Supersymmetry anomalies in N=1 conformal supergravity
PPrepared for submission to JHEP
KIAS-P19007
Supersymmetry anomalies in N = 1 conformal supergravity Ioannis Papadimitriou a School of Physics, Korea Institute for Advanced Study, 85 Hoegiro, Seoul 02455, Korea
E-mail: [email protected]
Abstract:
We solve the Wess-Zumino consistency conditions of N = 1 off-shell conformal su-pergravity in four dimensions and determine the general form of the superconformal anomaliesfor arbitrary a and c anomaly coefficients to leading non trivial order in the gravitino. Besidesthe 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 andonly if R -symmetry is anomalous. The Q - and S -supersymmetry anomalies give rise to an anoma-lous supersymmetry transformation for the supercurrent on curved backgrounds admitting Killingspinors, resulting in a deformed rigid supersymmetry algebra. Our results may have implicationsfor supersymmetric localization and supersymmetry phenomenology. Analogous results are ex-pected to hold in dimensions two and six and for other supergravity theories. The present analysisof the Wess-Zumino consistency conditions reproduces the holographic result of arXiv:1703.04299and generalizes it to arbitrary a and c anomaly coefficients. Keywords:
Supersymmetry, anomalies, Wess-Zumino conditions, QFT on curved backgrounds a r X i v : . [ h e p - t h ] M a y ontents N = 1 conformal supergravity 43 Classical Ward identities 64 Superconformal anomalies from the Wess-Zumino consistency conditions 85 Anomalous supercurrent transformation under Q - and S -supersymmetry 116 Concluding remarks 12A Spinor conventions and identities 13B Solving the Wess-Zumino consistency conditions 15 Supersymmetric quantum field theories have proven invaluable for probing strong coupling physicsdue to non-renormalization theorems and supersymmetric localization techniques [1–4]. They alsoplay a pivotal role in holographic dualities and beyond the Standard Model phenomenology. Giventhe enormous utility of supersymmetry, it is rather surprising that the question of whether it isanomalous at the quantum level is still not conclusively answered, despite the extensive literatureaddressing this question in various contexts. The consensus seems to be that standard super-symmetry (often termed Q -supersymmetry) is not anomalous. However, we demonstrate in thispaper that the Wess-Zumino consistency conditions [5] imply that Q -supersymmetry is necessarilyanomalous 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 relatedobservation 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 holographicdual 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 theoryand 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 intabletop experiments). They do mean, however, that the theory cannot be coupled consistentlyto dynamical gauge fields for the anomalous global symmetry. Specifically, a quantum anomalyin global supersymmetry implies that the theory cannot be coupled consistently to dynamicalsupergravity at the quantum level. It may also mean that certain conditions necessary to provenon-perturbative results are in fact not met.In flat space, global – or rigid – anomalies are typically visible only in higher-point functions ascontact terms that violate the classical Ward identities. For example, the lowest correlation func-tions where the Q -supersymmetry anomaly is visible in flat space are four-point functions involving– 1 –wo 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-pointfunctions when arbitrary sources for the current operators are turned on (i.e. when the theoryis coupled to background gauge fields for the global symmetries), or when the theory is put on acurved background admitting Killing symmetries. In particular, global supersymmetry anomaliesare related to supersymmetric index theorems and may affect observables such as partition func-tions, the Casimir energy, and Wilson loop expectation values of supersymmetric theories on curvedbackgrounds 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 attractedconsiderable interest. A systematic procedure for placing a supersymmetric theory on a curvedbackground was proposed in [16]. The first step is coupling the theory to a given off-shell back-ground supergravity, which corresponds to turning on arbitrary sources for the current multipletoperators and promoting the global symmetries – including supersymmetry – to local ones. TheKilling spinor equations obtained by setting the supersymmetry variations of the fermionic back-ground 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 forvarious spacetime dimensions [17–27] (see also [28, 29] for earlier work). However, this procedureis 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 anal-ysis of [30–32] and the holographic result of [33]. Based on the classical supersymmetry algebra oncurved backgrounds that admit a certain number of supercharges, the authors of [30–32] demon-strated that the supersymmetric partition function on such backgrounds should be independent ofspecific deformations of the supersymmetric background. However, an explicit evaluation of theon-shell action of minimal N = 2 gauged supergravity on supersymmetric asymptotically locallyAdS solutions using holographic renormalization [34, 35] in [33] demonstrated that the holographicpartition 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 thebosonic and fermionic superconformal Ward identities were derived holographically, including thecorresponding superconformal anomalies. It was then shown that the anomalies in the fermionicWard identities (the ones in the divergence and the gamma-trace of the supercurrent) lead to adeformed superconformal algebra on backgrounds admitting Killing spinors. Repeating the argu-ment of [30–32] using this deformed supersymmetry algebra reproduced exactly the dependence onthe deformations of the supersymmetric background seen in [33]. The Ward identities and quantum anomalies of four dimensional superconformal theories havebeen studied extensively over the years [36–47]. They are usually discussed in superspace languageand can be written compactly in the form (see e.g. appendix A of [40]) ∇ ˙ α J α ˙ α = ∇ α J, (1.1)where the supertrace superfield J is given by J = 124 π (cid:0) c W − a E (cid:1) , (1.2) At least for theories with a = c on supersymmetric backgrounds of the form S × M with M a Seifert manifoldit is possible to remove the rigid supersymmetry anomaly at the expense of breaking certain diffeomorphisms usingthe local but non-covariant counterterm found in [33]. We will elaborate on this point in section 5. – 2 –nd W = 12 W αβγ W αβγ , E = W + ( ∇ + R )( G + 2 RR ) , (1.3)are respectively the square of the superWeyl tensor and the chirally projected superEuler density.The chiral superfields W αβγ and R and the vector superfield G α ˙ α are the three superspace curvatures[48] and in the conventions of [40] G = G α ˙ α G α ˙ α . The components of the supertrace superfield J contain the trace of the stress tensor, the gamma-trace of the supercurrent, and the divergence ofthe R -current. The divergence of the stress tensor and of the supercurrent appear as componentsof 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 deriveare likely related to the fermionic components of the superspace cocycles found in [37, 39, 44], eventhough none of these earlier works concerns N = 1 conformal supergravity and so the field contentis somewhat different. Moreover, to the best of our knowledge these fermionic components havenot been written explicitly in the literature before and so a direct comparison with our resultsis 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-Zuminoconsistency 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 supersym-metry 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 theoryin 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 underlyingthe descent equations that relates the R -symmetry and supersymmetry anomalies is identical tothat 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]. Wedetermine the algebra of local symmetry transformations and derive the corresponding classicalWard identities. The main result of the paper is the solution of the Wess-Zumino consistencyconditions [5] associated with the N = 1 conformal supergravity algebra from which we obtainthe general form of the superconformal anomalies to leading non trivial order in the gravitino forarbitrary a and c anomaly coefficients. Our analysis is carried out in components and we explic-itly determine the fermionic Ward identities corresponding to the divergence and the gamma-traceof the supercurrent, including their anomalies. To the best of our knowledge these have not ap-peared in the literature before, at least explicitly. The bosonic Ward identities and anomaliesreproduce well known results [40] (corrected in [46]). We find that the divergence of the super-current, which corresponds to the Ward identity associated with Q -supersymmetry, is anomalouswhenever R -symmetry is anomalous. Moreover, the N = 1 supergravity algebra dictates thatthe Q -supersymmetry anomaly cannot be removed by a local counterterm without breaking dif-feomorphisms and/or local Lorentz transformations. The Ward identities and the superconformalanomalies we obtain by solving the Wess-Zumino conditions reproduce those found holographicallyin [8] in the special case when the a and c anomaly coefficients are equal.The paper is organized as follows. In section 2 we review relevant aspects of N = 1 off-shellconformal supergravity and determine the algebra of its local symmetry transformations. Thesetransformations are used in section 3 to derive the corresponding classical Ward identities. The main– 3 –esult is presented in section 4, where we obtain the general form of the superconformal anomalies bysolving the Wess-Zumino consistency conditions associated with the N = 1 conformal supergravityalgebra. The actual calculation is shown in considerable detail in appendix B. In section 5 wedetermine the anomalous transformation of the supercurrent under local supersymmetry and discussthe implications for the rigid supersymmetry algebra on curved backgrounds admitting Killingspinors of conformal supergravity. We conclude in section 6 and collect our conventions and severalgamma matrix identities in appendix A. N = 1 conformal supergravity In this section we review relevant aspects of N = 1 off-shell conformal supergravity in four di-mensions and determine its local off-shell symmetry algebra as a preparatory step for solving theWess-Zumino consistency conditions. N = 1 conformal supergravity can be constructed as a gaugetheory of the superconformal algebra [53–56] (see [57–60] and chapter 16 of [61] for pedagogicalreviews). Its field content consists of the vielbein e aµ , an Abelian gauge field A µ , and a Majoranagravitino ψ µ , 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, backgroundconformal supergravity is relevant for describing the Ward identities of superconformal theories andtheir quantum anomalies. Moreover, other supergravity theories can be obtained from conformalsupergravity by coupling it to compensator multiplets and gauge fixing via the so called tensor ormultiplet calculus [59, 62, 63]. When applied to background supergravity, this procedure may bethought of as the process of turning on local relevant couplings at the ultraviolet superconformalfixed point. Finally, off-shell N = 1 conformal supergravity in four dimensions is induced on theconformal boundary of five dimensional anti de Sitter space by minimal N = 2 on-shell gaugedsupergravity in the bulk [64]. This means that the Wess-Zumino consistency conditions for N = 1conformal supergravity should reproduce the superconformal anomalies obtained holographicallyin [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 superconformalalgebra, Q - and S -supersymmetry are on the same footing before the curvature constraints areimposed, each having its own independent gauge field, respectively ψ µ and φ µ . The covariantderivative acts on these gauge fields as D µ ψ ν ≡ (cid:16) ∂ µ + 14 ω µab ( e, ψ ) γ ab + iγ A µ (cid:17) ψ ν − Γ ρµν ψ ρ ≡ (cid:0) D µ + iγ A µ (cid:1) ψ ν , D µ φ ν = (cid:16) ∂ µ + 14 ω µab ( e, ψ ) γ ab − iγ A µ (cid:17) φ ν − Γ ρµν φ ρ = (cid:0) D µ − iγ A µ (cid:1) φ ν , (2.1)where ω µab ( e, ψ ) denotes the torsion-full spin connection ω µab ( e, ψ ) ≡ ω µab ( e ) + 14 (cid:0) ψ a γ µ ψ b + ψ µ γ a ψ b − ψ µ γ b ψ a (cid:1) . (2.2)Once the curvature constraints are imposed, however, the gauge field φ µ ceases to be an independentfield and it is expressed locally in terms of the physical fields as φ µ ≡ γ ν (cid:16) D ν ψ µ − D µ ψ ν − i γ (cid:15) νµρσ D ρ ψ σ (cid:17) = − (cid:0) δ [ ρµ δ σ ] ν + iγ (cid:15) µν ρσ (cid:1) γ ν D ρ ψ σ . (2.3) The purely bosonic part of the spin connection, ω µab ( e ), is torsion-free. For most part of the subsequent analysiswe will work to leading non trivial order in the gravitino, in which case the torsion-free part of the spin connectionsuffices. However, the full spin connection is necessary in order to determine e.g. the local symmetry algebra. – 4 –s we will see shortly, this quantity appears in the supersymmetry transformation of the gaugefield A µ as well as in the fermionic superconformal anomalies.Our spinor conventions are given in appendix A and follow those of [61]. Compared to [60], weuse Lorentzian signature instead of Euclidean and we have rescaled the gauge field A µ according to − A FT µ → A µ in order for its coefficient in the covariant derivatives (2.1) to be unity, as is standardin the field theory literature. Local symmetry transformations
The local symmetries of N = 1 conformal supergravity are diffeomorphisms ξ µ ( x ), Weyl transfor-mations σ ( x ), local frame rotations λ ab ( 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 covariantderivative acts on the spinor parameters ε and η as D µ ε ≡ (cid:16) ∂ µ + 14 ω µab ( e, ψ ) γ ab + iγ A µ (cid:17) ε ≡ (cid:0) D µ + iγ A µ (cid:1) ε, D µ η ≡ (cid:16) ∂ µ + 14 ω µab ( e, ψ ) γ ab − iγ A µ (cid:17) η ≡ (cid:0) D µ − iγ A µ (cid:1) η. (2.4)Under these local transformations the fields of N = 1 conformal supergravity transform as δe aµ = ξ λ ∂ λ e aµ + e aλ ∂ µ ξ λ − λ ab e bµ + σe aµ − ψ µ γ a ε,δψ µ = ξ λ ∂ λ ψ µ + ψ λ ∂ µ ξ λ − λ ab γ ab ψ µ + 12 σψ µ + D µ ε − γ µ η − iγ θψ µ ,δA µ = ξ λ ∂ λ A µ + A λ ∂ µ ξ λ + 3 i φ µ γ ε − i ψ µ γ η + ∂ µ θ. (2.5)These transformations imply that the quantity φ µ in (2.3) transforms as δφ µ = ξ λ ∂ λ φ µ + φ λ ∂ µ ξ λ − λ ab γ ab φ µ − σφ µ + 12 (cid:16) P µν + 2 i F µν γ − (cid:101) F µν (cid:17) γ ν ε + D µ η + iγ θφ µ , (2.6)where P µν ≡ (cid:16) R µν − Rg µν (cid:17) , (2.7)is the Schouten tensor in four dimensions and the dual fieldstrength is defined as (cid:101) F µν ≡ (cid:15) µν ρσ F ρσ . (2.8)Notice that the transformations (2.5) coincide with those induced on the boundary of fivedimensional anti de Sitter space by minimal N = 2 gauged supergravity in the bulk [8, 64]. Inorder to compare with the results of [8] one should take into account that we have rescaled thegauge field and the local symmetry parameters according to √ A there µ /(cid:96) → A µ , (2.9)and σ there /(cid:96) → σ, (cid:15) there+ /(cid:96) → ε, (cid:15) there − /(cid:96) → η, √ θ there /(cid:96) → 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 – ocal symmetry algebra The local transformations (2.5) determine the algebra of local symmetries, i.e. the commutators[ δ Ω , δ Ω (cid:48) ], where Ω and Ω (cid:48) denote any of the local parameters σ, ξ, λ, θ, ε, η . Off-shell closure of thealgebra requires that the parameters transform under the local symmetries as δξ µ = ξ (cid:48) ν ∂ ν ξ µ − ξ ν ∂ ν ξ (cid:48) µ , δλ ab = ξ µ ∂ µ λ ab , δσ = ξ µ ∂ µ σ, δθ = ξ µ ∂ µ θ,δε = ξ µ ∂ µ ε + 12 σε − λ ab γ ab ε − iθγ ε, δη = ξ µ ∂ µ η − ση − λ ab γ ab η + iθγ η. (2.11)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:[ δ ξ , δ ξ (cid:48) ] = δ ξ (cid:48)(cid:48) , ξ (cid:48)(cid:48) µ = ξ ν ∂ ν ξ (cid:48) µ − ξ (cid:48) ν ∂ ν ξ µ , [ δ λ , δ λ (cid:48) ] = δ λ (cid:48)(cid:48) , λ (cid:48)(cid:48) ab = λ (cid:48) ac λ cb − λ ac λ (cid:48) cb , [ δ ε , δ η ] = δ σ + δ λ + δ θ , σ = 12 εη, λ ab = − εγ ab η, θ = − i εγ η, [ δ ε , δ ε (cid:48) ] = δ ξ + δ λ + δ θ , ξ µ = 12 ε (cid:48) γ µ ε, λ ab = −
12 ( ε (cid:48) γ ν ε ) ω ν ab , θ = −
12 ( ε (cid:48) γ ν ε ) A ν . (2.12)Notice that the composite parameters resulting from the commutator of two Q -supersymmetrytransformations are field dependent, which means that the structure constants of the gauge algebraare field dependent. Such algebras are often termed soft algebras (see [65] for a recent discussionof soft algebras and their BRST cohomology) and supergravity theories are typically based onsoft algebras. The commutation relations (2.12) form the basis for the Wess-Zumino consistencycondition analysis to determine the superconformal anomalies in N = 1 conformal supergravity. We now turn to the derivation of the classical Ward identities of a local quantum field theory coupledto background N = 1 conformal supergravity. These identities can be thought of as Noether’sconservation laws following from the local symmetry transformations (2.5). Since these depend onlyon the structure of the background supergravity, the resulting Ward identities are independent ofthe specific field theory Lagrangian, provided the coupling of the theory to background supergravitypreserves the local supergravity symmetries at the classical level.The classical Ward identities can be expressed in the compact form δ Ω W [ 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 thegenerating functional of connected correlation functions of local current operators associated withthe background supergravity fields, namely T µa = e − δ W δe aµ , J µ = e − δ W δA µ , S µ = e − δ W δψ µ , (3.2)where e ≡ det( e aµ ). These definitions do not rely on a Lagrangian description of the quantum fieldtheory, 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 –here Z [ e, A, ψ ] is obtained from the path integral Z [ e, A, ψ ] = ˆ [dΦ] e iS [Φ; e,A,ψ ] , (3.4)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 ofthe background supergravity fields (2.5), classical invariance of W [ e, A, ψ ] leads to a conservationlaw – or Ward identity – for each local symmetry, which we will now derive. Diffeomorphisms
The transformation of the generating functional under diffeomorphisms is given by δ ξ W = ˆ d x e (cid:0) δ ξ e aµ T µa + δ ξ A µ J µ + δ ξ ψ µ S µ (cid:1) = ˆ d x e (cid:16) ∇ µ ( ξ ν e aν ) T µa − ξ ν ω ν ab e bµ T µa + (cid:0) ξ ν F νµ + ∂ µ ( A ν ξ ν ) (cid:1) J µ + ξ ν ( ψ µ ←−D ν − ψ ν ←−D µ ) S µ + ( ξ ν ψ ν ) ←−D µ S µ − iξ ν A ν ψ µ γ S µ + 14 ξ ν ω νab ψ µ γ ab S µ (cid:17) = ˆ d x e ξ ν (cid:16) − e aν ∇ µ T µa − ∇ µ ( ψ ν S µ ) + ( ψ µ ←−D ν ) S µ + F νµ J µ − A ν (cid:0) ∇ µ J µ + iψ µ γ S µ (cid:1) + ω ν ab (cid:0) e µ [ a T µb ] + 14 ψ µ γ ab S µ (cid:1)(cid:17) . (3.5)Setting this quantity to zero for arbitrary ξ ν ( x ) gives the classical diffeomorphism Ward identity e aµ ∇ ν T νa + ∇ ν ( ψ µ S ν ) − ψ ν ←−D µ S ν − F µν J ν + A µ (cid:0) ∇ ν J ν + iψ ν γ S ν (cid:1) − ω µab (cid:16) e ν [ a T νb ] + 14 ψ ν γ ab S ν (cid:17) = 0 . (3.6)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 δ σ W = ˆ d x e (cid:0) δ σ e aµ T µa + δ σ A µ J µ + δ σ ψ µ S µ (cid:1) = ˆ d x e σ (cid:16) e aµ T µa + 12 ψ µ S µ (cid:17) , (3.7)and, hence, the classical trace Ward identity takes the form e aµ T µa + 12 ψ µ S µ = 0 . (3.8) R -symmetry The transformation of the generating function under U (1) R gauge transformations is given by δ θ W = ˆ d x e (cid:0) δ θ e aµ T µa + δ θ A µ J µ + δ θ ψ µ S µ (cid:1) = ˆ d x e (cid:0) ∂ µ θ J µ − iθψ µ γ S µ (cid:1) = ˆ d x e θ (cid:0) − ∇ µ J µ − iψ µ γ S µ (cid:1) . (3.9)– 7 –ence, the classical R -symmetry Ward identity takes the form ∇ µ J µ + iψ µ γ S µ = 0 . (3.10) Local frame rotations
Under local frame rotations the generating function transforms according to δ λ W = ˆ d x e (cid:0) δ λ e aµ T µa + δ λ A µ J µ + δ λ ψ µ S µ (cid:1) = − ˆ d x eλ ab (cid:16) e µ [ b T µa ] + 14 ψ µ γ ba S µ (cid:17) . (3.11)Hence, the corresponding classical Ward identity is e µ [ a T µb ] + 14 ψ µ γ ab S µ = 0 . (3.12) Q -supersymmetry The Q -supersymmetry transformation of the generating function is δ ε W = ˆ d x e (cid:0) δ ε e aµ T µa + δ ε A µ J µ + δ ε ψ µ S µ (cid:1) = ˆ d x e (cid:16) − ψ µ γ a ε T µa + 3 i φ µ γ ε J µ + ε ←−D µ S µ (cid:17) = ˆ d x e ε (cid:16) γ a ψ µ T µa + 3 i γ φ µ J µ − D µ S µ (cid:17) . (3.13)Therefore, the classical Q -supersymmetry Ward identity takes the from D µ S µ = 12 γ a ψ µ T µa + 3 i γ φ µ J µ . (3.14) S -supersymmetry Finally, the transformation of the generating function under S -supersymmetry is given by δ η W = ˆ d x e (cid:0) δ η e aµ T µa + δ η A µ J µ + δ η ψ µ S µ (cid:1) = ˆ d x e (cid:16) − i ψ µ γ η J µ + ηγ µ S µ (cid:17) = ˆ d x e η (cid:16) − i γ ψ µ J µ + γ µ S µ (cid:17) , (3.15)and hence the classical Ward identity for S -supersymmetry is γ µ S µ − i γ ψ µ J µ = 0 . (3.16) At the quantum level the generating function W may not be invariant under all local symmetriesof background conformal supergravity, i.e. δ Ω W (cid:54) = 0 . (4.1)– 8 –he non-invariance of the generating function of four dimensional theories can be parameterized as δ Ω W = ˆ d x √− g (cid:0) σ A W − θ A R − ε A Q + η A S (cid:1) , (4.2)where A W , A R , A Q and A S are possible quantum anomalies under Weyl, R -symmetry, Q - and S -supersymmetry transformations, respectively. Recall that the gravitational and Lorentz (framerotation) anomalies are related by a local counterterm [66] and exist only in 4 k + 2 dimensions,with k = 0 , , . . . . 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 thisscheme diffeomorphisms remain a symmetry at the quantum level and the transformation of thegenerating 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) resultsin the same diffeomorphism and Lorentz Ward identities as those obtained respectively in (3.6) and(3.12), but the remaining Ward identities become e aµ T µa + 12 ψ µ S µ = A W , ∇ µ J µ + iψ µ γ S µ = A R , D µ S µ − γ a ψ µ T µa − i γ φ µ J µ = A Q ,γ µ S µ − i γ ψ µ J µ = A S . (4.3)The objective of this section is to determine the general form of the quantum anomalies A W , A R , A Q and A S by solving the Wess-Zumino consistency conditions [5] associated with the N = 1conformal supergravity algebra (2.12).The Wess-Zumino consistency conditions amount to the requirement that the local symmetryalgebra (2.12) is realized when successive infinitesimal local symmetry variations δ Ω (also knownas Ward operators) act on the generating functional W and read[ δ Ω , δ Ω (cid:48) ] W = δ [Ω , Ω (cid:48) ] W , (4.4)for any pair of local symmetries Ω = ( ξ, σ, λ, θ, ε, η ) and Ω (cid:48) = ( ξ (cid:48) , σ (cid:48) , λ (cid:48) , θ (cid:48) , ε (cid:48) , η (cid:48) ).In appendix B we determine general non trivial solution of the Wess-Zumino consistency con-ditions (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 solutionsrelated respectively to the a and c coefficients of the Weyl anomaly and they take the form A W = c π (cid:16) W − F (cid:17) − a π E + O ( ψ ) , A R = (5 a − c )27 π (cid:101) F F + ( c − a )24 π P , A Q = − (5 a − c ) i π (cid:101) F µν A µ γ φ ν + ( a − c )6 π ∇ µ (cid:0) A ρ (cid:101) R ρσµν (cid:1) γ ( ν ψ σ ) − ( a − c )24 π F µν (cid:101) R µνρσ γ ρ ψ σ + O ( ψ ) , A S = (5 a − c )6 π (cid:101) F µν (cid:16) D µ − i A µ γ (cid:17) ψ ν + ic π F µν (cid:0) γ µ [ σ δ ρ ] ν − δ [ σµ δ ρ ] ν (cid:1) γ D ρ ψ σ + 3(2 a − c )4 π P µν g µ [ ν γ ρσ ] D ρ ψ σ + ( a − c )8 π (cid:16) R µνρσ γ µν − Rg µν g µ [ ν γ ρσ ] (cid:17) D ρ ψ σ + O ( ψ ) , (4.5)– 9 –here W 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 W ≡ W µνρσ W µνρσ = R µνρσ R µνρσ − R µν R µν + 13 R ,E = R µνρσ R µνρσ − R µν R µν + R , P ≡ (cid:15) κλµν R κλρσ R µν ρσ = (cid:101) R µνρσ R µνρσ , (4.6)where the dual Riemann tensor is defined in analogy with the dual U(1) R fieldstrength in (2.8) as (cid:101) R µνρσ ≡ (cid:15) µν κλ R κλρσ . (4.7)Moreover, P µν is the Schouten tensor defined in (2.7) and we have introduced the shorthand notation F ≡ F µν F µν , F (cid:101) F ≡ (cid:15) µνρσ F µν F ρσ . (4.8)Finally, we have used the normalization of the central charges adopted in [40], according to whichthe a and c anomaly coefficients for free chiral and vector multiplets are given respectively by a = 148 ( N χ + 9 N v ) , c = 124 ( N χ + 3 N v ) . (4.9)Several comments are in order here. Firstly, we should point out that the anomalies (4.5), aswell as the current operators defined in (3.2), are the consistent ones. The corresponding covariantquantities can be obtained by adding the appropriate Bardeen-Zumino terms [66]. Secondly, thebosonic 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 – inthe literature before. Moreover, the Ward identities (3.6), (3.12) and (4.3) derived above, includingthe anomalies (4.5), are in complete agreement with the results of [8] for theories with a holographicdual that have a = c = π(cid:96) G , (4.10)where G is the Newton constant in five dimensions and (cid:96) is the AdS radius. Of course, suchan agreement was expected since minimal N = 2 gauged supergravity induces off-shell N = 1conformal supergravity on the four dimensional boundary of AdS [64] and the anomalies can becomputed through holographic renormalization [34] (see also [9, 10, 46, 68]).An interesting question is whether there exists a local counterterm W ct that removes the Q -supersymmetry anomaly, i.e. such that δ ε ( W + W ct ) = 0. Closure of the algebra requires that[ δ ε , δ ε (cid:48) ]( W + W ct ) = ( δ ξ + δ λ + δ θ )( W + W ct ) , (4.11)with the composite parameters for the bosonic transformations given in (2.12). It follows that ifsuch a counterterm exists, then it must also satisfy( δ ξ + δ λ + δ θ )( W + W ct ) = 0 . (4.12)Hence, either W ct removes also the R -symmetry anomaly, or it breaks diffeomorphisms and/orlocal 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/Lorentztransformations is a problem we hope to address in future work. In contrast to the Riemann tensor, (cid:101) R µνρσ is not symmetric under exchange of the first and second pair of indices. – 10 – Anomalous supercurrent transformation under Q - and S -supersymmetry The superconformal anomalies (4.5) lead to anomalous transformations for the current operatorsunder the corresponding local symmetries. In particular, the fermionic anomalies A Q and A S 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 thetransformations of the currents result in a deformed superalgebra.The classical (non-anomalous) part of the current transformations can be deduced directly fromthe 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 ε (cid:16) δδψ µ (cid:17) = 12 γ a ε δδe aµ + i (cid:0) δ [ µν δ ρ ] σ + iγ (cid:15) µν ρσ (cid:1) γ ν γ D ρ (cid:16) ε δδA σ (cid:17) ,δ η (cid:16) δδψ µ (cid:17) = 3 i γ η δδA µ . (5.1)It follows that the Q - and S -supersymmetry transformations of the supercurrent are given by δ ε S µ = e − δ ε (cid:16) δδψ µ (cid:17) W + e − δδψ µ δ ε W = 12 γ a ε T µa + i (cid:0) δ [ µν δ ρ ] σ + iγ (cid:15) µν ρσ (cid:1) γ ν γ D ρ (cid:104) ε (cid:16) J σ + 4(5 a − c )27 π (cid:101) F σκ A κ (cid:17)(cid:105) + ( a − c )6 π ∇ ρ (cid:0) A σ (cid:101) R σλρκ (cid:1) δ µ ( κ γ λ ) ε + ( a − c )24 π F ρσ (cid:101) R ρσµν γ ν ε, (5.2) δ η S µ = e − δ η (cid:16) δδψ µ (cid:17) W + e − δδψ µ δ η W = 3 i γ η (cid:16) J µ + 4(5 a − c )27 π (cid:101) F µν A ν (cid:17) + (5 a − c )6 π D ν ( (cid:101) F µν η ) − ic π (cid:0) γ [ µρ δ ν ] σ − δ [ µρ δ ν ] σ (cid:1) γ D ν ( F ρσ η ) − a − c )4 π D ν (cid:0) P ρσ g ρ [ σ γ µν ] η (cid:1) − ( a − c )8 π D ν (cid:104)(cid:16) R µνρσ γ ρσ − Rg ρσ g ρ [ σ γ µν ] (cid:17) η (cid:105) . (5.3)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 rewritingof the two fermionic Ward identities in (4.3) and are useful for e.g. determining the effect of thesuperconformal anomalies in correlation functions [6]. They also determine the rigid superalgebraon curved backgrounds that admit Killing spinors of conformal supergravity. Namely, when thelocal spinor parameters ε and η are restricted to solutions ( ε o , η o ) of the Killing spinor equation δ ε o ,η o ψ µ = D µ ε o − γ µ η o = 0 , (5.4)on a fixed bosonic background specified by g µν and A µ , the corresponding transformation of thesupercurrent under rigid supersymmetry is given by δ ( ε o ,η o ) S µ = {Q [ ε o , η o ] , S µ } = ( δ ε o + δ η o ) S µ , (5.5) An equivalent but more formal way to determine how the currents transform under the local symmetries isto utilize the symplectic structure underlying the space of couplings and local operators [69]. The Ward identitiescorrespond to first class constraints on this space, generating the local symmetry transformations under the Poissonbracket. In appendix B.1 of [8] this approach is used to obtain the anomalous transformation of the supercurrentunder Q - and S -supersymmetry in the case a = c . Note that ( ε o , η o ) are c -number parameters, while ( ε, η ) are Grassmann valued local parameters that transformnon trivially under the local symmetries according to (2.11). – 11 –here Q [ ε o , η o ] is the conserved supercharge associated with the Killing spinor ( ε o , η o ) and allbosonic 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 numericallyzero for a class of N = 1 conformal supergravity backgrounds admitting two real supercharges ofopposite R -charge [18, 19], the anomalous terms in the transformations of the supercurrent underrigid 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 tensionbetween the field theory results of [30–32] that used the classical superalgebra and the holographiccomputation of [33]. Besides the dependence of the supersymmetric partition function on thebackground, however, the transformation of the supercurrent determines also the spectrum of BPSstates. 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 localcounterterm that breaks diffeomorphisms and/or local Lorentz transformations as discussed in theprevious section, it is plausible that the anomaly in rigid supersymmetry may be removed by a localcounterterm that breaks certain (large) diffeomorphisms, but preserves the underlying structure ofthe supersymmetric background. An example of a somewhat analogous situation was discussedin [70], where supersymmetric gauge theories in three dimensions with both Maxwell and Chern-Simons terms coupled to background topological gravity were considered. The partition functionof such theories on Seifert manifolds, which admit two supercharges of opposite R -charge, can becomputed 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, whichimplies that the partition function does indeed depend on the metric at the quantum level. Thepuzzle, 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 invariantunder Seifert reparameterizations and Seifert-topological (i.e. independent of metric deformationsthat 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 beremoved by a local counterterm that breaks those large diffeomorphisms that are not compatiblewith 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 wasfound in [33]. It would be interesting to generalize this counterterm to non holographic theorieswith arbitrary a and c using the general form of the fermionic anomalies we obtained in this paper. In this paper we determined the local symmetry algebra of N = 1 off-shell conformal supergravityin four dimensions and obtained the general form of the superconformal anomalies by solving theassociated Wess-Zumino consistency conditions. To the best of our knowledge, the explicit formof 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, isanomalous whenever R -symmetry is anomalous. This anomaly cannot be removed by a localcounterterm without breaking diffeomorphisms and/or local Lorentz transformations.Several open questions remain. Our result that Q -supersymmetry is anomalous in any the-ory with an anomalous R -symmetry does not seem to depend on the specific supergravity theorywe used in this paper. Indeed, we expect this result to hold in non-superconformal theories with– 12 –n 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 section 4, such a counterterm would necessarily breakdiffeomorphisms and/or local Lorentz rotations. The related question for rigid supersymmetry onbackgrounds that admit Killing spinors is relevant for the validity of supersymmetric localizationcomputations on four-manifolds. An important example is the computation of generalized super-symmetric indices that count the microstates of supersymmetric AdS black holes [71–75]. Wehope to return to these questions in future work. Acknowledgments
I would like to thank Benjamin Assel, Roberto Auzzi, Loriano Bonora, Davide Cassani, CyrilClosset, Camillo Imbimbo, Heeyeon Kim, Zohar Komargodski, Dario Martelli, Sameer Murthyand Dario Rosa for illuminating discussions and email correspondence. I am also grateful to theUniversity of Southampton, King’s College London and the International Center for TheoreticalPhysics in Trieste for hospitality and partial financial support during the completion of this work.
AppendixA Spinor conventions and identities
Throughout this paper we follow the conventions of [61]. In particular, the tangent space metricis η = diag ( − , , ,
1) and the Levi-Civita symbol ε µνρσ = ± ε = 1. The Levi-Civitatensor is defined as usual as (cid:15) µνρσ = √− g ε µνρσ = e ε µνρσ . Moreover, the chirality matrix in fourdimensions is given by γ = iγ γ γ γ , (A.1)and we define the antisymmetrized products of gamma matrices as γ µ µ ...µ n ≡ γ [ µ γ µ · · · γ µ n ] , (A.2)where antisymmetrization is done with weight one.In the conventions we use here the gravitino ψ µ 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 λγ µ γ µ · · · γ µ p χ = ( − p χγ µ p · · · γ µ γ µ λ, [eq. (3.53) in [61]] . (A.3)For Majorana fermions we also have that( χγ µ ...µ r λ ) ∗ = χγ µ ...µ r λ, [eq. (3.82) in [61]] . (A.4)It is convenient to collect several identities involving antisymmetrized products of gamma– 13 –atrices in d dimensions, most of which can be found in section 3 of [61]: γ µνρ = 12 { γ µ , γ νρ } ,γ µνρσ = 12 [ γ µ , γ νρσ ] ,γ µν γ ρσ = γ µν ρσ + 4 γ [ µ [ σ δ ν ] ρ ] + 2 δ [ µ [ σ δ ν ] ρ ] ,γ µ γ ν ...ν p = γ µν ...ν p + pδ [ ν µ γ ν ...ν p ] ,γ ν ...ν p γ µ = γ ν ...ν p µ + pγ [ ν ...ν p − δ ν p ] µ ,γ µνρ γ στ = γ µνρστ + 6 γ [ µν [ τ δ ρ ] σ ] + 6 γ [ µ δ ν [ τ δ ρ ] σ ] ,γ µνρσ γ τλ = γ µνρστλ + 8 γ [ µνρ [ λ δ σ ] τ ] + 12 γ [ µν δ ρ [ λ δ σ ] τ ] ,γ µνρ γ στλ = γ µνρστλ + 9 γ [ µν [ τλ δ ρ ] σ ] + 18 γ [ µ [ λ δ ν τ δ ρ ] σ ] + 6 δ [ µ [ λ δ ν τ δ ρ ] σ ] ,γ µ ...µ r ν ...ν s γ ν s ...ν = ( d − r )!( d − r − s )! γ µ ...µ r ,γ µρ γ ρν = ( d − γ µν + ( d − δ µν ,γ µνρ γ ρσ = ( d − γ µν σ + 2( d − γ [ µ δ ν ] σ ,γ µν γ νρσ = ( d − γ µρσ + 2( d − δ [ ρµ γ σ ] ,γ µνλ γ λρσ = ( d − γ µν ρσ + 4( d − γ [ µ [ σ δ ν ] ρ ] + 2( d − δ [ µ [ σ δ ν ] ρ ] ,γ µρ γ ρστ γ τν = ( d − γ µσν + ( d − d −
3) ( γ µ δ σν − γ σ g µν )+ ( d − d − δ σµ γ ν − ( d − γ σ γ µν ,γ ρ γ µ µ ...µ p γ ρ = ( − p ( d − p ) γ µ µ ...µ p . (A.5)In particular, the following identities apply specifically to d = 4 and are used extensively: γ ρ γ µ γ σ + γ σ γ µ γ ρ = 2( g µρ γ σ + g µσ γ ρ − g ρσ γ µ ) ,γ ρ γ µ γ σ − γ σ γ µ γ ρ = 2 γ ρµσ ,γ ρ γ µ γ σ = g µρ γ σ + g µσ γ ρ − g ρσ γ µ + γ ρµσ γ µρ γ σ = γ µρσ + γ µ g ρσ − γ ρ g µσ γ µρσ = i(cid:15) µρσν γ ν γ ,γ µν = i (cid:15) µνρσ γ ρσ γ . (A.6)Finally, the following three identities in four dimensions help compare the superconformal Wardidentities (4.3) and the anomalies (4.5) with the corresponding results obtained in [8]:( γ µν − g µν ) γ νκλ = 4 δ µ [ κ γ λ ] − γ µκλ = 4 δ µ [ κ γ λ ] − i (cid:15) µνρσ γ σ γ ,γ µ γ νκλ = γ µνκλ + 3 g µ [ ν γ κλ ] , (2 γ νκ γ µ − γ νκµ ) γ µρσ = 2 γ νκ γ µ γ µρσ − γ νκµ γ µρσ = 4 γ νκ γ ρσ − (cid:0) γ [ ν [ σ δ κ ] ρ ] + δ [ ν [ σ δ κ ] ρ ] (cid:1) = 4 (cid:0) γ νκρσ + γ [ ν [ σ δ κ ] ρ ] − δ [ ν [ σ δ κ ] ρ ] (cid:1) . (A.7)– 14 – Solving the Wess-Zumino consistency conditions
In this appendix we provide the details of the proof that the superconformal anomalies (4.5) satisfythe 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 explicitlysince 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 arestraightforward to check. We shall therefore focus on the commutation relations involving at leastone fermionic symmetry transformation, except for the four commutators[ δ ξ , δ ε ] W = 0 , [ δ ξ , δ η ] W = 0 , [ δ λ , δ ε ] W = 0 , [ δ λ , δ η ] W = 0 , (B.1)which hold trivially. All computations in this appendix assume either a compact spacetime manifoldor 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 ψ µ .[ δ σ , δ ε ] W = 0 and [ δ σ , δ η ] W = 0:Let us first consider the two commutation relations[ δ σ , δ ε ] W , [ δ σ , δ η ] W . (B.2)Taking into account the Weyl transformation of the supersymmetry parameters in (2.11), it isstraightforward to check that both the Q - and S -supersymmetry anomalies are Weyl invariant, i.e. δ σ δ ε W = 0 , δ σ δ η W = 0 . (B.3)Moreover, the O ( ψ ) terms in the Weyl anomaly in (4.5) ensure that the Weyl anomaly density isinvariant under both Q - and S -supersymmetry, i.e. δ ε ( e A W ) = 0 , δ η ( e A W ) = 0 . (B.4)Combining these results we conclude that these two commutators satisfy the Wess-Zumino consis-tency conditions compatible with the local symmetry algebra, namely[ δ σ , δ ε ] W = 0 , [ δ σ , δ η ] W = 0 . (B.5)[ δ ε , δ θ ] W = 0:The only remaining commutation relations involving a bosonic symmetry are those betweenlocal gauge transformations and either Q - or S -supersymmetry transformations. Staring with Q - For the case a = c the O ( ψ ) 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 –upersymmetry, the anomalies in (4.5) determine δ ε δ θ W = − δ ε ˆ d x e θ A R = − (5 a − c )54 π ˆ d x e θ (cid:15) µνρσ δ ε ( F µν F ρσ ) + ( c − a )48 π ˆ d x e θ (cid:15) κλµν δ ε ( R ρσκλ R σρµν )= − a − c )27 π ˆ d x e θ (cid:15) µνρσ F µν ∂ ρ δ ε A σ + ( c − a )12 π ˆ d x e θ (cid:15) κλµν R ρσκλ ∇ µ δ ε Γ σρν = (5 a − c ) i π ˆ d x e ∂ ρ θ (cid:15) µνρσ F µν εγ φ σ − ( c − a )12 π ˆ d x e ∂ µ θ (cid:15) κλµν R ρσκλ ∇ ρ δ ε g σν , = (5 a − c ) i π ˆ d x e ∂ ρ θ (cid:15) µνρσ F µν εγ φ σ + ( c − a )12 π ˆ d x e ∇ ρ (cid:0) ∂ µ θ (cid:15) κλµν R ρσκλ (cid:1) εγ ( σ ψ ν ) , (B.6) δ θ δ ε W = − δ θ ˆ d x e ε A Q = (5 a − c ) i π ˆ d x e ∂ ρ θ (cid:15) µνρσ F µν εγ φ σ + ( c − a )12 π ˆ d x e (cid:15) µνρσ ∇ κ (cid:0) ∂ ρ θ R κλµν (cid:1) εγ ( λ ψ σ ) . (B.7)Hence, [ δ ε , δ θ ] W = 0 , (B.8)as required by the Wess-Zumino conditions.[ δ η , δ θ ] W = 0:For S -supersymmetry we have similarly δ η δ θ W = − δ η ˆ d x e θ A R (B.9)= − (5 a − c )54 π ˆ d x e θ (cid:15) µνρσ δ η ( F µν F ρσ )= − a − c )27 π ˆ d x e θ (cid:15) µνρσ F µν ∂ ρ δ η A σ = − (5 a − c ) i π ˆ d x e ∂ ρ θ (cid:15) µνρσ F µν ηγ ψ σ , (B.10) δ θ δ η W = δ θ ˆ d x e η A S = − (5 a − c ) i π ˆ d x e ∂ ρ θ (cid:15) µνρσ F µν ηγ ψ σ , (B.11)and hence, [ δ η , δ θ ] W = 0 . (B.12)[ δ ε , δ ε (cid:48) ] W = δ θ W , with θ = − ( ε (cid:48) γ λ ε ) A λ : – 16 –here remain only the four commutators among fermionic symmetries. Applying two successive Q -supersymmetry transformations on the generating function W gives δ ε (cid:48) δ ε W = − δ ε (cid:48) ˆ d x e ε A Q = (5 a − c )18 π i ˆ d x e (cid:15) µνρσ F µν A ρ εγ δ ε (cid:48) φ σ + ( c − a )12 π ˆ d x e (cid:15) µνρσ ∇ κ (cid:0) A ρ R κλµν (cid:1) εγ ( λ δ ε (cid:48) ψ σ ) + ( a − c )48 π ˆ d x e (cid:15) µνρσ F ρσ R κλµν εγ κ δ ε (cid:48) ψ λ = (5 a − c )18 π i ˆ d x e (cid:15) µνρσ F µν A ρ εγ (cid:16) P σλ + i F σλ γ − (cid:15) σλκτ F κτ (cid:17) γ λ ε (cid:48) − ( c − a )12 π ˆ d x e (cid:15) µνρσ A ρ R κλµν ∇ κ ( εγ ( λ D σ ) ε (cid:48) )+ ( a − c )48 π ˆ d x e (cid:15) µνρσ F ρσ R κλµν εγ κ D λ ε (cid:48) , (B.13)and hence [ δ ε , δ ε (cid:48) ] W = − (5 a − c )27 π ˆ d x e (cid:15) µνρσ F µν A ρ F σλ ε (cid:48) γ λ ε − ( c − a )48 π ˆ d x e (cid:15) µνρσ A ρ R κλµν R κλστ ( ε (cid:48) γ τ ε ) − ( c − a )24 π ˆ d x e (cid:15) µνρσ A ρ R κλµν ∇ κ ∇ σ ( ε (cid:48) γ λ ε )+ ( a − c )48 π ˆ d x e (cid:15) µνρσ F ρσ R κλµν ∇ λ ( ε (cid:48) γ κ ε ) . (B.14)The last two terms can be rearranged as (cid:15) µνρσ A ρ R κλµν ∇ κ ∇ σ ( ε (cid:48) γ λ ε ) − (cid:15) µνρσ F ρσ R κλµν ∇ κ ( ε (cid:48) γ λ ε )= (cid:15) µνρσ A ρ R κλµν [ ∇ κ , ∇ σ ]( ε (cid:48) γ λ ε ) + ∇ σ (cid:0) (cid:15) µνρσ A ρ R κλµν ∇ κ ( ε (cid:48) γ λ ε ) (cid:1) = ∇ σ (cid:0) (cid:15) µνρσ A ρ R κλµν ∇ κ ( ε (cid:48) γ λ ε ) (cid:1) + (cid:15) µνρσ A ρ R κλµν R κσλτ ( ε (cid:48) γ τ ε )= ∇ σ (cid:0) (cid:15) µνρσ A ρ R κλµν ∇ κ ( ε (cid:48) γ λ ε ) (cid:1) + 12 (cid:15) µνρσ A ρ R κλµν R κλστ ( ε (cid:48) γ τ ε ) , (B.15)so that [ δ ε , δ ε (cid:48) ] W = − (5 a − c )27 π ˆ d x e (cid:15) µνρσ F µν A ρ F σλ ε (cid:48) γ λ ε − ( c − a )24 π ˆ d x e (cid:15) µνρσ A ρ R κλµν R κλστ ( ε (cid:48) γ τ ε ) . (B.16)Moreover, the fact that F [ λσ F µν A ρ ] = 0 and R κλ [ µν R κλστ A ρ ] = 0 in four dimensions leads to thetwo identities (cid:15) µνρσ F µν F σλ A ρ = − (cid:15) µνρσ F µν F ρσ A λ ,(cid:15) µνρσ R µνκλ R στ κλ A ρ = − (cid:15) µνρσ R µνκλ R ρσκλ A τ . (B.17)Therefore, we finally get[ δ ε , δ ε (cid:48) ] W = (5 a − c )108 π ˆ d x e (cid:15) µνρσ F µν F ρσ ( A λ ε (cid:48) γ λ ε ) + ( c − a )96 π ˆ d x e (cid:15) µνρσ R κλµν R κλρσ ( A τ ε (cid:48) γ τ ε )= − ˆ d x e θ A R , (B.18)– 17 –ith θ = −
12 ( ε (cid:48) γ λ ε ) A λ , (B.19)as required by the Wess-Zumino consistency conditions.[ δ η , δ η (cid:48) ] W = 0:Two successive S -supersymmetry transformations on the generating function W give δ η (cid:48) δ η W = − (5 a − c ) i π ˆ d x e (cid:15) µνρσ F µν A ρ ηγ δ η (cid:48) ψ σ + (5 a − c )12 π ˆ d x e (cid:15) µν ρσ F µν η D ρ δ η (cid:48) ψ σ + ic π ˆ d x e F µν η (cid:0) γ µ [ σ δ ρ ] ν − δ [ σµ δ ρ ] ν (cid:1) γ D ρ δ η (cid:48) ψ σ + 3(2 a − c )4 π ˆ d x e P µν g µ [ ν ηγ ρσ ] D ρ δ η (cid:48) ψ σ + ( a − c )8 π ˆ d x e η (cid:16) R µνρσ γ µν − Rg µν g µ [ ν γ ρσ ] (cid:17) D ρ δ η (cid:48) ψ σ = (5 a − c ) i π ˆ d x e (cid:15) µνρσ F µν A ρ ηγ γ σ η (cid:48) − (5 a − c )12 π ˆ d x e (cid:15) µν ρσ F µν ηγ σ D ρ η (cid:48) − ic π ˆ d x e F µν η (cid:0) γ µ [ σ δ ρ ] ν − δ [ σµ δ ρ ] ν (cid:1) γ γ σ D ρ η (cid:48) − a − c )4 π ˆ d x e P µν g µ [ ν ηγ ρσ ] γ σ D ρ η (cid:48) − ( a − c )8 π ˆ d x e η (cid:16) R µνρσ γ µν − Rg µν g µ [ ν γ ρσ ] (cid:17) γ σ D ρ η (cid:48) . (B.20)We will now show that each of these terms vanishes once the corresponding expression with η and η (cid:48) interchanged is subtracted.The first term in the second equality in (B.20) vanishes trivially since ηγ γ σ η (cid:48) = η (cid:48) γ γ σ η. (B.21)For the second term we have ηγ σ D ρ η (cid:48) − η (cid:48) γ σ D ρ η = ∇ ρ ( ηγ σ η (cid:48) ) . (B.22)Integrating by parts and using the Bianchi identity (cid:15) µνρσ ∂ ρ F µν = 0 we find that the second termvanishes as well.The third term can be simplified as F νκ η (cid:0) γ ν [ σ δ κρ ] − δ ν [ σ δ κρ ] (cid:1) γ γ σ D ρ η (cid:48) = − F νκ η (cid:0) γ ν σ g ρκ − γ ν ρ g σκ − g σν g ρκ (cid:1) γ σ γ D ρ η (cid:48) = − F νκ η (cid:0) γ ν g ρκ − γ ν ρκ − γ ν g ρκ + γ ρ (cid:8)(cid:8)(cid:8)(cid:42) g νκ − g ρκ γ ν (cid:1) γ D ρ η (cid:48) = − F νκ ηγ ρνκ γ D ρ η (cid:48) . (B.23)Hence, subtracting the same quantity with η and η (cid:48) interchanged we obtain − F νκ ηγ ρνκ γ D ρ η (cid:48) + 12 F νκ η (cid:48) γ ρνκ γ D ρ η = − F νκ ηγ ρνκ γ D ρ η (cid:48) + 12 F νκ ∇ ρ (cid:0) η (cid:48) γ ρνκ γ η (cid:1) + 12 F νκ ηγ γ κν ρ D ρ η (cid:48) = 12 F νκ ∇ ρ (cid:0) η (cid:48) γ ρνκ γ η (cid:1) , (B.24)– 18 –hich again vanishes up to a total derivative term due to the Bianchi identity (cid:15) µνρσ ∂ ρ F µν = 0 .In order to evaluate the term proportional to the Schouten tensor P µν we note that ηγ µν γ λ D κ η (cid:48) − η (cid:48) γ µν γ λ D κ η = η { γ µν , γ λ }D κ η (cid:48) − ∇ κ ( η (cid:48) γ µν γ λ η )= 2 ηγ µν λ D κ η (cid:48) − ∇ κ (cid:0) η (cid:48) ( γ µν λ + γ µ g νλ − γ ν g µλ ) η (cid:1) . (B.25)Hence, P µν (cid:0) ηg µ [ ν γ κλ ] γ λ D κ η (cid:48) − η (cid:48) g µ [ ν γ κλ ] γ λ D κ η (cid:1) = 13 P µν (cid:0) g µν ηγ κλ γ λ D κ η (cid:48) + g µλ ηγ νκ γ λ D κ η (cid:48) + g µκ ηγ λν γ λ D κ η (cid:48) − η (cid:48) ↔ η (cid:1) = − P µν ∇ κ (cid:0) g µν η (cid:48) ( γ κ g λλ − γ λ g κλ ) η + g µλ η (cid:48) ( γ ν g κλ − γ κ g νλ ) η + g µκ η (cid:48) ( γ λ g νλ − γ ν g λλ ) η (cid:1) = − P µν ∇ κ (cid:0) g µν η (cid:48) γ κ η + η (cid:48) ( γ ν g µκ − γ κ g µν ) η − g µκ η (cid:48) γ ν η (cid:1) = − P µν ∇ κ (cid:0) g µν η (cid:48) γ κ η − g µκ η (cid:48) γ ν η (cid:1) = − R ∇ µ ( η (cid:48) γ µ η ) + 13 R µν ∇ µ ( η (cid:48) γ ν η ) = 13 ∇ µ (cid:104)(cid:16) R µν − Rg µν (cid:17) ( η (cid:48) γ ν η ) (cid:105) , (B.26)where in the last equality we have used the Bianchi identity ∇ µ (cid:16) R µν − Rg µν (cid:17) = 0 . (B.27)It follows that the term proportional to the Schouten tensor P µν in (B.20) also vanishes.Finally, for the last term in (B.20) we have R µνρσ ηγ µν γ σ D ρ η (cid:48) = R µνρσ η ( (cid:24)(cid:24)(cid:24)(cid:58) γ µνσ + γ µ g νσ − γ ν g µσ ) D ρ η (cid:48) = 2 R µρ ηγ µ D ρ η (cid:48) , (B.28)with R µν ηγ µ D ν η (cid:48) − R µν η (cid:48) γ µ D ν η = R µν ∇ ν ( ηγ µ η (cid:48) ) . (B.29)Hence, R µνρσ ηγ µν γ σ D ρ η (cid:48) − R µνρσ η (cid:48) γ µν γ σ D ρ η − Rg µν (cid:0) ηg µ [ ν γ κλ ] γ λ D κ η (cid:48) − η (cid:48) g µ [ ν γ κλ ] γ λ D κ η (cid:1) = 2 R µν ∇ ν ( ηγ µ η (cid:48) ) + R ∇ µ ( η (cid:48) γ µ η ) = 2 ∇ µ (cid:104)(cid:16) R µν − Rg µν (cid:17) ( ηγ ν η (cid:48) ) (cid:105) , (B.30)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 η (cid:48) interchanged, leading to the commutator [ δ η , δ η (cid:48) ] W = 0 , (B.31)in agreement with the Wess-Zumino consistency conditions.[ δ ε , δ η ] W = δ σ W + δ θ W , with σ = εη and θ = − i εγ η :– 19 –he final commutator we need to consider is the one between Q - and S -supersymmetry trans-formations. We start with the contribution of the A Q anomaly to the commutator, namely δ η δ ε W = − δ η ˆ d x e ε A Q = (5 a − c ) i π ˆ d x e (cid:15) µνρσ F µν A ρ εγ δ η φ σ − ( a − c )12 π ˆ d x e (cid:15) µνρσ ∇ κ (cid:0) A ρ R κλµν (cid:1) εγ ( λ δ η ψ σ ) + ( a − c )48 π ˆ d x e (cid:15) µνρσ F ρσ R κλµν εγ κ δ η ψ λ = (5 a − c ) i π ˆ d xe (cid:15) µνρσ F µν A ρ εγ D σ η + ( a − c )12 π ˆ d x e (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:58) (cid:15) µνρσ ∇ κ (cid:0) A ρ R κσµν (cid:1) εη − ( a − c )48 π ˆ d x e (cid:15) µνρσ F ρσ R κλµν εγ κλ η = (5 a − c ) i π ˆ d x e (cid:15) µνρσ F µν A ρ εγ D σ η − ( a − c )48 π ˆ d x e (cid:15) µνρσ F ρσ R κλµν εγ κλ η. (B.32)The last term in this expression can be simplified further using the last identity in (A.6) and theproduct of two Levi-Civita tensors − (cid:15) µνρσ (cid:15) κλτφ = g µκ g νλ g ρτ g σφ + (cid:0) g µλ g νκ g ρφ g στ + g µτ g νφ g ρκ g σλ + g µφ g ντ g ρλ g σκ (cid:1) − (cid:0) g µλ g νκ g ρτ g σφ + g µτ g νλ g ρκ g σφ + g µφ g νλ g ρτ g σκ + g µκ g ντ g ρλ g σφ + g µκ g νφ g ρτ g σλ + g µκ g νλ g ρφ g στ (cid:1) + (cid:0) g µτ g νκ g ρλ g σφ + g µλ g ντ g ρκ g σφ + g µφ g νκ g ρτ g σλ + g µλ g νφ g ρτ g σκ + g µφ g νλ g ρκ g στ + g µτ g νλ g ρφ g σκ + g µκ g νφ g ρλ g στ + g µκ g ντ g ρφ g σλ (cid:1) − (cid:0) g µφ g νκ g ρλ g στ + g µτ g νκ g ρφ g σλ + g µφ g ντ g ρκ g σλ + g µλ g νφ g ρκ g στ + g µτ g νφ g ρλ g σκ + g µλ g ντ g ρφ g σκ (cid:1) , (B.33)where we have grouped terms according to the conjugacy classes of the symmetric group S andterms of a given color give the same result when contracted with F µν R ρσκλ ηγ τφ γ ε . In particular, − (cid:15) µνρσ (cid:15) κλτφ F µν R ρσκλ ηγ τφ γ ε = 4 F µν R µνρσ ηγ ρσ γ ε − F µν R νρ ηγ µρ γ ε + 4 F µν R ρν ηγ ρµ γ ε + 4 F ν µ R νρ ηγ µρ γ ε − F ν µ R ρν ηγ ρµ γ ε + 4 F µν Rηγ µν γ ε = 4 (cid:0) F µν R µνρσ ηγ ρσ γ ε − F µν R νρ ηγ µρ γ ε + F µν Rηγ µν γ ε (cid:1) , (B.34)and so (cid:15) µνρσ F ρσ R κλµν ηγ κλ ε = i (cid:15) µνρσ (cid:15) κλτφ F µν R ρσκλ ηγ τφ γ ε = − i (cid:0) F µν R µνρσ ηγ ρσ γ ε − F µν R νρ ηγ µρ γ ε + F µν Rηγ µν γ ε (cid:1) . (B.35)– 20 –he contribution of the A S anomaly to the commutator is δ ε δ η W = δ ε ˆ d x e η A S = − (5 a − c )18 π i ˆ d x e (cid:15) µνρσ F µν A ρ ηγ δ ε ψ σ + (5 a − c )12 π ˆ d x e (cid:15) µνρσ F µν η D ρ δ ε ψ σ + ic π ˆ d x e F µν η (cid:0) γ µ [ σ δ ν ρ ] − δ µ [ σ δ ν ρ ] (cid:1) γ D ρ δ ε ψ σ + 3(2 a − c )4 π ˆ d x e P µν g µ [ ν ηγ ρσ ] D ρ δ ε ψ σ + ( a − c )8 π ˆ d x e η (cid:16) R µνρσ γ µν − Rg µν g µ [ ν γ ρσ ] (cid:17) D ρ δ ε ψ σ = − (5 a − c )18 π i ˆ d x e (cid:15) µνρσ F µν A ρ ηγ D σ ε + (5 a − c )12 π ˆ d x e (cid:15) µνρσ F µν η D ρ D σ ε + ic π ˆ d x e F µν η (cid:0) γ µ [ σ δ ν ρ ] − δ µ [ σ δ ν ρ ] (cid:1) γ D ρ D σ ε + 3(2 a − c )4 π ˆ d x e P µν g µ [ ν ηγ ρσ ] D ρ D σ ε + ( a − c )8 π ˆ d x e η (cid:16) R µνρσ γ µν − Rg µν g µ [ ν γ ρσ ] (cid:17) D ρ D σ ε. (B.36)Using the identity, 2 D [ µ D ν ] ε = (cid:16) R µνρσ γ ρσ + iγ F µν (cid:17) ε, (B.37)we now evaluate the four terms involving two covariant derivatives on the spinor parameter ε . Theeasiest term to evaluate is iF µν η (cid:0) γ µ [ σ δ ν ρ ] − δ µ [ σ δ ν ρ ] (cid:1) γ D ρ D σ ε = i F µν η (cid:0) γ µσ δ ν ρ − δ µσ δ ν ρ (cid:1) γ (cid:16) R ρσκλ γ κλ + iγ F ρσ (cid:17) ε = − F µν F µν ηε + i F µν R νσκλ ηγ µσ γ κλ γ ε + i F µν R µνκλ ηγ κλ γ ε = − F µν F µν ηε + i F µν R µνκλ ηγ κλ γ ε + i F µν R νσκλ η (cid:0) − i (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:58) (cid:15) µσκλ γ + 2 γ µλ g σκ − γ σλ g µκ + (cid:24)(cid:24)(cid:24)(cid:24)(cid:58) g µλ g σκ − (cid:24)(cid:24)(cid:24)(cid:24)(cid:58) g σλ g µκ (cid:1) γ ε = − F µν F µν ηε + i F µν ( R µνρσ + 2 R µρνσ − g µρ R νσ ) ηγ ρσ γ ε = − F µν F µν ηε + i F µν ( R µνρσ − g µρ R νσ ) ηγ ρσ γ ε. (B.38)We next consider the term (cid:15) µνρσ F µν η D ρ D σ ε = i (cid:15) µνρσ F µν F ρσ ηγ ε + 18 (cid:15) µνρσ F µν R ρσκλ ηγ κλ ε = i (cid:15) µνρσ F µν F ρσ ηγ ε − i (cid:0) F µν R µνρσ ηγ ρσ γ ε − F µν R νρ ηγ µρ γ ε + F µν Rηγ µν γ ε (cid:1) , (B.39)where in the last line we have utilized the identity (B.35).– 21 –ext we evaluate the term P µν g µ [ ν ηγ κλ ] D κ D λ ε = 16 P µν η (cid:0) g µν γ κλ + g µλ γ νκ + g µκ γ λν (cid:1)(cid:16) R κλρσ γ ρσ + iγ F κλ (cid:17) ε = 16 P µν η ( g µν γ κλ − g µκ γ νλ ) (cid:16) R κλρσ γ ρσ + iγ F κλ (cid:17) ε = 136 R (cid:16) R κλρσ ηγ κλ γ ρσ ε + iηγ κλ γ εF κλ (cid:17) − P µν (cid:16) R µλρσ ηγ νλ γ ρσ ε + iηγ νλ γ εF µλ (cid:17) = − (cid:16) R µν − Rg µν (cid:17) × (cid:16) iF µλ ηγ νλ γ ε + 14 R µλρσ η (cid:0) − i (cid:8)(cid:8)(cid:8)(cid:42) (cid:15) νλρσ γ + 2 (cid:24)(cid:24)(cid:24)(cid:24)(cid:58) γ νσ g λρ − (cid:24)(cid:24)(cid:24)(cid:24)(cid:58) γ λσ g νρ + g νσ g λρ − g λσ g νρ (cid:1) ε (cid:17) = − (cid:16) R µν − Rg µν (cid:17) iF µλ ηγ νλ γ ε + 112 (cid:16) R µν R µν − R (cid:17) ηε. (B.40)Finally, the last term gives (cid:16) R µνρσ γ µν − Rg µν g µ [ ν γ ρσ ] (cid:17) D ρ D σ ε = 12 (cid:16) R µνρσ γ µν − Rg µν g µ [ ν γ ρσ ] (cid:17)(cid:16) R ρσκλ γ κλ + iγ F ρσ (cid:17) ε = 18 R µνρσ R κλρσ (cid:0) − i(cid:15) µν κλ γ + 4 (cid:8)(cid:8)(cid:8)(cid:8)(cid:42) γ µλ g κν + 2 g λµ g κν (cid:1) ε + i R µνρσ F ρσ γ µν γ ε − (cid:0) iRF µν ηγ µν γ ε − R ηε (cid:1) = − i P γ ε − R µνρσ R µνρσ ε + i R µνρσ F ρσ γ µν γ ε − (cid:0) iRF µν ηγ µν γ ε − R ηε (cid:1) , (B.41)where P = (cid:15) µνρσ R µνκλ R ρσκλ 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[ δ ε , δ η ] W = (5 a − c ) i π ˆ d x e (cid:15) µνρσ F µν F ρσ ηγ ε − c π ˆ d x e F µν F µν ηε + (2 a − c )16 π ˆ d x e (cid:16) R µν R µν − R (cid:17) ηε − ( a − c )32 π ˆ d x e (cid:16) i P ηγ ε + R µνρσ R µνρσ ηε − R ηε (cid:17) = ˆ d x e ( − θ A R + σ A W ) , (B.42)with θ = − iεγ η, σ = 12 εη, (B.43)as required by the Wess-Zumino consistency conditions. 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