KKIAS-P19008
Anomalous supersymmetry
Georgios Katsianis,
1, 2
Ioannis Papadimitriou, Kostas Skenderis,
1, 2 and Marika Taylor
1, 2 STAG Research Centre, Highfield, University of Southampton, SO17 1BJ Southampton, UK Mathematical Sciences, Highfield, University of Southampton, SO17 1BJ Southampton, UK School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea (Dated: May 21, 2019)We show that supersymmetry is anomalous in N = 1 superconformal quantum field theories(SCFTs) with an anomalous R-symmetry. This anomaly was originally found in holographic SCFTsat strong coupling. Here we show that this anomaly is present in general and demonstrate it for themassless superconformal Wess-Zumino model via an one loop computation. The anomaly appearsfirst in four-point functions of two supercurrents with either two R-currents or with an R-currentand an energy momentum tensor. In fact, the Wess-Zumino consistency conditions together withthe standard R-symmetry anomaly imply the existence of the anomaly. We outline the implicationsof this anomaly. Anomalies of symmetries play an important role inquantum field theories. If a global symmetry is anoma-lous, classical selection rules are not respected in thequantum theory and classically forbidden processes mayoccur. This is a feature of the theory and it is linkedwith observable effects. For example, the axial anomalyexplains the π decay and leads to the resolution of the U (1) problem in QCD [1, 2]. On the other hand, anoma-lies in local (gauge) symmetries lead to inconsistencies,such as lack of unitarity, and they must be canceled.An important corollary is that anomalous global sym-metries cannot be consistently coupled to correspondinglocal symmetries. Reviews on anomalies in quantum fieldtheories may be found in [3, 4]. Anomalies in supersymmetric theories. — In this pa-per we discuss a new anomaly in four-dimensional su-persymmetric quantum field theories with an anomalousR-symmetry: global supersymmetry itself is anomalous.This anomaly was discovered in the context of supercon-formal theories that can be realised holographically [5].Here we show that the same anomaly arises in perturba-tion theory in the simplest supersymmetric model: thefree superconformal Wess-Zumino (WZ) model.An anomaly may be detected either by putting the the-ory on a non-trivial background, or by computing correla-tion functions on a flat background and checking whetherthe Ward identities are satisfied. The latter method wasthe one that led to the original discovery of anomaliesvia one-loop triangle diagrams [1, 2]. Here we will carryout the analogous computation for the supersymmetryanomaly. The anomaly is associated in particular withanomalous one-loop contributions to four-point correla-tion functions between two supersymmetry currents andtwo R-currents or an R-current and an energy momen-tum tensor. We will discuss the former in the free super-conformal WZ model but analogous contributions wouldarise in any supersymmetric theory with a (softly broken)anomalous R-symmetry. Actually, as will be sketched be-low and is shown in detail in the companion paper [6],the WZ consistency conditions [7] together with the stan- dard triangle anomalies imply that supersymmetry mustbe anomalous.Discussion of anomalies in 4d (super)conformal QFThas a long history. It has been known since the 1970s[8, 9] that the trace of the stress tensor T µµ is anomalousin the presence of a curved background metric g µν andbackground source A µ for a chiral current J µ , and theR-current is similarly anomalous. Moreover, there aregenerally mixed anomalies involving two energy momen-tum tensors and a chiral current [10, 11]. It has also beenknown since [12] that the currents sit in a supermultiplet,as do the anomalies. In particular, the trace anomaly andthe R-current anomaly are in the same multiplet as thegamma trace of the supercurrent, γ µ Q µ . The latter isan anomaly in the conservation of the special supersym-metry current, x ν γ ν Q µ . It follows that special super-symmetry (sometimes also called S-supersymmetry) isanomalous. It was believed however that supersymmetryitself (sometimes called Q-supersymmetry) is preserved,i.e. the conservation of Q µ is non-anomalous.There have been extensive studies in the past regard-ing anomalies in supersymmetry. It was realised early on[13–18] that one cannot maintain at the quantum level si-multaneously ∂ µ Q µ = 0 and γ µ Q µ = 0 and, if the modelis a gauge theory, gauge invariance: one of the three con-ditions must be relaxed and the standard choice is tohave a superconformal anomaly. This is the standard su-perconformal anomaly mentioned above and is distinctfrom the anomaly discussed here. Also distinct is theKonishi anomaly [19, 20], which is a superspace versionof the chiral anomaly in supersymmetric gauge theories.Another set of studies, reviewed in [21], considersthe effective action for elementary fields and examineswhether it is invariant under supersymmetry includingloop effects; it investigates the conservation of the su-percurrent inside correlators of elementary fields and/orsolves the WZ consistency conditions relevant for thissetup, and finds no supersymmetry anomaly. This doesnot contradict the results we present below: to find theanomaly one should either put the theory on a non- a r X i v : . [ h e p - t h ] M a y trivial background or consider correlation functions of(classically) conserved currents . Studies involving cor-relators of currents have also appeared but typically onlydiscuss 3-point functions of bosonic currents. As men-tioned above, the supersymmetry anomaly appears firstin 4-point functions involving two supercurrents and twobosonic currents and to our knowledge these have notbeen computed before.Anomalies associated with correlation functions of con-served currents can be analysed by coupling the currentsto external sources, which in our case form an N = 1 su-perconformal multiplet. As such, the anomaly we discusshere could be related to existing superspace results onanomaly candidates for D = 4, N = 1 supergravity the-ories [22–26] (in particular, in type II anomalies in [25]),though we emphasise that in our case the supergravityfields are external and thus non-dynamical (off-shell).A supersymmetry anomaly appears in super Yang-Mills (SYM) theory in the WZ gauge when there aregauge anomalies [27] (see also [28–30]). This anomaly iseasy to understand: in the WZ gauge, supersymmetrytransformations require a compensating gauge transfor-mation and this transfers the anomaly from the gaugesector to supersymmetry. When the SYM theory is con-sistent at the quantum level (i.e. the gauge anomaliescancel) then supersymmetry is also non-anomalous. Asupersymmetry anomaly appears in theories with gravi-tational anomalies [31–33], as one may anticipate basedon the fact that the energy momentum tensor and the su-percurrent are part of the same supermultiplet. Indeedthis supersymmetry anomaly sits in the same multipletas the gravitational anomaly.Here we will discuss a supersymmetry anomaly in con-sistent QFTs (no gauge anomalies) which have a con-served energy momentum tensor. We also emphasise thatwe are concerned with local anomalies, not with betafunctions. Holographic anomalies. — The anomaly we discuss herewas first computed holographically [5]. In holography,given a bulk action, one can use holographic renormalisa-tion [34, 35] to compute the Ward identities and anoma-lies of the dual QFT. AdS/CFT relates N = 1 SCFTin four dimensions to N = 2 gauged supergravity infive dimensions. Starting from gauged supergravity inan asymptotically locally AdS spacetime and turningon sources for all superconformal currents one can com-pute the complete set of superconformal anomalies. Thiscomputation is available for holographic CFTs, which in To illustrate this point, consider a free fermion in a complexrepresentation in flat spacetime. This theory has a standardaxial anomaly originating from the 3-point function of the axialcurrent. However, if one only looks at correlators of elementaryfields these are non-anomalous and the axial current inside suchcorrelators is conserved. particular means that the central charges should satisfy a = c as N → ∞ [34].Early attempts to compute the supertrace Ward iden-tity can be found in [36, 37] but these missed contribu-tions to the anomaly involving the R-symmetry currentand the Ricci tensor. Following the work of Pestun [38],there was renewed interest in supersymmetric theories oncurved spacetimes and their holographic duals. The holo-graphic anomalies for bosonic currents were computedin [39], reproducing (and correcting) known field theoryresults [40]. The full superconformal anomalies for the N = 1 current multiplet were computed holographicallyin [5], while [41] obtained the superconformal anomaliesin the presence of local supersymmetric scalar couplings.An analogous holographic computation relevant to two-dimensional SCFTs was reported in [42].The holographic results leave open the possibility thatthe anomaly is special to holographic theories at strongcoupling. In this Letter we show that this is not thecase. One could have anticipated the anomaly basedon the structure of the supersymmetric variation of thesupercurrent, which is of the schematic form δ Q µ ∼ γ ν T µν ε + C µνρ ∂ ν J ρ ε , where C µνρ is a tensor constructedfrom gamma matrices and the metric. The Ward identityfor the 4-point function involving two supercurrents andtwo R-currents would then involve terms of the form ∂ x µ (cid:104)Q µ ( x ) ¯ Q ν ( x ) J κ ( x ) J λ ( x ) (cid:105) (1) ∼ δ ( x − x ) (cid:104) δ ¯ Q ν ( x ) J κ ( x ) J λ ( x ) (cid:105) + · · · , where the dots denote additional terms (the exact Wardidentity is given (9)). Using the variation of the super-current we find that the r.h.s. contains the 3-point func-tion of three R-currents, which is anomalous, and cor-respondingly one may anticipate (1) will be anomalous.Similarly, the same 4-point function but with one of theR-currents replaced by an energy momentum tensor isexpected to be anomalous, since (cid:104)J T T (cid:105) is anomalous.To determine whether an anomaly appears or not weneed to carry out the computation explicitly. Before weturn to this, we discuss the consistency condition thatthe anomalies must satisfy. Wess-Zumino consistency. — Let e aµ , A µ and ψ µ denotethe sources (vierbein, gauge field and gravitino) that cou-ple to the superconformal currents and W [ e, A, ψ ] be thegenerating functional of connected graphs. We define thecurrents in the presence of sources (as usual) by T µa = e − δ W δe aµ , J µ = e − δ W δA µ , Q µ = e − δ W δψ µ , (2)where e ≡ det( e aµ ). In the presence of anomalies δ i W = (cid:90) d x e (cid:15) i A i , (3)where δ i denotes the superconformal transformations, (cid:15) i are the (local) parameters of the transformations and A i 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 + σψ µ + D µ ε − γ µ η − iγ θψ µ ,δA µ = ξ λ ∂ λ A µ + A λ ∂ µ ξ λ + i φ µ γ ε − i ψ µ γ η + ∂ µ θ, φ µ ≡ γ ν (cid:0) D ν ψ µ − D µ ψ ν − i γ (cid:15) νµρσ D ρ ψ σ (cid:1) [ δ ε , δ ε (cid:48) ] = δ ξ + δ λ + δ θ , ξ µ = ε (cid:48) γ µ ε, λ ab = − ( ε (cid:48) γ ν ε ) ω νab , θ = − ( ε (cid:48) γ ν ε ) A ν [ δ ε , δ η ] = δ σ + δ λ + δ θ , σ = εη, λ ab = − εγ ab η, θ = − i εγ η TABLE I. Transformation rules of the current sources and their algebra, to leading order in the gravitino. All other commutatorsvanish, except for that of two diffeomorphisms and two local Lorentz transformations, which take a standard form. are the corresponding anomalies. The variations form analgebra, [ δ i , δ j ] = f kij δ k , and using this in (3) we obtainthe WZ consistency condition (cid:90) d x (cid:0) δ i ( e (cid:15) j A j ) − δ j ( e (cid:15) i A i ) − f kij e (cid:15) k A k (cid:1) = 0 . (4)The transformation rules and the local algebra they sat-isfy are derived in [6] and are given in Table I.Assuming the R-symmetry current has the standardtriangle anomalies (i.e. assuming the from of A R in Ta-ble II) the WZ consistency conditions (4) may be viewedas equations to determine the remaining anomalies. Thiscomputation is presented in [6] and the results are sum-marised in Table II. Note in particular that all anoma-lies are given in terms of the central changes a and c .The anomalies of the bosonic currents are in agreementwith the results derived in [39, 40]. The supersymmetryanomaly A Q that we discuss here is related to the R-symmetry anomaly A R through the same descent equa-tion that relates the supersymmetry anomaly discussedin [27] to the corresponding gauge anomaly. However,as noted earlier, there are important differences in thephysics (in [27] the gauge anomalies must vanish for con-sistency of the model, while this is not so for the R-anomalies relevant for us), as well as in the context (theWZ conditions discussed in [27] are for a vector multipletin flat space, while the anomalies in Table II are those of N = 1 conformal supergravity [6]).Here we only discuss one of the WZ equations: the oneobtained by considering the commutator of R-symmetry(with parameter θ ) with Q-supersymmetry (with param-eter ε ): (cid:90) d x (cid:0) δ ε ( e θ A R ) − δ θ ( e ε A Q ) (cid:1) = 0 . (5)Using the explicit form of A R it is easy to see that δ ε A R (cid:54) = 0 and the WZ consistency condition requiresthat A Q (cid:54) = 0. This argument does not rely on the theoryhaving conformal invariance, and thus we expect any 4dsupersymmetric theory with an R-symmetry anomaly tohave a corresponding anomaly in the conservation of the supercurrent. One may wonder whether this anomaly can be removedby adding a local counterterm W ct to the action suchthat W ren = W + W ct is non-anomalous, i.e. δ ε W ren = 0.Using the commutator of two supersymmetry variations,[ δ ε , δ ε (cid:48) ], given in Table I we find( δ ξ + δ λ + δ θ ) W ren = 0 ⇒ ( δ ξ + δ λ ) W ren (cid:54) = 0 , (6)since δ θ W ren = A R (cid:54) = 0. It follows that if one wishesto preserve supersymmetry W ct must break diffeomor-phisms and/or local Lorentz transformations. Next, wecalculate this anomaly by one-loop computations withina specific model.
Model. — Consider the massless Wess-Zumino actionwith one complex bosonic field φ and one Majoranafermionic field χS = − (cid:90) d x (cid:16) ∂ µ φ∂ µ φ ∗ + 12 ¯ χ /∂χ (cid:17) . (7)The conserved currents are given in Table III. We haveincluded improvement terms so that classically T µµ =0 , γ µ Q µ = 0 and we are dealing with an N = 1 SCFT.From the form of the anomaly A Q in Table II followsthat the first anomalous contribution in flat space corre-lators appears in 4-point functions involving two super-currents and either two R-currents or an R-current andan energy momentum tensor. Here we discuss the former,referring to [44] for a detailed account of both cases.Since we seek to investigate the possibility of a super-symmetry anomaly, we should not assume the existenceof a supersymmetric regulator: the one-loop computa-tion should not be done in superspace. We will instead This expectation has been verified in the followup paper [43]. Note that since A R is a genuine anomaly it is not possible toset the r.h.s. of the second equation in (6) to zero using a localcounterterm. This implies that there are no further local coun-terterms that can restore diffeomeorphisms and local Lorentzinvariance. On the other hand, the form of anomalies respects the symme-tries they break and thus one may use superspace to analysepossible anomaly candidates. e aµ T µa + ψ µ Q µ = A W , ∇ µ J µ + iψ µ γ Q µ = A R Weyl square: W ≡ W µνρσ W µνρσ D µ Q µ − γ a ψ µ T µa − i γ φ µ J µ = A Q , γ µ Q µ − i γ ψ µ J µ = A S Euler density: E = R µνρσ R µνρσ − R µν R µν + R Pontryagin density: A W = c π (cid:0) W − F (cid:1) − a π E + O ( ψ ), A R = (5 a − c )27 π (cid:101) F F + ( c − a )24 π P P ≡ (cid:101) R µνρσ R µνρσ (cid:101) R µνρσ ≡ (cid:15) µνκλ R κλρσ A Q = − (5 a − c ) i π (cid:101) F µν A µ γ φ ν + ( a − c )6 π (cid:0) ∇ µ ( A ρ (cid:101) R ρσµν ) γ ( ν ψ σ ) − F µν (cid:101) R µνρσ γ ρ ψ σ (cid:1) + O ( ψ ) Schouten tensor: P µν ≡ (cid:0) R µν − Rg µν (cid:1) A S = (5 a − c )6 π (cid:101) F µν (cid:0) D µ − i A µ γ ) 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 U(1) R field strengths:+ a − c )4 π P µν g µ [ ν γ ρσ ] D ρ ψ σ + ( a − c )8 π (cid:0) R µνρσ γ µν − Rg µν g µ [ ν γ ρσ ] (cid:1) D ρ ψ σ + O ( ψ ) (cid:101) F µν ≡ (cid:15) µνρσ F ρσ F ≡ F µν F µν F (cid:101) F ≡ F µν (cid:101) F µν TABLE II. Anomalous Ward identities and corresponding anomalies [6]. ( D µ ψ ν ≡ ( ∂ µ + ω µab ( e, ψ ) γ ab + iγ A µ ) ψ ν − Γ ρµν ψ ρ with ω µab ( e, ψ ) ≡ ω µab ( e ) + (cid:0) ψ a γ µ ψ b + ψ µ γ a ψ b − ψ µ γ b ψ a (cid:1) ; ∇ µ is the Levi-Civita connection; φ µ is defined in Table I.) T µa = ( η µρ η σa + η µσ η ρa − η µα η ρσ ) ∂ ρ φ ∗ ∂ σ φ − (cid:0) ∂ µ ∂ a − η µa ∂ (cid:1) ( φ ∗ φ ) + χ ( γ µ ∂ a + γ a ∂ µ ) χ J µ = i (cid:0) φ ∗ ∂ µ φ − φ∂ µ φ ∗ + χγ µ γ χ (cid:1) Q µ = √ ( /∂φγ µ χ R + /∂φ ∗ γ µ χ L ) + √ γ µν ∂ ν ( φχ R + φ ∗ χ L ) , χ L ≡ (1 + γ ) χ, χ R ≡ (1 − γ ) χ. TABLE III. The (on-shell) energy-momentum tensor, T µa , the R-symmetry current, J µ , and the supersymmetry current, Q µ ,for the massless superconformal WZ model in flat space. do the computation in components and use the same reg-ulator as in the original triangle anomaly computation,namely momentum cut-off [1, 2]. We will consider the4-point correlation function (cid:10) Q µ ( x ) ¯ Q ν ( x ) J κ ( x ) J λ ( x ) (cid:11) . (8)Standard path integral manipulations show that this cor-relator classically satisfies the following Ward identity: − i∂ x µ (cid:10) Q µ ¯ Q ν J κ J λ (cid:11) = δ (4) ( x ) (cid:10) δ ¯ Q ν J κ J λ (cid:11) + (cid:104) δ (4) ( x ) (cid:10) δ J κ ¯ Q ν J λ (cid:11) − ∂ x ρ (cid:16) δ (4) ( x ) (cid:10) δ J (cid:48) ρκ ¯ Q ν J λ (cid:11)(cid:17) +(3 , κ ) ↔ (4 , λ ) (cid:105) − ∂ x ρ (cid:16) δ (4) ( x ) (cid:10) δ ¯ Q (cid:48) νρ J κ J λ (cid:11)(cid:17) , (9)where we have used the shorthand notation Q µ ( x i ) ≡Q µi , etc., x ij ≡ x i − x j , and the contributions on the r.h.s.are expressed in terms of the supersymmetry variationsof the currents: δ ε Q µ = εδ Q µ + ∂ ν εδ Q (cid:48) µν and idem for J µ .A similar Ward identity follows from R-invariance. One-loop computation. — We now compute (9). Sincethe theory is free the complete computation is one-loop.The 4-point function receives contributions from threeclasses of Feynman box diagrams, shown in Figure 1;this computation is straightforward but tedious.
FIG. 1. Box diagrams contributing to the 4-point correlationfunction (8). Zig-zag lines denote R-currents; wavy lines de-note supersymmetry currents; straight lines denote fermionicprogagators and dashed lines denote bosonic propagators.
One may verify that (9), as well as the correspondingR-symmetry Ward identity, are (naively) satisfied by asimple shift of the loop momentum, much the same wayas the triangle Ward identity is naively satisfied. Again inparallel with the triangle anomaly, (part of) the one-loopcontributions to the 4-point function are superficially lin-early divergent. This implies that there is a momentumrouting ambiguity when using a momentum cut-off reg-ulator (see for example Jackiw’s lectures in [3]).We proceed by taking the ∂ x κ of (9) and subtract-ing from it the ∂ x µ derivative of the corresponding R-symmetry Ward identity. By construction, the 4-pointfunctions cancel and one is left with an identity involv-ing 3-point functions only (namely the terms appearingon the r.h.s. of the Ward identities). Had these 3-pointfunctions been non-anomalous, this would be an iden-tity. However, the 3-point functions involve the anoma-lous (cid:104)J J J (cid:105) correlator and this implies that either (9)or the corresponding R-symmetry Ward identity shouldbe anomalous. Assuming the form of the bosonic Wardidentities is standard (i.e. given by the expressions in Ta-ble II) the R-symmetry 4-point function Ward identity isnot anomalous and therefore the supersymmetry Wardidentity is anomalous. This computation is the counter-part of (5) but now in terms of Feynman diagrams.One can then show that there is a momentum routingsuch that 1) the triangle R-symmetry anomaly is repro-duced; 2) the 4-point R-symmetry Ward identity is non-anomalous and 3) the supersymmetry Ward identity isanomalous, with the anomaly given in Table II and with c = 2 a = 1 /
24, which are the values in our model. Inaddition, upon taking the gamma trace of the same 4-point function, γ µ (cid:10) Q µ ¯ Q ν J κ J λ (cid:11) , one automatically re-produces the A S anomaly given in Table II.In general, changing the momentum routing one maymove the anomaly from one conserved current to another.This would be equivalent to adding local finite countert-erms and as argued earlier there is no choice of such coun-terterms that would remove the supersymmetry anomalywhile preserving diffeomorphisms/local Lorentz transfor-mations.It is also straightforward to check that the sameanomaly is present in the massive WZ model as well.As in the case of standard triangle anomalies, adding amass term modifies the Ward identities but the anomalyremains the same. This is as expected since the anomalyarises from the UV behaviour of Feynman diagrams andthe parts of the loop computation that give rise to theanomaly remain the same. Implications of the anomaly. — Let us conclude witha few comments about the implications of this anomaly.As mentioned earlier, an important consequence is thata SQFT with such a supersymmetry anomaly cannotbe coupled to dynamical supergravity. In the contextof supersymmetric model building, one does not usuallywork with theories with an R-symmetry, anomalous ornon-anomalous; non-anomalous R-symmetry is not com-patible with gaugino masses (see [45]). More generally,one does not expect a theory with continuous symmetry The anomalous R-symmetry alone implies that coupling to asupergravity that gauges the R-symmetry is inconsistent. Herewe see that couplings to supergravity that do not gauge the R-symmetry are also inconsistent. to emerge from a consistent quantum theory of gravity,such as string theory. However, such models may beconsidered in bottom-up approaches (see [46] for a re-cent example). Similar comments apply to bottom-upstring cosmology models. This anomaly also affects su-persymmetric localisation computations, as has alreadybeen noted in [5, 6, 41, 42]. However, it is possible that asuitable non-covariant local counterterm may cancel therigid supersymmetry anomaly at the expense of breakingcertain diffeomorphisms on a given supersymmetric back-ground. From a more formal perspective, it would be in-teresting to explore how the supersymmetry anomaly iscaptured in index theorems. It would also be interestingto investigate the existence of such an anomaly in otherdimensions and/or extended supersymmetry. Note Added:
While this paper was finalised, a re-lated work [48] appeared on the arXiv.
Acknowledgments. — We would like to thank Ben-jamin Assel, Roberto Auzzi, Friedmann Brandt, Lori-ano Bonora, Davide Cassani, Cyril Closset, Camillo Im-bimbo, Manthos Karydas, Heeyeon Kim, Zohar Komar-godski, Dario Martelli, Sunil Mukhi, Sameer Murthy,Parameswaran Nair, Dario Rosa, Stanislav Schmidt,Ashoke Sen and Peter West for illuminating discussionsand email correspondence. KS and MMT are supportedin part by the Science and Technology Facilities Coun-cil (Consolidated Grant “Exploring the Limits of theStandard Model and Beyond”). This research was sup-ported in part by the National Science Foundation un-der Grant No. NSF PHY-1748958 and this projecthas received funding/support from the European Union’sHorizon 2020 research and innovation programme underthe Marie Sklodowska-Curie grant agreement No 690575.IP would like to thank the University of Southampton,King’s College London, and the International Center forTheoretical Physics in Trieste for hospitality and par-tial financial support during the completion of this work.MMT would like to thank the Kavli Institute for thePhysics and Mathematics of the Universe for hospitalityduring the completion of this work. [1] S. L. Adler, Phys. Rev. , 2426 (1969).[2] J. S. Bell and R. Jackiw, Nuovo Cim.
A60 , 47 (1969).[3] S. B. Treiman, E. Witten, R. Jackiw, and B. Zumino,
Current algebra and anomalies (1986).[4] K. Fujikawa and H. Suzuki,
Path integrals and quantumanomalies (2004).[5] I. Papadimitriou, JHEP , 038 (2017),arXiv:1703.04299 [hep-th]. For theories with a = c such a counterterm evaluated on super-symmetric backgrounds of the form S × M , with M a Seifertmanifold, should agree with the counterterm used in [47]. [6] I. Papadimitriou, (2019), arXiv:1902.06717 [hep-th].[7] J. Wess and B. Zumino, Phys. Lett. , 95 (1971).[8] D. M. Capper and M. J. Duff, Nuovo Cim. A23 , 173(1974).[9] S. Deser, M. J. Duff, and C. J. Isham, Nucl. Phys.
B111 ,45 (1976).[10] R. Delbourgo and A. Salam, Phys. Lett. , 381 (1972).[11] L. Alvarez-Gaume and E. Witten, Nucl. Phys.
B234 , 269(1984).[12] S. Ferrara and B. Zumino, Nucl. Phys.
B87 , 207 (1975).[13] B. de Wit and D. Z. Freedman, Phys. Rev.
D12 , 2286(1975).[14] L. F. Abbott, M. T. Grisaru, and H. J. Schnitzer, Phys.Rev.
D16 , 2995 (1977).[15] L. F. Abbott, M. T. Grisaru, and H. J. Schnitzer, Phys.Lett. , 161 (1977).[16] L. F. Abbott, M. T. Grisaru, and H. J. Schnitzer, Phys.Lett. , 71 (1978).[17] K. Hieda, A. Kasai, H. Makino, and H. Suzuki, PTEP , 063B03 (2017), arXiv:1703.04802 [hep-lat].[18] Y. R. Batista, B. Hiller, A. Cherchiglia, and M. Sampaio,Phys. Rev.
D98 , 025018 (2018), arXiv:1805.08225 [hep-th].[19] K. Konishi, Phys. Lett. , 439 (1984).[20] K.-i. Konishi and K.-i. Shizuya, Nuovo Cim.
A90 , 111(1985).[21] O. Piguet and K. Sibold,
Renormalized supersymmetry.The perturbation theory of N=1 supersymmetric theoriesin flat space-time , Vol. 12 (1986).[22] L. Bonora, P. Pasti, and M. Tonin, Nucl. Phys.
B252 ,458 (1985).[23] I. L. Buchbinder and S. M. Kuzenko, Nucl. Phys.
B274 ,653 (1986).[24] F. Brandt, Class. Quant. Grav. , 849 (1994),arXiv:hep-th/9306054 [hep-th].[25] F. Brandt, Annals Phys. , 357 (1997), arXiv:hep-th/9609192 [hep-th].[26] L. Bonora and S. Giaccari, JHEP , 116 (2013),arXiv:1305.7116 [hep-th].[27] H. Itoyama, V. P. Nair, and H.-c. Ren, Nucl. Phys. B262 , 317 (1985).[28] O. Piguet and K. Sibold, Nucl. Phys.
B247 , 484 (1984). [29] E. Guadagnini and M. Mintchev, Nucl. Phys.
B269 , 543(1986).[30] B. Zumino, in
Symposium on Anomalies, Geometry,Topology Argonne, Illinois, March 28-30, 1985 (1985).[31] P. S. Howe and P. C. West, Phys. Lett. , 335 (1985).[32] Y. Tanii, Nucl. Phys.
B259 , 677 (1985).[33] H. Itoyama, V. P. Nair, and H.-c. Ren, Phys. Lett. ,78 (1986).[34] M. Henningson and K. Skenderis, JHEP , 023 (1998),arXiv:hep-th/9806087 [hep-th].[35] S. de Haro, S. N. Solodukhin, and K. Skenderis,Commun. Math. Phys. , 595 (2001), arXiv:hep-th/0002230 [hep-th].[36] M. Chaichian and W. F. Chen, Nucl. Phys. B678 , 317(2004), arXiv:hep-th/0304238 [hep-th].[37] M. Chaichian and W. F. Chen, in
Symmetries in grav-ity and field theory (2003) pp. 449–472, arXiv:hep-th/0312050 [hep-th].[38] V. Pestun, Commun. Math. Phys. , 71 (2012),arXiv:0712.2824 [hep-th].[39] D. Cassani and D. Martelli, JHEP , 025 (2013),arXiv:1307.6567 [hep-th].[40] D. Anselmi, D. Z. Freedman, M. T. Grisaru, and A. A.Johansen, Nucl. Phys. B526 , 543 (1998), arXiv:hep-th/9708042 [hep-th].[41] O. S. An, JHEP , 107 (2017), arXiv:1703.09607 [hep-th].[42] O. S. An, Y. H. Ko, and S.-H. Won, (2018),arXiv:1812.10209 [hep-th].[43] I. Papadimitriou, (2019), arXiv:1904.00347 [hep-th].[44] G. Katsianis, I. Papadimitriou, K. Skenderis, andM. Taylor, (to appear).[45] M. Drees, R. Godbole, and P. Roy, Theory andphenomenology of sparticles: An account of four-dimensional N=1 supersymmetry in high energy physics (2004).[46] C. Pallis, (2018), arXiv:1812.10284 [hep-ph].[47] P. Benetti Genolini, D. Cassani, D. Martelli, andJ. Sparks, JHEP02