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
1
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.
4
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)
7
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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