Anomalies in Quantum Field Theory

Home , Chiral anomaly, Gauge anomaly, Global anomaly, Gravitational anomaly

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 VierbeinsandSpinors ...... 8 1.3 CharacteristicClasses ...... 12

2 Lecture II 18 2.1 SupersymmetricQuantumMechanics...... 18 2.2 AddingGaugeFields...... 22 2.3 SymmetriesinQFT ...... 24 2.3.1 WTIdentitiesinQED...... 28

3 Lecture III 30 3.1 TheChiralAnomaly ...... 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 ExplicitformofGaugeAnomaly ...... 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 theSM ..... 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 anomalies 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 unusual, 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 matching 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 gravitational 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-Gaumèand 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 of dimension d, a tensor of type (p,q) is given by M

µ1...µp ∂ ∂ ν1 νq T = T ν ...νq . . . dx ...dx , (1.1.1) 1 ∂xµ1 ∂xµp where µi,νi = 1,...,d. A manifold is called Riemannian if it admits a positive definite µ ν symmetric (0,2) tensor. We call such tensor the metric g = gµν dx dx . 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 has no torsion (as will be assumed from now on), then ∂ V V , M µ ν → ∇µ ν where 1 V = ∂ V Γρ V , Γρ = gρσ(∂ g + ∂ g ∂ g ) . (1.1.2) ∇µ ν µ ν − µν ρ µν 2 µ νσ ν µσ − σ µν ρ The (not tensor) field Γµν is called the Levi-Civita connection. In terms of it, we can construct the fundamental curvature (or Riemann) tensor

Rµ = ∂ Γµ ∂ Γµ +Γα Γµ Γα Γµ . (1.1.3) νρσ ρ νσ − σ νρ σν αρ − ρν ασ

5 Other relevant tensors constructed from the Riemann tensor are the Ricci tensor Rνσ = µ νσ R νµσ and the scalar curvature R = g Rνσ. 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 1 ω = ω dxµ1 . . . dxµp . (1.1.4) p p! µ1...µp ∧ ∧ The wedge product in eq.(1.1.4) represents the completely antisymmetric product of the ∧ basic 1-forms dxµi :

dxµ1 . . . dxµp = sign (P )dxµP (1) ...dxµP (p) , (1.1.5) ∧ ∧ PX∈Sp 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 d! = . (1.1.6) p (d p)!p! ! − The total vector space spanned by all p-forms, p = 0,...,d, equals

d d = 2d . (1.1.7) p=0 p ! X Given a p-form ω and a q-form χ , their wedge product ω χ is a (p + q) form. From p q p ∧ q eq.(1.1.5) one has

pq ωp χq = ( ) χq ωp ∧ − ∧ (1.1.8) (ω χ ) η = ω (χ η ) . p ∧ q ∧ r p ∧ q ∧ r One has clearly ω χ =0 if p + q > d and ω ω =0 if p is odd. Given a p-form ω , p ∧ q p ∧ p p we deﬁne the exterior derivative dp as the (p + 1)-form given by 1 d ω = ∂ ω dxµp+1 dxµ1 . . . dxµp . (1.1.9) p p p! µp+1 µ1...µp ∧ ∧ ∧ Notice that in eq.(1.1.9) we have the ordinary, rather than the covariant derivative, because upon antisymmetrization their action is the same:

ω = ∂ ω Γα ω + . . . = ∂ ω , (1.1.10) ∇[µp+1 µ1...µp] [µp+1 µ1...µp] − [µp+1µ1 α|µ2...µp] [µp+1 µ1...µp] the Levi-Civita connection being symmetric in its two lower indices. Given eq.(1.1.8), we have d (ω χ )= d ω χ + ( )pω d χ . (1.1.11) p+q p ∧ q p p ∧ q − p ∧ q q

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 d ω = 0 and exact if ω = d χ for some (p 1)- p p p p−1 p−1 − form χ . Clearly, if ω = d χ , then d ω = 0 (exact closed) but not viceversa. p−1 p p−1 p−1 p p → Correspondingly, the image of dp−1, the space of p-forms such that ωp = dp−1χp−1, is included in the kernel of d , the space of p-forms such that d ω =0: Im d Ker d . p p p p−1 ⊂ p The coset space Ker dp/Im dp−1 is called the de Rham cohomology group on the manifold . By Stokes theorem, the integral of an exact p-form on a p-dimensional cycle of M Cp M with no boundaries necessarily vanish:

ωp = dpχp−1 = χp−1 = 0 . (1.1.12) ZCp ZCp Z∂Cp

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 µ one-form A = Aµdx , with 1 1 F = dA = (∂ A ∂ A )dxµ dxν = F dxµ dxν (1.1.13) 2 µ ν − ν µ ∧ 2 µν ∧ 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 a a a µ A = A T , A = Aµdx , (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

[ω ,χ ] ωa χb[T a, T b]= ωa χbT aT b ( )pqχb ωa T bT a . (1.1.15) p q ≡ p ∧ q p ∧ q − − q ∧ p 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 1 1 ω ω = [T a, T b]ωa ωb dxµ1 . . . dxµ2p = [ω ,ω ] . (1.1.16) p ∧ p 2 µ1...µp µp+1...µ2p ∧ ∧ 2 p p 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−1F g 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 obtained by looking at eq.(1.1.1). For instance, under a diffeomorphism xµ yµ(x), a vector → transforms as ∂xµ V µ(x) V ν(y) . (1.2.1) → ∂yν 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 a b gµν (x)= eµ(x)eν (x)δab , (1.2.2)

1 a where a, b = 1,...,d. The ﬁelds eµ(x) are called vierbeins. They are not uniquely deﬁned by eq.(1.2.2). If we perform the local Lorentz transformation

ea (x) (Λ−1)a (x)eb (x) , (1.2.3) µ → b µ 1 In 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 transformations, viceversa for the second: invariant under diffeomorphisms and transforms under local Lorentz transformations. They are sometimes denoted curved and flat indices, respectively. 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 g = (det ea )2. Since the metric is positive ≡ µν µ definite, this implies that ea is a non-singular m m matrix at any point. We denote its µ × µ µ b ab µ a µ µ a inverse by ea : ea eµ = δ , ea eν = δν . It is easy to see that ea is obtained from eµ by using µν µ µν a the inverse metric g : ea = g eν. Curved indices are lowered and raised by the metric µν gµν and its inverse g . In euclidean space the position of flat indices does not matter, 2 µ being raised and lowered by the identity matrix. The vierbeins ea can be seen as a set of orthonormal (1,0) vectors in the tangent space. As such, we can decompose any vector V µ µ µ a a in their basis: V = ea V . Similarly for the (0,1) vectors in cotangent space: Vµ = eµVa. 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 = T a1...ap E ...E eb1 ...ebq , (1.2.4) b1...bq a1 ap where ∂ E = eµ , ea = ea dxµ . (1.2.5) a a ∂xµ µ We can then trade, as already mentioned, diffeomorphisms for local Lorentz transforma- a tions. The vierbeins eµ 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 a of dimension d. The vierbein components eµ 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 ab ωµ the components of the connection. They transform in the adjoint representation of ab ba SO(d), which can be represented as an antisymmetric matrix ωµ = ωµ , a, b = 1,...d. − ρ It is called the spin connection and is the analogue of the Levi-Civita connection Γµν for diffeomorphisms. The Levi-Civita connection is a function of the metric, and thus we ex- ab pect that ωµ 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 + ωa eb = 0 . (1.2.6) b ∧ 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 a a the name of torsion-free condition. We will not need the explicit expression of ω b = ω b(e) obtained by solving eq.(1.2.6). Derivative of tensors with ﬂat indices are then promoted to covariant derivatives: a a a b DµT = ∂µT + ωµ,bT . (1.2.7) ρ If we denote by the covariant derivative T = ∂ T ΓµνT , the property below ∇µ ∇µ ν µ ν − ρ follows.

Exercise 2: Using the property that ea + ωa eb = 0, prove that ∇µ ν µ,b ν T = eaD T . (1.2.8) ∇µ ν ν µ a

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

Ra = Dωa = dωa + ωa ωc . (1.2.9) b b b c ∧ b ρ ρ σ a a b By using eq.(1.2.8) and the properties [ , ]V = R µνσV , [D , D ]V = R T , we ∇µ ∇ν µ ν µν,b have a a ρ σ Rµν,b = eσeb R µνρ . (1.2.10) A differential form notation can also be introduced to describe the Levi-Civita connection µ µ ρ (1.1.2) and the Riemann tensor (1.1.3). One defines the matrix of one-forms Γν =Γ ρνdx and the curvature two-form 1 Rµ = dΓµ +Γµ Γσ = Rµ dxρ dxσ . (1.2.11) ν ν σ ∧ ν 2 νρσ ∧ 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) { } a µ µ µ a Notice that we have written γ and not γ because in a curved space γ (x)= ea (x)γ = 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 1 γa1...ap = γa1 . . . γap (p 1)! perms (1.2.14) p! ± − is completely antisymmetrized in the indices a1 . . . ap. The total number of elements of the base is d d = 2d = 2d/2 2d/2 . (1.2.15) × p=0 p ! X 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 a deﬁne a matrix γd+1, commuting with all other γ ’s:

d n a γd+1 = i γ . (1.2.16) a=1 Y n 2 The factor i ensures that γd+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 γ ψ = ψ . Weyl spinors in d dimensions + − d+1 ± ± ± have then 2d/2−1 components. Using eq.(1.2.12), the matrices 1 1 J ab = [γa, γb]= γab (1.2.17) 4 2 are shown to be (anti-hermitian) generators of SO(d). Under a global Lorentz transformation, a spinor ψ transforms as follows:

1 λ Jab ψ e 2 ab ψ , (1.2.18) → where λ are the d(d 1)/2 parameters of the transformation. The covariant derivative ab − of a fermion in curved space is obtained by replacing 1 ∂ D = ∂ + ωabJ . (1.2.19) µ → µ µ 2 µ ab The Lagrangian density of a massive Dirac fermion coupled to gravity is ﬁnally given by

= e ψ¯ ieµγaD m ψ , (1.2.20) Lψ a µ − where e det ea . ≡ µ

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 Q (F ) tr F n. Using the results of Exercise 1, it is easy to 2n ≡ show that Q2n(F ) is closed: n−1 dQ (F )= tr F i[A, F ]F n−i−1 = tr [A, F n] = 0 . (1.3.1) 2n − − Xi=0 Given two field strengths F an F ′, the difference Q (F ) Q (F ′) = dχ for some 2n − 2n 2n−1 2n 1-form χ . The integral of Q (F ) over a 2n-dimensional sub-manifold without − 2n−1 2n C2n boundary of the manifold (or over the entire ) does not depend on F : M M ′ Q2n(F )= Q2n(F ) , (1.3.2) ZC2n ZC2n and defines a cohomology class, called the characteristic class of the polynomial Q2n(F ). Let us prove that Q (F ) Q (F ′) is an exact form. Although the proof is a bit involved, it 2n − 2n 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 A = A + t(A′ A) , t [0, 1] . (1.3.3) t − ∈ We clearly have A = A, A = A′. Let us also denote by θ = A′ A the difference of the 0 1 − two connections. We have (recall eq.(1.1.15))

2 2 2 2 Ft = dAt + At = dA + tdθ + (A + tθ) = F + t(dθ + [A, θ]) + t θ . (1.3.4) We can also write 1 1 ′ d dFt n−1 Q2n(F ) Q2(F )= dt Q2n(Ft)= n dt tr Ft − 0 dt 0 dt Z 1 Z (1.3.5) n−1 2 n−1 = n dt tr DθFt + 2t tr θ Ft . 0 Z On the other hand, we have n−2 dtr θF n−1 = tr dθF n−1 tr θF idF F n−i−2 (1.3.6) t t − t t t Xi=0 n−2 = tr DθF n−1 tr [A, θ]F n−1 tr θF iDF F n−i−2 tr θF i[A, F ]F n−i−2 . t − t − t t t − t t t Xi=0 12 The second and fourth terms in eq.(1.3.6) combine into a total commutator that vanishes n−1 inside a trace: tr [A, θFt ] = 0. The derivative DFt equals

DF = dF + [A, F ]= dF + [A A + A, F ]= D F t[θ, F ]= t[θ, F ] , (1.3.7) t t t t t − t t t t − t − t given that DtFt = 0. We also have

n−2 0 = tr [θ,θF n−1] = tr 2θ2F n−1 tr θF i[θ, F ]F n−i−2 . (1.3.8) t t − t t t Xi=0 Plugging eqs.(1.3.7) and (1.3.8) in eq.(1.3.6) gives

n−1 n−1 2 n−1 dtr θFt = tr DθFt + 2t tr θ Ft . (1.3.9)

We have then proved that Q (F ′) Q (F )= dχ , with 2n − 2 2n−1 1 n−1 χ2n−1 = n dt tr θFt . (1.3.10) Z0 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

iF ch (F ) tr e 2π . (1.3.11) ≡ The Taylor expansion of the exponential gives rise to the series of 2n-forms denoted by nth Chern characters 1 iF n ch (F )= tr . (1.3.12) n n! 2π 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

F = 0 . (1.3.13) 2 6 ZS 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

A = ig(1 cos θ)dφ , A = ig(1 + cos θ)dφ . (1.3.14) N − − S

2 More precisely, we split S =ΣN +ΣS,ΣN,S being the northern and southern hemispheres, and compute

F = dAN + dAS = (AN AS) , (1.3.15) 2 − ZS ZΣN ZΣS I where the closed integral is around the equator at θ = π/2. From the exercise 3 we have A A = 2igdφ and hence N − S − i 1 2π F = dφ(2g) = 2g . (1.3.16) 2π 2 2π ZS Z0 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 A F eHS1 = eRD , (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 H A R F − R F (R + R )F R F e S1 = e ΣN = e ΣS = e DN DS = e S2 = 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 a curvature two-form R b 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 R p(R) = det 1+ . (1.3.21) 2π 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

0 λ1

 λ1 0  − Rab =  ...  , (1.3.22)    0 λ   d/2     λd/2 0   −    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 1 1 2 p (R)= tr R2 , 1 −2 2π (1.3.23) 1 1 4 p (R)= (tr R2)2 2tr R4 . 2 8 2π − 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 det R/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

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

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

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

2 χ(S )= p1(R)=+2 . (1.3.25) 2 ZS 15 The characteristic class entering directly in the evaluation of anomalies is the so-called roof-genus, deﬁned as

d/2 xk/2 1 1 2 Aˆ(R)= = 1 p1(R)+ (7p1(R) 4p2(R)) + . . . (1.3.26) sinh(xk/2) − 24 5760 − kY=1 where xk = λk/2π. An important theorem by Atiyah and Singer, called index theorem, relates the spectral properties of differential operators on a manifold with its topology (fibers included). M 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 1 D/ = eµγa ∂ + ωabJ + AaT a . (1.3.27) R a µ 2 µ ab R In eq.(1.3.27) the subscript refers to the representation of the fermion under the (un- R a specified) gauge group G, and TR 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 = indexD/ = ch (F )Aˆ(R) , (1.3.28) + − − R R ZM where iF ch (F ) tr e 2π . (1.3.29) R ≡ R 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 = , one has ch (F ) =ch (F )ch (F ). For example, for G = SU(N), we have R R1 ⊗R2 R R1 R2 fund. fund. = adj. 1. Correspondingly, ⊗ ⊕ ch (F ) = ch(F )ch( F ) 1 , (1.3.30) Adj. − − where the Chern character without subscript refers to the fundamental representation and we used the fact that ch (F ) = ch( F ). Expanding eq.(1.3.30) we have fund. − 1 1 N 2 1 tr F 2 + . . . = (N tr F 2 + . . .)2 1 , (1.3.31) − − 2 Adj. − 2 −

16 from which we deduce the known result

a b a b trAdj.t t = 2N tr t t . (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):

2 H = (iD/ R) . (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 = ∂2 = = p2, namely twice − µ −∇ 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 p/2. The “spinor” structure is simply absent. This problem is ﬁxed by adding fermions. The resulting quantum 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 1 i L = x˙ µx˙ µ + ψµψ˙ µ . (2.1.2) 2 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 notice that the system described by the lagrangian (2.1.2) is invariant under the following (supersymmetric) transformation:

δxµ = iǫψµ , (2.1.3) δψµ = ǫx˙ µ , − where ǫ is a constant Grassmann variable, ǫ2 = 0, anticommuting with ψµ. It is easy to check that. We have i i i d δL =x ˙ µiǫψ˙µ + ( ǫx˙ µ)ψ˙µ + ψµ( ǫx¨µ)= ǫ (x ˙ µψµ) . (2.1.4) 2 − 2 − 2 dt 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 momentaR and impose quantization. We have dL pµ = =x ˙ µ , dx˙ µ dL i (2.1.5) πµ = = ψµ . dψ˙µ 2 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, ψν = πµ,πν = 0 , ψµ,πν = iδµν (2.1.6) { }P { }P { }P 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 i χµ πµ ψµ = 0 . (2.1.7) ≡ − 2 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 interpretation 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,χν cµν = δµν . A consistent quantization { }P ≡ 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, B A, B A, χM c−1 χN ,B , (2.1.8) { }D ≡ { }P − { }P MN { }P 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 1 i ψµ, ψµ = δµν , πµ,πµ = δµν , ψµ,πµ = δµν . (2.1.9) { }D { }D −4 { }D 2 The Hamiltonian of the system is given by (pay attention to the order of the fermion ﬁelds) 1 1 H = p x˙ µ + π ψ˙µ L = p2 = , (2.1.10) µ µ − 2 µ −2∇ namely it is just the standard free hamiltonian in absence of fermions! The fermion generator of SUSY is given by

Q = ψµx˙ µ = ψµpµ . (2.1.11) − − We can easily check that it leads to the correct transformation properties:

δxµ = [ǫQ,xµ]= ǫψν [pν,xµ]= iǫψµ , − (2.1.12) δψµ = [ǫQ,ψµ]= [ǫψνpν, ψµ]= ǫpν ψν , ψµ = ǫpµ = ǫx˙ µ . − − { } − − 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: 1 ψµ γµ . (2.1.13) → √2 The anti-commutator of two supercharges equal

Q,Q = ψµpµ, ψν pν = p2 = 2H = Q2 = H . (2.1.14) { } { } µ ⇒ Plugging eq.(2.1.13) in the supercharge Q in eq.(2.1.11) gives

i µ 1 Q = γ ∂µ = i∂/ , (2.1.15) √2 √2

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 | i | i bosonic state B , Q B is a fermionic one. Viceversa, if F is a fermionic state, then | i | i | i Q F is a bosonic one. States are grouped in multiplets that rotate as spinors do in d | i 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 γ = 1 and γ = 1 will be denoted bosons and d+1 d+1 − 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 = δψµx˙ µ ψµδx˙ µ = ǫx˙ µx˙ µ ψµ(iǫψ˙µ) = 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 . The SUSY transfor- M mations (2.1.3) do not depend on the metric of and are unchanged. Correspondingly M Q should be given by the straightforward generalization of eq.(2.1.11):

Q = g (x)ψµx˙ ν . (2.1.17) − µν 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

1 i L = g (x)x ˙ µx˙ ν + g (x)ψµ ψ˙ ν +Γν (x)ψρx˙ σ , (2.1.18) g 2 µν 2 µν ρσ ν where Γρσ 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 µ µ a ψ = ea ψ , so that d d ψν = (eν ψa) = (∂ eν)x ˙ µψa + eν ψ˙a , (2.1.19) dt dt a µ a a 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

µ ˙ν ν ρ σ a a ν µ b ν ˙b ν ρ b µ gµν ψ ψ +Γρσ(x)ψ x˙ = ψ eν (∂µeb )x ˙ ψ + eb ψ +Γρµeb ψ x˙ ˙a b µ a ν a ρ ν = ψaψ + ψaψ x˙ (eν∂µeb + eρΓνµeb ) (2.1.20) = ψ ψ˙a + ψ ψbx˙ µeν( ∂ ea +Γν eρ) a a b − µ ν ρµ b 1 = ψ ψ˙a + [ψ , ψ ]x ˙ µωab , a 2 a b µ where in the last step we used the vielbein postulate ea + ωa eb = 0 (see exercise 2). ∇µ ν µ,b ν The Lagrangian (2.1.18) becomes

1 i i L = g (x)x ˙ µx˙ ν + ψ ψ˙a + [ψ , ψ ]ωabx˙ µ . (2.1.21) g 2 µν 2 a 4 a b µ The momentum conjugate to xµ is now modiﬁed: i p = g x˙ ν + [ψ , ψ ]ωab . (2.1.22) µ µν 4 a b µ

Using eq.(2.1.22) and replacing p i∂ , ψa γa/√2 in the expression for Q in µ → − µ → eq.(2.1.17), we get the natural curved space generalization of eq.(2.1.15):

1 µ a 1 ab 1 Q = ie γ ∂µ + ω [γa, γb] = iD/ . (2.1.23) √2 a 8 µ √2 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 ∗ A fermionic degrees of freedom cA and c , where A runs from one up to the dimension of the representation associated to eq.(1.3.27). The fields cA and c∗ transform then under the R A representation and its complex conjugate of the gauge group G, respectively. The R R Lagrangian associated to these fields is i 1 L = (c∗ c˙A c˙∗ cA)+ ic∗ AA x˙ µcB + c∗ cBψµψν F A , (2.2.1) A 2 A − A A µ B 2 A µνB A α α A AB α α A where Aµ B = Aµ(T ) B, Fµν = Fµν (T ) B are the connection and field strength associated to the group G, with generators T α. In order to simplify the notation, it is quite

22 convenient to deﬁne the one-form A A ψµ and two-form F = F ψµψν and suppress ≡ µ µν the gauge indices. In this more compact notation the Lagrangian (2.2.1) reads

i L = (c∗c˙ c˙∗c)+ ic∗A c x˙ µ + c∗F c . (2.2.2) A 2 − µ 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

i ∗ i π = c , π ∗ = c . (2.2.4) c 2 c 2 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 π and π ∗ : χ = π ic /2 = 0, χ ∗ = π ∗ ic/2 = c c c c − c c − 0. Using eq.(2.1.8), it is straightforward to find the consistent Dirac (anti)commutation B ∗ relations between the fields c and cA:

⋆ B B cA, c = δA . (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⋆ 0 , 2-particle states c⋆ c⋆ 0 , | i A| i A B| i etc. Among all these states, only the 1-particle states correspond to the representation R of the gauge group G, the vacuum being a singlet of G and multiparticle states leading to tensor products of the representation . On such particle states, the operator c∗ (tα)A cB R A B α A acts eﬀectively as (t ) B. The supercharge is given by eq.(2.1.17), but the momentum µ conjugate to x gets another term, coming from LA: i p = g x˙ ν + [ψ , ψ ]ωab + ic∗A c . (2.2.6) µ µν 4 a b µ µ 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

1 µ ν i ˙ a i ab µ L = gµν x˙ x˙ + ψaψ + ψa, ψb ωµ x˙ 2 2 4 (2.2.7) i 1 + (c⋆ c˙A c˙⋆ cA)+ ic∗ AA x˙ µcB + c⋆ cBψaψbF A , 2 A − A A µ B 2 A ab B and it is invariant under a supersymmetry transformation Q that acts on the Hilbert space as i µ a 1 ab A α A 1 Q = e γ ∂µ + ω γa, γb δ + A T = iD/ . (2.2.8) √2 a 8 µ B µ α B √2 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 important 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 . An inﬁnitesimal L transformation φ(x) φ′(x)= φ(x)+ ǫ∆φ(x) (2.3.1) 7→ parameterized by a parameter ǫ is a symmetry of the theory if, correspondingly, the action S = d4x (φ, ∂ φ) associated with does not change. In order to be so, the variation L µ L of theR 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” µ such that J (φ′, ∂ φ′)= (φ, ∂ φ)+ ǫ∂ µ + (ǫ2). (2.3.2) L µ L µ µJ O Note that µ can also be a Lorentz tensor. A direct calculation of the variation of the J Lagrangian induced by the transformation shows that

µ δ δ ∂µ = L∆φ + L ∂µ∆φ. (2.3.3) J δφ δ∂µφ

24 By rewriting the second term on the r.h.s. as ∂ [∆φδ /δ∂ φ] ∆φ∂ δ /δ∂ φ one ﬁnds µ L µ − µ L µ that δ δ ∂ jµ = ∆φ L ∂ L , (2.3.4) µ − δφ − µ δ∂ φ µ where we introduced the current δ jµ(x) L ∆φ µ. (2.3.5) ≡ δ∂µφ −J

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

′ ′ µ δ 2 (φ , ∂µφ )= (φ, ∂µφ)+ ǫ(x)∂µ + L ∆φ∂µǫ(x)+ (ǫ ), L L J δ∂µφ O (2.3.6) = (φ, ∂ φ)+ jµ(x)∂ ǫ(x)+ (ǫ2), L µ µ O where = indicates that theR equality is valid after integration over the space, i.e., at the level of theR 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

φ (x) φ′(x)= φ (x)+ iǫα(T )mφ (x), (2.3.7) l 7→ l l α l m 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 function Z[J] deﬁned as usual by

4 0 0 Z[J]= φ eiS(φ)+i R d xJ(x)φ(x) , (2.3.8) h | iJ ≡ D Z

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

′ 4 ′ 4 µ 4 Z[J]= φ′ eiS(φ )+i R d xJ(x)φ (x) = φ′ eiS(φ)+i R d xj ∂µǫ(x)+i R d xJ(x)[φ(x)+ǫ∆φ(x)] D D Z Z 4 = φ 1+ i d4xǫ(x) [ ∂ jµ(x)+ J(x)∆φ(x)] + (ǫ2) eiS(φ)+i R d xJ(x)φ(x) D − µ O Z Z = Z[J]+ i d4xǫ(x) ∂ jµ(x) + J(x) ∆φ(x) + (ǫ2), − h µ iJ h iJ O Z (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′ φ in 7→ 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

∂ jµ(x) = J(x) ∆φ(x) . (2.3.10) h µ iJ h iJ This relation is an example of a Ward-Takahashi (WT) identity associated with the symmetry (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- m ternal symmetry (2.3.7), where the associated current is given by jα,µ = i(∂µφ)l(Tα)l φm.

By diﬀerentiating eq.(2.3.10) with respect to Jn1 (x1) and Jn2 (x2) and by then setting J = 0 we get

µ ∂µ jα(x)φn1 (x1)φn2 (x2) = h i (2.3.11) = δ(x x )(T )m φ (x )φ (x ) + δ(x x )(T )m φ (x )φ (x ) . − 1 α n1 h m 1 n2 2 i − 2 α n2 h m 2 n1 1 i 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) φ′(x)= φ(x)+ ǫF [φ(x)], (2.3.12) 7→ 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 φ′ = φ. In terms of the generating function Z[J] → D D

26 one ﬁnds

′ 4 ′ 4 Z[J]= φ′ eiS(φ )+i R d xJ(x)φ (x) = φ eiS(φ)+i R d xJ(x){φ(x)+ǫF [φ(x)]} D D Z Z (2.3.13) 4 = φ 1+ iǫ d4x J(x)F [φ(x)] + (ǫ2) eiS(φ)+i R d xJ(x)φ(x), D O Z Z which implies d4x F [φ(x)] J(x) = 0 . (2.3.14) h iJ Z Recall now the deﬁnition of the functional Γ[Φ], generating 1-particle irreducible amplitudes. The source Φ(x) is deﬁned as

δW [J] Φ(x) (2.3.15) ≡ δJ(x) and the 1-particle irreducible generator is deﬁned as

Γ[Φ] = W [J] d4x Φ(x)J(x) . (2.3.16) − Z 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 δW δJ(y) = d4y Φ(y) J(x)= J(x) . (2.3.17) δΦ(x) δJ(y) − δΦ(x) − − Z Using then eq.(2.3.17), we can rewrite eq.(2.3.14) as

δΓ d4x F [φ(x)] = 0, (2.3.18) h iJ(Φ) δΦ(x) Z which implies that the 1PI action is invariant for Φ Φ+ ǫ F [φ(x)] : → h iJ(Φ) δΓ Γ(Φ + ǫ F [φ(x)] ) = Γ(Φ)+ ǫ d4x F [φ(x)] = Γ(Φ) . (2.3.19) h iJ(Φ) h iJ(Φ) δΦ(x) Z Notice that, in general, F [φ(x)] = F (Φ) . (2.3.20) h iJ(Φ) 6 They coincide only for transformations that are at most linear in the ﬁelds, namely for3

F (φ)= s(x)+ d4yt(x,y)φ(y) . (2.3.21) Z 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)] = s(x)+ d4yt(x,y) φ(x) = s(x)+ d4yt(x,y)Φ(y)= F (Φ) , h iJ(Φ) h iJ(Φ) Z Z (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 1 = (∂ Aµ)2 , (2.3.23) Lg.f. −2ξ µ where ξ is an arbitrary real parameter. The total Lagrangian density is then

(A , ψ, ψ¯)= (A , ψ, ψ¯)+ (A ) , (2.3.24) L µ LQED µ Lg.f. µ where 1 = F F µν + ψ¯ iD/ (A) m ψ + counterterms . (2.3.25) LQED −4 µν −

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

µ J (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) , A (x) A (x)+ ∂ ǫ(x), → → − µ → µ µ (2.3.27) the measure and remain invariant, while and the source term will change: LQED Lg.f. ∂µA (A , ψ, ψ¯) (A , ψ, ψ¯) µ ✷ǫ . L µ →L µ − ξ J µA + Jψ¯ + ψJ¯ J µA + Jψ¯ + ψJ¯ + J µ∂ ǫ + ieJψǫ¯ ieψJǫ.¯ (2.3.28) µ → µ µ − 4 In 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 (ǫ) terms from the exponential and taking a functional derivative O with respect to ǫ(x), we get the identity

i i∂ J µ(x)Z = e ψ¯(x) J(x) eJ¯(x) ψ(x) ✷∂ Aµ(x) = 0 . (2.3.29) µ h iJ − h iJ − ξ µh iJ

We also have5

δZ δZ µ δZ ψ¯(x) J = i , ψ(x) J = i , A (x) J = i (2.3.30) h i δJ(x) h i − δJ¯(x) h i − δJµ(x) and thus we can rewrite eq.(2.3.28) in terms of W = i log Z as −

µ δW δW i δW i∂µJ (x)= e J(x) eJ¯(x) ✷∂µ = 0 . (2.3.31) − δJ(x) − δJ¯(x) − ξ δJµ(x)

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 vertex, 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 J µ. 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,

2 µ q q Gµα1...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 δ (µ,ν = 1,..., 2n), and in an arbitrary representation of µν R a gauge group G (A = 1,..., dim ). The minimal coupling of the fermion to the gauge R and gravitational ﬁelds is described by the Lagrangian

= e ψ¯(x) i(D/ )A ψB , (3.1.1) L A r B where e is the determinant of the vielbein andD/ r 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+1 ψ , ψ¯ ψe¯ iαγ2n+1 , (3.1.2) → → where γ2n+1 is the chirality matrix (1.2.16) in 2n dimensions, and α is a constant param- µ ¯ µ A eter. This implies that the chiral current J2n+1 = ψAγ2n+1γ ψ 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 2n e−Γ(e,ω,A) = ψ ψ¯ e− R d x L , (3.1.3) D D Z 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¯ = iαψγ¯ 2n+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 δ = J µ ∂ α. The variation of the measure is instead always non-vanishing, α L 2n+1 µ becauseR the transformationR (3.1.4) leads to a non-trivial Jacobian factor, which has the form δ [ ψ ψ¯] = [ ψ ψ¯]exp 2i α , as we will see below. In total, the eﬀective α D D D D {− A} action therefore transforms as R

δ Γ= d2nxe α(x) 2i (x) ∂ J µ (x) . (3.1.5) α A − h µ 2n+1 i Z h i The condition δαΓ = 0 then implies the anomalous Ward identity:

∂ J µ = 2i . (3.1.6) h µ 2n+1i A In order to compute the anomaly , we need to deﬁne the integration measure more A precisely. This is best done by considering the eigenfunctions of the Dirac operator iD/ r.

Since the latter is Hermitian, the set of its eigenfunctions ψk(x) with eigenvalues λk, deﬁned by iDψ/ n = λnψn, form an orthonormal and complete basis of spinor modes:

d2nx e ψ† (x)ψ (x)= δ , e ψ† (x)ψ (y)= δ(2n)(x y) . (3.1.7) k l k,l k k − Z Xk The fermion ﬁelds ψ and ψ¯, which are independent from each other in Euclidean space, can be decomposed as ¯ ¯ † ψ = akψk , ψ = bkψk , (3.1.8) Xk Xk so that the measure becomes ψ ψ¯ = da d¯b . (3.1.9) D D k l Yk,l Under the chiral transformation (3.1.4), we have

2n † ¯ 2n ¯ † δαak = i d x e ψkαγ2n+1ψlal , δαbk = i d x e blψl αγ2n+1ψk , (3.1.10) Z Xl Z Xl and the measure (3.1.9) transforms as

δ ψ ψ¯ = ψ ψ¯ exp 2i d2nx e ψ† αγ ψ . (3.1.11) α D D D D − k 2n+1 k h i Xk Z 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 ( 1= ). A convenient way of regularizing this k ∞ expression is to introduce a gauge-invariantP Gaussian cut-off. The integrated anomaly = can then be defined as Z A R 2 † −βλn/2 = lim ψ γ2n+1ψke Z β→0 k Xk −β(iD/)2/2 = lim Tr γ2n+1e , (3.1.12) β→0 h i 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 = (iD/ )2/2 and as density matrix ρ = γ e−βH : = Tr ρ. is not clearly the ordinary 2n+1 Z Z 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 | i paired with its supersymmetric partner E˜ : Q E = √E E˜ . Of course, E and E˜ | i | i | i | i | i 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 of E and E˜ is equal and | i Z | i | i opposite 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 | i paired. We conclude that = n n (3.1.13) Z + − − independently of β, where n± are the number of zero energy bosonic and fermionic states. Interestingly enough, does not depend on other possible deformation of the system. Z Indeed, if a state E with E > 0 approaches E = 0, so has to do its partner E˜ . | i | i 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 coincides with the index of the Dirac operator defined in eq.(1.3.28). Using the Atiyah- Z 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 . This computation Z can then be seen as a physical way to prove the Atiyah-Singer theorem applied to the Dirac operator [9].

32 The integrated anomaly could be derived by computing directly the partition func- Z 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 operators c⋆ and cA. As discussed below eq.(2.2.5), only the 1-particle states c⋆ 0 correspond A A| i to the representation of the gauge group G. In order to compute the anomaly for a R single spinor in the representation , it is therefore necessary to restrict the partition R 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 ⋆ A is Hamiltonian with respect to the fields cA and c , 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 can then be written as Z β µ a µ a ⋆ A = Trc,c⋆ x ψ exp dτ x (τ), ψ (τ), cA, c . (3.1.14) Z P D P D − 0 R Z Z Z The subscript P on the functional integrals stands for periodic boundary conditions along ⋆ the closed time direction τ, Tr ⋆ represents the trace over the 1-particle states c 0 , and c,c A| i the Euclidean Routhian is given by R 1 1 1 = g x˙ µx˙ ν + ψ ψ˙ a + ψ , ψ ωabx˙ µ R 2 µν 2 a 4 a b µ 1 + c⋆ AA x˙ µcB c⋆ cBψ aψbF A . (3.1.15) A µ B − 2 A ab B Equation (3.1.14) should be understood as follows: after integrating over the fields xµ and a ˆ ⋆ ⋆ A ψ , one gets an effective Hamiltonian H(c, c ) for the operators cA and c , from which −βHˆ (c,c⋆) one finally computes Trc,c⋆ e . 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 x . It is also convenient to rescale τ βτ and define 0 → µ µ µ a a 1 a a x (τ)= x + β e (x0)ξ (τ) , ψ (τ)= ψ + λ (τ) . (3.1.16) 0 a √β 0 p 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

ωab(x)x ˙ µ(τ) = βωab(x )eµ(x )ξ˙d(τ)+ β∂ ωab(x )eµ(x )ξ˙d(τ)eν (x ) ξc(τ)+ (β3/2) µ µ 0 d 0 ν µ 0 d 0 c 0 O =p 0+ β∂ ωab(x )eµ(x )eν (x )ξ˙d(τ)ξc(τ)+ (β3/2) (3.1.17) [ν µ] 0 d 0 c 0 O = βRab(x )ξ˙d(x )ξc(x )+ (β3/2) . cd 0 0 0 O

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: → 1 eff = ξ˙ ξ˙a + λ λ˙ a + R (x , ψ )ξaξ˙b c⋆ F A (x , ψ )cB , (3.1.18) R 2 a a ab 0 0 − A B 0 0 h i where 1 R (x , ψ )= R (x )ψcψd , ab 0 0 2 abcd 0 0 0 1 F A (x , ψ )= F A (x )ψaψb . (3.1.19) B 0 0 2 ab B 0 0 0 a 2 Since the fermionic zero modes ψ0 anticommute with each other, 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⋆ 0 , and the latter by the determinants arising from the Gaussian path A| i integral over the bosonic and fermionic fluctuation fields:

2n 2n c⋆ F A cB −1/2 2 1/2 = d x d ψ Tr ⋆ e A B det ∂ δ + R ∂ det ∂ δ . (3.1.20) Z 0 0 c,c P − τ ab ab τ P τ ab Z Z h i h i h i The trace yields simply c⋆ F A cB F Trc,c⋆ e A B = trRe . (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:

t free − m R x˙ 2dτ Z = xe 2 0 . (3.1.22) B D ZP 2 a The anticommuting ψ0 ’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 = xn and rewrite

∞ 2 ∞ −mt ∞ 2πn |x |2 t Zfree = dx dx dx∗ e Pn=1 t n = dx . (3.1.23) B 0 n n 0 2πmn2 Z nY=1 Z nY=1 Using now the identity ∞ ∞ d −s fn = exp lim fn , (3.1.24) − s→0 ds n=1 n=1 Y X we compute

∞ ∞ t d t s t = exp n2s = exp ζ(0) log +2ζ′(0) , (3.1.25) 2πmn2 − ds 2πm s=0 2πm nY=1 nX=1 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 m 1/2 Zfree = L , (3.1.26) B 2πt 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 t free i m R ψψdτ˙ −1/2 Z = ψ ψ e 2 0 = m . (3.1.27) F D ZP 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

n −1/2 2 −n λi/2 detP ∂τ δab + Rab∂τ = (2π) , (3.1.28) − sin(λi/2) h i Yi=1 det1/2 ∂ δ = ( i)n . (3.1.29) P τ ab − h i

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

d2nψ ψa1 ...ψa2n = ( 1)n ǫa1...a2n . (3.1.30) 0 0 0 − Z 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:

= ch (F ) Aˆ(R) , (3.1.31) Z R ZM2n where ch(F ) and Aˆ(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

∂ J µ = 2i ch (F )Aˆ(R) . (3.1.32) h µ 2n+1i R In 4 space-time dimensions, for instance, eq.(3.1.32) gives

1 dim ∂ J µ = iǫµνρσ tr F F + RR αβR . (3.1.33) h µ 5 i − 16π2 µν ρσ 384π2 µν ρσαβ 3.2 Consistency Conditions for Gauge and Gravitational Anomalies

Anomalies can also affect currents related to space-time dependent symmetries. For simplicity, 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, modulo 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 Dǫ → − 1 δΓ(A) δ Γ(A)= d2nx tr D ǫ = d2nx tr D J µǫ = d2nx tr ǫ (ǫ ) , (3.2.1) ǫ1 − δA µ 1 µ 1 A 1 ≡I 1 Z µ Z Z where is the gauge anomaly and we have defined its integrated form (including the Aα gauge parameter ǫ) by . The WT identities associated to the conservation of currents I 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 that would now appear in the WT identities associated to the (would-be) A 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 defined in eq.(3.2.1): I δ (ǫ ) δ (ǫ )= ([ǫ ,ǫ ]) . (3.2.3) ǫ1 I 2 − ǫ2 I 1 I 1 2 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), connection 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- (0) lence class of Chern–Simons (2n + 1)-forms Ω2n+1(a, f) through the local decomposition (0) (0) Ω2n+2 = dΩ2n+1, like in section 1.3. In this way, we specify Ω2n+1 only modulo exact 2n-forms, implementing the redundancy associated to the local symmetry under consider- (1) ation. One can then define yet another equivalence class of 2n-forms Ω2n (ǫ, a, f), modulo exact (2n 1)-forms, through the transformation properties of the Chern–Simons form − (0) (1) under a local symmetry transformation: δǫΩ2n+1 = dΩ2n . It is the unique integral of this (1) class of 2n-forms Ω2n that gives the relevant general solution of eq.(3.2.3):

(ǫ) = 2πi Ω(1)(ǫ) . (3.2.4) I 2n ZM2n 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 manifolds M2n+1 whose boundary is M2n: M2n = ′ ′′ ∂M2n+1, and similarly the equivalence class of (2n+2)-dimensional manifolds M2n+2 such ′ ′′ ′ ′′ that M2n+1 = ∂M2n+2. Both M2n+1 and M2n+2 are deﬁned modulo components with no boundary, on which exact forms integrate to zero. Since

(1) (1) (0) (0) Ω2n (ǫ)= dΩ2n (ǫ)= δǫ Ω2n+1 = δǫ dΩ2n+1 = δǫ Ω2n+2, M M ′ M ′ M ′′ M ′′ Z 2n Z 2n+1 Z 2n+1 Z 2n+2 Z 2n+2 (3.2.5) eq. (3.2.4) can be rewritten as

(0) (ǫ) = 2πi δǫ Ω2n+1 = 2πi δǫ Ω2n+2 . (3.2.6) I M ′ M ′′ Z 2n+1 Z 2n+2

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 connection 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 = Ω2n+2 provides precisely the 2n + 2- Z M2n+2 form defined above through which we can solveR 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 diffeomorphisms (or local Lorentz transformations), as = 2πi Ω(1). I M2n 2n R

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)

µ We can then deﬁne, besides the usual exterior derivative d = dx ∂µ with respect to the µ α coordinates x , an additional exterior derivative dˆ= dθ ∂α with respect to the parameters θα. The range of values of the index α is for the moment undetermined. The operators d and dˆ anticommute and are both nilpotent, d2 = dˆ2 = 0. This implies that their sum ∆ = d + dˆ is also nilpotent: ∆2 = 0. The two operators d and dˆ naturally deﬁne two transformation parameters v andv ˆ through the expressions:

v(x,θ)= g−1(x,θ)dg(x,θ) , (4.1.3) vˆ(x,θ)= g−1(x,θ)dgˆ (x,θ) . (4.1.4)

39 Exercise 8: Verify that

dˆvˆ = vˆ2 , dˆA¯ = D¯ vˆ , dˆF¯ = [ˆv, F¯] . (4.1.5) − − −

Equations (4.1.5) show that dˆ generates an infinitesimal gauge transformation with pa- rameterv ˆ 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 withv ˆ. At this point, it is possible to define yet another connection and field-strength as A F = g−1 A +∆ g = A¯ +v ˆ , (4.1.6) A = ∆ + 2 = g−1F g = F¯ . (4.1.7) F A A The last relation is easily proved. Using eqs.(4.1.5) one has

= (d + dˆ)(A¯ +v ˆ) + (A¯ +v ˆ)2 F = dA¯ + dvˆ D¯ vˆ vˆ2 +v ˆ2 + A¯vˆ +v ˆA¯ + A¯2 (4.1.8) − − = dA¯ + A¯2 = F¯ .

The crucial point is now that and are deﬁned with respect to ∆ exactly in the same A F way as A¯ and F¯ are deﬁned with respect to d. Therefore, the corresponding Chern–Simons decompositions must have the same form:

Q ( )=∆Q ( , ) , (4.1.9) 2n+2 F 2n+1 A F 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 + dˆ)Q2n+1(A¯ +v, ˆ 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, ˆ F¯) in powers ofv ˆ as

¯ ¯ (0) ¯ ¯ (1) ¯ ¯ (2n+1) ¯ ¯ Q2n+1(A +v, ˆ F )= Q2n+1(A, F )+ Q2n (ˆv, A, F )+ . . . + Q0 (ˆv, A, F ) , (4.1.12)

40 where the superscripts denote the powers ofv ˆ 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 ofv ˆ, we ﬁnally ﬁnd the Stora–Zumino descent relations [12]:

ˆ (0) (1) dQ2n+1 + dQ2n = 0 , ˆ (1) (2) dQ2n + dQ2n−1 = 0 , . . . ˆ (2n) (2n+1) dQ1 + dQ0 = 0 , ˆ (2n+1) dQ0 = 0 . (4.1.13)

The operator dˆmakes it possible to understand in a simple way why the Stora–Zumino descent represents the most general non-trivial solution of the Wess–Zumino consistency condition (3.2.3). For the case of gauge anomalies we are considering here, (v)= tr(v a(A)), I the latter reads R

δ tr(v a(A)) δ tr(v a(A)) tr([v , v ] a(A)) = 0 . (4.1.14) v1 2 − v2 1 − 1 2 Z Z Z The two transformations with parameters v1 and v2 can be incorporated into a family of transformations parametrized by θ1 and θ2, with parameter

1 2 α vˆ = v1dθ + v2dθ = vαdθ . (4.1.15)

−1 α In this way, vα = g ∂αg. At θ = 0, g(x, 0) = 1 and therefore A¯(x, 0) = A(x) and F¯(x, 0) = F (x). At that point, dˆ generates ordinary gauge transformations on A and F , ˆ α with d = dθ δvα . For instance, eq.(3.2.1) can be rewritten as

dˆΓ= trv ˆ (4.1.16) A Z The condition (4.1.14) can then be multiplied by dθ1dθ2 and rewritten as

0 = dθ1dθ2 tr(v δ ) trv δ ) tr[v , v ] 2 v1 A − 1 v2 A − 1 2 A Z Z Z = dθαdθβ tr(v δ ) tr[v , v ] (4.1.17) β vα A − α β A Z Z = tr(ˆvdˆ ) trˆv2 . − A − A Z Z Since dˆvˆ = vˆ2, this can be rewritten simply as: − dˆ tr(ˆv ) = 0 . (4.1.18) A Z 41 The Wess–Zumino consistency condition is therefore the statement that the anomaly is dˆ-closed, dˆ (ˆv) = 0 . (4.1.19) I It is clear that the trivial dˆ-exact solutions (ˆv)= dˆ f(A) in terms of a local functional I f(A) of the gauge field correspond to the gauge variationR 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 dˆ. 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 (ˆv) = Q(1). With the above I 2n definitions, we have Q = dQ(0) and δ Q(0) = dQ(1). We can therefore identify 2n+2 2n+1 v 2n+1 − 2n R Ω Q , Ω(0) Q(0) , Ω(1) Q(1), (4.1.20) 2n+2 ↔ 2n+2 2n+1 ↔ 2n+1 2n ↔− 2n 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 of a gauge group G, ψ¯ should R transform in the complex conjugate representation ⋆, otherwise the mass term ψψ¯ would R 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

= e ψ¯(x) i(D/ )A ψB , (4.2.1) L A r BPη where 1 = 1+ ηγ , η = 1 , (4.2.2) Pη 2 2n+1 ± 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 , iD/ η = iD/ η, and η P is not Hermitian. For this reason, we have to use the eigenfunctions φn of the Hermitian † η † operator (iD/ η) iD/ η to expand ψ and the eigenfunctions ϕn of iD/ η(iD/ η) to expand ψ¯. Let us first consider the case of gauge anomalies. Under an infinitesimal gauge trans- A α A formation with parameter vα, or v B = v Tα B, the fermion fields transform as

δ ψA = vA ψB , δ ψ¯A = vA ψ¯B . (4.2.3) v − B v B This induces a variation of the integration measure given by

δ ψ ψ¯ = ψ ψ¯ exp d2nx e φη†v T αφη ϕη†v T αϕη . (4.2.4) v D D D D k α k − k α k h i Xk Z 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

† † gauge † α −β(iD/η ) iD/η/2 † α −βiD/η(iD/η ) /2 (v)= lim φ vαT e φk ϕ vαT e ϕk I − β→0 k − k Xk gauge −β(iD/)2/2 = η lim Tr γ2n+1Q e . (4.2.5) − β→0 h i gauge α The operator Q is defined in such a way to act as vαT on the Hilbert space. A concrete realization of it within the supersymmetric quantum mechanics introduced in the previous subsection is Qgauge c∗ vA cB . (4.2.6) → A B 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 correspond 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 information, and there is a well-defined procedure to switch from one to the other [13]. To put the covariant anomaly emerging from a computation àla Fujikawa into a consistent form, it

43 is sufficient to take the leading-order part of it, which contains n +1 fields, add the symmetrization factor required for a Bose-symmetric result, and then use the Wess–Zumino consistency condition to reconstruct the subleading part. All this procedure is automatically 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

(1) gauge (v) = 2πiη chR(F ) Aˆ(R) . (4.2.7) I M n Z 2 h i The case of gravitational anomalies is similar. Under an infinitesimal diffeomorphism with parameter ǫµ, the fermion fields transform, modulo a local Lorentz transformation, as δ ψ = ǫµD ψ , δ ψ¯ = ǫµD ψ¯ . (4.2.8) ǫ − µ ǫ µ This induces the following variation of the integration measure:

δ ψ ψ¯ = ψ ψ¯ exp d2nx e φη†ǫµD φη ϕη†ǫµD ϕη . (4.2.9) ǫ D D D D k µ k − k µ k h i n Xk Z o The regularized expression for the integrated gravitational anomaly is then:

† † grav † −β(iD/η ) iD/η /2 µ † −βiD/η(iD/η ) /2 µ (ǫ)= lim φ e ǫ Dµφk ϕ e ǫ Dµϕk I − β→0 k − k Xk grav −β(iD/)2/2 = η lim Tr γ2n+1Q e . (4.2.10) − β→0 h i The operator Qgrav must act as ǫµD on the Hilbert space. Since ix˙ µ Dµ upon canonical µ → quantization, we can identify Qgrav iǫ x˙ µ . (4.2.11) → µ We have then (recall the rotation to euclidean time τ iτ) →− µ β grav µ a dx (τ) µ a ⋆ A (ǫ)= ηTrc,c⋆ x ψ ǫµ(x) exp dτ x (τ), ψ (τ), cA, c . I P D P D dτ − 0 R Z Z Z (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)

dxµ(τ) 1 β dxµ(τ) ǫ (x) dτ ǫ (x) . (4.2.13) µ dτ → β µ dτ Z0

44 Rescaling τ βτ, using eq.(3.1.16) and expanding in normal coordinates gives, modulo → irrelevant higher order terms in β,

1 β dxµ(τ) 1 1 dξµ(τ) dτ ǫ (x) = dτ ǫ (x )+ ǫ (x ) βξν (τ)+ (β) β β µ dτ β µ 0 ∇ν µ 0 O dτ Z0 Z0 1 µ p 1 µp 1 dξ (τ) ν dξ (τ) = ǫµ(x0) dτ + νǫµ(x0) dτξ (4.2.14) √β dτ ∇ dτ Z0 Z0 1 1 = D ǫ (x ) dτ(ξaξ˙b ξbξ˙a) , 2 a b 0 − Z0 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 R R D ǫ + D ǫ in the effective ab → ab − a b b a 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

(1) grav (ǫ) = 2πiη chR(F ) Aˆ(R) . (4.2.15) I M n Z 2 h i 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

2n δΓ(g) 2n µν ν grav δǫΓ(g)= d x ( µǫν + νǫµ)= d x√g µT ǫν = ǫν (ǫ) , − δgµν ∇ ∇ ∇ A ≡I Z Z Z (4.2.16) where δ g = ( ǫ + ǫ ) and we have deﬁned the symmetric energy-momentum ǫ µν − ∇µ ν ∇ν µ tensor 2 δΓ(g) T µν . (4.2.17) ≡ √g δgµν 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 a Lorentz symmetry. In this case we see Γ as a functional of the vielbeins eµ. Under a local Lorentz transformation for which δ ea = αa eb , we have L µ − b µ δΓ(e) δ Γ(e)= d2nx α eb = d2nxe T abα = abα L(α) , (4.2.18) L − δe ab µ ab A ab ≡I Z µa Z Z where we have used the relation

b δΓ(e) b δΓ(g) a ab eµ = eµ2 eν = eT . (4.2.19) δeµa δgµν

45 Since α = α , an anomaly in a local Lorentz transformation implies a conserved, but ab − ba 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: i n+1 Ω (F )= Q (F ) , Q (F )=tr F n+1 . (4.3.1) 2n+2 2π 2n+2 2n+2 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) (A, F ) is easily obtained by setting F = A = 0 in eq.(1.3.10) (with n n + 1). One 2n+1 → has F = tF + (t2 t)A2 and hence t − 1 (0) n Q2n+1(A, F ) = (n + 1) dt tr A Ft . (4.3.2) 0 Z h i (1) To compute the second descendent, Q2n (v, A, F ), we use the result (4.3.2) to determine the left-hand side of eq.(4.1.12). Defining for convenience F¯˜ = tF¯ + (t2 t)(A¯ +v ˆ)2, this t − reads 1 ¯ ¯ ¯ ¯ñ Q2n+1(A +v, ˆ F ) = (n + 1) dt tr (A +v ˆ)Ft . (4.3.3) 0 Z h i The quantity we are after is then found by expanding in powers ofv ˆ and retaining the linear order. We have F¯˜ = F¯ +(t2 t)[A,¯ vˆ]+ (ˆv2). It is useful to define the symmetrized t t − O

46 αi αi trace of generic matrix-valued forms ωi = ωi T as

1 α1 αp α1 αp str (ω ...ω )= ω . . . ωp tr (T ...T ) . (4.3.4) 1 p p! 1 ∧ ∧ perms. X In this way we find 1 Q(1)(ˆv, A,¯ F¯) = (n + 1) dt str vˆF¯n + n(t2 t)A¯ A,¯ vˆ F¯n−1 2n t − t Z0 1 h i = (n + 1) dt str vˆ F¯n + n(t 1) t A,¯ A¯ F¯n−1 A¯ A¯ , F¯n−1 t − t − t t Z0 1 h i = (n + 1) dt str vˆ F¯n + n(t 1) ∂ F¯ dA¯ F¯n−1 + Ad¯ F¯n−1 t − t t − t t Z0 1 h i ¯n ¯n ¯ ¯n−1 = (n + 1) dt str vˆ Ft + (t 1)∂tFt + n(1 t)d AFt . (4.3.5) 0 − − Z h i ¯n−1 ¯n−1 ¯ ¯n−1 In the third equality we have used the identities DtFt = dFt + [At, Ft ] = 0 and ∂tF¯t = 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)):

n+1 1 (1) i n−1 Ω2n (v, A, F )= n(n + 1) dt (1 t)str vd(AFt ) . (4.3.6) − 2π 0 − Z h i 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: η 1 δ Γ(A)= d4x str vd AF A3 . (4.3.7) v −24π2 − 2 Z h i 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 1 1 Dαβγ tr( T α, T β T γ )= tr (T αT βT γ) . (4.3.8) ≡ 2 { } 3! perms. X 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 consider 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:

Γ (k , k ) J ( k k )J (k )J (k ) , (5.1.1) µνρ 1 2 ≡ h µ,5 − 1 − 2 ν 1 ρ 2 i where Jµ,5 = ψγ¯ µγ5ψ, Jν = ψγ¯ νψ, Jρ = ψγ¯ ρψ. For simplicity, we will consider massless fermions. In this case, the classical conservation of the axial and vector currents would imply µ ν ρ (k1 + k2) Γµνρ = k1 Γµνρ = k2 Γµνρ = 0 . (5.1.2)

48 ρ,k2 ν,k1

µ, −k1 − k2 µ, −k1 − k2 +

ν,k1 ρ,k2

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 γ in d = 4 dimensions 5 6 (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 1 kµΓ (k , k )= ǫ kωkγ . (5.1.3) µνρ 1 2 2π2 γρων 1 2 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 derivatives 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

J α = ψγ¯ T αψ, (5.1.4) µ µPη where is the chiral projector defined in eq.(4.2.2). The resulting computation is very Pη 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 χt σ2Mχ + h.c. (5.1.5) Lm ⊃ 1 2 where χ and χ are in irreducible representations and of the group G, and M is 1 2 R1 R2 a non-singular mass matrix. The term (5.1.5) is gauge-invariant if

T tM = MT , (5.1.6) − 1 2 where T and T are the generators in the representations and . Eq. (5.1.6) implies 1 2 R1 R2 that T t and T are related by a similarity transformation, since T t = MT M −1. We − 1 2 − 1 2 have then 1 1 D1 = tr T , T T = ( 1)3 tr T t , T t T t = D2 . (5.1.7) αβγ 2 { 1α 1β} 1γ − 2 { 2α 2β} 2γ − αβγ

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 χ with charge q and obviously admits the mass term mχ σ2χ + h.c.. The 2 − 1 2 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 conjugate representation T = T t and thus do not lead to anomalies. In manifestly parity- r¯ − r 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 anomalies 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 G = SU(3) SU(2) U(1). Its fermion content, in terms of left- SM × × handed ﬁelds, is composed by three copies (generations) of the following representations:

(3, 2) 1 + (3¯, 1) 2 + (3¯, 1) 1 + (1, 2) 1 + (1, 1)1 (5.2.1) 6 − 3 3 − 2 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: tr T = 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 = 1/2tr T α, T β T γ = δαβtr T 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 ⋆ = AT A−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

αβγ αβγ αβγ αβγ αβγ αβγ αβγ D = 2D3 + D + D = 2D3 D3 D3 = 0 , (5.2.2) ccc 3¯ 3¯ − − αβγ αβγ 2 using that D = D3 . For SU(3) U(1) we have 3¯ − × αβ α β 1 α β 2 α β 1 α β 1 2 1 D = 2tr3 T T + tr3¯ T T + tr3¯ T T = tr3 T T + = 0 , ccY × 6 × − 3 × 3 3 − 3 3 (5.2.3)

51 α β α β 2 with tr3 T T = tr3¯ T T . For SU(2) U(1), only doublets contribute. We get × αβ α β 1 α β 1 α β 1 1 D = 3tr2 T T + tr2 T T = tr2 T T = 0 . (5.2.4) wwY × 6 × − 2 2 − 2 For U(1)3 anomalies all quarks and leptons contribute and one gets 1 3 2 3 1 3 1 3 D = 3 2 + 3 + 3 + 2 + (1)3 = 0 . (5.2.5) YYY × × 6 × − 3 × 3 × − 2 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 T α, 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)-gravitationalP anomaly is then proportional to 1 2 1 1 3 2 + 3 + 3 + 2 +1=0 . (5.2.6) × × 6 × − 3 × 3 × − 2 Notice how the fermion charges nicely combine to give a vanishing result for all the anomalies, 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) is manifestly × EM 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 spontaneously 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(n ) SU(n ) U(1) U(1) global symmetry, spontaneously f V × f A × V × A broken to SU(n ) U(1) by the quark condensate. Let us compute the possible anoma- f V × V 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 tr T = 0. The only possible non-vanishing anomalies are of the form SU(3)2 U(1) and SU(3)2 U(1) . In × V × A terms of left-handed quarks, a quark and its conjugate have opposite charges with respect to U(1) and equal charges with respect to U(1) , thus the SU(3)2 U(1) anomaly V A × V vanishes while SU(3)2 U(1) does not. In QCD, then, the U(1) symmetry, in addition × A A 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 n = 2 (up and down quarks only) and consider the U(1)2 f EM × U(1) axial anomaly, where U(1) SU(2) is the abelian subgroup generated by T 3. A A ⊂ A We easily get e e N 2e 2 e 2 ∂ J µ = c ǫµνρσF F 1+ − ( 1) µ A −16π2 µν ρσ 3 × 3 × − N e2 = c ǫµνρσF F . (5.3.1) − 48π2 µν ρσ

Under the U(1)A chiral transformation above, the low-energy eﬀective chiral Lagrangian

π describing the dynamics of the π mesons interacting with photons should then not L e vanish, but rather reproduce the axial anomaly (5.3.1). Considering that δ π0 = ǫf , ǫ π Lπ should include the coupling

2 Nce µνρσ 0 π 2 ǫ Fµν Fρσπ . (5.3.2) L ⊃−48π fπ 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 Lπ but 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 N = 3. The chiral Lagrangian should be described in c Lπ a a terms of the matrix of fields U = exp(2π T /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 (1) generators are anti-hermitian, and hence Ω2n 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 euclidean 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 complex 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 ω in d dimensions, ⋆ω is the m p-form given by p p − g √ µ1...µp µ1 µd−p ⋆ω = ω ǫ ν ...ν dx . . . dx , (5.4.1) p p!(d p)! µ1...µp 1 d−p ∧ ∧ − where ǫµ1...µd is the completely antisymmetric tensor, with ǫ1...d = +1. It is easy to see 2 that ⋆⋆ω = ( 1)p(d−p)ω . Specializing for d = 2n and p = n, gives ⋆⋆ω = ( 1)n ω . p − p n − n 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 under 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 ab representation of the group, the connection being ωµ and the group O(d). The associated quantum mechanical model is then given by eq.(3.1.15), with AA iωa and µ B → µ b F AB iRab . In terms of the skew-eigenvalues x of the curvature two-form R defined µν → µν i ab in eq.(1.3.26), the chiral anomaly in 2n dimensions is found to be

n n R xk/2 π ˆ 3/2 = (tre 2 1) A(R)= 2 cosh xj 1 (5.4.2) Z M2n − M2n − sinh(xk/2) Z Z Xj=1 kY=1 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 n 1 xk A = . (5.4.3) Z −8 M2n tanh xk Z kY=1 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

= + , (5.4.4) ZIIB ZA Z3/2 −Z1/2 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 Bibliography

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