Supersymmetry Anomalies in N = 1 Conformal Supergravity

Prepared for submission to JHEP KIAS-P19007

Supersymmetry anomalies in N = 1 conformal supergravity

Ioannis Papadimitriou aSchool of Physics, Korea Institute for Advanced Study, 85 Hoegiro, Seoul 02455, Korea

Abstract: We solve the Wess-Zumino consistency conditions of N = 1 off-shell conformal supergravity in four dimensions and determine the general form of the superconformal anomalies for arbitrary a and c anomaly coefficients to leading non trivial order in the gravitino. Besides the well known Weyl and R-symmetry anomalies, we compute explicitly the fermionic Q- and S-supersymmetry anomalies. In particular, we show that Q-supersymmetry is anomalous if and only if R-symmetry is anomalous. The Q- and S-supersymmetry anomalies give rise to an anomalous supersymmetry transformation for the supercurrent on curved backgrounds admitting Killing spinors, resulting in a deformed rigid supersymmetry algebra. Our results may have implications for supersymmetric localization and supersymmetry phenomenology. Analogous results are expected to hold in dimensions two and six and for other supergravity theories. The present analysis of the Wess-Zumino consistency conditions reproduces the holographic result of arXiv:1703.04299 and generalizes it to arbitrary a and c anomaly coefficients.

Keywords: Supersymmetry, anomalies, Wess-Zumino conditions, QFT on curved backgrounds arXiv:1902.06717v3 [hep-th] 4 May 2019 Contents

1 Introduction 1

2 The local symmetry algebra of N = 1 conformal supergravity4

3 Classical Ward identities6

4 Superconformal anomalies from the Wess-Zumino consistency conditions8

5 Anomalous supercurrent transformation under Q- and S-supersymmetry 11

6 Concluding remarks 12

A Spinor conventions and identities 13

B Solving the Wess-Zumino consistency conditions 15

1 Introduction

Supersymmetric quantum field theories have proven invaluable for probing strong coupling physics due to non-renormalization theorems and supersymmetric localization techniques [1–4]. They also play a pivotal role in holographic dualities and beyond the Standard Model phenomenology. Given the enormous utility of supersymmetry, it is rather surprising that the question of whether it is anomalous at the quantum level is still not conclusively answered, despite the extensive literature addressing this question in various contexts. The consensus seems to be that standard supersymmetry (often termed Q-supersymmetry) is not anomalous. However, we demonstrate in this paper that the Wess-Zumino consistency conditions [5] imply that Q-supersymmetry is necessarily anomalous in theories with an anomalous R-symmetry. The same conclusion is reached in the com- panion paper [6] by means of an one loop calculation in the free Wess-Zumino model. A related observation was made using the R-multiplet in the recent paper [7]. These results confirm the Q- supersymmetry anomaly discovered in the context of supersymmetric theories with a holographic dual in [8], and the related anomalies in rigid supersymmetry [8–10]. Global anomalies do not render the theory inconsistent – they are a property of the theory and affect physical observables, such as decay channels [11, 12] and transport coefficients (see [13] for a recent review and [14] for an observation of the mixed chiral-gravitational anomaly in tabletop experiments). They do mean, however, that the theory cannot be coupled consistently to dynamical gauge fields for the anomalous global symmetry. Specifically, a quantum anomaly in global supersymmetry implies that the theory cannot be coupled consistently to dynamical supergravity at the quantum level. It may also mean that certain conditions necessary to prove non-perturbative results are in fact not met. In flat space, global – or rigid – anomalies are typically visible only in higher-point functions as contact terms that violate the classical Ward identities. For example, the lowest correlation functions where the Q-supersymmetry anomaly is visible in flat space are four-point functions involving

– 1 – two supercurrents and either two R-currents or one R-current and one stress tensor [6]. However, global anomalies become manifest at the level of the quantum effective action and in one-point functions when arbitrary sources for the current operators are turned on (i.e. when the theory is coupled to background gauge fields for the global symmetries), or when the theory is put on a curved background admitting Killing symmetries. In particular, global supersymmetry anomalies are related to supersymmetric index theorems and may affect observables such as partition functions, the Casimir energy, and Wilson loop expectation values of supersymmetric theories on curved backgrounds admitting rigid supersymmetry. Following the recent advances in supersymmetric localization techniques [4] (see [15] for a com- prehensive review), supersymmetric quantum field theories on curved backgrounds have attracted considerable interest. A systematic procedure for placing a supersymmetric theory on a curved background was proposed in [16]. The first step is coupling the theory to a given off-shell background supergravity, which corresponds to turning on arbitrary sources for the current multiplet operators and promoting the global symmetries – including supersymmetry – to local ones. The Killing spinor equations obtained by setting the supersymmetry variations of the fermionic background fields to zero determine the curved backgrounds that admit a notion of rigid supersymmetry. Such backgrounds have been largely classified for a number of off-shell supergravity theories and for various spacetime dimensions [17–27] (see also [28, 29] for earlier work). However, this procedure is classical and does not account for possible quantum anomalies. It was in that context that the anomalies in Q-supersymmetry [8] and rigid supersymmetry [8,9] were discovered, providing a resolution to an apparent tension between the field theory analysis of [30–32] and the holographic result of [33]. Based on the classical supersymmetry algebra on curved backgrounds that admit a certain number of supercharges, the authors of [30–32] demonstrated that the supersymmetric partition function on such backgrounds should be independent of specific deformations of the supersymmetric background. However, an explicit evaluation of the on-shell action of minimal N = 2 gauged supergravity on supersymmetric asymptotically locally

AdS5 solutions using holographic renormalization [34, 35] in [33] demonstrated that the holographic partition function on the same supersymmetric backgrounds does in fact depend on the deforma- tion parameters. The resolution to this apparent contradiction was provided in [8], where both the bosonic and fermionic superconformal Ward identities were derived holographically, including the corresponding superconformal anomalies. It was then shown that the anomalies in the fermionic Ward identities (the ones in the divergence and the gamma-trace of the supercurrent) lead to a deformed superconformal algebra on backgrounds admitting Killing spinors. Repeating the argu- ment of [30–32] using this deformed supersymmetry algebra reproduced exactly the dependence on the deformations of the supersymmetric background seen in [33].1 The Ward identities and quantum anomalies of four dimensional superconformal theories have been studied extensively over the years [36–47]. They are usually discussed in superspace language and can be written compactly in the form (see e.g. appendix A of [40])

α˙ ∇ Jαα˙ = ∇αJ, (1.1) where the supertrace superﬁeld J is given by 1 J = cW2 − aE, (1.2) 24π2

1 1 At least for theories with a = c on supersymmetric backgrounds of the form S × M3 with M3 a Seifert manifold it is possible to remove the rigid supersymmetry anomaly at the expense of breaking certain diﬀeomorphisms using the local but non-covariant counterterm found in [33]. We will elaborate on this point in section5.

– 2 – and 1 2 W2 = W W αβγ, E = W2 + (∇ + R)(G2 + 2RR), (1.3) 2 αβγ are respectively the square of the superWeyl tensor and the chirally projected superEuler density.

The chiral superfields Wαβγ and R and the vector superfield Gαα˙ are the three superspace curvatures 2 1 αα˙ [48] and in the conventions of [40] G = 2 G Gαα˙ . The components of the supertrace superfield J contain the trace of the stress tensor, the gamma-trace of the supercurrent, and the divergence of the R-current. The divergence of the stress tensor and of the supercurrent appear as components of the superspace conservation equation (1.1). The superspace analyses of [37, 44] and [39] seem especially related to our results in this paper. In particular, the anomalies in the divergence and in the gamma-trace of the supercurrent we derive are likely related to the fermionic components of the superspace cocycles found in [37, 39, 44], even though none of these earlier works concerns N = 1 conformal supergravity and so the field content is somewhat different. Moreover, to the best of our knowledge these fermionic components have not been written explicitly in the literature before and so a direct comparison with our results is not straightforward. Another result that may be related to the anomalies we find here is [47], where it was shown that in the presence of anomalous Abelian flavor symmetries the Wess-Zumino consistency conditions require additional – non holomorphic – terms to the supertrace anomaly. These terms seem related to terms we find here in the case of an anomalous R-symmetry. Finally, we should mention that there is an extensive body of literature discussing supersymmetry anomalies in the presence of gauge anomalies in supersymmetric gauge theories, reviewed in [49]. Such anomalies involve the dynamical fields in the Lagrangian description of the gauge theory in flat space and are distinct from the supersymmetry anomalies we identify in the present paper, which involve the background supergravity fields. However, the mathematical structure underlying the descent equations that relates the R-symmetry and supersymmetry anomalies is identical to that relating gauge anomalies to supersymmetry anomalies [50] (see also [51, 52]). In this paper we consider N = 1 off-shell conformal supergravity in four dimensions [53–56], which provides a suitable background for superconformal theories via the construction of [16]. We determine the algebra of local symmetry transformations and derive the corresponding classical Ward identities. The main result of the paper is the solution of the Wess-Zumino consistency conditions [5] associated with the N = 1 conformal supergravity algebra from which we obtain the general form of the superconformal anomalies to leading non trivial order in the gravitino for arbitrary a and c anomaly coefficients. Our analysis is carried out in components and we explicitly determine the fermionic Ward identities corresponding to the divergence and the gamma-trace of the supercurrent, including their anomalies. To the best of our knowledge these have not appeared in the literature before, at least explicitly. The bosonic Ward identities and anomalies reproduce well known results [40] (corrected in [46]). We find that the divergence of the supercurrent, which corresponds to the Ward identity associated with Q-supersymmetry, is anomalous whenever R-symmetry is anomalous. Moreover, the N = 1 supergravity algebra dictates that the Q-supersymmetry anomaly cannot be removed by a local counterterm without breaking diffeomorphisms and/or local Lorentz transformations. The Ward identities and the superconformal anomalies we obtain by solving the Wess-Zumino conditions reproduce those found holographically in [8] in the special case when the a and c anomaly coefficients are equal. The paper is organized as follows. In section2 we review relevant aspects of N = 1 off-shell conformal supergravity and determine the algebra of its local symmetry transformations. These transformations are used in section3 to derive the corresponding classical Ward identities. The main

– 3 – result is presented in section4, where we obtain the general form of the superconformal anomalies by solving the Wess-Zumino consistency conditions associated with the N = 1 conformal supergravity algebra. The actual calculation is shown in considerable detail in appendixB. In section5 we determine the anomalous transformation of the supercurrent under local supersymmetry and discuss the implications for the rigid supersymmetry algebra on curved backgrounds admitting Killing spinors of conformal supergravity. We conclude in section6 and collect our conventions and several gamma matrix identities in appendixA.

2 The local symmetry algebra of N = 1 conformal supergravity

In this section we review relevant aspects of N = 1 off-shell conformal supergravity in four dimensions and determine its local off-shell symmetry algebra as a preparatory step for solving the Wess-Zumino consistency conditions. N = 1 conformal supergravity can be constructed as a gauge theory of the superconformal algebra [53–56] (see [57–60] and chapter 16 of [61] for pedagogical a reviews). Its field content consists of the vielbein eµ, an Abelian gauge field Aµ, and a Majorana gravitino ψµ, which comprise 5+3 bosonic and 8 fermionic off-shell degrees of freedom. The reason for focusing on N = 1 conformal supergravity here is threefold. Firstly, background conformal supergravity is relevant for describing the Ward identities of superconformal theories and their quantum anomalies. Moreover, other supergravity theories can be obtained from conformal supergravity by coupling it to compensator multiplets and gauge fixing via the so called tensor or multiplet calculus [59, 62, 63]. When applied to background supergravity, this procedure may be thought of as the process of turning on local relevant couplings at the ultraviolet superconformal fixed point. Finally, off-shell N = 1 conformal supergravity in four dimensions is induced on the conformal boundary of five dimensional anti de Sitter space by minimal N = 2 on-shell gauged supergravity in the bulk [64]. This means that the Wess-Zumino consistency conditions for N = 1 conformal supergravity should reproduce the superconformal anomalies obtained holographically in [8] for the case a = c. We will see in the subsequent sections that this is indeed the case. In the construction of N = 1 conformal supergravity as a gauge theory of the superconformal algebra, Q- and S-supersymmetry are on the same footing before the curvature constraints are imposed, each having its own independent gauge field, respectively ψµ and φµ. The covariant derivative acts on these gauge fields as

1 ab 5 ρ 5 Dµψν ≡ ∂µ + ωµ (e, ψ)γ + iγ Aµ ψν − Γ ψρ ≡ Dµ + iγ Aµ ψν, 4 ab µν 1 ab 5 ρ 5 Dµφν = ∂µ + ωµ (e, ψ)γ − iγ Aµ φν − Γ φρ = Dµ − iγ Aµ φν, (2.1) 4 ab µν

ab 2 where ωµ (e, ψ) denotes the torsion-full spin connection 1 ω ab(e, ψ) ≡ ω ab(e) + ψ γ ψ + ψ γ ψ − ψ γ ψ . (2.2) µ µ 4 a µ b µ a b µ b a

Once the curvature constraints are imposed, however, the gauge field φµ ceases to be an independent field and it is expressed locally in terms of the physical fields as 1 i 1 φ ≡ γν D ψ − D ψ − γ5 ρσD ψ = − 4δ[ρδσ] + iγ5 ρσγνD ψ . (2.3) µ 3 ν µ µ ν 2 νµ ρ σ 6 µ ν µν ρ σ

2 ab The purely bosonic part of the spin connection, ωµ (e), is torsion-free. For most part of the subsequent analysis we will work to leading non trivial order in the gravitino, in which case the torsion-free part of the spin connection suﬃces. However, the full spin connection is necessary in order to determine e.g. the local symmetry algebra.

– 4 – As we will see shortly, this quantity appears in the supersymmetry transformation of the gauge

field Aµ as well as in the fermionic superconformal anomalies. Our spinor conventions are given in appendixA and follow those of [61]. Compared to [60], we use Lorentzian signature instead of Euclidean and we have rescaled the gauge field Aµ according to 3 FT − 4 Aµ → Aµ in order for its coefficient in the covariant derivatives (2.1) to be unity, as is standard in the field theory literature.

Local symmetry transformations The local symmetries of N = 1 conformal supergravity are diﬀeomorphisms ξµ(x), Weyl transformations σ(x), local frame rotations λab(x), U(1) gauge transformations θ(x), as well as Q- and S-supersymmetry, parameterized respectively by the local spinors ε(x) and η(x). The covariant derivative acts on the spinor parameters ε and η as

1 ab 5 5 Dµε ≡ ∂µ + ωµ (e, ψ)γ + iγ Aµ ε ≡ Dµ + iγ Aµ ε, 4 ab 1 ab 5 5 Dµη ≡ ∂µ + ωµ (e, ψ)γ − iγ Aµ η ≡ Dµ − iγ Aµ η. (2.4) 4 ab Under these local transformations the ﬁelds of N = 1 conformal supergravity transform as 1 δea = ξλ∂ ea + ea∂ ξλ − λa eb + σea − ψ γaε, µ λ µ λ µ b µ µ 2 µ 1 1 δψ = ξλ∂ ψ + ψ ∂ ξλ − λ γabψ + σψ + D ε − γ η − iγ5θψ , µ λ µ λ µ 4 ab µ 2 µ µ µ µ 3i 3i δA = ξλ∂ A + A ∂ ξλ + φ γ5ε − ψ γ5η + ∂ θ. (2.5) µ λ µ λ µ 4 µ 4 µ µ

These transformations imply that the quantity φµ in (2.3) transforms as

λ λ 1 ab 1 1 2i 5 1 ν 5 δφµ = ξ ∂λφµ +φλ∂µξ − λabγ φµ − σφµ + Pµν + Fµνγ − Feµν γ ε+Dµη+iγ θφµ, (2.6) 4 2 2 3 3 where 1 1 P ≡ R − Rg , (2.7) µν 2 µν 6 µν is the Schouten tensor in four dimensions and the dual ﬁeldstrength is deﬁned as

1 ρσ Feµν ≡ µν Fρσ. (2.8) 2 Notice that the transformations (2.5) coincide with those induced on the boundary of ﬁve dimensional anti de Sitter space by minimal N = 2 gauged supergravity in the bulk [8, 64]. In order to compare with the results of [8] one should take into account that we have rescaled the gauge ﬁeld and the local symmetry parameters according to √ there 3Aµ /` → Aµ, (2.9) and √ there there σthere/` → σ, + /` → ε, − /` → η, 3 θthere/` → θ, (2.10) where “there” refers to the variables used in [8]. Moreover, we use the Majorana formulation of N = 1 conformal supergravity here instead of the Weyl formulation used in [8].

– 5 – Local symmetry algebra The local transformations (2.5) determine the algebra of local symmetries, i.e. the commutators 0 [δΩ, δΩ0 ], where Ω and Ω denote any of the local parameters σ, ξ, λ, θ, ε, η. Oﬀ-shell closure of the algebra requires that the parameters transform under the local symmetries as

µ 0ν µ ν 0µ a µ a µ µ δξ = ξ ∂νξ − ξ ∂νξ , δλ b = ξ ∂µλ b, δσ = ξ ∂µσ, δθ = ξ ∂µθ, 1 1 1 1 δε = ξµ∂ ε + σε − λ γabε − iθγ5ε, δη = ξµ∂ η − ση − λ γabη + iθγ5η. (2.11) µ 2 4 ab µ 2 4 ab Applying the transformations (2.5) repeatedly we then ﬁnd that (to leading order in the gravitino) the only non vanishing commutators and the corresponding composite symmetry parameters are:

00µ ν 0µ 0ν µ [δξ, δξ0 ] = δξ00 , ξ = ξ ∂νξ − ξ ∂νξ ,

00a 0a c a 0c [δλ, δλ0 ] = δλ00 , λ b = λ cλ b − λ cλ b, 1 1 3i [δ , δ ] = δ + δ + δ , σ = εη, λa = − εγa η, θ = − εγ5η, ε η σ λ θ 2 b 2 b 4 µ 1 0 µ a 1 0 ν a 1 0 ν [δ , δ 0 ] = δ + δ + δ , ξ = ε γ ε, λ = − (ε γ ε) ω , θ = − (ε γ ε)A . (2.12) ε ε ξ λ θ 2 b 2 ν b 2 ν Notice that the composite parameters resulting from the commutator of two Q-supersymmetry transformations are ﬁeld dependent, which means that the structure constants of the gauge algebra are ﬁeld dependent. Such algebras are often termed soft algebras (see [65] for a recent discussion of soft algebras and their BRST cohomology) and supergravity theories are typically based on soft algebras. The commutation relations (2.12) form the basis for the Wess-Zumino consistency condition analysis to determine the superconformal anomalies in N = 1 conformal supergravity.

3 Classical Ward identities

We now turn to the derivation of the classical Ward identities of a local quantum field theory coupled to background N = 1 conformal supergravity. These identities can be thought of as Noether’s conservation laws following from the local symmetry transformations (2.5). Since these depend only on the structure of the background supergravity, the resulting Ward identities are independent of the specific field theory Lagrangian, provided the coupling of the theory to background supergravity preserves the local supergravity symmetries at the classical level. The classical Ward identities can be expressed in the compact form

δΩW [e, A, ψ] = 0, (3.1) where Ω = (ξ, σ, λ, θ, ε, η) denotes any of the local transformations (2.5) and W [e, A, ψ] is the generating functional of connected correlation functions of local current operators associated with the background supergravity ﬁelds, namely

µ −1 δW µ −1 δW µ −1 δW Ta = e a , J = e , S = e , (3.2) δeµ δAµ δψµ

a where e ≡ det(eµ). These deﬁnitions do not rely on a Lagrangian description of the quantum ﬁeld theory, but if such a description exists, then the generating functional W [e, A, ψ] is expressed as

W [e, A, ψ] = −i log Z [e, A, ψ], (3.3)

– 6 – where Z [e, A, ψ] is obtained from the path integral

Z [e, A, ψ] = [dΦ]eiS[Φ;e,A,ψ], (3.4) ˆ over the microscopic fields Φ. At the classical level, therefore, the generating function W [e, A, ψ] corresponds to the classical action S[Φ; e, A, ψ], with the microscopic fields Φ evaluated on-shell. Given the definition of the current operators (3.2) and the local symmetry transformations of the background supergravity fields (2.5), classical invariance of W [e, A, ψ] leads to a conservation law – or Ward identity – for each local symmetry, which we will now derive.

Diﬀeomorphisms The transformation of the generating functional under diﬀeomorphisms is given by

4 a µ µ µ δ W = d x e δ e T + δ AµJ + δ ψ S ξ ˆ ξ µ a ξ ξ µ = d4x e ∇ (ξνea)T µ − ξνω a eb T µ + ξνF + ∂ (A ξν)J µ ˆ µ ν a ν b µ a νµ µ ν ←− ←− ←− 1 + ξν(ψ D − ψ D )Sµ + (ξνψ )D Sµ − iξνA ψ γ5Sµ + ξνω ψ γabSµ µ ν ν µ ν µ ν µ 4 νab µ ←− = d4x e ξν − ea∇ T µ − ∇ (ψ Sµ) + (ψ D )Sµ + F J µ ˆ ν µ a µ ν µ ν νµ 1 − A ∇ J µ + iψ γ5Sµ + ω abe T µ + ψ γ Sµ . (3.5) ν µ µ ν µ[a b] 4 µ ab Setting this quantity to zero for arbitrary ξν(x) gives the classical diﬀeomorphism Ward identity

a ν ν ←− ν ν eµ∇νTa + ∇ν(ψµS ) − ψν D µS − FµνJ 1 (3.6) + A ∇ J ν + iψ γ5Sν − ω ab e T ν + ψ γ Sν = 0. µ ν ν µ ν[a b] 4 ν ab We will see shortly that the terms in the second line correspond to the classical Ward identities for

U(1)R gauge transformations and local frame rotations respectively.

Weyl symmetry Under local Weyl rescalings the generating function transforms as

4 a µ µ µ δσW = d x e δσe T + δσAµJ + δσψ S ˆ µ a µ 1 = d4x e σ ea T µ + ψ Sµ , (3.7) ˆ µ a 2 µ and, hence, the classical trace Ward identity takes the form 1 ea T µ + ψ Sµ = 0. (3.8) µ a 2 µ R-symmetry

The transformation of the generating function under U(1)R gauge transformations is given by

4 a µ µ µ δ W = d x e δ e T + δ AµJ + δ ψ S θ ˆ θ µ a θ θ µ = d4x e∂ θJ µ − iθψ γ5Sµ ˆ µ µ = d4x e θ − ∇ J µ − iψ γ5Sµ. (3.9) ˆ µ µ

– 7 – Hence, the classical R-symmetry Ward identity takes the form

µ 5 µ ∇µJ + iψµγ S = 0. (3.10)

Local frame rotations Under local frame rotations the generating function transforms according to

4 a µ µ µ δ W = d x e δ e T + δ AµJ + δ ψ S λ ˆ λ µ a λ λ µ 1 = − d4x eλab e T µ + ψ γ Sµ . (3.11) ˆ µ[b a] 4 µ ba Hence, the corresponding classical Ward identity is 1 e T µ + ψ γ Sµ = 0. (3.12) µ[a b] 4 µ ab Q-supersymmetry The Q-supersymmetry transformation of the generating function is

4 a µ µ µ δεW = d x e δεe T + δεAµJ + δεψ S ˆ µ a µ 1 3i ←− = d4x e − ψ γaεT µ + φ γ5εJ µ + εD Sµ ˆ 2 µ a 4 µ µ 1 3i = d4x e ε γaψ T µ + γ5φ J µ − D Sµ . (3.13) ˆ 2 µ a 4 µ µ Therefore, the classical Q-supersymmetry Ward identity takes the from 1 3i D Sµ = γaψ T µ + γ5φ J µ. (3.14) µ 2 µ a 4 µ S-supersymmetry Finally, the transformation of the generating function under S-supersymmetry is given by

4 a µ µ µ δηW = d x e δηe T + δηAµJ + δηψ S ˆ µ a µ 3i = d4x e − ψ γ5ηJ µ + ηγ Sµ ˆ 4 µ µ 3i = d4x e η − γ5ψ J µ + γ Sµ , (3.15) ˆ 4 µ µ and hence the classical Ward identity for S-supersymmetry is 3i γ Sµ − γ5ψ J µ = 0. (3.16) µ 4 µ

4 Superconformal anomalies from the Wess-Zumino consistency conditions

At the quantum level the generating function W may not be invariant under all local symmetries of background conformal supergravity, i.e.

δΩW 6= 0. (4.1)

– 8 – The non-invariance of the generating function of four dimensional theories can be parameterized as

4 √ δΩW = d x −g σAW − θAR − εAQ + ηAS , (4.2) ˆ where AW , AR, AQ and AS are possible quantum anomalies under Weyl, R-symmetry, Q- and S-supersymmetry transformations, respectively. Recall that the gravitational and Lorentz (frame rotation) anomalies are related by a local counterterm [66] and exist only in 4k + 2 dimensions, with k = 0, 1,.... Moreover, the mixed anomaly can be moved entirely to the conservation of the R-current by a choice of local counterterms (that is setting α = 0 in eq. (2.43) of [67]). In this scheme diﬀeomorphisms remain a symmetry at the quantum level and the transformation of the generating function under all local symmetries can be parameterized as in (4.2). Repeating the exercise of the previous section with the anomalous transformation (4.2) results in the same diﬀeomorphism and Lorentz Ward identities as those obtained respectively in (3.6) and (3.12), but the remaining Ward identities become 1 ea T µ + ψ Sµ = A , µ a 2 µ W µ 5 µ ∇µJ + iψµγ S = AR, 1 3i D Sµ − γaψ T µ − γ5φ J µ = A , µ 2 µ a 4 µ Q 3i γ Sµ − γ5ψ J µ = A . (4.3) µ 4 µ S

The objective of this section is to determine the general form of the quantum anomalies AW , AR, AQ and AS by solving the Wess-Zumino consistency conditions [5] associated with the N = 1 conformal supergravity algebra (2.12). The Wess-Zumino consistency conditions amount to the requirement that the local symmetry algebra (2.12) is realized when successive inﬁnitesimal local symmetry variations δΩ (also known as Ward operators) act on the generating functional W and read

[δΩ, δΩ0 ]W = δ[Ω,Ω0]W , (4.4) for any pair of local symmetries Ω = (ξ, σ, λ, θ, ε, η) and Ω0 = (ξ0, σ0, λ0, θ0, ε0, η0). In appendixB we determine general non trivial solution of the Wess-Zumino consistency conditions (4.4) for the N = 1 conformal supergravity algebra (2.12) in the scheme where diﬀeomor- phisms and local Lorentz transformations are non anomalous. There are two non trivial solutions related respectively to the a and c coeﬃcients of the Weyl anomaly and they take the form c 8 a A = W 2 − F 2 − E + O(ψ2), W 16π2 3 16π2 (5a − 3c) (c − a) AR = FFe + P, 27π2 24π2 (5a − 3c)i µν 5 (a − c) ρσµν (a − c) µνρσ 3 AQ = − Fe Aµγ φν + ∇µ AρRe γ(νψσ) − FµνRe γρψσ + O(ψ ), 9π2 6π2 24π2 (5a − 3c) µν 2i 5 ic µν [σ ρ] [σ ρ] 5 AS = Fe Dµ − Aµγ ψν + F γµ δ − δ δ γ Dρψσ 6π2 3 6π2 ν µ ν 3(2a − c) (a − c) 1 + P gµ[νγρσ]D ψ + Rµνρσγ − Rg gµ[νγρσ] D ψ + O(ψ3), (4.5) 4π2 µν ρ σ 8π2 µν 2 µν ρ σ

– 9 – where W 2 is the square of the Weyl tensor, E is the Euler density and P is the Pontryagin density. Their expressions in terms of the Riemann tensor are 1 W 2 ≡ W W µνρσ = R Rµνρσ − 2R Rµν + R2, µνρσ µνρσ µν 3 µνρσ µν 2 E = RµνρσR − 4RµνR + R ,

1 κλµν ρσ µνρσ P ≡ RκλρσRµν = Re Rµνρσ, (4.6) 2 3 where the dual Riemann tensor is deﬁned in analogy with the dual U(1)R ﬁeldstrength in (2.8) as

1 κλ Reµνρσ ≡ µν Rκλρσ. (4.7) 2

Moreover, Pµν is the Schouten tensor deﬁned in (2.7) and we have introduced the shorthand notation

2 µν 1 µνρσ F ≡ FµνF ,F Fe ≡ FµνFρσ. (4.8) 2 Finally, we have used the normalization of the central charges adopted in [40], according to which the a and c anomaly coefficients for free chiral and vector multiplets are given respectively by 1 1 a = (N + 9N ), c = (N + 3N ). (4.9) 48 χ v 24 χ v Several comments are in order here. Firstly, we should point out that the anomalies (4.5), as well as the current operators defined in (3.2), are the consistent ones. The corresponding covariant quantities can be obtained by adding the appropriate Bardeen-Zumino terms [66]. Secondly, the bosonic Ward identities and anomalies we obtain reproduce well known results [40] (corrected in [46]), but the fermionic Ward identities and anomalies have not appeared – at least explicitly – in the literature before. Moreover, the Ward identities (3.6), (3.12) and (4.3) derived above, including the anomalies (4.5), are in complete agreement with the results of [8] for theories with a holographic dual that have π`3 a = c = , (4.10) 8G5 where G5 is the Newton constant in five dimensions and ` is the AdS5 radius. Of course, such an agreement was expected since minimal N = 2 gauged supergravity induces off-shell N = 1 conformal supergravity on the four dimensional boundary of AdS5 [64] and the anomalies can be computed through holographic renormalization [34] (see also [9, 10, 46, 68]). An interesting question is whether there exists a local counterterm Wct that removes the Q- supersymmetry anomaly, i.e. such that δε(W + Wct) = 0. Closure of the algebra requires that

[δε, δε0 ](W + Wct) = (δξ + δλ + δθ)(W + Wct), (4.11) with the composite parameters for the bosonic transformations given in (2.12). It follows that if such a counterterm exists, then it must also satisfy

(δξ + δλ + δθ)(W + Wct) = 0. (4.12)

Hence, either Wct removes also the R-symmetry anomaly, or it breaks diffeomorphisms and/or local frame rotations. This means that for theories with an R-symmetry anomaly, either Q- supersymmetry or diffeomorphisms/Lorentz transformations are anomalous as well. The identi- fication of a possible local counterterm that moves the Q-anomaly to diffeomorphisms/Lorentz transformations is a problem we hope to address in future work.

3 In contrast to the Riemann tensor, Reµνρσ is not symmetric under exchange of the ﬁrst and second pair of indices.

– 10 – 5 Anomalous supercurrent transformation under Q- and S-supersymmetry

The superconformal anomalies (4.5) lead to anomalous transformations for the current operators under the corresponding local symmetries. In particular, the fermionic anomalies AQ and AS contribute to the transformation of the supercurrent under respectively Q- and S-supersymmetry [8]. When restricted to rigid symmetries of a speciﬁc background, the anomalous terms in the transformations of the currents result in a deformed superalgebra. The classical (non-anomalous) part of the current transformations can be deduced directly from the supergravity transformations (2.5) and the deﬁnition of the currents in (3.2). For example, (2.5) imply that the functional derivative with respect to the gravitino transforms according to

δ 1 a δ i [µ ρ] 5 µ ρ ν 5 δ δε = γ ε a + 4δν δσ + iγ ν σ γ γ Dρ ε , δψµ 2 δeµ 8 δAσ

δ 3i 5 δ δη = γ η . (5.1) δψµ 4 δAµ It follows that the Q- and S-supersymmetry transformations of the supercurrent are given by4

µ −1 δ −1 δ δεS = e δε W + e δεW δψµ δψµ 1 a µ i [µ ρ] 5 µ ρ ν 5 h σ 4(5a − 3c) σκ i = γ εT + 4δ δ + iγ ν σ γ γ Dρ ε J + Fe Aκ 2 a 8 ν σ 27π2 (a − c) σλρκ µ (a − c) ρσµν + ∇ρ AσRe δ γλ)ε + FρσRe γνε, (5.2) 6π2 (κ 24π2

µ −1 δ −1 δ δηS = e δη W + e δηW δψµ δψµ 3i 5 µ 4(5a − 3c) µν (5a − 3c) µν ic [µ ν] [µ ν] 5 ρσ = γ η J + Fe Aν + Dν(Fe η) − γ ρδ − δ δ γ Dν(F η) 4 27π2 6π2 6π2 σ ρ σ 3(2a − c) (a − c) h 1 i − D P gρ[σγµν]η − D Rµνρσγ − Rg gρ[σγµν] η . (5.3) 4π2 ν ρσ 8π2 ν ρσ 2 ρσ Notice that these transformations coincide with those in eq. (5.9) of [8] in the special case a = c. The anomalous transformations of the supercurrent in (5.2) and (5.3) are essentially a rewriting of the two fermionic Ward identities in (4.3) and are useful for e.g. determining the eﬀect of the superconformal anomalies in correlation functions [6]. They also determine the rigid superalgebra on curved backgrounds that admit Killing spinors of conformal supergravity. Namely, when the local spinor parameters ε and η are restricted to solutions (εo, ηo) of the Killing spinor equation

δεo,ηo ψµ = Dµεo − γµηo = 0, (5.4) on a ﬁxed bosonic background speciﬁed by gµν and Aµ, the corresponding transformation of the supercurrent under rigid supersymmetry is given by5

µ µ µ δ(εo,ηo)S = {Q[εo, ηo], S } = (δεo + δηo )S , (5.5) 4An equivalent but more formal way to determine how the currents transform under the local symmetries is to utilize the symplectic structure underlying the space of couplings and local operators [69]. The Ward identities correspond to ﬁrst class constraints on this space, generating the local symmetry transformations under the Poisson bracket. In appendix B.1 of [8] this approach is used to obtain the anomalous transformation of the supercurrent under Q- and S-supersymmetry in the case a = c. 5 Note that (εo, ηo) are c-number parameters, while (ε, η) are Grassmann valued local parameters that transform non trivially under the local symmetries according to (2.11).

– 11 – where Q[εo, ηo] is the conserved supercharge associated with the Killing spinor (εo, ηo) and all bosonic fields in the transformations (5.2) and (5.3) are evaluated on the specific background. In [8,9] it was shown that even though the Weyl and R-symmetry anomalies are numerically zero for a class of N = 1 conformal supergravity backgrounds admitting two real supercharges of opposite R-charge [18, 19], the anomalous terms in the transformations of the supercurrent under rigid supersymmetry do not vanish, leading to a deformed rigid superalgebra on such backgrounds. As reviewed in the Introduction, this observation was the key to resolving the apparent tension between the field theory results of [30–32] that used the classical superalgebra and the holographic computation of [33]. Besides the dependence of the supersymmetric partition function on the background, however, the transformation of the supercurrent determines also the spectrum of BPS states. The anomalous transformation of the supercurrent results in a shifted spectrum [8,9]. Although it may not be desirable – or even possible – to eliminate the Q-anomaly by a local counterterm that breaks diffeomorphisms and/or local Lorentz transformations as discussed in the previous section, it is plausible that the anomaly in rigid supersymmetry may be removed by a local counterterm that breaks certain (large) diffeomorphisms, but preserves the underlying structure of the supersymmetric background. An example of a somewhat analogous situation was discussed in [70], where supersymmetric gauge theories in three dimensions with both Maxwell and Chern- Simons terms coupled to background topological gravity were considered. The partition function of such theories on Seifert manifolds, which admit two supercharges of opposite R-charge, can be computed via supersymmetric localization and depends explicitly on the Seifert structure modulus b. In principle, this dependence could be explained by the presence of a framing anomaly, which implies that the partition function does indeed depend on the metric at the quantum level. The puzzle, however, is that for Seifert manifolds specifically the framing anomaly is numerically zero. The authors of [70] resolve the puzzle by arguing that in order to make the quantum theory invariant under Seifert reparameterizations and Seifert-topological (i.e. independent of metric deformations that preserve the Seifert structure) they need to add a local but non fully covariant counterterm. This counterterm does depend on the Seifert structure modulus b, which resolves the paradox. It is plausible that similarly the rigid supersymmetry anomaly in four dimensions can be removed by a local counterterm that breaks those large diffeomorphisms that are not compatible with the structure of the supersymmetric background. In fact, for holographic theories (i.e. a = c at large N) defined on trivial circle fibrations over Seifert manifolds such a local counterterm was found in [33]. It would be interesting to generalize this counterterm to non holographic theories with arbitrary a and c using the general form of the fermionic anomalies we obtained in this paper.

6 Concluding remarks

In this paper we determined the local symmetry algebra of N = 1 off-shell conformal supergravity in four dimensions and obtained the general form of the superconformal anomalies by solving the associated Wess-Zumino consistency conditions. To the best of our knowledge, the explicit form of the fermionic Ward identities and their anomalies have not appeared in the literature before. We find that the divergence of the supercurrent, which is associated with Q-supersymmetry, is anomalous whenever R-symmetry is anomalous. This anomaly cannot be removed by a local counterterm without breaking diffeomorphisms and/or local Lorentz transformations. Several open questions remain. Our result that Q-supersymmetry is anomalous in any theory with an anomalous R-symmetry does not seem to depend on the specific supergravity theory we used in this paper. Indeed, we expect this result to hold in non-superconformal theories with

– 12 – an anomalous R-symmetry as well. This expectation is supported by the recent analysis of [7]. Another interesting question is whether there exists a local counterterm that eliminates the Q- supersymmetry anomaly. As we saw in section4, such a counterterm would necessarily break diﬀeomorphisms and/or local Lorentz rotations. The related question for rigid supersymmetry on backgrounds that admit Killing spinors is relevant for the validity of supersymmetric localization computations on four-manifolds. An important example is the computation of generalized supersymmetric indices that count the microstates of supersymmetric AdS5 black holes [71–75]. We hope to return to these questions in future work.

Acknowledgments

I would like to thank Benjamin Assel, Roberto Auzzi, Loriano Bonora, Davide Cassani, Cyril Closset, Camillo Imbimbo, Heeyeon Kim, Zohar Komargodski, Dario Martelli, Sameer Murthy and Dario Rosa for illuminating discussions and email correspondence. I am also grateful to the University of Southampton, King’s College London and the International Center for Theoretical Physics in Trieste for hospitality and partial ﬁnancial support during the completion of this work.

Appendix

A Spinor conventions and identities

Throughout this paper we follow the conventions of [61]. In particular, the tangent space metric is η = diag (−1, 1, 1, 1) and the Levi-Civita symbol ε = ±1 satisfies ε = 1. The Levi-Civita √ µνρσ 0123 tensor is defined as usual as µνρσ = −g εµνρσ = e εµνρσ. Moreover, the chirality matrix in four dimensions is given by 5 γ = iγ0γ1γ2γ3, (A.1) and we define the antisymmetrized products of gamma matrices as

γµ1µ2...µn ≡ γ[µ1 γµ2 ··· γµn], (A.2) where antisymmetrization is done with weight one.

In the conventions we use here the gravitino ψµ is a Majorana spinor (see section 3.3 of [61] for the deﬁnition) and we make extensive use of the spinor bilinear identity in four dimensions

λγµ1 γµ2 ··· γµp χ = (−1)pχγµp ··· γµ2 γµ1 λ, [eq. (3.53) in [61]]. (A.3)

For Majorana fermions we also have that

∗ (χγµ1...µr λ) = χγµ1...µr λ, [eq. (3.82) in [61]]. (A.4)

It is convenient to collect several identities involving antisymmetrized products of gamma

– 13 – matrices in d dimensions, most of which can be found in section 3 of [61]: 1 γµνρ = {γµ, γνρ}, 2 1 γµνρσ = [γµ, γνρσ], 2 µν µν [µ ν] [µ ν] γ γρσ = γ ρσ + 4γ [σδ ρ] + 2δ [σδ ρ],

ν1...νp ν1...νp [ν1 ν2...νp] γµγ = γµ + pδµ γ ,

ν1...νp ν1...νp [ν1...νp−1 νp] γ γµ = γ µ + pγ δµ , µνρ µνρ [µν ρ] [µ ν ρ] γ γστ = γ στ + 6γ [τ δ σ] + 6γ δ [τ δ σ], µνρσ µνρσ [µνρ σ] [µν ρ σ] γ γτλ = γ τλ + 8γ [λδ τ] + 12γ δ [λδ τ], µνρ µνρ [µν ρ] [µ ν ρ] [µ ν ρ] γ γστλ = γ στλ + 9γ [τλδ σ] + 18γ [λδ τ δ σ] + 6δ [λδ τ δ σ], (d − r)! γµ1...µrν1...νs γ = γµ1...µr , νs...ν1 (d − r − s)! µρ µ µ γ γρν = (d − 2)γ ν + (d − 1)δν , µνρ µν [µ ν] γ γρσ = (d − 3)γ σ + 2(d − 2)γ δ σ, νρσ ρσ [ρ σ] γµνγ = (d − 3)γµ + 2(d − 2)δµ γ , µνλ µν [µ ν] [µ ν] γ γλρσ = (d − 4)γ ρσ + 4(d − 3)γ [σδ ρ] + 2(d − 2)δ [σδ ρ], ρστ 2 σ σ σ γµργ γτν = (d − 4) γµ ν + (d − 4)(d − 3) (γµδν − γ gµν) σ σ + (d − 3)(d − 2)δµγν − (d − 3)γ γµν, µ1µ2...µp ρ p µ1µ2...µp γργ γ = (−1) (d − 2p)γ . (A.5)

In particular, the following identities apply speciﬁcally to d = 4 and are used extensively:

γργµγσ + γσγµγρ = 2(gµργσ + gµσγρ − gρσγµ), γργµγσ − γσγµγρ = 2γρµσ, γργµγσ = gµργσ + gµσγρ − gρσγµ + γρµσ γµργσ = γµρσ + γµgρσ − γρgµσ µρσ µρσν 5 γ = i γνγ , i γµν = µνρσγ γ5. (A.6) 2 ρσ Finally, the following three identities in four dimensions help compare the superconformal Ward identities (4.3) and the anomalies (4.5) with the corresponding results obtained in [8]:

νκλ [κ λ] κλ (γµν − 2gµν)γ = 4δµ γ − γµ [κ λ] νρσ = 4δµ γ − i µ γσγ5, γµγνκλ = γµνκλ + 3gµ[νγκλ], νκ µ νκµ νκ µ νκµ (2γ γ − 3γ ) γµρσ = 2γ γ γµρσ − 3γ γµρσ νκ [ν κ] [ν κ] = 4γ γρσ − 12 γ [σδ ρ] + δ [σδ ρ] νκ [ν κ] [ν κ] = 4 γ ρσ + γ [σδ ρ] − δ [σδ ρ] . (A.7)

– 14 – B Solving the Wess-Zumino consistency conditions

In this appendix we provide the details of the proof that the superconformal anomalies (4.5) satisfy the Wess-Zumino consistency conditions (4.4) associated with the local symmetry algebra (2.12) of N = 1 conformal supergravity. Only a subset of the algebra relations need be checked explicitly since all commutators between any two non-anomalous symmetries are trivially satisﬁed. More- over, the Wess-Zumino conditions for purely bosonic symmetries are known to hold [76] and are straightforward to check. We shall therefore focus on the commutation relations involving at least one fermionic symmetry transformation, except for the four commutators

[δξ, δε]W = 0, [δξ, δη]W = 0, [δλ, δε]W = 0, [δλ, δη]W = 0, (B.1) which hold trivially. All computations in this appendix assume either a compact spacetime manifold or that the ﬁelds go to zero at inﬁnity so that total derivative terms can be dropped. Moreover, we only keep the leading non trivial terms in the gravitino ψµ.

[δσ, δε]W = 0 and [δσ, δη]W = 0:

Let us ﬁrst consider the two commutation relations

[δσ, δε]W , [δσ, δη]W . (B.2)

Taking into account the Weyl transformation of the supersymmetry parameters in (2.11), it is straightforward to check that both the Q- and S-supersymmetry anomalies are Weyl invariant, i.e.

δσδεW = 0, δσδηW = 0. (B.3)

Moreover, the O(ψ2) terms in the Weyl anomaly in (4.5) ensure that the Weyl anomaly density is invariant under both Q- and S-supersymmetry, i.e.6

δε(e AW ) = 0, δη(e AW ) = 0. (B.4)

Combining these results we conclude that these two commutators satisfy the Wess-Zumino consistency conditions compatible with the local symmetry algebra, namely

[δσ, δε]W = 0, [δσ, δη]W = 0. (B.5)

[δε, δθ]W = 0:

The only remaining commutation relations involving a bosonic symmetry are those between local gauge transformations and either Q- or S-supersymmetry transformations. Staring with Q-

6For the case a = c the O(ψ2) terms in the Weyl anomaly can be found in [8]. For generic a and c the conditions (B.4) can be used to derive the fermionic terms in the Weyl anomaly.

– 15 – supersymmetry, the anomalies in (4.5) determine

4 δεδ W = −δε d x e θAR θ ˆ (5a − 3c) (c − a) = − d4x e θ µνρσδ (F F ) + d4x e θ κλµνδ (Rρ Rσ ) 54π2 ˆ ε µν ρσ 48π2 ˆ ε σκλ ρµν 2(5a − 3c) (c − a) = − d4x e θ µνρσF ∂ δ A + d4x e θ κλµνRρ ∇ δ Γσ 27π2 ˆ µν ρ ε σ 12π2 ˆ σκλ µ ε ρν (5a − 3c)i (c − a) = d4x e ∂ θ µνρσF εγ5φ − d4x e ∂ θ κλµνRρσ ∇ δ g , 18π2 ˆ ρ µν σ 12π2 ˆ µ κλ ρ ε σν (5a − 3c)i (c − a) = d4x e ∂ θ µνρσF εγ5φ + d4x e ∇ ∂ θ κλµνRρσ εγ ψ , (B.6) 18π2 ˆ ρ µν σ 12π2 ˆ ρ µ κλ (σ ν)

4 δ δεW = −δ d x e εAQ θ θ ˆ (5a − 3c)i (c − a) = d4x e ∂ θ µνρσF εγ5φ + d4x e µνρσ∇ ∂ θ Rκλ εγ ψ . (B.7) 18π2 ˆ ρ µν σ 12π2 ˆ κ ρ µν (λ σ) Hence,

[δε, δθ]W = 0, (B.8) as required by the Wess-Zumino conditions.

[δη, δθ]W = 0:

For S-supersymmetry we have similarly

4 δηδ W = − δη d x e θAR (B.9) θ ˆ (5a − 3c) = − d4x e θ µνρσδ (F F ) 54π2 ˆ η µν ρσ 2(5a − 3c) = − d4x e θ µνρσF ∂ δ A 27π2 ˆ µν ρ η σ (5a − 3c)i = − d4x e ∂ θ µνρσF ηγ5ψ , (B.10) 18π2 ˆ ρ µν σ

4 δ δηW = δ d x e ηAS θ θ ˆ (5a − 3c)i = − d4x e ∂ θ µνρσF ηγ5ψ , (B.11) 18π2 ˆ ρ µν σ and hence,

[δη, δθ]W = 0. (B.12)

1 0 λ [δε, δε0 ]W = δθW , with θ = − 2 (ε γ ε)Aλ:

– 16 – There remain only the four commutators among fermionic symmetries. Applying two successive Q-supersymmetry transformations on the generating function W gives

4 δ 0 δεW = − δ 0 d x e εAQ ε ε ˆ (5a − 3c) 4 µνρσ 5 (c − a) 4 µνρσ κλ = i d x e F A εγ δ 0 φ + d x e ∇ A R εγ δ 0 ψ 18π2 ˆ µν ρ ε σ 12π2 ˆ κ ρ µν (λ ε σ)

(a − c) 4 µνρσ κλ + d x e F R εγ δ 0 ψ 48π2 ˆ ρσ µν κ ε λ (5a − 3c) 1 i 1 = i d4x e µνρσF A εγ5 P + F γ5 − κτ F γλε0 18π2 ˆ µν ρ 2 σλ 3 σλ 12 σλ κτ (c − a) − d4x e µνρσA Rκλ ∇ (εγ D ε0) 12π2 ˆ ρ µν κ (λ σ) (a − c) + d4x e µνρσF Rκλ εγ D ε0, (B.13) 48π2 ˆ ρσ µν κ λ and hence

(5a − 3c) 4 µνρσ 0 λ [δε, δ 0 ]W = − d x e FµνAρF ε γ ε ε 27π2 ˆ σλ (c − a) − d4x e µνρσA Rκλ R (ε0γτ ε) 48π2 ˆ ρ µν κλστ (c − a) − d4x e µνρσA Rκλ ∇ ∇ (ε0γ ε) 24π2 ˆ ρ µν κ σ λ (a − c) + d4x e µνρσF Rκλ ∇ (ε0γ ε). (B.14) 48π2 ˆ ρσ µν λ κ The last two terms can be rearranged as 1 µνρσA Rκλ ∇ ∇ (ε0γ ε) − µνρσF Rκλ ∇ (ε0γ ε) ρ µν κ σ λ 2 ρσ µν κ λ µνρσ κ 0 λ µνρσ κλ 0 = AρR λµν[∇κ, ∇σ](ε γ ε) + ∇σ AρR µν∇κ(ε γλε) µνρσ κλ 0 µνρσ κλ 0 τ = ∇σ AρR µν∇κ(ε γλε) + AρR µνRκσλτ (ε γ ε) 1 = ∇ µνρσA Rκλ ∇ (ε0γ ε) + µνρσA Rκλ R (ε0γτ ε), (B.15) σ ρ µν κ λ 2 ρ µν κλστ so that

(5a − 3c) 4 µνρσ 0 λ [δε, δ 0 ]W = − d x e FµνAρF ε γ ε ε 27π2 ˆ σλ (c − a) − d4x e µνρσA Rκλ R (ε0γτ ε). (B.16) 24π2 ˆ ρ µν κλστ κλ Moreover, the fact that F[λσFµνAρ] = 0 and Rκλ[µνR στ Aρ] = 0 in four dimensions leads to the two identities 1 µνρσF F A = − µνρσF F A , µν σλ ρ 4 µν ρσ λ 1 µνρσR R κλA = − µνρσR R κλA . (B.17) µνκλ στ ρ 4 µνκλ ρσ τ Therefore, we ﬁnally get

(5a − 3c) 4 µνρσ 0 λ (c − a) 4 µνρσ κλ 0 τ [δε, δ 0 ]W = d x e FµνFρσ(A ε γ ε) + d x e R µνR (Aτ ε γ ε) ε 108π2 ˆ λ 96π2 ˆ κλρσ = − d4x e θA , (B.18) ˆ R

– 17 – with 1 θ = − (ε0γλε)A , (B.19) 2 λ as required by the Wess-Zumino consistency conditions.

[δη, δη0 ]W = 0:

Two successive S-supersymmetry transformations on the generating function W give

(5a − 3c)i 4 µνρσ 5 (5a − 3c) 4 ρσ µν δ 0 δηW = − d x e FµνAρηγ δ 0 ψσ + d x e µν F ηDρδ 0 ψσ η 18π2 ˆ η 12π2 ˆ η ic 4 µν [σ ρ] [σ ρ] 5 3(2a − c) 4 µ[ν ρσ] + d x e F η γ δ − δ δ γ D δ 0 ψ + d x e P g ηγ D δ 0 ψ 6π2 ˆ µ ν µ ν ρ η σ 4π2 ˆ µν ρ η σ

(a − c) 4 µνρσ 1 µ[ν ρσ] + d x e η R γ − Rg g γ D δ 0 ψ 8π2 ˆ µν 2 µν ρ η σ (5a − 3c)i (5a − 3c) = d4x e µνρσF A ηγ5γ η0 − d4x e ρσF µνηγ D η0 18π2 ˆ µν ρ σ 12π2 ˆ µν σ ρ ic 3(2a − c) − d4x e F µνηγ [σδρ] − δ[σδρ]γ5γ D η0 − d4x e P gµ[νηγρσ]γ D η0 6π2 ˆ µ ν µ ν σ ρ 4π2 ˆ µν σ ρ (a − c) 1 − d4x e η Rµνρσγ − Rg gµ[νγρσ] γ D η0. (B.20) 8π2 ˆ µν 2 µν σ ρ We will now show that each of these terms vanishes once the corresponding expression with η and η0 interchanged is subtracted. The ﬁrst term in the second equality in (B.20) vanishes trivially since

5 0 0 5 ηγ γση = η γ γση. (B.21)

For the second term we have

0 0 0 ηγσDρη − η γσDρη = ∇ρ(ηγση ). (B.22)

µνρσ Integrating by parts and using the Bianchi identity ∂ρFµν = 0 we ﬁnd that the second term vanishes as well. The third term can be simpliﬁed as

νκ [σ ρ] [σ ρ] 5 0 F η γν δκ − δν δκ γ γσDρη 1 = − F νκηγ σgρ − γ ρgσ − 2gσgργ γ5D η0 2 ν κ ν κ ν κ σ ρ 1 0 = − F νκη3γ gρ − γ ρ − γ gρ + γρg ¨¨*− 2gργ γ5D η0 2 ν κ ν κ ν κ ¨νκ κ ν ρ 1 = − F νκηγρ γ5D η0. (B.23) 2 νκ ρ Hence, subtracting the same quantity with η and η0 interchanged we obtain 1 1 − F νκηγρ γ5D η0 + F νκη0γρ γ5D η 2 νκ ρ 2 νκ ρ 1 1 1 = − F νκηγρ γ5D η0 + F νκ∇ η0γρ γ5η + F νκηγ5γ ρD η0 2 νκ ρ 2 ρ νκ 2 κν ρ 1 = F νκ∇ η0γρ γ5η, (B.24) 2 ρ νκ

– 18 – µνρσ which again vanishes up to a total derivative term due to the Bianchi identity ∂ρFµν = 0 . In order to evaluate the term proportional to the Schouten tensor Pµν we note that

µν 0 0 µν ηγ γλDκη − η γ γλDκη µν 0 0 µν = η{γ , γλ}Dκη − ∇κ(η γ γλη) µν 0 0 µν µ ν ν µ = 2ηγ λDκη − ∇κ η (γ λ + γ gλ − γ gλ )η . (B.25)

Hence,

µ[ν κλ] 0 0 µ[ν κλ] Pµν ηg γ γλDκη − η g γ γλDκη 1 = P gµνηγκλγ D η0 + gµληγνκγ D η0 + gµκηγλνγ D η0 − η0 ↔ η 3 µν λ κ λ κ λ κ 1 = − P ∇ gµνη0(γκgλ − γλgκ)η + gµλη0(γνgκ − γκgν)η + gµκη0(γλgν − γνgλ)η 3 µν κ λ λ λ λ λ λ 1 = − P ∇ 3gµνη0γκη + η0(γνgµκ − γκgµν)η − 3gµκη0γνη 3 µν κ 2 = − P ∇ gµνη0γκη − gµκη0γνη 3 µν κ 1 1 1 h 1 i = − R∇ (η0γµη) + R ∇µ(η0γνη) = ∇µ R − Rg (η0γνη) , (B.26) 6 µ 3 µν 3 µν 2 µν where in the last equality we have used the Bianchi identity 1 ∇µ R − Rg = 0. (B.27) µν 2 µν

It follows that the term proportional to the Schouten tensor Pµν in (B.20) also vanishes. Finally, for the last term in (B.20) we have

µνρσ 0 µνρσ 0 0 µρ 0 R ηγµνγσDρη = R η(γµνσ:+ γµgνσ − γνgµσ)Dρη = 2R ηγµDρη , (B.28) with µν 0 µν 0 µν 0 R ηγµDνη − R η γµDνη = R ∇ν(ηγµη ). (B.29) Hence, 1 Rµνρσηγ γ D η0 − Rµνρση0γ γ D η − Rg ηgµ[νγκλ]γ D η0 − η0gµ[νγκλ]γ D η µν σ ρ µν σ ρ 2 µν λ κ λ κ h 1 i = 2Rµν∇ (ηγ η0) + R∇ (η0γµη) = 2∇µ R − Rg (ηγνη0) , (B.30) ν µ µ µν 2 µν where we have again used the Bianchi identity (B.27) in the last equality. To summarize, all terms in (B.20) vanish upon subtracting the same quantities with η and η0 interchanged, leading to the commutator

[δη, δη0 ]W = 0, (B.31) in agreement with the Wess-Zumino consistency conditions.

1 3i 5 [δε, δη]W = δσW + δθW , with σ = 2 εη and θ = − 4 εγ η:

– 19 – The ﬁnal commutator we need to consider is the one between Q- and S-supersymmetry transformations. We start with the contribution of the AQ anomaly to the commutator, namely

4 δηδεW = − δη d x e εAQ ˆ (5a − 3c)i (a − c) = d4x e µνρσF A εγ5δ φ − d4x e µνρσ∇ A Rκλ εγ δ ψ 18π2 ˆ µν ρ η σ 12π2 ˆ κ ρ µν (λ η σ) (a − c) + d4x e µνρσF Rκλ εγ δ ψ 48π2 ˆ ρσ µν κ η λ : 0 (5a − 3c)i 4 µνρσ 5 (a − c) 4 µνρσ κ = d xe FµνAρεγ Dση + d x e ∇κAρR σµν εη 18π2 ˆ 12π2 ˆ (a − c) − d4x e µνρσF Rκλ εγ η 48π2 ˆ ρσ µν κλ (5a − 3c)i (a − c) = d4x e µνρσF A εγ5D η − d4x e µνρσF Rκλ εγ η. (B.32) 18π2 ˆ µν ρ σ 48π2 ˆ ρσ µν κλ

The last term in this expression can be simpliﬁed further using the last identity in (A.6) and the product of two Levi-Civita tensors

− µνρσκλτφ = gµκgνλgρτ gσφ + gµλgνκgρφgστ + gµτ gνφgρκgσλ + gµφgντ gρλgσκ − gµλgνκgρτ gσφ + gµτ gνλgρκgσφ + gµφgνλgρτ gσκ + gµκgντ gρλgσφ + gµκgνφgρτ gσλ + gµκgνλgρφgστ + gµτ gνκgρλgσφ + gµλgντ gρκgσφ + gµφgνκgρτ gσλ + gµλgνφgρτ gσκ + gµφgνλgρκgστ + gµτ gνλgρφgσκ + gµκgνφgρλgστ + gµκgντ gρφgσλ − gµφgνκgρλgστ + gµτ gνκgρφgσλ + gµφgντ gρκgσλ + gµλgνφgρκgστ + gµτ gνφgρλgσκ + gµλgντ gρφgσκ, (B.33) where we have grouped terms according to the conjugacy classes of the symmetric group S4 and 5 terms of a given color give the same result when contracted with FµνRρσκληγτφγ ε. In particular,

µνρσ κλτφ 5 − FµνRρσκληγτφγ ε µν ρσ 5 ν µρ 5 ν ρµ 5 =4 F Rµνρσηγ γ ε − 4Fµ Rνρηγ γ ε +4 Fµ Rρνηγ γ ε ν µρ 5 ν ρµ 5 µν 5 +4 F µRνρηγ γ ε − 4F µRρνηγ γ ε +4 FµνRηγ γ ε µν ρσ 5 ν µρ 5 µν 5 = 4 F Rµνρσηγ γ ε − 4Fµ Rνρηγ γ ε + FµνRηγ γ ε , (B.34) and so i µνρσF Rκλ ηγ ε = µνρσκλτφF R ηγ γ5ε ρσ µν κλ 2 µν ρσκλ τφ µν ρσ 5 ν µρ 5 µν 5 = − 2i F Rµνρσηγ γ ε − 4Fµ Rνρηγ γ ε + FµνRηγ γ ε . (B.35)

– 20 – The contribution of the AS anomaly to the commutator is

4 δεδηW = δε d x e ηAS ˆ (5a − 3c) (5a − 3c) = − i d4x e µνρσF A ηγ5δ ψ + d4x e µνρσF ηD δ ψ 18π2 ˆ µν ρ ε σ 12π2 ˆ µν ρ ε σ ic 3(2a − c) + d4x e F µνηγ [σδ ρ] − δ [σδ ρ]γ5D δ ψ + d4x e P gµ[νηγρσ]D δ ψ 6π2 ˆ µ ν µ ν ρ ε σ 4π2 ˆ µν ρ ε σ (a − c) 1 + d4x e η Rµνρσγ − Rg gµ[νγρσ] D δ ψ 8π2 ˆ µν 2 µν ρ ε σ (5a − 3c) (5a − 3c) = − i d4x e µνρσF A ηγ5D ε + d4x e µνρσF ηD D ε 18π2 ˆ µν ρ σ 12π2 ˆ µν ρ σ ic 3(2a − c) + d4x e F µνηγ [σδ ρ] − δ [σδ ρ]γ5D D ε + d4x e P gµ[νηγρσ]D D ε 6π2 ˆ µ ν µ ν ρ σ 4π2 ˆ µν ρ σ (a − c) 1 + d4x e η Rµνρσγ − Rg gµ[νγρσ] D D ε. (B.36) 8π2 ˆ µν 2 µν ρ σ Using the identity, 1 2D D ε = R γρσ + iγ5F ε, (B.37) [µ ν] 4 µνρσ µν we now evaluate the four terms involving two covariant derivatives on the spinor parameter ε. The easiest term to evaluate is

µν [σ ρ] [σ ρ] 5 iF η γµ δν − δµ δν γ DρDσε i 1 = F µνηγ σδ ρ − δ σδ ργ5 R γκλ + iγ5F ε 2 µ ν µ ν 4 ρσκλ ρσ 1 i i = − F F µνηε + F νR ηγµσγκλγ5ε + F µνR ηγκλγ5ε 2 µν 8 µ νσκλ 8 µνκλ 1 i = − F F µνηε + F µνR ηγκλγ5ε 2 µν 8 µνκλ i : 0 : 0 : 0 + F νR η − iµσκλγ5 + 2γµλgσκ − 2γσλgµκ +gµλgσκ −gσλgµκγ5ε 8 µ νσκλ 1 i = − F F µνηε + F µν(R + 2R − 2g R )ηγρσγ5ε 2 µν 8 µνρσ µρνσ µρ νσ 1 i = − F F µνηε + F µν(R − g R )ηγρσγ5ε. (B.38) 2 µν 4 µνρσ µρ νσ We next consider the term

µνρσ FµνηDρDσε i 1 = µνρσF F ηγ5ε + µνρσF R ηγκλε 2 µν ρσ 8 µν ρσκλ i i = µνρσF F ηγ5ε − F µνR ηγρσγ5ε − 4F νR ηγµργ5ε + F Rηγµνγ5ε, (B.39) 2 µν ρσ 4 µνρσ µ νρ µν where in the last line we have utilized the identity (B.35).

– 21 – Next we evaluate the term

µ[ν κλ] Pµνg ηγ DκDλε 1 1 = P ηgµνγκλ + gµλγνκ + gµκγλν R γρσ + iγ5F ε 6 µν 4 κλρσ κλ 1 1 = P η(gµνγκλ − 2gµκγνλ) R γρσ + iγ5F ε 6 µν 4 κλρσ κλ 1 1 1 1 = R R ηγκλγρσε + iηγκλγ5εF − P Rµ ηγνλγρσε + iηγνλγ5εF µ 36 4 κλρσ κλ 3 µν 4 λρσ λ 1 1 = − R − Rg × 6 µν 3 µν 0 µ νλ 5 1 µ νλρσ¨* 5 νσ λρ: 0 λσ νρ: 0 νσ λρ λσ νρ iF ηγ γ ε + R η − i¨ ¨ γ + 2γ g − 2γ g + g g − g g ε λ 4 λρσ 1 1 1 1 = − R − Rg iF µ ηγνλγ5ε + R Rµν − R2 ηε. (B.40) 6 µν 3 µν λ 12 µν 3 Finally, the last term gives 1 Rµνρσγ − Rg gµ[νγρσ] D D ε µν 2 µν ρ σ 1 1 1 = Rµνρσγ − Rg gµ[νγρσ] R γκλ + iγ5F ε 2 µν 2 µν 4 ρσκλ ρσ 1 ¨¨* 0 = RµνρσR − i κλγ5 + 4γ ¨λgκ + 2gλgκε 8 κλρσ µν ¨µ ν µ ν i 1 + RµνρσF γ γ5ε − 2iRF ηγµνγ5ε − R2ηε 2 ρσ µν 12 µν i 1 i 1 = − Pγ5ε − R Rµνρσε + RµνρσF γ γ5ε − 2iRF ηγµνγ5ε − R2ηε, (B.41) 4 4 µνρσ 2 ρσ µν 12 µν

1 µνρσ κλ where P = 2 RµνκλRρσ is the Pontryagin density. Substituting (B.35) in (B.32), and (B.38), (B.39), (B.40) and (B.41) in (B.36) we ﬁnally obtain

(5a − 3c)i 4 µνρσ 5 c 4 µν [δε, δη]W = d x e FµνFρσηγ ε − d x e FµνF ηε 72π2 ˆ 12π2 ˆ (2a − c) 1 + d4x e R Rµν − R2 ηε 16π2 ˆ µν 3 (a − c) 1 − d4x e iPηγ5ε + R Rµνρσηε − R2ηε 32π2 ˆ µνρσ 3 = d4x e (−θA + σA ), (B.42) ˆ R W with 3 1 θ = − iεγ5η, σ = εη, (B.43) 4 2 as required by the Wess-Zumino consistency conditions.

References

[1] E. Witten, Supersymmetry and Morse theory, J. Diﬀ. Geom. 17 (1982), no. 4 661–692. [2] E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353. [3] N. A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), no. 5 831–864, [hep-th/0206161].

– 22 – [4] V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71–129, [0712.2824]. [5] J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. 37B (1971) 95–97. [6] G. Katsianis, I. Papadimitriou, K. Skenderis, and M. Taylor, Anomalous Supersymmetry, 1902.06715. [7] O. S. An, J. U. Kang, J. C. Kim, and Y. H. Ko, Quantum consistency in supersymmetric theories with R-symmetry in curved space, 1902.04525. [8] I. Papadimitriou, Supercurrent anomalies in 4d SCFTs, JHEP 07 (2017) 038, [1703.04299]. [9] O. S. An, Anomaly-corrected supersymmetry algebra and supersymmetric holographic renormalization, JHEP 12 (2017) 107, [1703.09607]. [10] O. S. An, Y. H. Ko, and S.-H. Won, Super-Weyl Anomaly from Holography and Rigid Supersymmetry Algebra on Two-Sphere, 1812.10209. [11] S. L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426–2438. [12] J. S. Bell and R. Jackiw, A PCAC puzzle: π0 → γγ in the σ model, Nuovo Cim. A60 (1969) 47–61. [13] K. Landsteiner, Notes on Anomaly Induced Transport, Acta Phys. Polon. B47 (2016) 2617, [1610.04413]. [14] J. Gooth et. al., Experimental signatures of the mixed axial-gravitational anomaly in the Weyl semimetal NbP, Nature 547 (2017) 324–327, [1703.10682]. [15] V. Pestun et. al., Localization techniques in quantum field theories, J. Phys. A50 (2017), no. 44 440301, [1608.02952]. [16] G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114, [1105.0689]. [17] H. Samtleben and D. Tsimpis, Rigid supersymmetric theories in 4d Riemannian space, JHEP 05 (2012) 132, [1203.3420]. [18] C. Klare, A. Tomasiello, and A. Zaffaroni, Supersymmetry on Curved Spaces and Holography, JHEP 08 (2012) 061, [1205.1062]. [19] T. T. Dumitrescu, G. Festuccia, and N. Seiberg, Exploring Curved Superspace, JHEP 08 (2012) 141, [1205.1115]. [20] J. T. Liu, L. A. Pando Zayas, and D. Reichmann, Rigid Supersymmetric Backgrounds of Minimal Off-Shell Supergravity, JHEP 10 (2012) 034, [1207.2785]. [21] T. T. Dumitrescu and G. Festuccia, Exploring Curved Superspace (II), JHEP 01 (2013) 072, [1209.5408]. [22] A. Kehagias and J. G. Russo, Global Supersymmetry on Curved Spaces in Various Dimensions, Nucl. Phys. B873 (2013) 116–136, [1211.1367]. [23] C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. Komargodski, Supersymmetric Field Theories on Three-Manifolds, JHEP 05 (2013) 017, [1212.3388]. [24] H. Samtleben, E. Sezgin, and D. Tsimpis, Rigid 6D supersymmetry and localization, JHEP 03 (2013) 137, [1212.4706]. [25] D. Cassani, C. Klare, D. Martelli, A. Tomasiello, and A. Zaffaroni, Supersymmetry in Lorentzian Curved Spaces and Holography, Commun. Math. Phys. 327 (2014) 577–602, [1207.2181]. [26] P. de Medeiros, Rigid supersymmetry, conformal coupling and twistor spinors, JHEP 09 (2014) 032, [1209.4043]. [27] K. Hristov, A. Tomasiello, and A. Zaffaroni, Supersymmetry on Three-dimensional Lorentzian Curved Spaces and Black Hole Holography, JHEP 05 (2013) 057, [1302.5228].

– 23 – [28] M. Blau, Killing spinors and SYM on curved spaces, JHEP 11 (2000) 023, [hep-th/0005098]. [29] S. M. Kuzenko, Symmetries of curved superspace, JHEP 03 (2013) 024, [1212.6179]. [30] C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. Komargodski, The Geometry of Supersymmetric Partition Functions, JHEP 01 (2014) 124, [1309.5876]. [31] C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. Komargodski, From Rigid Supersymmetry to Twisted Holomorphic Theories, Phys. Rev. D90 (2014), no. 8 085006, [1407.2598]. [32] B. Assel, D. Cassani, and D. Martelli, Localization on Hopf surfaces, JHEP 08 (2014) 123, [1405.5144]. [33] P. Benetti Genolini, D. Cassani, D. Martelli, and J. Sparks, Holographic renormalization and supersymmetry, JHEP 02 (2017) 132, [1612.06761]. [34] M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023, [hep-th/9806087]. [35] S. de Haro, S. N. Solodukhin, and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence, Commun. Math. Phys. 217 (2001) 595–622, [hep-th/0002230]. [36] I. N. McArthur, Super b(4) Coefficients in Supergravity, Class. Quant. Grav. 1 (1984) 245. [37] L. Bonora, P. Pasti, and M. Tonin, Cohomologies and Anomalies in Supersymmetric Theories, Nucl. Phys. B252 (1985) 458–480. [38] I. L. Buchbinder and S. M. Kuzenko, Matter Superfields in External Supergravity: Green Functions, Effective Action and Superconformal Anomalies, Nucl. Phys. B274 (1986) 653–684. [39] F. Brandt, Anomaly candidates and invariants of D = 4, N=1 supergravity theories, Class. Quant. Grav. 11 (1994) 849–864, [hep-th/9306054]. [40] D. Anselmi, D. Z. Freedman, M. T. Grisaru, and A. A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B526 (1998) 543–571, [hep-th/9708042]. [41] O. Piguet and S. Wolf, The Supercurrent trace identities of the N=1, D = 4 superYang-Mills theory in the Wess-Zumino gauge, JHEP 04 (1998) 001, [hep-th/9802027]. [42] J. Erdmenger and C. Rupp, Geometrical superconformal anomalies, hep-th/9809090. [43] J. Erdmenger and C. Rupp, Superconformal Ward identities for Green functions with multiple supercurrent insertions, Annals Phys. 276 (1999) 152–187, [hep-th/9811209]. [44] L. Bonora and S. Giaccari, Weyl transformations and trace anomalies in N=1, D=4 supergravities, JHEP 08 (2013) 116, [1305.7116]. [45] D. Butter and S. M. Kuzenko, Nonlocal action for the super-Weyl anomalies: A new representation, JHEP 09 (2013) 067, [1307.1290]. [46] D. Cassani and D. Martelli, Supersymmetry on curved spaces and superconformal anomalies, JHEP 10 (2013) 025, [1307.6567]. [47] R. Auzzi and B. Keren-Zur, Superspace formulation of the local RG equation, JHEP 05 (2015) 150, [1502.05962]. [48] J. Wess and J. Bagger, Supersymmetry and supergravity. Princeton University Press, Princeton, NJ, USA, 1992. [49] O. Piguet and K. Sibold, Renormalized supersymmetry. The perturbation theory of N=1 supersymmetric theories in flat space-time, vol. 12. 1986. [50] H. Itoyama, V. P. Nair, and H.-c. Ren, Supersymmetry Anomalies and Some Aspects of Renormalization, Nucl. Phys. B262 (1985) 317–330.

– 24 – [51] E. Guadagnini and M. Mintchev, CHIRAL ANOMALIES AND SUPERSYMMETRY, Nucl. Phys. B269 (1986) 543–556. [52] O. Piguet and K. Sibold, The Anomaly in the Slavnov Identity for N = 1 Supersymmetric Yang-Mills Theories, Nucl. Phys. B247 (1984) 484. [53] M. Kaku, P. K. Townsend, and P. van Nieuwenhuizen, Gauge Theory of the Conformal and Superconformal Group, Phys. Lett. 69B (1977) 304–308. [54] M. Kaku, P. K. Townsend, and P. van Nieuwenhuizen, Superconformal Uniﬁed Field Theory, Phys. Rev. Lett. 39 (1977) 1109. [55] M. Kaku, P. K. Townsend, and P. van Nieuwenhuizen, Properties of Conformal Supergravity, Phys. Rev. D17 (1978) 3179. [,853(1978)]. [56] P. K. Townsend and P. van Nieuwenhuizen, Simpliﬁcations of Conformal Supergravity, Phys. Rev. D19 (1979) 3166. [57] P. Van Nieuwenhuizen, Supergravity, Phys. Rept. 68 (1981) 189–398. [58] B. de Wit, CONFORMAL INVARIANCE IN EXTENDED SUPERGRAVITY, in First School on Supergravity Trieste, Italy, April 22-May 6, 1981, p. 0267, 1981. [59] B. de Wit, MULTIPLET CALCULUS, in September School on Supergravity and Supersymmetry Trieste, Italy, September 6-18, 1982, 1983. [60] E. S. Fradkin and A. A. Tseytlin, CONFORMAL SUPERGRAVITY, Phys. Rept. 119 (1985) 233–362. [61] D. Z. Freedman and A. Van Proeyen, Supergravity. Cambridge Univ. Press, Cambridge, UK, 2012. [62] M. Kaku and P. K. Townsend, POINCARE SUPERGRAVITY AS BROKEN SUPERCONFORMAL GRAVITY, Phys. Lett. 76B (1978) 54–58. [63] K. S. Stelle and P. C. West, Tensor Calculus for the Vector Multiplet Coupled to Supergravity, Phys. Lett. 77B (1978) 376. [64] V. Balasubramanian, E. G. Gimon, D. Minic, and J. Rahmfeld, Four-dimensional conformal supergravity from AdS space, Phys. Rev. D63 (2001) 104009, [hep-th/0007211]. [65] B. de Wit, S. Murthy, and V. Reys, BRST quantization and equivariant cohomology: localization with asymptotic boundaries, JHEP 09 (2018) 084, [1806.03690]. [66] W. A. Bardeen and B. Zumino, Consistent and Covariant Anomalies in Gauge and Gravitational Theories, Nucl. Phys. B244 (1984) 421–453. [67] K. Jensen, R. Loganayagam, and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088, [1207.5824]. [68] Y. Nakayama, CP-violating CFT and trace anomaly, Nucl. Phys. B859 (2012) 288–298, [1201.3428]. [69] I. Papadimitriou, Lectures on Holographic Renormalization, Springer Proc. Phys. 176 (2016) 131–181. [70] C. Imbimbo and D. Rosa, Topological anomalies for Seifert 3-manifolds, JHEP 07 (2015) 068, [1411.6635]. [71] J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209–254, [hep-th/0510251]. [72] A. Cabo-Bizet, D. Cassani, D. Martelli, and S. Murthy, Microscopic origin of the Bekenstein-Hawking

entropy of supersymmetric AdS5 black holes, 1810.11442. [73] S. Choi, J. Kim, S. Kim, and J. Nahmgoong, Large AdS black holes from QFT, 1810.12067. [74] F. Benini and P. Milan, Black holes in 4d N = 4 Super-Yang-Mills, 1812.09613. [75] M. Honda, Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, 1901.08091. [76] L. Bonora, P. Pasti, and M. Bregola, WEYL COCYCLES, Class. Quant. Grav. 3 (1986) 635.

– 25 –