn−i−2t . (1.3.8)

Plugging eqs.(1.3.7) and (1.3.8) in eq.(1.3.6) gives

dtr θFn−1t = tr DθFn−1

t + 2t tr θ2Fn−1t . (1.3.9)

We have then proved that Q2n(F′)−Q2(F ) = dχ2n−1, with

χ2n−1 = n

∫ 1

0dt tr θFn−1

t . (1.3.10)

The invariant polynomials we will be interested in comes from a generating functions of

polynomials defined in any number of dimensions. We will briefly mention here the ones

featuring in anomalies.

One series of invariant polynomials are given by the Chern character defined as

ch (F ) ≡ tr eiF2π . (1.3.11)

The Taylor expansion of the exponential gives rise to the series of 2n-forms denoted by

nth Chern characters

chn(F ) =1

n!tr( iF2π

)n. (1.3.12)

It is clear that for any given d, chn(F ) = 0 for n > d/2.

It is useful to discuss in some detail what is the simplest example of non-trivial Chern

character: a monopole in a U(1) gauge theory. We know that in presence of a monopole

it is impossible to globally define a gauge connection. Indeed, given a two-sphere S2

surrounding the monopole, we should have

∫

S2

F 6= 0 . (1.3.13)

and proportional to the monopole charge. On the other hand, if F = dA applies globally,

the above integral should vanish by Stokes theorem. A monopole induces a magnetic field

of the form ~B = g~r/r3. Correspondingly, we no longer have a globally defined connection

on S2.

13

Exercise 3: Show that in radial coordinates the connection 1-forms in the

northern and southern hemispheres can be taken as

AN = −ig(1− cos θ)dφ , AS = ig(1 + cos θ)dφ . (1.3.14)

More precisely, we split S2 = ΣN + ΣS, ΣN,S being the northern and southern hemi-

spheres, and compute

∫

S2

F =

∫

ΣN

dAN +

∫

ΣS

dAS =

∮(AN −AS) , (1.3.15)

where the closed integral is around the equator at θ = π/2. From the exercise 3 we have

AN −AS = −2igdφ and hence

i

2π

∫

S2

F =1

2π

∫ 2π

0dφ(2g) = 2g . (1.3.16)

We now show that 2g must be an integer in order to have a well-defined quantum theory. If

a charged particle moves along the equator S1, its action will contain the minimal coupling

of the particle with the gauge field. In a path-integral formulation, it corresponds to a

term

e∮S1 A = e

∫DF , (1.3.17)

where we used Stokes’ theorem and D is any two-dimensional space with S1 as boundary:

∂D = S1. Since the choice of D in eq.(1.3.17) is irrelevant (provided that ∂D = S1), we

get that

e∮S1 A = e

∫ΣN

F= e

−∫ΣS

F=⇒ e

(∫DN

+∫DS

)F= e

∫S2 F = 1 . (1.3.18)

Combining with eq.(1.3.16), we get the condition

g = 2πn, n ∈ Z . (1.3.19)

We have assumed here the electric charge to be unit normalized. For a generic charge q,

eq.(1.3.19) gives the known Dirac quantization condition between electric and magnetic

charges:

qg = 2πn, n ∈ Z . (1.3.20)

Let us now come back to characteristic classes and consider those that involve the

curvature two-form Rab defined in eq.(1.2.9). For simplicity of notation we will denote it

14

just by R, omitting the flat indices a and b. It should not be confused with the scalar

curvature that will never enter in our considerations! An interesting characteristic class is

the Pontrjagin class defined by

p(R) = det(1 +

R

2π

). (1.3.21)

The expansion of p(R) in invariant polynomials is obtained by bringing the curvature

2-form Rab into a block-diagonal form (this can always be done by an appropriate local

Lorentz rotation) of the type

Rab =

0 λ1

−λ1 0

...

0 λd/2

−λd/2 0

, (1.3.22)

where λi are 2-forms. In this way it is not difficult to find the terms in the expansion

of p(R). Due to the antisymmetry of R, there are only even terms in R and hence the

invariant polynomials are 4n-forms. The first two terms with n = 1, 2 are

p1(R) = −1

2

( 1

2π

)2trR2 ,

p2(R) =1

8

( 1

2π

)4((trR2)2 − 2trR4

).

(1.3.23)

Like for the Chern class, on a manifold with dimension d all 4n-forms with 4n > d are

trivially vanishing. There is however a way in which we can define an other class from

the formally vanishing p2d(R) term. We see from eq.(1.3.21) that the 2d-form equals just

detR/2π (if the factor 1 is taken in any diagonal entry in evaluating the determinant, one

necessarily gets a form with lower degree). The determinant of an antisymmetric matrix

is always a square of a polynomial called the Pfaffian. Combining these two facts, we can

then define a d-form e(R) called the Euler class and defined as

p2d(R) = e(R)2 . (1.3.24)

The integral over the manifold of e(R) is an important invariant called the Euler charac-

teristic of the manifold.

Exercise 4: Compute p1(R) for a two-sphere S2 and show that

χ(S2) =

∫

S2

p1(R) = +2 . (1.3.25)

15

The characteristic class entering directly in the evaluation of anomalies is the so-called

roof-genus, defined as

A(R) =

d/2∏

k=1

xk/2

sinh(xk/2)= 1− 1

24p1(R) +

1

5760(7p1(R)

2 − 4p2(R)) + . . . (1.3.26)

where xk = λk/2π.

An important theorem by Atiyah and Singer, called index theorem, relates the spectral

properties of differential operators on a manifold M with its topology (fibers included).

This is a remarkable property, because at first glance the two concepts might seem unre-

lated. We will not discuss index theorems here, but states the result for a specific operator

that will feature in the following: the dirac operator

/DR = eµaγa(∂µ +

1

2ωabµ Jab +AaT aR

). (1.3.27)

In eq.(1.3.27) the subscript R refers to the representation of the fermion under the (un-

specified) gauge group G, and T aR are the generators in the corresponding representation.

The index of the Dirac operator is nothing else that the difference between the number of

zero energy eigenfunctions of positive and negative chirality. One has

n+ − n− = index /DR =

∫

MchR(F )A(R) , (1.3.28)

where

chR(F ) ≡ trReiF2π . (1.3.29)

As we will see in great detail in the next lectures, eq.(1.3.28) gives the contribution of a

Dirac fermion to the so called chiral anomaly in d dimensions. It is a remarkable compact

formula that should be properly understood. For any given dimension d, one should

expand the integrand in eq.(1.3.28) and selects the form of degree d, which is the only one

that can be meaningfully integrated over the manifold M.

Let us conclude by spending a few words on the character chR(F ). If a representation

R = R1⊗R2, one has chR(F ) = chR1(F )chR2(F ). For example, for G = SU(N), we have

fund.⊗ fund. = adj.⊕ 1. Correspondingly,

chAdj.(F ) = ch(F )ch(−F )− 1 , (1.3.30)

where the Chern character without subscript refers to the fundamental representation and

we used the fact that chfund.(F ) = ch(−F ). Expanding eq.(1.3.30) we have

N2 − 1− 1

2trAdj.F

2 + . . . = (N − 1

2trF 2 + . . .)2 − 1 , (1.3.31)

16

from which we deduce the known result

trAdj.tatb = 2N tr tatb . (1.3.32)

17

Chapter 2

Lecture II

2.1 Supersymmetric Quantum Mechanics

Let us come back to physics. For reasons that will be clear in the next lecture, the most

elegant computation of anomalies, that makes clear its connection with index theorems

and eq.(1.3.28), is essentially mapped to a computation in a quantum mechanical model

whose Hamiltonian equals the square of the Dirac operator (1.3.27):

H = (i/DR)2 . (2.1.1)

Let us start by considering the simplest situation of flat space Rd with vanishing gauge

potential. The hamiltonian (2.1.1) reduces to H = (i/∂)2 = −∂2µ = −∇ = p2, namely twice

the hamiltonian of a free particle moving in Rd with unit mass. Despite the hamiltonian

is the correct one, there is no obvious way to see p2 as coming from /p2. The “spinor”

structure is simply absent. This problem is fixed by adding fermions. The resulting quan-

tum mechanical model turns out to enjoy a particular symmetry, called supersymmetry

(SUSY). We will not assume any prior knowledge of SUSY, but keep in mind that SUSY

and its application is a huge topic, that would deserve entire courses on its own to be

properly explained. We will here focus on the bare minimum notions required for our

purposes. En passant, quantum mechanics is the easiest system where to learn the basics,

since the algebra is reduced to the minimum.

SUSY is essentially a symmetry relating a boson to a fermion and viceversa. Then,

contrary to standard symmetries, the charge associated to this symmetry is fermionic,

rather than bosonic. Coming back to the free theory of a particle on Rd, let us add

18

fermionic variables ψµ to the ordinary bosonic coordinates xµ and write the lagrangian1

L =1

2xµxµ +

i

2ψµψµ . (2.1.2)

The first term is the usual free kinetic term for a particle, the second is a somewhat

more unusual “free kinetic term” for fermion coordinates.2 Notice that ψµ are Grassmann

variables in one dimension but transform as vectors on Rd. Formally speaking, we no-

tice that the system described by the lagrangian (2.1.2) is invariant under the following

(supersymmetric) transformation:

δxµ = iǫψµ ,

δψµ = −ǫxµ ,(2.1.3)

where ǫ is a constant Grassmann variable, ǫ2 = 0, anticommuting with ψµ. It is easy to

check that. We have

δL = xµiǫψµ +i

2(−ǫxµ)ψµ + i

2ψµ(−ǫxµ) = i

2ǫd

dt(xµψµ) . (2.1.4)

Since the Lagrangian transforms as a total derivative the action S =∫dtL is invariant

and hence eq.(2.1.3) is a good symmetry. Let us define canonical momenta and impose

quantization. We have

pµ =dL

dxµ= xµ ,

πµ =dL

dψµ=i

2ψµ .

(2.1.5)

In the bosonic sector we can straightforwardly proceed with the usual replacement of

Poisson brackets with quantum commutators, i.e. [xµ, xν ] = [pµ, pν ] = 0, [xµ, pν ] = iδµν .

In the fermion sector one has to be more careful because the naive anti-commutators

ψµ, ψνP = πµ, πνP = 0 , ψµ, πνP = iδµν (2.1.6)

are clearly incompatible with the actual form of πµ in eq.(2.1.5). This problem is solved

by looking at the definition of πµ in eq.(2.1.3) as a constraint

χµ ≡ πµ − i

2ψµ = 0 . (2.1.7)

The quantization of theories with constrains is known (see e.g. section 7.6 of ref.[6] for an

excellent introduction). If the matrix of the Poisson brackets among the constraints χµ

1We are in flat euclidean space, so upper and lower vector indices are equivalent.2Notice that we are not discussing a QFT here, so ψµ do not represent fermion particles. The interpre-

tation of ψµ will be given shortly.

19

is non-singular (computed using the brackets (2.1.4)), we say that the constraints are of

second class. This is our case, since χµ, χνP ≡ cµν = δµν . A consistent quantization

in presence of second class constraints is obtained by replacing the Poison bracket with

the so called Dirac bracket. More in general, for two anti-commuting operators A and B

subject to a series of second class constraints of the form χM = 0, we have

A,BD ≡ A,BP − A,χMP c−1MNχN , BP , (2.1.8)

where c−1 is the inverse of the matrix of brackets cMN = χM , χN. A similar formula

applies for bosonic fields, with commutators replacing anti-commutators. Coming back to

our case, if we take A = χρ or B = χρ in eq.(2.1.8) we get a vanishing result, consistently

with the constraints χµ = 0. The consistent anti-commutators for fermions are then

ψµ, ψµD = δµν , πµ, πµD = −1

4δµν , ψµ, πµD =

i

2δµν . (2.1.9)

The Hamiltonian of the system is given by (pay attention to the order of the fermion

fields)

H = pµxµ + πµψ

µ − L =1

2p2µ = −1

2∇ , (2.1.10)

namely it is just the standard free hamiltonian in absence of fermions! The fermion

generator of SUSY is given by

Q = −ψµxµ = −ψµpµ . (2.1.11)

We can easily check that it leads to the correct transformation properties:

δxµ = [ǫQ, xµ] = −ǫψν [pν , xµ] = iǫψµ ,

δψµ = [ǫQ,ψµ] = −[ǫψνpν , ψµ] = −ǫpνψν , ψµ = −ǫpµ = −ǫxµ .(2.1.12)

The fermion Dirac brackets in eq.(2.1.9) generate the same Clifford algebra of gamma

matrices in d dimensions. The action in the Hilbert space of ψµ is then that of a gamma

matrix:

ψµ → 1√2γµ . (2.1.13)

The anti-commutator of two supercharges equal

Q,Q = ψµpµ, ψνpν = p2µ = 2H =⇒ Q2 = H . (2.1.14)

Plugging eq.(2.1.13) in the supercharge Q in eq.(2.1.11) gives

Q =i√2γµ∂µ =

1√2i/∂ , (2.1.15)

20

namely the supercharge acts in the Hilbert space as the Dirac operator. Thanks to the

fermion operators, we can now reinterpret the hamiltonian (2.1.10) as the square of the

Dirac operator. In quantum mechanics, the existence of SUSY is the statement about

the possibility of classifying the spectrum of the system using a fermion number operator.

If |B〉 is a bosonic state, then the state ψµ|B〉 is fermionic. Correspondingly, given a

bosonic state |B〉, Q|B〉 is a fermionic one. Viceversa, if |F 〉 is a fermionic state, then

Q|F 〉 is a bosonic one. States are grouped in multiplets that rotate as spinors do in d

dimensions upon the action of the operator ψµ. For d even, we might then split the

spectrum into “boson” and “fermion” states, according to the action of the matrix γd+1

defined in eq.(1.2.16). States with γd+1 = 1 and γd+1 = −1 will be denoted bosons and

fermions, respectively. The chiral matrix γd+1 is then equivalent to a fermion number

operator (−)F .

Let us finally compute the SUSY transformation of the generator Q itself:

δQ = −δψµxµ − ψµδxµ = ǫxµxµ − ψµ(iǫψµ) = 2ǫL . (2.1.16)

We then notice two important properties of Q: its square is proportional to the Hamilto-

nian and its SUSY variation is proportional to the Lagrangian. In turn, the variation of

the Lagrangian is proportional to the time derivative of Q, see eq.(2.1.4).

Let us now consider a particle moving on a general manifold M. The SUSY transfor-

mations (2.1.3) do not depend on the metric of M and are unchanged. Correspondingly

Q should be given by the straightforward generalization of eq.(2.1.11):

Q = −gµν(x)ψµxν . (2.1.17)

The simplest way to get the curved space generalization of the Lagrangian (2.1.2) is to

use eq.(2.1.4).

Exercise 5: Compute the curved space lagrangian Lg by using eqs.(2.1.16) and

(2.1.17) and show that

Lg =1

2gµν(x)x

µxν +i

2gµν(x)ψ

µ(ψν + Γνρσ(x)ψ

ρxσ), (2.1.18)

where Γνρσ is the Levi-Civita connection (1.1.2).

21

It is important for our purposes to rewrite eq.(2.1.18) in terms of vierbeins. We have

ψµ = eµaψa, so thatd

dtψν =

d

dt(eνaψ

a) = (∂µeνa)x

µψa + eνaψa , (2.1.19)

where from now on we omit the dependence of vierbeins, metric, etc. on the coordinates

xµ. The terms in the Lagrangian (2.1.18) involving the fermions can then be rewritten as

gµνψµ(ψν + Γνρσ(x)ψ

ρxσ)= ψaeaν

((∂µe

νb )x

µψb + eνb ψb + Γνρµe

ρbψ

bxµ)

= ψaψa + ψaψ

bxµ(eaν∂µeνb + eaρΓ

ρνµe

νb )

= ψaψa + ψaψ

bxµeνb (−∂µeaν + Γνρµeρb )

= ψaψa +

1

2[ψa, ψb]x

µωabµ ,

(2.1.20)

where in the last step we used the vielbein postulate ∇µeaν + ωaµ,be

bν = 0 (see exercise 2).

The Lagrangian (2.1.18) becomes

Lg =1

2gµν(x)x

µxν +i

2ψaψ

a +i

4[ψa, ψb]ω

abµ x

µ . (2.1.21)

The momentum conjugate to xµ is now modified:

pµ = gµν xν +

i

4[ψa, ψb]ω

abµ . (2.1.22)

Using eq.(2.1.22) and replacing pµ → −i∂µ, ψa → γa/√2 in the expression for Q in

eq.(2.1.17), we get the natural curved space generalization of eq.(2.1.15):

Q =1√2ieµaγ

a(∂µ +

1

8ωabµ [γa, γb]

)=

1√2i/D . (2.1.23)

2.2 Adding Gauge Fields

We have so far succeeded in reproducing the curved space covariant derivative (1.3.27),

but we still miss the gauge field terms. In order to reproduce these terms, we can add new

fermionic degrees of freedom c∗A and cA, where A runs from one up to the dimension of the

representation R associated to eq.(1.3.27). The fields cA and c∗A transform then under the

representation R and its complex conjugate R of the gauge group G, respectively. The

Lagrangian associated to these fields is

LA =i

2(c∗Ac

A − c∗AcA) + ic∗AA

AµBx

µcB +1

2c∗Ac

BψµψνFAµνB , (2.2.1)

where AAµ B = Aαµ(Tα)AB, F

ABµν = Fαµν(T

α)AB are the connection and field strength asso-

ciated to the group G, with generators Tα. In order to simplify the notation, it is quite

22

convenient to define the one-form A ≡ Aµψµ and two-form F = Fµνψ

µψν and suppress

the gauge indices. In this more compact notation the Lagrangian (2.2.1) reads

LA =i

2(c∗c− c∗c) + ic∗Aµc x

µ + c∗Fc . (2.2.2)

The Lagrangian (2.2.2) is invariant under SUSY transformations of the form

δc = −iǫAc , δc∗ = −iǫc∗A , (2.2.3)

together with the transformations (2.1.3) that are left unchanged.

Exercise 6: Show that LA is invariant under the transformations (2.1.3) and

(2.2.3). Hint: use a compact notation and recall that DF = dF + [A,F ] = 0.

The canonical momenta of the fields c∗ and c read

πc =i

2c∗ , πc∗ =

i

2c . (2.2.4)

Like the fermions ψµ before, eq.(2.2.4) should be interpreted as a set of constraints among

the fields c, c∗ and their momenta πc and πc∗: χc = πc − ic∗/2 = 0, χc∗ = πc∗ − ic/2 =

0. Using eq.(2.1.8), it is straightforward to find the consistent Dirac (anti)commutation

relations between the fields cB and c∗A:

c⋆A, c

B= δBA . (2.2.5)

Considering c∗ and c as creation and annihilation operators, respectively, the Hilbert space

of the system consists of the vacuum |0〉, 1-particle states c⋆A|0〉, 2-particle states c⋆Ac⋆B |0〉,

etc. Among all these states, only the 1-particle states correspond to the representation Rof the gauge group G, the vacuum being a singlet of G and multiparticle states leading to

tensor products of the representation R. On such particle states, the operator c∗A(tα)ABc

B

acts effectively as (tα)AB . The supercharge is given by eq.(2.1.17), but the momentum

conjugate to xµ gets another term, coming from LA:

pµ = gµν xν +

i

4[ψa, ψb]ω

abµ + ic∗Aµc . (2.2.6)

It is straightforward to check that the transformations (2.2.3) are reproduced by acting

with Q on the fields c∗ and c.

23

Summarizing, the total Lagrangian is given by L = Lg + LA, with

L =1

2gµν x

µxν +i

2ψaψ

a +i

4

[ψa, ψb

]ωabµ x

µ

+i

2(c⋆Ac

A − c⋆AcA) + ic∗AA

AµB x

µcB +1

2c⋆Ac

BψaψbFAab B ,

(2.2.7)

and it is invariant under a supersymmetry transformation Q that acts on the Hilbert space

as

Q =i√2eµaγ

a

[(∂µ +

1

8ωabµ[γa, γb

])δAB +Aαµ T

Aα B

]=

1√2i/D . (2.2.8)

We have finally determined a SUSY quantum mechanical model with Hamiltonian given

by eq.(2.1.1).

2.3 Symmetries in QFT

We mentioned in the introduction that an anomaly is essentially an obstruction to keep

a classical symmetry at the quantum level. Before discussing about anomalies, it is im-

portant to understand how symmetries are realized at the quantum level. We will assume

here that it is possible to regularize the theory in such a way that the original symmetry

of the theory is preserved, namely there are no anomalies! Symmetries are important in

QFT because they constrain the form of amplitudes, correlation functions, and establish

relationships among them. This can be rather easily seen within the functional formalism,

as we discuss below in generality.

Let us consider a local field theory characterized by a Lagrangian L. An infinitesimal

transformation

φ(x) 7→ φ′(x) = φ(x) + ǫ∆φ(x) (2.3.1)

parameterized by a parameter ǫ is a symmetry of the theory if, correspondingly, the action

S =∫d4xL(φ, ∂µφ) associated with L does not change. In order to be so, the variation

of the Lagrangian induced by eq. (2.3.1) has to take the form of a total derivative. This

requirement can be translated in requiring that there exists some “current” J µ such that

L(φ′, ∂µφ′) = L(φ, ∂µφ) + ǫ∂µJ µ +O(ǫ2). (2.3.2)

Note that J µ can also be a Lorentz tensor. A direct calculation of the variation of the

Lagrangian induced by the transformation shows that

∂µJ µ =δLδφ

∆φ+δLδ∂µφ

∂µ∆φ. (2.3.3)

24

By rewriting the second term on the r.h.s. as ∂µ[∆φδL/δ∂µφ] −∆φ∂µδL/δ∂µφ one finds

that

∂µjµ = −∆φ

[δLδφ

− ∂µδLδ∂µφ

], (2.3.4)

where we introduced the current

jµ(x) ≡ δLδ∂µφ

∆φ− J µ. (2.3.5)

The r.h.s. of eq. (2.3.4) vanishes on the field configuration which minimizes the action (i.e.,

without quantum fluctuations) and which satisfies Euler-Lagrange equation: accordingly

jµ is nothing but the classical Noether’s current associated with the symmetry (2.3.1)

and jµ is the corresponding conserved current from which one can derive the conserved

charge. In the case of QFT, where quantum fluctuations are inevitably present, relations

such as eq. (2.3.4) have to be properly interpreted and acquire a definite meaning only

when inserted in correlation functions. For the purpose of understanding the consequence

of the symmetry (2.3.1) on the generating functions, it is convenient to consider how

the Lagrangian density changes when the parameter ǫ in eq. (2.3.1) is assumed to be

space-dependent (and, in general, the local transformation is no longer a symmetry of the

theory): by direct calculation one can verify that

L(φ′, ∂µφ′) = L(φ, ∂µφ) + ǫ(x)∂µJ µ +δLδ∂µφ

∆φ∂µǫ(x) +O(ǫ2),

∫= L(φ, ∂µφ) + jµ(x)∂µǫ(x) +O(ǫ2),

(2.3.6)

where∫= indicates that the equality is valid after integration over the space, i.e., at the level

of the action (under the assumption of vanishing boundary terms). Though the discussion

so far is completely general, we shall focus below on the case in which the symmetry is

linearly realized, meaning that ∆φ is an arbitrary linear function of φ. This is the case,

for example of global U(1) or other internal symmetries for multiplets such as

φl(x) 7→ φ′l(x) = φl(x) + iǫα(Tα)ml φm(x), (2.3.7)

where Tα are the generators of the symmetry in a suitable representation.

In order to work out the consequences of the symmetry, consider the generating func-

tion Z[J ] defined as usual by

〈0|0〉J ≡ Z[J ] =

∫Dφ eiS(φ)+i

∫d4xJ(x)φ(x) , (2.3.8)

25

Under the transformation (2.3.1) with a space-dependent parameter ǫ = ǫ(x) we have

Z[J ] =

∫Dφ′ eiS(φ′)+i

∫d4xJ(x)φ′(x) =

∫Dφ′ eiS(φ)+i

∫d4x jµ∂µǫ(x)+i

∫d4x J(x)[φ(x)+ǫ∆φ(x)]

=

∫Dφ(1 + i

∫d4x ǫ(x) [−∂µjµ(x) + J(x)∆φ(x)] +O(ǫ2)

)eiS(φ)+i

∫d4xJ(x)φ(x)

= Z[J ] + i

∫d4x ǫ(x)

(− 〈∂µjµ(x)〉J + J(x)〈∆φ(x)〉J

)+O(ǫ2),

(2.3.9)

where on the second line we expanded up to first order in ǫ and used the fact that,

being the symmetry realized linearly, the Jacobian of the change of variable φ′ 7→ φ in

the functional integration measure is independent of the fields and can be absorbed in

the overall normalization of the measure. The previous equation has to be valid for an

arbitrary choice of ǫ(x) and therefore it implies that

〈∂µjµ(x)〉J = J(x)〈∆φ(x)〉J . (2.3.10)

This relation is an example of a Ward-Takahashi (WT) identity associated with the sym-

metry (2.3.1) and constitutes the equivalent of the Noether theorem in QFT. By taking

functional derivatives with respect to J , this equation implies an infinite set of relationships

between correlation functions of the fields φ on the r.h.s. and those of the fields with an

insertion of the current operator jµ(x) on the l.h.s. As an explicit example, consider the in-

ternal symmetry (2.3.7), where the associated current is given by jα,µ = i(∂µφ)l(Tα)ml φm.

By differentiating eq.(2.3.10) with respect to Jn1(x1) and Jn2(x2) and by then setting

J = 0 we get

∂µ〈jµα(x)φn1(x1)φn2(x2)〉 == δ(x − x1)(Tα)

mn1〈φm(x1)φn2(x2)〉+ δ(x− x2)(Tα)

mn2〈φm(x2)φn1(x1)〉.

(2.3.11)

It is important to point out here that the effective action Γ is not always invariant under

the same field transformations for which the classical action S is. The transformations of

S and Γ coincide only when they act at most linearly on fields. Let us discuss in more

detail this important point. Consider a transformation of the form

φ(x) 7→ φ′(x) = φ(x) + ǫF [φ(x)], (2.3.12)

where F [φ] is a generic functional of the field φ, not necessarily linear. Assume now that

both the action S(φ) and the integration measure are invariant under the transformation

φ → φ′, such that S(φ) = S(φ′) and Dφ′ = Dφ. In terms of the generating function Z[J ]

26

one finds

Z[J ] =

∫Dφ′ eiS(φ′)+i

∫d4xJ(x)φ′(x) =

∫Dφ eiS(φ)+i

∫d4xJ(x)φ(x)+ǫF [φ(x)]

=

∫Dφ

1 + iǫ

∫d4xJ(x)F [φ(x)] +O(ǫ2)

eiS(φ)+i

∫d4x J(x)φ(x),

(2.3.13)

which implies ∫d4x 〈F [φ(x)]〉JJ(x) = 0 . (2.3.14)

Recall now the definition of the functional Γ[Φ], generating 1-particle irreducible ampli-

tudes. The source Φ(x) is defined as

Φ(x) ≡ δW [J ]

δJ(x)(2.3.15)

and the 1-particle irreducible generator is defined as

Γ[Φ] =W [J ]−∫d4xΦ(x)J(x) . (2.3.16)

It is a sort of Legendre transform of W [J ], very much like the Lagrangian is the Legendre

transform of the Hamiltonian. In eq. (2.3.16) it is understood that we have inverted

eq. (2.3.15) so that J = J [Φ]. We have

δΓ

δΦ(x)=

∫d4y( δW

δJ(y)− Φ(y)

) δJ(y)δΦ(x)

− J(x) = −J(x) . (2.3.17)

Using then eq.(2.3.17), we can rewrite eq.(2.3.14) as

∫d4x 〈F [φ(x)]〉J(Φ)

δΓ

δΦ(x)= 0, (2.3.18)

which implies that the 1PI action is invariant for Φ → Φ+ ǫ〈F [φ(x)]〉J(Φ):

Γ(Φ + ǫ〈F [φ(x)]〉J(Φ)) = Γ(Φ) + ǫ

∫d4x 〈F [φ(x)]〉J(Φ)

δΓ

δΦ(x)= Γ(Φ) . (2.3.19)

Notice that, in general,

〈F [φ(x)]〉J(Φ) 6= F (Φ) . (2.3.20)

They coincide only for transformations that are at most linear in the fields, namely for3

F (φ) = s(x) +

∫d4y t(x, y)φ(y) . (2.3.21)

3The usual purely linear transformations of the fields are obtained from eq.(2.3.21) by taking s(x) = 0

and t(x, y) = tδ(x− y).

27

In this case, and only in this case, we have

〈F [φ(x)]〉J(Φ) = s(x) +

∫d4y t(x, y)〈φ(x)〉J(Φ) = s(x) +

∫d4y t(x, y)Φ(y) = F (Φ) ,

(2.3.22)

where the second identity is a consequence of the definition of Φ.

When a symmetry is local, the WT identities among amplitudes are crucial in order to

ensure that possible unphysical states decouple from the theory. It is useful to consider in

more detail the simplest case of U(1) gauge symmetry of QED . This will be the subject

of next section.

2.3.1 WT Identities in QED

Let us discuss in some more detail the specific case of WT Identities in QED. Gauge

invariance in QED requires the addition of a gauge-fixing term 4

Lg.f. = − 1

2ξ(∂µA

µ)2 , (2.3.23)

where ξ is an arbitrary real parameter. The total Lagrangian density is then

L(Aµ, ψ, ψ) = LQED(Aµ, ψ, ψ) + Lg.f.(Aµ) , (2.3.24)

where

LQED = −1

4FµνF

µν + ψ(i /D(A)−m

)ψ + counterterms . (2.3.25)

The functional integration is done both on the vector field Aµ and on the spinors ψ and

ψ. The source term at the exponential has a density of the form

Jµ(x)Aµ(x) + J(x)ψ(x) + ψ(x)J(x) , (2.3.26)

where J and J are Grassmann-valued (and hence anticommuting) functions. If we make

the infinitesimal change of variable associated to a gauge transformation,

ψ(x) → ψ(x) + ieǫ(x)ψ(x) , ψ(x) → ψ(x)− ieǫ(x)ψ(x) , Aµ(x) → Aµ(x) + ∂µǫ(x),

(2.3.27)

the measure and LQED remain invariant, while Lg.f. and the source term will change:

L(Aµ, ψ, ψ) → L(Aµ, ψ, ψ)−∂µAµξ

ǫ .

JµAµ + Jψ + ψJ → JµAµ + Jψ + ψJ + Jµ∂µǫ+ ieJψǫ− ieψJǫ. (2.3.28)

4In this section, and only in this, we use the more physical convention for which Aµ is a real, rather

than an imaginary, field.

28

By bringing down the O(ǫ) terms from the exponential and taking a functional derivative

with respect to ǫ(x), we get the identity

i∂µJµ(x)Z = e〈ψ(x)〉JJ(x)− eJ(x)〈ψ(x)〉J − i

ξ∂µ〈Aµ(x)〉J = 0 . (2.3.29)

We also have5

〈ψ(x)〉J = iδZ

δJ(x), 〈ψ(x)〉J = −i δZ

δJ(x), 〈Aµ(x)〉J = −i δZ

δJµ(x)(2.3.30)

and thus we can rewrite eq.(2.3.28) in terms of W = −i logZ as

i∂µJµ(x) = −e δW

δJ(x)J(x)− eJ(x)

δW

δJ (x)− i

ξ∂µ

δW

δJµ(x)= 0 . (2.3.31)

By taking arbitrary functional derivatives of eqs.(2.3.29) or (2.3.31) with respect to the

sources we get an infinite set of WT identities between connected and 1PI amplitudes,

respectively. In particular, we can relate the electron two-point function with the ver-

tex, show the transversality of the photon propagator, and prove that unphysical photon

polarization states decouple.

Let us consider the last property. Recall that S-matrix amplitudes are associated to

amputated connected Green functions. Let us focus on n-correlation functions of photons

only, in absence of fermion fields. We can then set J = J = 0 in eq.(2.3.31) and take n

functional derivatives of eq.(2.3.31) with respect to Jµ. In this way the l.h.s. and the first

two terms in the r.h.s. of eq.(2.3.29) vanish and we get, in momentum space,

q2qµGµα1...αn(q, p1, . . . , pn) = 0 , (2.3.32)

for the n+1 connected amplitudes involving photons only. By amputating the amplitude

(which includes to divide eq.(2.3.32) by q2), we see that the matrix elements involving

n + 1 photons must vanish, when one (or more) of them has the unphysical polarization

ǫµ(q) = qµ. The analysis can be easily extended in presence of external fermions.

We conclude that the validity of the WT identities is crucial to ensure a well-defined

consistent theory. A similar statement applies for non-abelian gauge theories and for local

Lorentz transformations in effective field theories of gravity.

5Pay attention to the anticommuting nature of the Grassmann fields to get the signs right!

29

Chapter 3

Lecture III

3.1 The Chiral Anomaly

In the path-integral formulation of QFT, anomalies arise from the transformation of the

measure used to define the fermion path integral [?].

Let ψA(x) be a massless Dirac fermion defined on a 2n-dimensional manifold M2n

with Euclidean signature δµν (µ, ν = 1, . . . , 2n), and in an arbitrary representation R of

a gauge group G (A = 1, . . . ,dimR). The minimal coupling of the fermion to the gauge

and gravitational fields is described by the Lagrangian

L = e ψ(x)Ai(/Dr)AB ψ

B , (3.1.1)

where e is the determinant of the vielbein and /Dr is the Dirac operator defined in eq.(1.3.27).

The classical Lagrangian (3.1.1) is invariant under the global chiral transformation

ψ → eiαγ2n+1ψ , ψ → ψeiαγ2n+1 , (3.1.2)

where γ2n+1 is the chirality matrix (1.2.16) in 2n dimensions, and α is a constant param-

eter. This implies that the chiral current Jµ2n+1 = ψAγ2n+1γµψA is classically conserved.

At the quantum level, however, this conservation law can be violated and turns into an

anomalous Ward identity. To derive it, we consider the quantum effective action Γ defined

by

e−Γ(e,ω,A) =

∫DψDψ e−

∫d2nxL , (3.1.3)

and study its behavior under an infinitesimal chiral transformation of the fermions, with

a space-time-dependent parameter α(x), given by1

δαψ = iαγ2n+1ψ , δαψ = iαψγ2n+1 . (3.1.4)

1For simplicity of the notation, we omit the gauge index A in the following equations. It will be

reintroduced later on in this section.

30

Since the external fields e, ω and A are inert, the transformation (3.1.4) represents a

redefinition of dummy integration variables, and should not affect the effective action:

δαΓ = 0. This statement carries however a non-trivial piece of information, since neither

the action nor the integration measure is invariant under eq.(3.1.4). The variation of the

classical action under eq.(3.1.4) is non-vanishing only for non-constant α, and has the

form δα∫L =

∫Jµ2n+1∂µα. The variation of the measure is instead always non-vanishing,

because the transformation (3.1.4) leads to a non-trivial Jacobian factor, which has the

form δα[DψDψ] = [DψDψ] exp−2i∫αA, as we will see below. In total, the effective

action therefore transforms as

δαΓ =

∫d2nxeα(x)

[2iA(x) − 〈∂µJµ2n+1(x)〉

]. (3.1.5)

The condition δαΓ = 0 then implies the anomalous Ward identity:

〈∂µJµ2n+1〉 = 2iA . (3.1.6)

In order to compute the anomaly A, we need to define the integration measure more

precisely. This is best done by considering the eigenfunctions of the Dirac operator i/Dr.

Since the latter is Hermitian, the set of its eigenfunctions ψk(x) with eigenvalues λk,

defined by i/Dψn = λnψn, form an orthonormal and complete basis of spinor modes:∫d2nx eψ†

k(x)ψl(x) = δk,l ,∑

k

eψ†k(x)ψk(y) = δ(2n)(x− y) . (3.1.7)

The fermion fields ψ and ψ, which are independent from each other in Euclidean space,

can be decomposed as

ψ =∑

k

akψk , ψ =∑

k

bkψ†k , (3.1.8)

so that the measure becomes

DψDψ =∏

k,l

dakdbl . (3.1.9)

Under the chiral transformation (3.1.4), we have

δαak = i

∫d2nx e

∑

l

ψ†kαγ2n+1ψlal , δαbk = i

∫d2nx e

∑

l

blψ†l αγ2n+1ψk , (3.1.10)

and the measure (3.1.9) transforms as

δα

[DψDψ

]= DψDψ exp

− 2i

∑

k

∫d2nx eψ†

kαγ2n+1ψk

. (3.1.11)

For simplicity we can take α to be constant. This formal expression is ill-defined as

it stands, since it decomposes into a vanishing trace over spinor indices (tr γ2n+1 = 0)

31

times an infinite sum over the modes (∑

k 1 = ∞). A convenient way of regularizing this

expression is to introduce a gauge-invariant Gaussian cut-off. The integrated anomaly

Z =∫A can then be defined as

Z = limβ→0

∑

k

ψ†kγ2n+1ψke

−βλ2n/2

= limβ→0

Tr[γ2n+1e

−β(i/D)2/2], (3.1.12)

where the trace has to be taken over the mode and the spinor indices, as well as over

the gauge indices. Equation (3.1.12) finally provides the connection between anomalies

and quantum mechanics. Indeed, it represents the high-temperature limit (T = 1/β) of

the partition function of the quantum mechanical model (2.2.7) that has as Hamiltonian

H = (i/D)2/2 and as density matrix ρ = γ2n+1e−βH : Z = Tr ρ. Z is not clearly the ordinary

thermal partition function of the system, because of the presence of the chirality matrix

γ2n+1. Its effect is rather interesting. As we mentioned in the previous lecture, γ2n+1

acts as the fermion counting operator (−)F . In supersymmetric quantum mechanics, with

H = Q2, we notice that any state |E〉 with strictly positive energy E > 0 is necessarily

paired with its supersymmetric partner |E〉: Q|E〉 =√E|E〉. Of course, |E〉 and |E〉

have the same energy but a fermion number F differing by one unit. Independently of

the bosonic or fermionic nature of |E〉, the contribution to Z of |E〉 and |E〉 is equal andopposite and always cancels. The only states that might escape this pairing are the ones

with zero energy. In this case Q|E = 0〉 = 0 and hence these states are not necessarily

paired. We conclude that

Z = n+ − n− (3.1.13)

independently of β, where n± are the number of zero energy bosonic and fermionic states.

Interestingly enough, Z does not depend on other possible deformation of the system.

Indeed, if a state |E〉 with E > 0 approaches E = 0, so has to do its partner |E〉.Viceversa, zero energy states can be deformed to E > 0 only in pairs. So, while n+

and n− individually might depend on the details of the theory, their difference is an

invariant object, called Witten index [8]. Since n± correspond to the number of zero

energy eigenfunctions of positive and negative chirality of the Dirac operator, we see that

Z coincides with the index of the Dirac operator defined in eq.(1.3.28). Using the Atiyah-

Singer theorem, i.e. the right hand side of eq.(1.3.28), we might just read the result for the

chiral anomaly. However, it is actually not so difficult to compute Z. This computation

can then be seen as a physical way to prove the Atiyah-Singer theorem applied to the

Dirac operator [9].

32

The integrated anomaly Z could be derived by computing directly the partition func-

tion of the above supersymmetric model in a Hamiltonian formulation. In order to obtain

the correct result, however, one has to pay attention when tracing over the fermionic oper-

ators c⋆A and cA. As discussed below eq.(2.2.5), only the 1-particle states c⋆A|0〉 correspondto the representation R of the gauge group G. In order to compute the anomaly for a

single spinor in the representation R, it is therefore necessary to restrict the partition

function to such 1-particle states. On the other hand, the partition function of a system

at finite temperature T = 1/β can be conveniently computed in the canonical formalism

by considering an euclidean path integral where the (imaginary) time direction is compact-

ified on a circle of radius 1/(2πT ). The insertion of the fermion operator (−)F amounts

to change the fermion boundary conditions from anti-periodic (the usual one in canonical

formalism) to periodic. So the best way to proceed is to use a hybrid formulation, which

is Hamiltonian with respect to the fields c⋆A and cA, and Lagrangian with respect to the

remaining fields xµ and ψa. Starting from the hamiltonian H, we then define a modified

Lagrange transform (technically called Routhian R) with respect to the fields xµ and ψa

only. The index Z can then be written as

Z = Trc,c⋆

∫

PDxµ

∫

PDψa exp

−∫ β

0dτ R

(xµ(τ), ψa(τ), c⋆A, c

A)

. (3.1.14)

The subscript P on the functional integrals stands for periodic boundary conditions along

the closed time direction τ , Trc,c⋆ represents the trace over the 1-particle states c⋆A|0〉, andthe Euclidean Routhian R is given by

R =1

2gµν x

µxν +1

2ψaψ

a +1

4

[ψa, ψb

]ωabµ x

µ

+ c⋆AAAµ Bx

µcB − 1

2c⋆Ac

BψaψbFAab B . (3.1.15)

Equation (3.1.14) should be understood as follows: after integrating over the fields xµ and

ψa, one gets an effective Hamiltonian H(c, c⋆) for the operators c⋆A and cA, from which

one finally computes Trc,c⋆e−βH(c,c⋆). Although this procedure looks quite complicated,

we will see that it drastically simplifies in the high-temperature limit we are interested in.

The computation of the path-integral is greatly simplified by using the background-field

method and expanding around constant bosonic and fermionic configurations in normal

coordinates [10]. These are defined as the coordinates for which around any point x0

the spin-connection (or equivalently, the Christoffel symbol), as well as its symmetric

derivatives, vanish at x0. It is also convenient to rescale τ → βτ and define

xµ(τ) = xµ0 +√β eµa(x0)ξ

a(τ) , ψa(τ) =1√βψa0 + λa(τ) . (3.1.16)

33

In this way, it becomes clear that it is sufficient to keep only quadratic terms in the

fluctuations, which have a β-independent integrated Routhian, since higher-order terms

in the fluctuations come with growing powers of β. In normal coordinates one has

ωabµ (x)xµ(τ) =√βωabµ (x0)e

µd (x0)ξ

d(τ) + β∂νωabµ (x0)e

µd (x0)ξ

d(τ)eνc (x0) ξc(τ) +O(β3/2)

= 0 + β∂[νωabµ] (x0)e

µd (x0)e

νc (x0)ξ

d(τ)ξc(τ) +O(β3/2) (3.1.17)

= βRabcd(x0)ξd(x0)ξ

c(x0) +O(β3/2) .

The term proportional to Aµ in eq.(3.1.15) vanishes in this limit because it is of order β.

Using these results, eq. (3.1.15) reduces to the following effective quadratic Routhian, in

the limit β → 0:

Reff =1

2

[ξaξ

a + λaλa +Rab(x0, ψ0)ξ

aξb]− c⋆AF

AB(x0, ψ0)c

B , (3.1.18)

where

Rab(x0, ψ0) =1

2Rabcd(x0)ψ

c0ψ

d0 ,

FAB(x0, ψ0) =1

2FAab B(x0)ψ

a0ψ

b0 . (3.1.19)

Since the fermionic zero modes ψa0 anticommute with each other,2 they define a basis of

differential forms on M2n, and the above quantities behave as curvature 2-forms.

From eq.(3.1.18) we see that the gauge and gravitational contributions to the chiral

anomaly are completely decoupled. The former is determined by the trace over the 1-

particle states c⋆A|0〉, and the latter by the determinants arising from the Gaussian path

integral over the bosonic and fermionic fluctuation fields:

Z =

∫d2nx0

∫d2nψ0 Trc,c⋆

[ec

⋆AF

ABc

B]det

−1/2P

[−∂2τ δab +Rab∂τ

]det

1/2P

[∂τ δab

]. (3.1.20)

The trace yields simply

Trc,c⋆ec⋆AF

ABc

B

= trReF . (3.1.21)

The determinants can be computed by decomposing the fields on a complete basis of

periodic functions of τ on the circle with unit radius, and using the standard ζ-function

regularization. Let us quickly recall how ζ-function regularization works by computing

the partition function of a free periodic real boson x on a line:

Z freeB =

∫

PDxe−m

2

∫ t0 x

2dτ . (3.1.22)

2The anticommuting ψa0 ’s are simply Grassmann variables in a path integral and should not be confused

with the operator-valued fields ψa entering eq.(2.2.7), which satisfy the anticommutation relations (2.1.9).

34

We expand x in Fourier modes xn, x−n = x∗n and rewrite

Z freeB =

∫dx0

∞∏

n=1

dxndx∗ne

−mt∑∞

n=1

(2πnt

)2|xn|2 =

∫dx0

∞∏

n=1

( t

2πmn2

). (3.1.23)

Using now the identity∞∏

n=1

fn = exp(− lims→0

d

ds

∞∑

n=1

f−sn

), (3.1.24)

we compute

∞∏

n=1

( t

2πmn2

)= exp

(− d

ds

( t

2πm

)s ∞∑

n=1

n2s)s=0

= exp(ζ(0) log

t

2πm+2ζ ′(0)

), (3.1.25)

where ζ(0) = −1/2, ζ ′ = − log 2π/2 are the value in zero of the ζ function and its first

derivative. We then find

Z freeB = L

( m2πt

)1/2, (3.1.26)

where L is the length of the line. The partition function of a free periodic fermion ψ on

a line vanishes because of the integration over the Grassmannian zero mode ψ0. Inserting

one ψ0 in the path integral and doing the same manipulation as for the bosonic case, one

gets

Z freeF =

∫

PDψ ψ e i

2m

∫ t0 ψψdτ = m−1/2 . (3.1.27)

Let us come back to the evaluation of the determinants in eq.(3.1.20). It is useful to

bring the curvature 2-form Rab into the block-diagonal form (1.3.22), so that the bosonic

determinant decomposes into n distinct determinants with trivial matrix structure.

Exercise 7: Compute the determinants appearing in eq.(3.1.20) and show that

det−1/2P

[−∂2τ δab +Rab∂τ

]= (2π)−n

n∏

i=1

λi/2

sin(λi/2), (3.1.28)

det1/2P

[∂τ δab

]= (−i)n . (3.1.29)

The final result for the anomaly is obtained by putting together (3.1.21), (3.1.28) and

(3.1.29), and integrating over the zero modes. The Berezin integral over the fermionic zero

modes vanishes unless all of them appear in the integrand, in which case it yields∫d2nψ0 ψ

a10 . . . ψa2n0 = (−1)n ǫa1...a2n . (3.1.30)

35

This amounts to selecting the 2n-form component from the expansion of the integrand

in powers of the 2-forms F and R. Since this is a homogeneous polynomial of degree n

in F and R, the factor (i/(2π))n arising from the normalization of (3.1.28), (3.1.29) and

(3.1.30) amounts to multiplying F and R by i/(2π). The final result for the integrated

anomaly can therefore be rewritten more concisely as:

Z =

∫

M2n

chR(F ) A(R) , (3.1.31)

where ch(F ) and A(R) have been defined in eqs.(1.3.26) and (1.3.29). In (3.1.31), as well

as in all the analogous expressions that will follow, it is understood that only the 2n-form

component of the integrand has to be considered.

The anomalous Ward identity (3.1.6) for the chiral symmetry can then be formally

written, in any even dimensional space, as

〈∂µJµ2n+1〉 = 2i chR(F )A(R) . (3.1.32)

In 4 space-time dimensions, for instance, eq.(3.1.32) gives

〈∂µJµ5 〉 = −iǫµνρσ[

1

16π2trFµνFρσ +

dimR384π2

R αβµν Rρσαβ

]. (3.1.33)

3.2 Consistency Conditions for Gauge and Gravitational Anomalies

Anomalies can also affect currents related to space-time dependent symmetries. For sim-

plicity, we will consider in the following anomalies related to spin gauge fields only, i.e.

gauge anomalies. Gravitational anomalies can be analyzed in almost the same way, mod-

ulo some subtleties we will mention later on. In presence of a gauge anomaly, the theory

is no longer gauge-invariant. This is best seen by considering the effective action Γ in

eq.(3.1.3). Under an infinitesimal gauge transformation A→ A−Dǫ1

δǫ1Γ(A) = −∫d2nx tr

δΓ(A)

δAµDµǫ1 =

∫d2nx trDµJ

µǫ1 =

∫d2nx trAǫ1 ≡ I(ǫ1) , (3.2.1)

where Aα is the gauge anomaly and we have defined its integrated form (including the

gauge parameter ǫ) by I. The WT identities associated to the conservation of currents

in a gauge theory allows us to define a consistent theory, where unphysical states are

decoupled, as we have seen in some detail in the QED U(1) case in subsection 2.3.1. The

anomaly term A that would now appear in the WT identities associated to the (would-be)

conservation of the current would have the devastating effect of spoiling this decoupling.

Gauge anomalies lead to inconsistent theories, where unitarity is broken at any scale.

36

The structure of the gauge anomalies that can occur in local symmetries is strongly

constrained by the group structure of these symmetry transformations. In particular, two

successive transformations δǫ1 and δǫ2 with parameters ǫ1 and ǫ2 must satisfy the basic

property:[δǫ1 , δǫ2

]= δ[ǫ1,ǫ2]. We should then have

[δǫ1 , δǫ2 ]Γ(A) = δ[ǫ1,ǫ2]Γ(A) , (3.2.2)

or in terms of the anomaly I defined in eq.(3.2.1):

δǫ1I(ǫ2)− δǫ2I(ǫ1) = I([ǫ1, ǫ2]) . (3.2.3)

The above relations are the so-called Wess–Zumino consistency conditions [11].

The general solution of this consistency condition can be characterized in an elegant

way in terms of a (2n + 2)-form with the help of the so-called Stora–Zumino descent

relations [12]. For any local symmetry with transformation parameter ǫ (a 0-form), con-

nection a (a 1-form) and curvature f (a 2-form), these are defined as follows. Starting

from a generic closed and invariant (2n + 2)-form Ω2n+2(f), one can define an equiva-

lence class of Chern–Simons (2n + 1)-forms Ω(0)2n+1(a, f) through the local decomposition

Ω2n+2 = dΩ(0)2n+1, like in section 1.3. In this way, we specify Ω

(0)2n+1 only modulo exact

2n-forms, implementing the redundancy associated to the local symmetry under consider-

ation. One can then define yet another equivalence class of 2n-forms Ω(1)2n (ǫ, a, f), modulo

exact (2n − 1)-forms, through the transformation properties of the Chern–Simons form

under a local symmetry transformation: δǫΩ(0)2n+1 = dΩ

(1)2n . It is the unique integral of this

class of 2n-forms Ω(1)2n that gives the relevant general solution of eq.(3.2.3):

I(ǫ) = 2πi

∫

M2n

Ω(1)2n (ǫ) . (3.2.4)

To understand this, it is convenient to introduce the analog of the above descent relations

for manifolds. Starting from the 2n-dimensional space-time manifold M2n, we define the

equivalence class of (2n+1)-dimensional manifoldsM ′2n+1 whose boundary isM2n: M2n =

∂M ′2n+1, and similarly the equivalence class of (2n+2)-dimensional manifoldsM ′′

2n+2 such

that M ′2n+1 = ∂M ′′

2n+2. Both M′2n+1 and M ′′

2n+2 are defined modulo components with no

boundary, on which exact forms integrate to zero. Since∫

M2n

Ω(1)2n (ǫ) =

∫

M ′2n+1

dΩ(1)2n (ǫ) = δǫ

∫

M ′2n+1

Ω(0)2n+1 = δǫ

∫

M ′′2n+2

dΩ(0)2n+1 = δǫ

∫

M ′′2n+2

Ω2n+2,

(3.2.5)

eq. (3.2.4) can be rewritten as

I(ǫ) = 2πi δǫ

∫

M ′2n+1

Ω(0)2n+1 = 2πi δǫ

∫

M ′′2n+2

Ω2n+2 . (3.2.6)

37

It is clear that eq.(3.2.6) provides a solution of eq.(3.2.3), since it is manifestly in the form

of the variation of some functional under the symmetry transformation. In fact, the descent

construction also insures that eq.(3.2.6) actually characterizes the most general non-trivial

solution of eq.(3.2.3), modulo possible local counterterms. This can be understood more

precisely within the BRST formulation of the Stora–Zumino descent relations, which we

describe in the next lecture.

The above reasoning applies equally well to local gauge symmetries, associated to a

connection A with curvature F , and to diffeomorphisms, associated to the Levi-Civita con-

nection Γ or spin-connection ω, with curvature R, and shows that gauge and gravitational

anomalies in a 2n-dimensional theory are characterized by a gauge-invariant (2n+2)-form,

which suggests a (2n+2)-dimensional interpretation. We shall see in the next lecture that

the chiral anomaly in 2n + 2 dimensions Z =∫M2n+2

Ω2n+2 provides precisely the 2n + 2-

form defined above through which we can solve the Wess-Zumino consistency relations.

The gauge and gravitational anomalies in 2n dimensions are then obtained through the

descent procedure, defined respectively with respect to gauge transformations and diffeo-

morphisms (or local Lorentz transformations), as I = 2πi∫M2n

Ω(1)2n .

38

Chapter 4

Lecture IV

4.1 The Stora–Zumino Descent Relations

The Stora–Zumino descent relations are best analyzed by using the convenient differential-

form notation introduced in section 1.1. The gauge field A is a 1-form and its field-strength

is the 2-form F = dA + A2, which satisfies the Bianchi identity DF = dF + [A,F ] = 0.

As we have seen in the previous lecture, the relevant starting point is a gauge-invariant

(2n+2)-form Ω2n+2(F ). This is exact and therefore defines a Chern–Simons (2n+1)-form

through the local decomposition Ω2n+2(F ) = dΩ2n+1(A,F ). Finally, the gauge variation

of the latter defines a 2n-form Ω2n(A,F ) through the transformation law δvΩ2n+1(A,F ) =

dΩ2n(v,A, F ).

To begin with, let us introduce the relevant formalism. We shall need to define a family

of gauge transformations g(x, θ) depending on the coordinates xµ and on some (ordinary,

not Grassmann) parameters θα. These gauge transformations change A and F into

A(x, θ) = g−1(x, θ)(A(x) + d

)g(x, θ) , (4.1.1)

F (x, θ) = g−1(x, θ)F (x)g(x, θ) . (4.1.2)

We can then define, besides the usual exterior derivative d = dxµ∂µ with respect to the

coordinates xµ, an additional exterior derivative d = dθα∂α with respect to the parameters

θα. The range of values of the index α is for the moment undetermined. The operators

d and d anticommute and are both nilpotent, d2 = d2 = 0. This implies that their sum

∆ = d + d is also nilpotent: ∆2 = 0. The two operators d and d naturally define two

transformation parameters v and v through the expressions:

v(x, θ) = g−1(x, θ)dg(x, θ) , (4.1.3)

v(x, θ) = g−1(x, θ)dg(x, θ) . (4.1.4)

39

Exercise 8: Verify that

dv = −v2 , dA = −D v , dF = −[v, F ] . (4.1.5)

Equations (4.1.5) show that d generates an infinitesimal gauge transformation with pa-

rameter v on the gauge field A and its field-strength F . Interestingly enough, these can

also be interpreted as BRST transformations, the ghost fields being identified with v. At

this point, it is possible to define yet another connection A and field-strength F as

A = g−1(A+∆

)g = A+ v , (4.1.6)

F = ∆A+A2 = g−1Fg = F . (4.1.7)

The last relation is easily proved. Using eqs.(4.1.5) one has

F = (d+ d)(A+ v) + (A+ v)2

= dA+ dv − Dv − v2 + v2 + Av + vA+ A2 (4.1.8)

= dA+ A2

= F .

The crucial point is now thatA and F are defined with respect to ∆ exactly in the same

way as A and F are defined with respect to d. Therefore, the corresponding Chern–Simons

decompositions must have the same form:

Q2n+2(F) = ∆Q2n+1(A,F) , (4.1.9)

Q2n+2(F ) = dQ2n+1(A, F ) . (4.1.10)

On the other hand, eq.(4.1.7) implies that the left-hand sides of these two equations are

identical. Equating the right-hand sides and using eq.(4.1.6) yields:

(d+ d)Q2n+1(A+ v, F ) = dQ2n+1(A, F ) . (4.1.11)

In order to extract the information carried by this equation, it is convenient to expand

Ω2n+1(A+ v, F ) in powers of v as

Q2n+1(A+ v, F ) = Q(0)2n+1(A, F ) +Q

(1)2n (v, A, F ) + . . .+Q

(2n+1)0 (v, A, F ) , (4.1.12)

40

where the superscripts denote the powers of v and the subscript the dimension of the form

in real space. Substituting this expansion in eq.(4.1.11) and equating terms with the same

power of v, we finally find the Stora–Zumino descent relations [12]:

dQ(0)2n+1 + dQ

(1)2n = 0 ,

dQ(1)2n + dQ

(2)2n−1 = 0 ,

. . .

dQ(2n)1 + dQ

(2n+1)0 = 0 ,

dQ(2n+1)0 = 0 . (4.1.13)

The operator d makes it possible to understand in a simple way why the Stora–Zumino de-

scent represents the most general non-trivial solution of the Wess–Zumino consistency con-

dition (3.2.3). For the case of gauge anomalies we are considering here, I(v) =∫tr(v a(A)),

the latter reads

δv1

∫tr(v2 a(A)) − δv2

∫tr(v1 a(A))−

∫tr([v1, v2] a(A)) = 0 . (4.1.14)

The two transformations with parameters v1 and v2 can be incorporated into a family of

transformations parametrized by θ1 and θ2, with parameter

v = v1dθ1 + v2dθ

2 = vαdθα. (4.1.15)

In this way, vα = g−1∂αg. At θα = 0, g(x, 0) = 1 and therefore A(x, 0) = A(x) and

F (x, 0) = F (x). At that point, d generates ordinary gauge transformations on A and F ,

with d = dθαδvα . For instance, eq.(3.2.1) can be rewritten as

dΓ =

∫tr vA (4.1.16)

The condition (4.1.14) can then be multiplied by dθ1dθ2 and rewritten as

0 = dθ1dθ2(∫

tr(v2δv1 A)−∫

trv1δv2 A)−∫

tr[v1, v2]A)

= dθαdθβ(∫

tr(vβδvα A)−∫

tr[vα, vβ ]A)

(4.1.17)

= −∫

tr(vdA)−∫

trv2A .

Since dv = −v2, this can be rewritten simply as:

d

∫tr(vA) = 0 . (4.1.18)

41

The Wess–Zumino consistency condition is therefore the statement that the anomaly is

d-closed,

dI(v) = 0 . (4.1.19)

It is clear that the trivial d-exact solutions I(v) = d∫f(A) in terms of a local functional

f(A) of the gauge field correspond to the gauge variation of local counterterms that can

be added to the theory, and the non-trivial anomalies emerging from the non-local part

of the effective action are therefore encoded in the cohomology of d. From the second

relation appearing in the Stora–Zumino descent relations (4.1.13), we see that the general

non-trivial element of this cohomology is of the form I(v) =∫Q

(1)2n . With the above

definitions, we have Q2n+2 = dQ(0)2n+1 and δvQ

(0)2n+1 = −dQ(1)

2n . We can therefore identify

Ω2n+2 ↔ Q2n+2, Ω(0)2n+1 ↔ Q

(0)2n+1 , Ω

(1)2n ↔ −Q(1)

2n , (4.1.20)

where Ω are the forms defined a the end of the last lecture.

4.2 Path Integral for Gauge and Gravitational Anomalies

Let us now come back to gauge and gravitational anomalies in 2n dimensions and let us

show that they can be computed starting from the chiral anomaly in 2n + 2 dimensions

using the Stora-Zumino descent relations.

Gauge and gravitational anomalies can be computed with the same method as used

above for chiral anomalies. In this context, their potential origin comes again from the

Jacobian of the transformation in the integration measure, since the classical action is

invariant. Differently from the chiral anomaly, gauge and gravitational anomalies can

arise only from massless chiral fermions, with given chiralities ±. For a Dirac fermion in

euclidean space, if ψ transforms under a representation R of a gauge group G, ψ should

transform in the complex conjugate representation R⋆, otherwise the mass term ψψ would

not be invariant. In this case, the Jacobian factors associated to ψ and ψ exactly cancel,

resulting in no anomaly. Similarly for local Lorentz transformations. The Lagrangian to

start with is then that of a chiral fermion

L = e ψ(x)Ai(/Dr)ABPη ψB , (4.2.1)

where

Pη =1

2

(1 + ηγ2n+1

), η = ±1 , (4.2.2)

is the chiral projector. The computation is technically analogous to the one we performed

for the chiral anomaly, except that the transformation law is now different and acts with

42

opposite signs on ψ and ψ. Moreover, the full Dirac operator is now , i/Dη = i/DPη, andis not Hermitian. For this reason, we have to use the eigenfunctions φηn of the Hermitian

operator (i/Dη)†i/Dη to expand ψ and the eigenfunctions ϕηn of i/Dη(i/Dη)

† to expand ψ.

Let us first consider the case of gauge anomalies. Under an infinitesimal gauge trans-

formation with parameter vα, or vAB = vαTAα B , the fermion fields transform as

δvψA = −vABψB , δvψ

A = vABψB . (4.2.3)

This induces a variation of the integration measure given by

δv

[DψDψ

]= DψDψ exp

∑

k

∫d2nx e

(φη†k vαT

αφηk − ϕη†k vαTαϕηk

). (4.2.4)

As in the case of the chiral anomaly, this formal expression needs to be regularized, and

we can define the integrated gauge anomaly to be

Igauge(v) = − limβ→0

∑

k

(φ†kvαT

αe−β(i/Dη)†i/Dη/2φk − ϕ†kvαT

αe−βi/Dη(i/Dη)†/2ϕk

)

= −η limβ→0

Tr[γ2n+1Q

gaugee−β(i/D)2/2]. (4.2.5)

The operator Qgauge is defined in such a way to act as vαTα on the Hilbert space. A

concrete realization of it within the supersymmetric quantum mechanics introduced in the

previous subsection is

Qgauge → c∗AvABc

B . (4.2.6)

The computation of eq.(4.2.5) is similar to that of eq.(3.1.12). The only difference is

the insertion of the operator (4.2.6) into the trace (3.1.21). It is clear from eq.(3.1.21)

that making this insertion is equivalent to substituting F → F + v in the trace without

insertion and taking the linear part in v of the result. We clearly see in this way the

close connection between chiral anomalies in 2n + 2 dimensions and gauge anomalies in

2n dimensions. However, despite being similar, the above prescription does not corre-

spond to the Stora–Zumino descent relations discussed above to get a consistent form

for the anomaly. Computed in this way, the anomaly does not automatically satisfy the

Wess–Zumino consistency conditions. The reason can be traced to the violation of the

Bose symmetry among the external states in the path integral derivation. The anomaly

computed in this form is called “covariant”, because it transforms covariantly under the

local symmetry. The correct anomaly satisfying the Wess–Zumino consistency conditions

is instead called “consistent”. These two forms of the anomaly contain the same informa-

tion, and there is a well-defined procedure to switch from one to the other [13]. To put the

covariant anomaly emerging from a computation a la Fujikawa into a consistent form, it

43

is sufficient to take the leading-order part of it, which contains n+ 1 fields, add the sym-

metrization factor required for a Bose-symmetric result, and then use the Wess–Zumino

consistency condition to reconstruct the subleading part. All this procedure is automat-

ically obtained by taking the Stora–Zumino descents of the chiral anomaly. Taking into

account the factor i/(2π) entering the definition of the Chern character, the consistent

form of the integrated gauge anomaly is therefore

Igauge(v) = 2πiη

∫

M2n

[chR(F )

](1)A(R) . (4.2.7)

The case of gravitational anomalies is similar. Under an infinitesimal diffeomorphism

with parameter ǫµ, the fermion fields transform, modulo a local Lorentz transformation,

as

δǫψ = −ǫµDµψ , δǫψ = ǫµDµψ . (4.2.8)

This induces the following variation of the integration measure:

δǫ

[DψDψ

]= DψDψ exp

∑

k

∫d2nx e

(φη†k ǫ

µDµφηk − ϕη†k ǫ

µDµϕηk

). (4.2.9)

The regularized expression for the integrated gravitational anomaly is then:

Igrav(ǫ) = − limβ→0

∑

k

(φ†ke

−β(i/Dη)†i/Dη/2ǫµDµφk − ϕ†ke

−βi/Dη(i/Dη)†/2ǫµDµϕk

)

= −η limβ→0

Tr[γ2n+1Q

grave−β(i/D)2/2]. (4.2.10)

The operator Qgrav must act as ǫµDµ on the Hilbert space. Since ixµ → Dµ upon canonical

quantization, we can identify

Qgrav → iǫµxµ . (4.2.11)

We have then (recall the rotation to euclidean time τ → −iτ)

Igrav(ǫ) = ηTrc,c⋆

∫

PDxµ

∫

PDψaǫµ(x)

dxµ(τ)

dτexp

−∫ β

0dτ R

(xµ(τ), ψa(τ), c⋆A, c

A)

.

(4.2.12)

Thanks to the periodic boundary conditions, the path integral does not depend on the

point where we insert Qgrav, so we can replace in eq.(4.2.12)

ǫµ(x)dxµ(τ)

dτ→ 1

β

∫ β

0dτ ǫµ(x)

dxµ(τ)

dτ. (4.2.13)

44

Rescaling τ → βτ , using eq.(3.1.16) and expanding in normal coordinates gives, modulo

irrelevant higher order terms in β,

1

β

∫ β

0dτ ǫµ(x)

dxµ(τ)

dτ=

1

β

∫ 1

0dτ(ǫµ(x0) +∇νǫµ(x0)

√βξν(τ) +O(β)

)√βdξµ(τ)

dτ

=1√βǫµ(x0)

∫ 1

0dτdξµ(τ)

dτ+∇νǫµ(x0)

∫ 1

0dτξν

dξµ(τ)

dτ(4.2.14)

=1

2Daǫb(x0)

∫ 1

0dτ(ξaξb − ξbξa) ,

where in the second row the first term and the symmetric part of the second vanish, being

total derivatives of periodic functions. It is now convenient to exponentiate the action of

Qgrav and notice that it amounts to the shift Rab → Rab − Daǫb + Dbǫa in the effective

Routhian (3.1.18). The original expression (4.2.10) is recovered by keeping only the linear

piece in ǫ. After adding the appropriate symmetrization factors and following the same

procedure as for the gauge anomaly to switch to a consistent form of the anomaly, this

implements the Stora–Zumino descent with respect to diffeomorphisms. The consistent

form of the integrated gravitational anomaly is finally found to be

Igrav(ǫ) = 2πiη

∫

M2n

chR(F )[A(R)

](1). (4.2.15)

We see from eq.(4.2.15) that purely gravitational anomalies, i.e. F = 0 in eq.(4.2.15), can

only arise in d = 4n+2 dimensions. The analogue of eq.(3.2.1) for diffeomorphisms reads

δǫΓ(g) = −∫d2nx

δΓ(g)

δgµν(∇µǫν +∇νǫµ) =

∫d2nx

√g∇µT

µνǫν =

∫Aνǫν ≡ Igrav(ǫ) ,

(4.2.16)

where δǫgµν = −(∇µǫν + ∇νǫµ) and we have defined the symmetric energy-momentum

tensor

T µν ≡ 2√g

δΓ(g)

δgµν. (4.2.17)

A gravitational anomaly leads then to a violation of the conservation of the energy-

momentum tensor. It is also possible to see the consequences of a violation of the local

Lorentz symmetry. In this case we see Γ as a functional of the vielbeins eaµ. Under a local

Lorentz transformation for which δLeaµ = −αabebµ, we have

δLΓ(e) = −∫d2nx

δΓ(e)

δeµaαabe

bµ =

∫d2nxeT abαab =

∫Aabαab ≡ IL(α) , (4.2.18)

where we have used the relation

ebµδΓ(e)

δeµa= ebµ2

δΓ(g)

δgµνeaν = eT ab . (4.2.19)

45

Since αab = −αba, an anomaly in a local Lorentz transformation implies a conserved, but

non-symmetric, energy-momentum tensor. It is possible to show that these anomalies

are not independent from those affecting diffeomorphisms, and it is always possible to

add a local functional to the action to switch to a situation in which one or the other of

the anomalies vanish, but not both [13]. Correspondingly, in presence of a gravitational

anomaly, the energy-momentum tensor can be chosen to be symmetric or conserved at

the quantum level, but not both simultaneously. The resulting theory will be inconsistent,

since unphysical graviton modes will not decouple in scattering amplitudes.

We see from eqs.(4.2.7) and (4.2.15) that we can also have mixed gauge/gravitational

anomalies, namely anomalies involving both gauge fields and gravity. Contrary to the

purely gravitational ones, they can arise in both 4n and 4n+2 dimensions. These anomalies

lead to both the non-conservation of the gauge current and of the energy momentum tensor.

However, by adding suitable local counterterms to the effective action, one can in general

bring the whole anomaly in either the gauge current or the energy momentum tensor.

The mixed gauge/gravitational anomaly signals the obstruction in having both currents

conserved at the same time.

4.3 Explicit form of Gauge Anomaly

It is useful to explicitly compute the expressions of the forms that are relevant to gauge

anomalies. The starting point is the (2n + 2)-form characterizing the chiral anomaly in

2n+ 2 dimensions:

Ω2n+2(F ) =( i

2π

)n+1Q2n+2(F ) , Q2n+2(F ) = trFn+1 . (4.3.1)

In orde to not carry the unnecessary (i/2π) factors, we will consider in what follows the

form Q, rather than Ω. As shown in section 1.3, Q is a closed form. Its first descendent

Q(0)2n+1(A,F ) is easily obtained by setting F = A = 0 in eq.(1.3.10) (with n→ n+1). One

has Ft = tF + (t2 − t)A2 and hence

Q(0)2n+1(A,F ) = (n+ 1)

∫ 1

0dt tr

[AFnt

]. (4.3.2)

To compute the second descendent, Q(1)2n (v,A, F ), we use the result (4.3.2) to determine

the left-hand side of eq.(4.1.12). Defining for convenience ˜Ft = tF + (t2 − t)(A+ v)2, this

reads

Q2n+1(A+ v, F ) = (n+ 1)

∫ 1

0dt tr

[(A+ v) ˜Fnt

]. (4.3.3)

The quantity we are after is then found by expanding in powers of v and retaining the

linear order. We have ˜Ft = Ft+(t2−t)[A, v]+O(v2). It is useful to define the symmetrized

46

trace of generic matrix-valued forms ωi = ωαi

i Tαi as

str (ω1 . . . ωp) =1

p!ωα11 ∧ . . . ∧ ωαp

p

∑

perms.

tr (Tα1 . . . Tαp) . (4.3.4)

In this way we find

Q(1)2n (v, A, F ) = (n+ 1)

∫ 1

0dt str

[vFnt + n(t2 − t)A

[A, v

]Fn−1t

]

= (n+ 1)

∫ 1

0dt str

[v(Fnt + n(t− 1)

(t[A, A

]Fn−1t − A

[At, F

n−1t

]))]

= (n+ 1)

∫ 1

0dt str

[v(Fnt + n(t− 1)

((∂tFt − dA

)Fn−1t + AdFn−1

t

))]

= (n+ 1)

∫ 1

0dt str

[v(Fnt + (t− 1)∂tF

nt + n(1− t)d

(AFn−1

t

))]. (4.3.5)

In the third equality we have used the identities DtFn−1t = dFn−1

t + [At, Fn−1t ] = 0 and

∂tFt = dA+ t[A, A]. After integrating by parts, the first and second term in the last line

of eq.(4.3.5) cancel. Setting θ = 0 and going back to the Ω-forms finally gives (keep in

mind the sign flip coming from eq.(4.1.20)):

Ω(1)2n (v,A, F ) = −

( i

2π

)n+1n(n+ 1)

∫ 1

0dt (1− t) str

[vd(AFn−1

t )]. (4.3.6)

The generalization of the above relations to the gravitational case is straightforward.

We substitute A and F with the connection Γ and the curvature R in eq.(1.2.11), and

consider infinitesimal diffeomorphisms. Alternatively, we substitute A and F with the

spin connection ω and the curvature R, and consider infinitesimal SO(2n) local rotations,

when dealing with local Lorentz symmetry.

In order to give a concrete example, let us derive the contribution of a Weyl fermion

with chirality η = ±1 to the non-Abelian gauge anomaly in 4 dimensions. Integrating in

t gives the following anomalous variation of the effective action:

δvΓ(A) = − η

24π2

∫d4x str

[vd(AF − 1

2A3)]. (4.3.7)

In model building and phenomenological applications one is mostly interested to see

whether a given set of currents (global or local) is anomalous or not. This is best seen by

looking at the term of the anomaly with the lowest number of gauge fields, the first term

in square brackets in eq.(4.3.7), the full form of the anomaly being reconstructed using

the Wess-Zumino consistency conditions. In 4 dimensions this term is proportional to the

factor

Dαβγ ≡ 1

2tr(Tα, T βT γ) = 1

3!

∑

perms.

tr (TαT βT γ) . (4.3.8)

When this factor vanishes, the whole gauge anomaly must then vanish.

47

Chapter 5

Lecture V

In this last lecture we will discuss anomalies from a more physical perspective and con-

sider some applications in concrete models. We will mention how gauge anomalies show

up in Feynman diagrams in Minkowski space, discuss the so called ’t Hooft anomaly

matching conditions, and provides a couple of relevant examples: the cancellation of

gauge/gravitational anomalies in the Standard Model and the computation of the axial

anomaly responsible for the π0 decay in QCD coupled to electromagnetism.

Finally we will report the form of gravitational anomalies for non-spin 1/2 fields and

see how they cancel in a certain 10 dimensional suprsymmetric QFT coupled to gravity

(supergravity field theory) coming from string theory.

5.1 Chiral and Gauge Anomalies from One-Loop Graphs

Anomalies were originally discovered by evaluating three-point functions between external

currents at one-loop level [1]. It is clear from the previous path integral derivation that

anomalies are expected from loops of internal fermion lines from current correlations that

involve the chiral matrix γ2n+1. For concreteness let us focus on the four-dimensional

Minkowski space-time. In d = 4 we are led to compute the three point functions between

two vector and one axial abelian current:

Γµνρ(k1, k2) ≡ 〈Jµ,5(−k1 − k2)Jν(k1)Jρ(k2)〉 , (5.1.1)

where Jµ,5 = ψγµγ5ψ, Jν = ψγνψ, Jρ = ψγρψ. For simplicity, we will consider massless

fermions. In this case, the classical conservation of the axial and vector currents would

imply

(k1 + k2)µΓµνρ = kν1Γµνρ = kρ2Γµνρ = 0 . (5.1.2)

48

µ,−k1 − k2

ν, k1

ρ, k2

+

µ,−k1 − k2

ρ, k2

ν, k1

Figure 5.1: One-loop graphs contributing to the anomaly. All external momenta are

incoming. The wavy and dashed lines represent the vector and axial currents, respectively.

Due to the anomaly, it turns out to be impossible to impose all three conditions (5.1.2)

simultaneously. The anomaly arises from the one-loop graphs depicted in fig.5.1. In any

given regularization, the anomaly appears because of some obstruction. For instance, in

DR the anomaly arises from subtleties related to the definition of γ5 in d 6= 4 dimensions

(see, e.g., ref. [14] for a computation of the anomaly in DR). In Pauli-Villars regularization,

where a heavy (PV) fermion is added, the anomaly is related to the explicit breaking of the

axial symmetry given by the PV fermion mass term. There is no regulator that preserves

at the same time the vector and axial symmetries and as a result an anomaly always

appears. Since the correlation among three vector currents is not anomalous, while that

of three axial currents is, it is customary to bring the anomaly in the axial current, keeping

the conservation of the vector currents. In this way, one gets

kµΓµνρ(k1, k2) =1

2π2ǫγρωνk

ω1 k

γ2 . (5.1.3)

The connection between eq. (5.1.3) and the gauge contribution to the anomaly in eq. (3.1.33)

(in the abelian case) is obtained by taking the Fourier transform, two functional deriva-

tives with respect to the gauge field Aµ, and the analytic continuation from euclidean

to minkowskian signature, of eq.(3.1.33). In the non-abelian case, additional one-loop

(square and pentagon) diagrams contribute to the anomaly and will reconstruct the full

non-abelian field strength appearing in the right-hand side of eq. (3.1.33). In 2n space-

time dimensions the chiral anomaly arises from one-loop diagrams involving at least n+1

currents with internal fermion lines running in the loop. Notice that the non-conservation

of a current due to an anomaly only occurs if some (or all) of the external currents are

coupled to gauge fields. If the currents are all associated to global symmetries, relations

like eq.(5.1.3) still hold, but effectively do not lead to any non-conservation of currents.

49

This is evident in the path integral formulation, where the non-conservation of the current

manifestly vanishes in the limit in which the gauge couplings vanish.

Similarly, gauge anomalies occur in the one-loop contribution where chiral fermions

run into the loop of n+ 1-point functions involving the chiral currents

Jαµ = ψγµPηTαψ, (5.1.4)

where Pη is the chiral projector defined in eq.(4.2.2). The resulting computation is very

similar to that obtained using Feynman diagrams for chiral anomalies mentioned before.

The gauge anomaly is proportional to the coefficient Dαβγ defined in eq.(4.3.8). As we

mentioned, massive fermions cannot lead to any gauge anomaly. For simplicity, let us take

the four dimensional case, n = 2. In terms of two-components spinors χ1 and χ2, a mass

term reads

Lm ⊃ χt1σ2Mχ2 + h.c. (5.1.5)

where χ1 and χ2 are in irreducible representations R1 and R2 of the group G, and M is

a non-singular mass matrix. The term (5.1.5) is gauge-invariant if

− T t1M =MT2 , (5.1.6)

where T1 and T2 are the generators in the representations R1 and R2. Eq. (5.1.6) implies

that −T t1 and T2 are related by a similarity transformation, since −T t1 = MT2M−1. We

have then

D1αβγ =

1

2trT1α, T1βT1γ = (−1)3

1

2trT t2α, T t2βT t2γ = −D2

αβγ . (5.1.7)

The contribution to the gauge anomaly given by χ1 is exactly cancelled by that of χ2.

Consider a U(1) gauge theory with a Dirac fermion with charge q. In terms of left-handed

fields, the Dirac fermion consists of one left-handed fermion χ1 with charge q and its

conjugate χ2 with charge −q and obviously admits the mass term mχ1σ2χ2 + h.c.. The

total gauge anomaly is proportional to

+ q3 + (−q)3 = 0 . (5.1.8)

Similarly, Dirac fermions in any representation Tr of a gauge group consist of one left-

handed fermion multiplet in the representation Tr and its conjugate in the complex con-

jugate representation Tr = −T tr and thus do not lead to anomalies. In manifestly parity-

invariant theories the absence of any anomaly is pretty obvious since all currents are

manifestly vector-like (i.e no γ5 appears in a 4-component Dirac notation). Non-trivial

anomalies can only arise in non-parity-invariant theories, namely in so called chiral gauge

50

theories, where a fermion and its complex conjugate transform in representations of the

gauge group that are not complex conjugate between each other. The absence of anoma-

lies in chiral gauge theories requires a non-trivial cancellation between different fermion

multiplets. The most important example of a theory of this sort is the SM, where gauge

anomalies cancel between quarks and leptons, as we will see in the next section.

5.2 A Relevant Example: Cancellation of Gauge Anomalies in the SM

The SM gauge group is GSM = SU(3)×SU(2)×U(1). Its fermion content, in terms of left-

handed fields, is composed by three copies (generations) of the following representations:

(3,2) 16+ (3,1)− 2

3+ (3,1) 1

3+ (1,2)− 1

2+ (1,1)1 (5.2.1)

corresponding to the quark doublet, up quark singlet, down quark singlet, lepton doublet

and charged lepton singlet, respectively. In principle there can be ten possible kinds of

gauge anomalies, associated to all possible combinations of SU(3) SU(2) and U(1) currents

in the trangular graph. Five of them, where a non-abelian group factor (SU(3) or SU(2))

appears only once, SU(3)2×SU(2), SU(3)×SU(2)2, SU(3)×SU(2)×U(1), SU(3)×U(1)2

and SU(2) × U(1)2 are trivially vanishing, since for SU(n) groups the generators are

traceless: trT = 0. The SU(2)3 anomaly is also manifestly vanishing because for SU(2)

the symmetric factor Dαβγ vanishes. In the case at hand, with doublets only, this is

easily seen: Dαβγ = 1/2trTα, T βT γ = δαβtrT γ = 0. In the general case, Dαβγ vanishes

because all SU(2) representations are equivalent to their complex conjugates, namely there

exists a matrix A such that T t = T ⋆ = −ATA−1. Using this relation, one immediately

sees that Dαβγ = 0 for any representation.

The remaining combinations SU(3)3, SU(3)2×U(1), SU(2)2×U(1) and U(1)3 have to

be checked. Let us then compute the values of the symmetric coefficients Dαβγ in each of

the above 4 cases and show that they always vanish. In order to distinguish the different

coefficients, we denote by a subscript c, w and Y the SU(3), SU(2) and U(1) factors,

respectively. It is enough to consider a single generation of quarks and doublets, because

the cancellation occurs generation per generation. Let us start with the SU(3)3 anomaly.

Only quarks contribute to it. We get

Dαβγccc = 2Dαβγ

3+Dαβγ

3+Dαβγ

3= 2Dαβγ

3−Dαβγ

3−Dαβγ

3= 0 , (5.2.2)

using that Dαβγ3

= −Dαβγ3

. For SU(3)2 × U(1) we have

DαβccY = 2tr3 T

αT β × 1

6+ tr3 T

αT β ×(− 2

3

)+tr3 T

αT β × 1

3= tr3 T

αT β(13− 2

3+

1

3

)= 0 ,

(5.2.3)

51

with tr3 TαT β = tr3 T

αT β. For SU(2)2 × U(1), only doublets contribute. We get

DαβwwY = 3tr2 T

αT β × 1

6+ tr2 T

αT β ×(− 1

2

)= tr2 T

αT β(12− 1

2

)= 0 . (5.2.4)

For U(1)3 anomalies all quarks and leptons contribute and one gets

DY Y Y = 3× 2×(16

)3+ 3×

(− 2

3

)3+ 3×

(13

)3+ 2×

(− 1

2

)3+ (1)3 = 0 . (5.2.5)

Let us now check the cancellation of anomalies involving both gauge currents and energy-

momentum tensors, namely mixed gauge/gravitational anomalies. In four dimensions,

the only possibly anomalous set of currents is obtained by taking one gauge current and

two energy-momentum tensors. In this case Dαβγ reduces to the trace for a single gauge

generator tr Tα, being the Lorentz indices all the same for left-handed spin 1/2 fermions.

The only non-trivial anomaly can arise for U(1) factors. It is proportional to∑

n qn, where

n runs over all fermions with charges qn. In the SM, the mixed U(1)-gravitational anomaly

is then proportional to

3× 2× 1

6+ 3×

(− 2

3

)+ 3× 1

3+ 2×

(− 1

2

)+ 1 = 0 . (5.2.6)

Notice how the fermion charges nicely combine to give a vanishing result for all the anoma-

lies, in particular the U(1)3 and the mixed U(1)-gravitational ones.

The SM is then a chiral, anomaly-free gauge theory. More precisely, we have shown

that the SM is anomaly-free in the unbroken phase where all the gauge group is linearly

realized (i.e. no Higgs mechanism is at work). Since the anomaly does not depend on

the Higgs field, our results automatically imply that the SM is anomaly-free in the broken

phase as well. In particular, the unbroken gauge group SU(3) × U(1)EM is manifestly

anomaly-free being all currents vector-like.

5.3 ’t Hooft Anomaly Matching and the Wess-Zumino-Witten Term

Asymptotically free gauge theories are strongly coupled in the IR and can give rise to

confinement of its fermion constituents, like quarks in QCD.

At low energies the propagating degrees of freedom are bound states of the elementary

high energy (UV) states, such as mesons and hadrons in QCD. What is the fate at low

energies of possible global anomalies coming from the elementary constituents at high

energy? t’Hooft has answered to this question by arguing that anomalies must arise in

the low energy effective theory as well. More precisely, the so called ’t Hooft anomaly

matching conditions state that if the UV theory has a set of global symmetries – neither

broken by gauge anomalies nor spontaneously broken (unlike chiral symmetries in QCD)

52

– with a non-vanishing anomaly coefficient Dαβγ induced by the elementary UV fermion

fields, then the IR theory must contain in its low energy spectrum massless fermion bound

states that exactly reproduce the UV global anomaly. ’t Hooft’s argument is very simple.

Let us assume of (weakly) gauging the unbroken global symmetries of the UV theory. In

this way the theory will have gauge anomalies and would then be inconsistent. We might

cancel the anomaly by adding suitable additional massless “spectator” fermions, neutral

under the strong gauge group but charged under the weakly gauged symmetries. At low

energies, when the strong gauge group confines, the spectrum of the theory will include

the IR bound states plus the “spectator” fermions that, being neutral, are unaffected

by the condensation of the strong gauge group. The UV theory with the spectators is,

by construction, consistent and it has to remain so for all values of the gauge coupling

constant. Hence, at low energies, there must be an anomaly contribution canceling that

of the fermion spectators. Since, by assumption, no symmetry is spontaneously broken,

no Goldstone bosons appear and only massless fermion bound states can possibly cancel

the anomaly. The argument is valid for an arbitrarily weak gauging and thus apply also

for global symmetries.

’t Hooft anomaly matching conditions do not apply if the global symmetries are spon-

taneously broken. Indeed, when (approximate) exact global symmetries are spontaneously

broken, Goldstone’s theorem ensures that (light) massless scalar particles appear at low

energies and these can replace in t’Hooft argument the massless fermion bound states

that are no longer required. In this situation the anomaly of the UV fermions must be

reproduced by the effective action of the (pseudo) Goldstone bosons.

The latter situation is actually realized in Nature in QCD. In presence of nf massless

quarks, QCD has an SU(nf )V ×SU(nf)A×U(1)V ×U(1)A global symmetry, spontaneously

broken to SU(nf )V ×U(1)V by the quark condensate. Let us compute the possible anoma-

lies that we can have by looking at the Dαβγ coefficients associated to the various groups.

Since quarks are vector-like, the gauge SU(3)3 anomalies all cancel. Anomalies with one

SU(3) non-abelian current of any kind also manifestly vanish, because trT = 0. The only

possible non-vanishing anomalies are of the form SU(3)2×U(1)V and SU(3)2×U(1)A. In

terms of left-handed quarks, a quark and its conjugate have opposite charges with respect

to U(1)V and equal charges with respect to U(1)A, thus the SU(3)2 × U(1)V anomaly

vanishes while SU(3)2×U(1)A does not. In QCD, then, the U(1)A symmetry, in addition

of being spontaneously broken by the condensate, is also explicitly broken by the anomaly.

The latter breaking is large when the theory becomes strongly coupled (due to instantons)

and is responsible for the absence of a light η′ in the QCD scalar meson spectrum.

Other anomalies are possible if one considers also the electromagnetic interactions.

53

They explicit break the U(nf )A axial symmetries and thus anomalies involving one non-

abelian SU(nf )A factor are no longer necessarily vanishing (U(nf )V is still anomaly free).

Consider for simplicity nf = 2 (up and down quarks only) and consider the U(1)2EM ×U(1)A axial anomaly, where U(1)A ⊂ SU(2)A is the abelian subgroup generated by T 3.

We easily get

∂µJµA = − Nc

16π2ǫµνρσFµνFρσ

[(2e3

)2× 1 +

(−e3

)2× (−1)

]

= −Nce2

48π2ǫµνρσFµνFρσ . (5.3.1)

Under the U(1)A chiral transformation above, the low-energy effective chiral Lagrangian

Lπ describing the dynamics of the π mesons interacting with photons should then not

vanish, but rather reproduce the axial anomaly (5.3.1). Considering that δǫπ0 = ǫfπ, Lπ

should include the coupling

Lπ ⊃ − Nce2

48π2fπǫµνρσFµνFρσπ

0 . (5.3.2)

The axial anomaly (5.3.1) has allowed to resolve the puzzle of the π0 → 2γ decay. In

absence of any anomaly, the term (5.3.2) would still appear in the chiral Lagrangian Lπbut with a much more suppressed coupling. On the contrary, anomaly considerations

uniquely fix its coefficient and it turns out that the experimental rate Γ(π0 → 2γ) is

successfully reproduced with Nc = 3. The chiral Lagrangian Lπ should be described in

terms of the matrix of fields U = exp(2πaT a/fπ), rather than by the π’s mesons themselves.

The anomalous term (5.3.2) should then be rewritten in terms of the U ’s. We will not

discuss the details of how to perform this rewriting. The ending result goes under the

name of the gauged version of the Wess-Zumino-Witten term. The latter includes many

other couplings, in addition to the term (5.3.2).

5.4 Gravitational Anomalies for Spin 3/2 and Self-Dual Tensors

Chiral spin 1/2 fermions are not the only fields giving rise to gravitational anomalies.

In order to understand which other fields might possibly lead to anomalies, let us come

back to the explicit form of the gauge anomaly in eq.(4.3.6). In our notations the gauge

generators are anti-hermitian, and hence Ω(1)2n is real in any dimension. Using eqs.(3.2.1)

and (3.2.4), this implies that gauge anomalies affect only the imaginary part of the eu-

clidean effective action Γ. Vector-like fermions do not give rise to anomalies and lead to

a real effective action. So we conclude that only the imaginary part of Γ can be affected

by anomalies. The same reasoning applies to gravitational anomalies (consider them as

54

O(2n) gauge theories). From this reasoning we conclude that fields transforming in a

complex representation of O(2n) might lead to anomalies. The group O(N) admits com-

plex representations only for N = 4n + 2, and hence only in d = 4n + 2 dimensions we

might expect purely gravitational anomalies, in agreeement with our result (4.2.15) for

spin 1/2 fermions. Chiral spin 3/2 fields also transform under complex representations of

O(4n + 2). Interestingly enough, (anti)self-dual tensors, 2n + 1-forms, also have complex

representations in 4n + 2 dimensions. This is best seen by looking at the euclidean self-

duality condition for forms: ⋆Fn = Fn, where ⋆ is the Hodge operation on forms. Given a

form ωp in d dimensions, ⋆ωp is the m− p-form given by

⋆ ωp =

√g

p!(d− p)!ωµ1...µpǫ

µ1...µpν1...νd−pdx

µ1 ∧ . . . ∧ dxµd−p , (5.4.1)

where ǫµ1...µd is the completely antisymmetric tensor, with ǫ1...d = +1. It is easy to see

that ⋆ ⋆ ωp = (−1)p(d−p)ωp. Specializing for d = 2n and p = n, gives ⋆ ⋆ ωn = (−1)n2ωn.

This is 1 for n even and −1 for n odd, hence in 4n and 4n+ 2 dimensions (anti)self-dual

tensors transform in real and complex representations of O(d), respectively.

It is not clear how to consistently get a QFT with (anti)self-dual tensors charged un-

der gauge fields. Also, in most relevant theories, spin 3/2 fields are neutral under gauge

symmetries. We will then consider the contribution of these fields to pure gravitational

anomalies only. We very briefly outline the derivation, referring the reader to ref.[3] for

further details. As for spin 1/2 fermions, it turns out that the gravitational anomalies

induced by fields can be obtained starting from chiral anomalies in two dimensions higher

and then apply the Stora-Zumino descent relations. The chiral anomaly can again be

efficiently computed using an auxiliary supersymmetric quantum mechanical model. For

spin 3/2 fermions, the vector index µ can be seen as a gauge field in the fundamental

representation of the group, the connection being ωabµ and the group O(d). The asso-

ciated quantum mechanical model is then given by eq.(3.1.15), with AAµ B → iωaµ b and

FABµν → iRabµν . In terms of the skew-eigenvalues xi of the curvature two-form Rab defined

in eq.(1.3.26), the chiral anomaly in 2n dimensions is found to be

Z3/2 =

∫

M2n

(treR2π − 1) A(R) =

∫

M2n

(2

n∑

j=1

coshxj − 1) n∏

k=1

xk/2

sinh(xk/2)(5.4.2)

where the −1 comes from the need to subtract the spin 1/2-contribution. Anomalies

of (anti)self-dual forms are more involved. Roughly speaking, one adds the anomaly

contribution of all other antisymmetric fields. This does not change the anomaly, since the

total contribution of these extra states sum up to zero. The whole series of antisymmetric

fields is equivalent to a bifermion field ψαβ . This is understood by recalling that a matrix

55

Aαβ can be expanded in the basis of matrices given by antisymmetric product of gamma

matrices, eq.(1.2.13). The analogue of the chiral matrix γ2n+1 in eq.(3.1.12) is played by

the Hodge operator ⋆ and the Hamiltonian is the Laplacian operator on generic p-forms.

By computing the path integral associated to the resulting super quantum mechanical

model one gets

ZA = −1

8

∫

M2n

n∏

k=1

xktanhxk

. (5.4.3)

With these result at hands we can verify that purely gravitational anomalies do indeed

cancel in a non-trivial way in a ten-dimensional gravitational theory, low-energy effective

action of a string theory, called IIB theory. The spectrum of this theory includes a chiral

spin 1/2 fermion, a spin 3/2 fermion with opposite chirality and a self-dual five-form

antisymmetric field. The total gravitational anomaly is obtained by taking the sum

ZIIB = ZA + Z3/2 −Z1/2 , (5.4.4)

evaluating each term for n = 5 and keeping in the expansion of the series only the form of

degree twelve. This is the relevant form that, after the Stora-Zumino descent procedure,

would give rise to the gravitational anomaly ten-form in ten dimensions. Remarkably, the

coefficient of this form exactly vanishes in the sum (5.4.4), proving that the IIB theory is

free of any gravitational anomaly.

56

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