JHEP04(2019)040 - and Q Springer April 1, 2019 April 4, 2019 : : March 11, 2019 : arXiv:1703.0429 = 1 off-shell confor- Accepted Published N Received conformal = 1 Published for SISSA by -symmetry anomalies, we compute ex- https://doi.org/10.1007/JHEP04(2019)040 -symmetry is anomalous. The R N R anomaly coefficients. anomaly coefficients to leading non trivial order in c c and -supersymmetry anomalies. In particular, we show that and a S a - and . Q 3 1902.06717 The Authors. Anomalies in Field and String Theories, Global Symmetries, Supergravity c

We solve the Wess-Zumino consistency conditions of , [email protected] -supersymmetry is anomalous if and only if 85 Hoegiro, Seoul 02455, Korea E-mail: School of Physics, Korea Institute for Advanced Study, -supersymmetry anomalies give rise to an anomalous supersymmetry transformation for Open Access Article funded by SCOAP Keywords: Models, Supersymmetry Breaking ArXiv ePrint: the supercurrent on curved backgroundsrigid admitting Killing supersymmetry spinors, algebra. resulting incalization Our a and deformed results supersymmetry may phenomenology. havedimensions Analogous implications two results for and are supersymmetric six expectedWess-Zumino lo- consistency and to conditions hold for reproduces the in other holographic resultand supergravity of generalizes theories. it to The arbitrary present analysis of the mal anomalies for arbitrary the gravitino. Besides theplicitly well the known fermionic Weyl and Q S Abstract: mal supergravity in four dimensions and determine the general form of the superconfor- Ioannis Papadimitriou Supersymmetry anomalies in supergravity JHEP04(2019)040 ] 6 12 9 ]. 10 – 8 ] and transport 4 12 , -supersymmetry) is Q 11 -supersymmetry S ]. These results confirm the 7 - and 16 Q ] for an observation of the mixed chiral- 14 gauge fields for the anomalous global sym- = 1 conformal supergravity – 1 – 15 N dynamical -supersymmetry is necessarily anomalous in theories with 7 Q -multiplet in the recent paper [ 14 R ], and the related anomalies in rigid supersymmetry [ 8 ] for a recent review and [ ] imply that 1 5 13 -symmetry. The same conclusion is reached in the companion paper [ R ]. They also play a pivotal role in holographic dualities and beyond the Standard 4 – 1 Global anomalies do not render the theory inconsistent — they are a property of -supersymmetry anomaly discovered in the context of supersymmetric theories with a mean that certain conditions necessary to prove non-perturbative results are in fact not met. coefficients (see [ gravitational anomaly in tabletop experiments).cannot be They do coupled mean, consistentlymetry. however, to that Specifically, a the theory quantumcannot anomaly be in coupled global consistently to supersymmetry dynamical supergravity implies at that the quantum the level. theory It may also tion was made usingQ the holographic dual in [ the theory and affect physical observables, such as decay channels [ consensus seems to benot that anomalous. standard supersymmetry (often However,tency termed conditions we [ demonstrate inan this anomalous paper thatby means the of Wess-Zumino an consis- one loop calculation in the free Wess-Zumino model. A related observa- Supersymmetric quantum fieldpling theories physics have due proven to invaluableniques non-renormalization [ theorems for and probing supersymmetric strongModel localization phenomenology. cou- tech- Given the enormous utilitythat of the supersymmetry, it question is of rather surprising whetheranswered, despite it the is extensive anomalous literature addressing at this the question quantum in level various contexts. is still The not conclusively 1 Introduction 6 Concluding remarks A Spinor conventions and identities B Solving the Wess-Zumino consistency conditions 3 Classical Ward identities 4 Superconformal anomalies from the Wess-Zumino consistency5 conditions Anomalous supercurrent transformation under Contents 1 Introduction 2 The local symmetry algebra of JHEP04(2019)040 ] for 1 15 ]. a Seifert -current 3 33 R M = 2 gauged ] (see [ 4 N with 3 M gauge fields for the × 1 ] and rigid supersym- S 8 ]. We will elaborate on this -currents or one 33 ]. Based on the classical su- R solutions using holographic 5 33 ], where both the bosonic and background 8 -supersymmetry [ ]. The first step is coupling the theory to Q 16 – 2 – -supersymmetry anomaly is visible in flat space are Q ] for earlier work). However, this procedure is classical 29 , on supersymmetric backgrounds of the form ] demonstrated that the holographic partition function on the 28 ] and the holographic result of [ c 33 32 = ] using this deformed supersymmetry algebra reproduced exactly – a ]. However, global anomalies become manifest at the level of the 32 30 6 – ] in [ ] demonstrated that the supersymmetric partition function on such 30 35 32 , – ] (see also [ 34 30 27 . – 5 17 ] were discovered, providing a resolution to an apparent tension between the 9 , 8 It was in that context that the anomalies in Following the recent advances in supersymmetric localization techniques [ In flat space, global — or rigid — anomalies are typically visible only in higher-point At least for theories with 1 diffeomorphisms using the localpoint in but section non-covariant counterterm found in [ to a deformed superconformaling algebra the on argument backgrounds of [ admitting Killingthe spinors. dependence on Repeat- the deformations of the supersymmetric backgroundmanifold seen it in is [ possible to remove the rigid supersymmetry anomaly at the expense of breaking certain renormalization [ same supersymmetric backgrounds does in factresolution depend on to the this deformation parameters. apparent contradictionfermionic The superconformal was Ward provided identities in were derived [ sponding holographically, including superconformal the anomalies. corre- It wasWard then identities shown (the that ones the in anomalies the in divergence the and fermionic the gamma-trace of the supercurrent) lead persymmetry algebra on curved backgroundsthe that admit authors a of certain number [ backgrounds of supercharges, should be independentground. of specific However, deformations an of explicitsupergravity the evaluation on supersymmetric of supersymmetric back- the asymptotically on-shell locally action AdS of minimal dimensions [ and does not account for possible quantum anomalies. metry [ field theory analysis of [ for the currentsupersymmetry multiplet — operators to andsupersymmetry local promoting variations ones. of the the global fermionic Thebackgrounds background symmetries that Killing fields admit to — spinor zero a including equationslargely determine notion classified the obtained of for curved by a rigid setting number supersymmetry. the of off-shell Such supergravity backgrounds theories have and been for various spacetime a comprehensive review), supersymmetrichave quantum attracted considerable field interest. theories Atheory on systematic on curved procedure a curved backgrounds for background placing was aa proposed given supersymmetric in off-shell [ background supergravity, which corresponds to turning on arbitrary sources operators are turned on (i.e.global when symmetries), the or theory when issymmetries. coupled the to In theory particular, is global put supersymmetryindex on theorems anomalies and a are may related curved affect to backgroundand observables supersymmetric such admitting Wilson as Killing partition loop functions,admitting expectation the rigid Casimir values energy, supersymmetry. of supersymmetric theories on curved backgrounds functions as contact termslowest correlation that functions where violate the thefour-point functions classical involving two supercurrents Ward and identities. eitherand two one For stress example, tensorquantum the [ effective action and in one-point functions when arbitrary sources for the current JHEP04(2019)040 ]) ). 40 = 1 (1.1) (1.2) (1.3) 1.1 N -symmetry. R are the three = 1 conformal ˙ α α , N ) . The components G ˙ α α anomaly coefficients. ], where it was shown RR G c ˙ 47 α + 2 α 2 G and 1 2 G a )( = R 2 ]. They are usually discussed in G + , 47 ] 2
– E ∇ 40 36 a ] associated with the 5 J, + ( ]). − α ] seem especially related to our results in 2 2 ∇ 52 , 39 W W and the vector superfield = c 51 ˙ = α R – 3 – 2 α -current. The divergence of the stress tensor and E π J 1 ˙ R α ] and [ 24 and ∇ 44 = = 1 off-shell conformal supergravity in four dimen- , , is given by αβγ J 37 ] (see also [ N αβγ J W 50 W contain the trace of the stress tensor, the gamma-trace of the ], even though none of these earlier works concerns J αβγ 44 ] and in the conventions of [ , ]. Such anomalies involve the dynamical fields in the Lagrangian W 48 1 2 49 39 , = ]. We determine the algebra of local symmetry transformations and 37 2 16 W ], which provides a suitable background for superconformal theories via the 56 – 53 -symmetry and supersymmetry anomalies is identical to that relating gauge anomalies In this paper we consider Finally, we should mention that there is an extensive body of literature discussing The superspace analyses of [ The Ward identities and quantum anomalies of four dimensional superconformal the- R construction of [ derive the corresponding classical Wardlution identities. of The the Wess-Zumino main consistency resultsupergravity conditions of algebra [ the from which paper we islies obtain the to the so- general leading form non of trivial the order superconformal in anoma- the gravitino for arbitrary anomalies we identify infields. the However, the present mathematical paper,the structure underlying which the involve descent the equationsto background that supersymmetry relates anomalies supergravity [ sions [ These terms seem related to terms we find heresupersymmetry in anomalies the case in oftheories, the an reviewed presence anomalous in of [ description gauge of anomalies the in gauge supersymmetric theory gauge in flat space and are distinct from the supersymmetry best of ourthe knowledge literature these before fermionic andAnother components result so that have a may be not direct relatedthat to comparison been in the the with anomalies written presence we our find of explicitlyconditions here results anomalous is in Abelian is require [ flavor not additional symmetries the straightforward. — Wess-Zumino consistency non holomorphic — terms to the supertrace anomaly. this paper. In particular,supercurrent the we anomalies derive in arecocycles the likely divergence found related and in to in [ conformal the the supergravity gamma-trace fermionic and of components so the of the the field superspace content is somewhat different. Moreover, to the density. The chiral superfields superspace curvatures [ of the supertrace superfield supercurrent, and the divergence ofof the the supercurrent appear as components of the superspace conservation equation ( and are respectively the square of the superWeyl tensor and the chirally projected superEuler ories have been studiedsuperspace extensively language over and the can be years written [ compactly in the form (see e.g. appendix A of [ where the supertrace superfield JHEP04(2019)040 = 1 , an a µ ] and e N 60 – 57 = 1 confor- N ]. When applied ] (see [ 63 56 , – 62 53 , 59 , where we obtain the general anomaly coefficients are equal. = 1 conformal supergravity can 4 c N = 2 on-shell gauged supergravity in , which comprise 5+3 bosonic and 8 to derive the corresponding classical and µ we review relevant aspects of N 3 ψ a = 1 off-shell conformal supergravity in 2 -supersymmetry, is anomalous whenever we determine the anomalous transforma- Q 5 N = 1 supergravity algebra dictates that the = 1 conformal supergravity N N and collect our conventions and several gamma – 4 – 6 . In section = 1 conformal supergravity here is threefold. Firstly, B N . A ]). We find that the divergence of the supercurrent, which cor- 46 , and a Majorana gravitino = 1 conformal supergravity algebra. The actual calculation is shown µ ] in the special case when the A N 8 ] for pedagogical reviews). Its field content consists of the vielbein 61 ]. This means that the Wess-Zumino consistency conditions for ] (corrected in [ 64 40 The reason for focusing on The paper is organized as follows. In section = 1 conformal supergravity in four dimensions is induced on the conformal boundary -supersymmetry anomaly cannot be removed by a local counterterm without breaking dif- -symmetry is anomalous. Moreover, the to background supergravity, thison procedure local may relevant be couplings at thoughtN the of ultraviolet as superconformal the fixedof process five point. dimensional of Finally, anti turning off-shell dethe Sitter bulk space [ by minimal background conformal supergravity is relevantconformal for describing theories the and Ward their identitiescan of quantum be super- anomalies. obtained Moreover,and from gauge other conformal fixing supergravity supergravity via theories by the so coupling called it tensor to or compensator multiplet calculus multiplets [ four dimensions and determinefor its solving local the off-shell Wess-Zumino symmetrybe consistency algebra constructed conditions. as as a achapter preparatory gauge 16 step theory of [ ofAbelian the gauge superconformal field algebra [ fermionic off-shell degrees of freedom. 2 The local symmetry algebraIn of this section we review relevant aspects of in considerable detail in appendix tion of the supercurrentrigid under supersymmetry local algebra supersymmetry onmal and curved supergravity. discuss backgrounds We conclude the admitting in implications Killingmatrix section for identities spinors in the of appendix confor- off-shell conformal supergravity and determinemations. the These algebra transformations of are itsWard used local identities. in symmetry section The transfor- mainform of result the is superconformal presentedassociated anomalies in with by the section solving the Wess-Zumino consistency conditions R Q feomorphisms and/or local Lorentz transformations. Theformal Ward anomalies identities and we the obtain supercon- byholographically solving in the [ Wess-Zumino conditions reproduce those found identities corresponding to the divergence anding the their gamma-trace anomalies. of the To supercurrent, the includ- before, best at of least our explicitly. knowledge The theseresults bosonic have Ward [ not identities appeared and anomalies in reproduce theresponds well literature known to the Ward identity associated with Our analysis is carried out in components and we explicitly determine the fermionic Ward JHEP04(2019)040 ), x and ( (2.4) (2.5) (2.1) (2.2) (2.3) ab µ λ ψ . , σ , , ν ν ψ µ ρ ψ φ ]. Compared

D θψ µ µ 61 ν ceases to be an 5 A γ A 5 µ iγ η. 5
ε, .
φ

iγ iγ ρσ µ − µ a = 1 conformal super- A η ψ A + µν − 5 b 5 µ  µ γ µ 5 γ N iγ iγ µ D D iγ − ψ − + θ. ε + µ µ ≡ µ − = µ ] ∂ ), local frame rotations b ρ D σ ν D ρ D x δ ψ + ψ φ ε, ( ρ a + [ a µ 2 σ η ≡ γ ≡ ρ µν ρ µν δ γ = 1 conformal supergravity trans- 5 µ µ -supersymmetry, parameterized re- 4 Γ µ Γ ε η γ ψ S and follow those of [ µ   ψ N σψ 1 6 − − µ µ 1 2 ψ + 1 2 ν A ν i − A A b 4 3 ψ φ 5 5 ), is torsion-free. For most part of the subsequent − + ψ e =   ( a µ - and µ iγ iγ µ − µ µ γ  ab ψ Q ε A A a σe µ + − σ 5 ). The covariant derivative acts on the spinor 5 5 ab ω ψ ψ γ x – 5 – + ab ab γ ρ ( iγ iγ µ γ γ b µ The local symmetries of 1 4 η D ab ) ) φ e + − i λ b in order for its coefficient in the covariant deriva- 4 ρσ 3 a 1 4 ab ab ) + e, ψ e, ψ λ µ νµ γ γ e ( ( +  ) ) − ( A ) and − 5 ab ab λ λ x ab γ ξ µ µ λ ), Weyl transformations ξ ( ), as well as µ i µ 2 e, ψ e, ψ ξ → ω ω x ε µ x ( ( ω ∂ ( µ = 1 conformal supergravity as a gauge theory of the super- ( ∂ 1 4 1 4 λ µ ∂ − θ λ ab ab -supersymmetry are on the same footing before the curvature ≡ ξ λ a FT µ µ µ A + + ν ψ N S e ) ω ω A ψ µ µ + + 3 4 1 4 1 4 µ + ∂ ∂ µ µ   . We will see in the subsequent sections that this is indeed the case. µ a e, ψ − + + c ( ψ A e - and λ λ λ ≡ ≡ − D µ µ as ab = ∂ ∂ ∂ ∂ ∂ Q µ ε µ η λ λ λ η   µ a µ ω ψ ξ ξ ξ ν D D ) denotes the torsion-full spin connection ≡ = as well as in the fermionic superconformal anomalies. = = = D ν ν and µ  µ µ µ a φ ψ ν e, ψ ε A µ µ ( δe γ δψ δA D 1 3 D ab ) to be unity, as is standard in the field theory literature. according to µ µ ≡ ω 2.1 ], we use Lorentzian signature instead of Euclidean and we have rescaled the gauge A µ ] for the case Under these local transformations the fields of Our spinor conventions are given in appendix In the construction of 60 φ The purely bosonic part of the spin connection, 8 2 . The covariant derivative acts on these gauge fields as µ analysis we will work tospin leading connection non suffices. trivial order However,symmetry the in algebra. full the gravitino, spin in connection which is case necessary the in torsion-free order part to of determine the e.g. the local form as parameters tives ( Local symmetry transformations. gravity are diffeomorphisms U(1) gauge transformations spectively by the local spinors As we will seegauge shortly, this field quantity appears in the supersymmetry transformationto of [ the field Once the curvature constraintsindependent are field imposed, and however, it the is gauge expressed field locally in terms of the physical fields as where in [ conformal algebra, constraints are imposed, each havingφ its own independent gauge field, respectively mal supergravity should reproduce the superconformal anomalies obtained holographically JHEP04(2019)040 , . µ ν θφ A (2.6) (2.7) (2.8) (2.9) η. ) 5 (2.10) (2.11) (2.12) 5 ε η, iγ ν 5 γ ]. iθγ + 0 8 ε εγ η + ( i µ 4 η 2 3 1 D θ, ab − − µ θ, + γ ∂ ε = = µ ab ν → ξ λ γ 1 4 = /`  , θ − b µν denote any of the local a e F ν 0 there ση 1 3 θ ω 1 2 ) ) determine the algebra of 3 − ε − η, θ 5 ν √ b σ, δθη 2.5 γ γ a 0 µ µ ε µν ∂ ∂ ) transforms as = 2 gauged supergravity in the , εγ ( µ µ F  2 1 2 1 ξ ξ i ] one should take into account that 2.3 3 N 2 8 η, − − , µν . = = µ + ρσ = = in ( , → Rg A , F b b µ µν 1 6 µ b a a ξ c P /` φ ρσ → 0 ], where Ω and Ω ν  0 − λ , δσ ∂ µν Ω c b 1 2 ν ]. Moreover, we use the Majorana formulation /` 0  a a µν 8 there − ξ , δ 1 2 λ + λ – 6 – R Ω µ µ ) coincide with those induced on the boundary − δ  there µ − ∂ ≡ µ 1 2 µ b ε, λ A 0 σφ 2.5 ε, δη c ξ µ 3 ξ µν 1 2 5 λ ≡ ν γ e c ε,  F = 0 √ ∂ − a εη,ε λ b 0 ) repeatedly we then find that (to leading order in the ν iθγ µ µν a 1 2 1 2 ξ λ → φ P − 2.5 The local transformations ( ε = = = = ab /` γ b ab µ µ a 00 γ ab 00 there + λ ab , δλ 1 4 λ µ . Off-shell closure of the algebra requires that the parameters 0 1 4 , σ , ξ ξ − θ θ ν λ δ − δ ∂ ξ ν µ + + σε σ,  ξ ∂ λ 2 1 λ λ − δ δ → φ µ + , λ , ξ ξ + ε + + 00 00 ν /` µ ξ λ σ ξ µ σ, ξ, λ, θ, ε, η ∂ ∂ δ δ δ δ φ ν ]. In order to compare with the results of [ 0 µ λ ξ ξ there ∂ ] = ] = ] = ] = σ 64 λ = = 0 0 0 , η ξ ε ξ λ 8 µ = 1 conformal supergravity here instead of the Weyl formulation used in [ , δ δε Notice that the transformations ( = , δ , δ , δ ε δξ ε ξ λ µ δ N δ δ [ -supersymmetry transformations are field dependent, which means that the structure δ [ [ [ δφ Notice thatQ the composite parameters resulting from the commutator of two parameters are: Applying the transformations ( gravitino) the only non vanishing commutators and the corresponding composite symmetry Local symmetry algebra. local symmetries, i.e. theparameters commutators [ transform under the local symmetries as and where “there” refers to the variables usedof in [ of five dimensional antibulk [ de Sitter spacewe have by rescaled minimal the gauge field and the local symmetry parameters according to is the Schouten tensor in four dimensions and the dual fieldstrength is defined as where These transformations imply that the quantity JHEP04(2019)040 ] ] ) ). soft 2.5 (3.2) (3.3) (3.4) (3.1) 2.12 e, A, ψ e, A, ψ [ [ W W ] leads to a ) and e, A, ψ [ , 2.5 W µ W ψ δ δ 1 − e = , , ] µ ] ], with the microscopic fields Φ S e,A,ψ , ) and the local symmetry transforma- e, A, ψ [Φ; [ , e, A, ψ iS µ 3.2 e Z W [Φ; ] = 0 δ δA S 1 ), classical invariance of log [dΦ] − i e 2.5 − Z – 7 – e, A, ψ = [ µ ] = ] = W J Ω δ e, A, ψ e, A, ψ [ [ , a µ W Z W δe δ 1 = 1 conformal supergravity. = 1 conformal supergravity. These identities can be thought of − ) denotes any of the local transformations ( e N N = µ a T ). These definitions do not rely on a Lagrangian description of the quan- ] for a recent discussion of soft algebras and their BRST cohomology) and ] is obtained from the path integral a µ e 65 ξ, σ, λ, θ, ε, η det( e, A, ψ ] corresponds to the classical action [ (see [ ≡ Z e Given the definition of the current operators ( The classical Ward identities can be expressed in the compact form e, A, ψ [ W evaluated on-shell. tions of the background supergravity fieldsconservation ( law — or Ward identity — for each local symmetry, which we will now derive. where over the microscopic fields Φ. At the classical level, therefore, the generating function where tum field theory, but ifis such expressed a as description exists, then the generating functional associated with the background supergravity fields, namely where Ω = ( is the generating functional of connected correlation functions of local current operators Since these depend only on theidentities structure of are the independent background supergravity, of the resulting thethe Ward specific theory field to theory background Lagrangian, supergravityclassical provided preserves level. the the coupling local of supergravity symmetries at the 3 Classical Ward identities We now turn to the derivation ofcoupled the classical to Ward identities background of a localas quantum Noether’s field theory conservation laws following from the local symmetry transformations ( algebras supergravity theories are typically based on softform algebras. the The basis commutation for relations the ( conformal Wess-Zumino anomalies consistency condition in analysis to determine the super- constants of the gauge algebra are field dependent. Such algebras are often termed JHEP04(2019)040  (3.7) (3.8) (3.9) (3.5) (3.6) µ (3.10) S ab γ µ gauge trans- ψ . R νab ω = 0 ν µ ξ  J 1 4 ν µ
) S .

J + ν µ µ ab ξ µ S νµ γ ν S   µ S ν F µ A µ 5 ψ . ψ ( ψ γ S +
σ θ µ 1 4 µ δ µ δ ∂ ab µ , ψ S γ +
S + ν + 5 + µ )  µ ] ν γ µ b . A ν µ µ ψ S ν µ ν . T νµ 5 S 1 4 J J D a ←− ψ [ F γ µ J iξ µ µ i = 0 µ ν ν µ + ψ = 0 e A A ξ ψ µν µ − ψ − ] 1 2 σ θ µ  µ F b S δ δ µ µ iθ ) gives the classical diffeomorphism Ward T 5
S + + S ab − a x γ µ + + µ J [ ) + ( µ µ − µ ( µ µ µ ν µ a a S µ ψ µ µ ν ω µ e a a D T T ψ µ S ←− ξ 1 2 S i T T ) b µ – 8 – a µ µ  J ν − ψ a µ a µ ν e e ξ θ − ∇ + b + e e ψ
D ab ←− δ ψ µ  ( a θ ν σ µ ν ν ν µ δ ∂ a ν µ δ S ξ + ω ψ T ω J 5 a µ µ ν µ γ + − e ξ + ( x e x e x e θ ν x e σ x e − ∇ J ∇
) 4 4 4 4 4 µ µ ψ ν µ − µ d d d d d i a S A S S µ T ) a ξ 5 µ + Z Z Z µ Z Z µ δ T γ ψ ) ν ∇ µ ( D = = = a ν = = ←− + a ν ν J e ψ ν e µ i ν ν a ∇ ψ W W ξ T ∇ − θ ( + σ The transformation of the generating functional under diffeomor- a µ + δ − -symmetry Ward identity takes the form µ δ µ e Under local Weyl rescalings the generating function transforms as  µ ν ν gauge transformations and local frame rotations respectively. ξ R a ν ∇ J A δ T D ←−  µ R ν The transformation of the generating function under U(1) µ + ∇ ∇ ψ x e x e ξ x e a µ ( 4 4 4 ν e ν d d d ξ A Z Z Z + − = = = W ξ δ -symmetry. Hence, the classical formations is given by and, hence, the classical trace Ward identity takes the form R Weyl symmetry. We will see shortlyidentities for that U(1) the terms in the second line correspond to the classical Ward Setting this quantity toidentity zero for arbitrary phisms is given by Diffeomorphisms. JHEP04(2019)040 - S (4.2) (4.1) (3.16) (3.12) (3.13) (3.14) (3.15) (3.11)  µ . , S
 µ

S µ . µ µ D ←− A S S S  ε , µ η µ µ µ  + ψ
ψ  S + µ λ . η µ µ µ ba δ − D S δ µ Q S S γ µ J µ µ µ A J + µ + ε . γ ε µ 5 ψ . ηγ J ψ µ µ ε φ γ µ 1 4 + δ − 5 J + µ J φ = 0 γ µ = 0 µ µ 5 φ R may not be invariant under all local µ + + i µ i γ A 4 A µ J 4 ] 3 A µ 3 µ i J S η λ a µ . 4 θ 3 η J δ δ W J T ab ψ + 5 + µ b µ 5 − [ γ γ + + + µ ψ µ µ 6= 0 γ a µ A µ a 5 e µ µ µ i ε W T a a a T ψ 4 γ ψ 3 δ -supersymmetry is µ  ε T T T i i W – 9 – 1 4 A 4 4 a S ψ a µ a µ µ 3 3 − Ω σ + ab γ a e e δ ψ +  η − µ λ µ γ a − a ] δ δ µ g η 2 1 ψ  γ b T µ T x eλ − 1 2 a µ 1 2 S 4 a e = [ µ √  x e x e x e ε − x e d µ 4 4 4 µ γ δ 4 x ε e  4 d d d d S Z d µ -supersymmetry transformation of the generating function is Z Z Z − Z x e x e x e D Z 4 4 4 Q Under local frame rotations the generating function transforms d d d ======-supersymmetry Ward identity takes the from Z Z Z W W Q The Finally, the transformation of the generating function under η W λ = = = δ δ Ω δ W ε δ conditions -supersymmetry. -supersymmetry. The non-invariance of the generatingeterized function as of four dimensional theories can be param- 4 Superconformal anomalies from the Wess-Zumino consistency At the quantum levelsymmetries of the background generating conformal supergravity, function i.e. and hence the classical Ward identity for supersymmetry is given by Therefore, the classical S Hence, the corresponding classical Ward identity is Q according to Local frame rotations. JHEP04(2019)040 , ) ) 3 = 0 4.2 ψ (4.4) (4.5) (4.3) , ( α ) 3 O ψ + ( σ O -symmetry, ). ψ ] associated 0 ρ 5 R + γ , η σ 0 ) in the scheme ψ ρ , ε 0 ] and exist only in and read µνρσ D e σ 2.12 , θ R 66 0  ] ψ W µν ρ , λ ρσ 0 F D γ ) 5 2 ν c , σ [ γ 0 , π , , µ .
ξ − ] g W R Q S ρ ν a 24 ( δ A A A A µν = ( σ [ µ coefficients of the Weyl anomaly − 0 = = = = δ ) Rg c σ µ µ µ µ 1 2 − , ] ψ S S ρ ν J J ν − 5 ). µ ( W δ µ µ and ] γ γ 0 σ ψ φ [ µν ψ
µ a Ω ) and Ω 5 5 1 2 µ , γ ψ 2.12 γ γ γ i [Ω i i + δ 4 4 3 3 ρσµν + µνρσ µ µν e a R = µ − − R T ρ F a µ 2 µ J µ  , A – 10 – a e W ) ) µ π ] ic S T 0 2 c 6 2 µ µ µ ∇ Ω ψ π γ − ξ, σ, λ, θ, ε, η ψ ∇ + ( 8 , δ a a ) ν ( O . Moreover, the mixed anomaly can be moved entirely c γ Ω 2 ψ δ 1 2 π + [ + −  = 1 conformal supergravity algebra ( 6 σ a 5 ,... E − ( γ ψ 1 2 are possible quantum anomalies under Weyl, N ρ µ µ , , + π a S D ν A S ] P -current by a choice of local counterterms (that is setting µ i 16 φ ) 3 = 0 A 2 5 2 R ρσ D a − γ k γ π ). − ), but the remaining Ward identities become − µ ) is realized when successive infinitesimal local symmetry variations by solving the Wess-Zumino consistency conditions [ ν  [ µ c 2 24 µ A and ( 4.2 S ) for the D g F 3.12 µν A 2.12 +  Q 3 8 we determine general non trivial solution of the Wess-Zumino consis- ]). In this scheme diffeomorphisms remain a symmetry at the quantum µν e 4.4 F A µν P − i B 67 e ) ) 2 e FF , F c c and R ) ) 2 3 2 W c c − ) and ( π A Q 2 = 1 conformal supergravity algebra ( π 3 3 − a 2  9 4 π , a A 2 π − − 3.6 -supersymmetry transformations, respectively. Recall that the gravitational and N 6 π , a a W c 27 3(2 (5 S R A (5 16 (5 − + A In appendix The Wess-Zumino consistency conditions amount to the requirement that the local Repeating the exercise of the previous section with the anomalous transformation ( = = = = , (also known as Ward operators) act on the generating functional + 2 dimensions, with S R Q - and W W k Ω A A A A tency conditions ( where diffeomorphisms and localtwo Lorentz non trivial transformations solutions are related respectively nonand to they anomalous. the take There the form are symmetry algebra ( δ for any pair of local symmetries Ω = ( The objective of thisA section is towith determine the the general form of the quantum anomalies level and the transformation ofparameterized the as in generating ( function under all local symmetriesresults can in be the same diffeomorphismtively in and ( Lorentz Ward identities as those obtained respec- Q Lorentz (frame rotation) anomalies are related4 by a local counterterm [ to the conservation of the in eq. (2.43) of [ where JHEP04(2019)040 ] 34 (4.6) (4.7) (4.8) (4.9) ), are (4.12) (4.10) (4.11) 4.5 ones. The ], according radius. Of ). It follows fieldstrength that removes 40 5 R ct 2.12 is the Pontryagin W , P 2 consistent R 1 3 , . ) ) + . is the AdS ct v = 2 gauged supergravity W ρσ N ` µν F + R N ) = 0. Closure of the algebra + 3 . µν ), are the , µν ct F W χ R W . )( N 3.2 2 θ ( µνρσ , ) = 0 + δ µνρσ − 2 1 R  ct 24 κλρσ + , ) and we have introduced the shorthand R 1 2 W W 5 R 3 λ ( = ]), but the fermionic Ward identities and + ε δ µνρσ ≡ µνρσ 2.7 G + δ κλ π` 8 e 46 R R + e is the Euler density and µν F µν W ξ  R = = δ E ] for theories with a holographic dual that have )( 1 2 c µνρσ 8 θ µν , c – 11 – δ ρσ ) R ) derived above, including the anomalies ( ≡ R = v 4 is not symmetric under exchange of the first and second pair µν + ) = ( ,F a N = 4.3 ct λ R − δ µν µνρσ W µνρσ + 9 F e + R e µνρσ R + κλρσ χ ξ µνρσ µν δ R N ] (corrected in [ W F ) and ( ( R ( W ]( 40 1 ≡ 0 48 ε κλµν µνρσ 2 3.12 µνρσ  , δ = F 1 2 R W ε ), ( a δ [ anomaly coefficients for free chiral and vector multiplets are given = ≡ ]). = 1 conformal supergravity on the four dimensional boundary of 3.6 2 c P ≡ E 68 , ]. Secondly, the bosonic Ward identities and anomalies we obtain re- W N 66 and 46 is the Schouten tensor defined in ( , a µν 10 is the Newton constant in five dimensions and , is the square of the Weyl tensor, P 3 9 ] and the anomalies can be computed through holographic renormalization [ 2 5 ), as well as the current operators defined in ( G 64 W ) as [ -supersymmetry anomaly, i.e. such that 4.5 An interesting question is whether there exists a local counterterm Several comments are in order here. Firstly, we should point out that the anoma- 5 Q 2.8 In contrast to the Riemann tensor, 3 with the composite parametersthat for if the such bosonic a transformations counterterm given exists, in then ( it must also satisfy of indices. (see also [ the requires that where course, such aninduces agreement off-shell wasAdS expected since minimal produce well known results [ anomalies have not appearedthe — Ward identities at ( least explicitlyin — complete agreement in with the the results literature of before. [ Moreover, lies ( corresponding covariant quantities canZumino be terms obtained [ by adding the appropriate Bardeen- notation Finally, we have used theto normalization which of the therespectively central by charges adopted in [ where the dual Riemannin ( tensor is defined in analogy with the dualMoreover, U(1) density. Their expressions in terms of the Riemann tensor are where JHEP04(2019)040 ). 3.2 (5.1) (5.2) (5.3) - and ]. The Q 69  κ -anomaly to A
, Q η σκ ]  e σ F µν . ) δ c γ c δA σ -symmetry anomaly, 3 [ 2 = ε ρ R π − a g  a ) ρ 27 - and ρσ η D ε, P 5 4(5 ν Q µν γ γ ν e + F ν ( D σ γ . ν )
 ρσµν J c σ D η e ρ 2 )  R − ν c π ε  a ] µ ρσ 3 4 2   F ρ π 5 µν − ) 3(2 6 γ a 2 D c iγ σ 5 ] this approach is used to obtain the anomalous π − [ (5 − 8 ) and the definition of the currents in ( γ ρ ) + a ν g η 24 ] + ( γ ρ σ 2.5 ρσ ρσ
 δ + ν σ -supersymmetry in the case µ W F W [ ν ε ρ S η ( ε ) Rg – 12 – δ A ν δ δ ν λ 4 1 2 µ γ µ µν µ  D -symmetry anomaly, or it breaks diffeomorphisms κ − i δ e 5 δ 8 5 ψ µ ψ ( F - and R δ δ γ δ . ) ρσ iγ ) lead to anomalous transformations for the current

1 c + 1 Q µ ] γ 3 − + ν − 2 σ δ a µ ] e e 4.5 δ π δ − ρ σ δA µ σλρκ δe δ [ + ρ a + -supersymmetry transformations of the supercurrent are η µνρσ 27 e δ µ ε 5 R [ ν S a R W σ δ γ W − 4(5 γ ] i 4 A 4 ν ! σ ! 1 2 3  + δ i µ µ 8 ρ ν µ ρ δ δ = = ψ ψ µ D ∇ + [ J - and δ δ ) ) µ γ   a  c c Q 2 2

]. When restricted to rigid symmetries of a specific background, the µ µ T η 2 ε η 8 π π − − removes also the ε δ δ 5 ψ ψ δ δ 8 6 π a a a ic δ δ ) imply that the functional derivative with respect to the gravitino trans- 1 1 γ 6 ( ( γ ct i − −   4 1 2 3 e e − − + W ε 2.5 η δ δ = = = = contribute to the transformation of the supercurrent under respectively µ µ S S S -supersymmetry or diffeomorphisms/Lorentz transformations are anomalous as ε η 4 A δ δ Q -supersymmetry The classical (non-anomalous) part of the current transformations can be deduced di- S and An equivalent but more formal way to determine how the currents transform under the local symme- 4 Q -supersymmetry [ transformation of the supercurrent under tries is to utilizeWard the identities symplectic correspond structure to underlying firstmations class the under constraints space the on of Poisson this couplings bracket. space, and In generating local the appendix operators local B.1 [ symmetry of transfor- [ It follows thatgiven the by forms according to S anomalous terms in the transformations of the currents resultrectly in from a the deformed supergravity transformations superalgebra. ( For example, ( The superconformal anomaliesoperators ( under the corresponding localA symmetries. In particular, the fermionic anomalies and/or local frame rotations.either This means that forwell. theories with an The identificationdiffeomorphisms/Lorentz transformations of is a a problem possible we hope local to counterterm address5 in that future moves work. the Anomalous supercurrent transformation under Hence, either JHEP04(2019)040 ]. 33 . In (5.4) (5.5) ]. b ) and o 9 , , η 8 o ε -anomaly by Q supersymmetry ] in the special 8 ) are essentially a 5.3 rigid -symmetry anomalies R ]. They also determine , 6 µ are restricted to solutions S ) and ( η ) o η 5.2 δ 5 ). , and + ε -charge, can be computed via su- o 2.11 , the corresponding transformation ε = 0 R ) are evaluated on the specific back- δ µ o ], the anomalous terms in the trans- A η ) and are useful for e.g. determining the 5.3 µ ], where supersymmetric gauge theories = ( ) are Grassmann valued local parameters that γ 19 } 4.3 70 , ε, η and µ − S 18 o , ε µν ] ) and ( g µ o – 13 – D , η 5.2 o = ε [ = 1 conformal supergravity backgrounds admitting µ -charge [ Q ψ { R o N ] resolve the puzzle by arguing that in order to make the ,η = o ε supersymmetry is given by 70 µ δ S ) o ,η rigid o ε ( -number parameters, while ( δ c ) are ] it was shown that even though the Weyl and o , η 9 , ] is the conserved supercharge associated with the Killing spinor ( o ε o 8 , η o . ] that used the classical superalgebra and the holographic computation of [ c ε [ 32 = Q ) of the Killing spinor equation – o a Although it may not be desirable — or even possible — to eliminate the The anomalous transformations of the supercurrent in ( Note that ( 30 , η 5 o ε implies that the partition functionThe does puzzle, indeed depend however, on is themerically that metric zero. for at The Seifert the authors quantum manifolds of level. specifically [ the framing anomalytransform is non nu- trivially under the local symmetries according to ( in three dimensions withtopological both gravity Maxwell were and considered.manifolds, Chern-Simons which terms admit The coupled two to partitionpersymmetric supercharges background function localization of opposite and of depends suchprinciple, this explicitly theories dependence on on could the be Seifert Seifert explained by structure the modulus presence of a framing anomaly, which a local counterterm thatdiscussed breaks in diffeomorphisms the and/or previous local section,may Lorentz it be transformations is removed as plausible by thatpreserves a the the local anomaly underlying counterterm in that structuresomewhat breaks of analogous certain the situation (large) was supersymmetric discussed diffeomorphisms, background. in but [ An example of a observation was the key toof resolving [ the apparent tensionBesides between the the dependence field of theory theever, results supersymmetric the partition transformation function of on theThe the background, anomalous supercurrent how- transformation determines of also the the supercurrent spectrum results of in BPS a states. shifted spectrum [ ground. In [ are numerically zero fortwo a real class supercharges of offormations opposite of thedeformed supercurrent rigid under superalgebra rigid on supersymmetry such backgrounds. do not As vanish, reviewed in leading the to Introduction, a this of the supercurrent under where all bosonic fields in the transformations ( supergravity. Namely, when the local( spinor parameters on a fixed bosonic background specified by case rewriting of the two fermionic Ward identitieseffect in ( of the superconformalthe anomalies rigid in superalgebra correlation on functions [ curved backgrounds that admit Killing spinors of conformal Notice that these transformations coincide with those in eq. (5.9) of [ JHEP04(2019)040 , such a 4 Seifert-topological (i.e. -symmetry is anomalous. R and = 1 off-shell conformal super- ]. We hope to return to these N 75 – 71 -supersymmetry is anomalous in any Q using the general form of the fermionic c and a – 14 – black holes [ 5 ]. It would be interesting to generalize this counterterm 33 -symmetry as well. This expectation is supported by the -supersymmetry anomaly. As we saw in section R ) defined on trivial circle fibrations over Seifert manifolds such Q -symmetry does not seem to depend on the specific supergravity N R supersymmetry on backgrounds that admit Killing spinors is rel- -supersymmetry, is anomalous whenever ]. Another interesting question is whether there exists a local coun- , which resolves the paradox. rigid 7 b Q at large c = a Several open questions remain. Our result that It is plausible that similarly the rigid supersymmetry anomaly in four dimensions can Cyril Closset, Camillo Imbimbo, Heeyeon Kim, ZoharMurthy Komargodski, and Dario Dario Martelli, Sameer Rosagrateful for illuminating to discussions the and UniversityCenter email of for correspondence. Theoretical Southampton, Physics I King’s in Trieste am College forthe hospitality also London completion and of partial and financial this the support work. during International the microstates of supersymmetricquestions AdS in future work. Acknowledgments I would like to thank Benjamin Assel, Roberto Auzzi, Loriano Bonora, Davide Cassani, terterm that eliminates the counterterm would necessarily break diffeomorphismsrelated and/or question local for Lorentz rotations.evant The for the validity ofimportant supersymmetric example localization is computations the on computation four-manifolds. of An generalized supersymmetric indices that count theory with an anomalous theory we used in thistheories paper. with Indeed, an we anomalous expectrecent this analysis result of to [ hold in non-superconformal edge, the explicitappeared form in of the literature theis before. fermionic associated We Ward find with identities thatThis the anomaly and cannot divergence be their of removed the anomalies byand/or a supercurrent, local have local which Lorentz counterterm not without transformations. breaking diffeomorphisms 6 Concluding remarks In this paper we determined thegravity local in symmetry four algebra dimensions of andby obtained solving the the general associated form of Wess-Zumino consistency the superconformal conditions. anomalies To the best of our knowl- compatible with the structure oftheories the (i.e. supersymmetric background. Ina local fact, counterterm for was holographic found into [ non holographic theories withanomalies arbitrary we obtained in this paper. independent of metric deformationsa that local preserve but the non Seifertstructure fully structure) modulus covariant they counterterm. need This to counterterm add does depend onbe the Seifert removed by a local counterterm that breaks those large diffeomorphisms that are not quantum theory invariant under Seifert reparameterizations JHEP04(2019)040 = 1. (A.5) (A.1) (A.2) (A.3) (A.4) , ] σ 0123 ] ]: ε ρ , δ ] 61 τ ρ . ] ν ν ]] δ δ . Moreover, the λ [ σ 61 [ µ [ µ [ ) δ δ µνρσ 1 satisfies µν . 2)  g + 6 e ε ]] , σ ] ] − σ τ = γ 61 = ] ] d ρ σ , − δ , δ ] τ λ σ µν ν σ [ ν ] µνρσ µνρσ δ + 2( γ ρ ρ δ ε [eq. (3.53) in [ µ ] σ δ δ λ , ρ , [ γ ] γ ] ] g ε τ [ ρ µν µ , ν n ] [ [ ν ]. In particular, the tangent space − 3) σ δ µ ν , ] γ γ δ ] 3) ( σ δ γ ν [ √ σ µ 61 − is a Majorana spinor (see section 3.3 [ σ δ µ [ λ, − γ , [ γ µ d 1 , µ = ρ [ 3 µ [eq. (3.82) in [ ··· [ ] γ [ µ ( d µ + 12 + 18 γ . , p γ δ ψ δ 2 , ] ] γ ν 2 p µ µ 3) ν ] µ − + 6 τ σ 2 δ 2) γ p 4)( ] δ 2) ] γ ] µ 1 1 ν ρ σ µνρσ − ...µ + 2 1 , σ − γ γ 1) γ δ −  2 ] δ ...ν − − ] µ r p d [ 0 ρ 2 µ σ λ µ ρ λ, d [ ] d d ν 1 δ τλ γ − δ r [ ν iγ ··· ...µ ...ν µ τ γ – 15 – [ 1 1 d δ 1 2) p γ ≡ ν µνρ µν µ ...µ + ( ν + 4( σ ) [ [ [ = µ [ [ µ µν 1 + 2( γ + 2( [ p n − γ γ ν µ 5 µ + ( [ , , 2 γ pγ σ ρσ pδ σ ] )! χγ γ d } γ ν ρσ ...µ µ s p χγ − 2 µν µν µ )! + µ + 8 + 9 + γ νρ µ νρσ r + 6 1) γ γ γ 2 3)( γ − d p µ 1 = + 4 ( τλ p , γ µ − r 3) 3) 4) 4) 2) − , γ p − ∗ στ στλ ...ν µ γ µ ) ρσ 1 d ...ν d − γ 1) − − − − − ν γ λ ( 1 = ( { [ µν ν µνρ µνρσ µνρ r µ d d d d d − d 1) and the Levi-Civita symbol ( 1 2 1 2 γ γ γ γ γ γ χ + ( , p ...µ 1 1 µ = ( = ( = ( = ( = ( ======( = = , µ γ 1 1 ρ p µ , ρν τν ρσ ρσ τλ στ γ γ dimensions, most of which can be found in section 3 of [ χγ 1 µνρ νρσ λρσ γ στλ γ γ γ γ ...ν p γ ...ν p ··· ( µνρσ s 1 γ γ γ γ d − 2 ν ν µρ µν γ ...ν ...µ µ γ ρστ µνρ µνρ γ µν γ 1 γ 2 s γ µνρσ µνρ ν µνλ µ γ γ γ γ µ 1 1 γ γ γ γ γ µ ...ν µρ µ 1 γ γ ν λγ ρ r = diag ( γ ...µ η 1 µ γ ] for the definition) and we make extensive use of the spinor bilinear identity in four It is convenient to collect several identities involving antisymmetrized products of In the conventions we use here the gravitino 61 gamma matrices in of [ dimensions For Majorana fermions we also have that and we define the antisymmetrized products of gamma matrices as where antisymmetrization is done with weight one. Throughout this paper wemetric follow the is conventions of [ The Levi-Civita tensor is definedchirality matrix as in usual four as dimensions is given by A Spinor conventions and identities JHEP04(2019)040 ) ]: 8 4.5 (B.1) (A.7) (A.6) . µ ψ ,
] = 0 ρ ] . κ
= 4 and are used ] W δ ] ρ σ ] η [ κ d ρµσ ν [ δ , δ , γ δ σ ) λ [ δ µ ν + [ + [ γ ] δ µ , ρ µρσ 5 ] γ ρσ γ − γ κ µσ g ] δ ρσ σ g , ρ ρ σ g γ , ] − [ ] νκµ κ γ ν ρ γ [ δ − κλ = 0 γ νρσ 3 γ σ − ρ [ γ κλ µ ) associated with the local symmetry . ν γ ν µσ − µ [ [ 5 ρσ W g γ i  , µ 12 ] γ γ g µσ 5 4.4 ε g µ g + − − γ − µρσ + ρσ , δ γ ] ] ν γ γ σ + λ λ λ γ + 3 µ , ρσ ρσ γ δ + γ γ σ [ γ γ κ κ γ νκ µρ ) with the corresponding results obtained in [ [ [ µνρσ ρµσ νκ νκ g γ µ µ – 16 – µρσν  µρσ µρ γ µνκλ γ γ δ δ i 2 4.5 g γ i γ , = 2( = 2 = = = = = 2 = 4 = 4 = 4 = 4 = ρ ρ σ σ = 0 µν γ γ γ γ µρσ νκλ νκλ µρσ γ µ µ µ γ µρ γ W γ γ γ γ γ ] ) ) γ µ σ ρ σ η γ γ γ γ µν , δ g νκµ ξ + − 2 γ δ [ σ σ 3 − γ γ µ µ − ] and are straightforward to check. We shall therefore focus on the µν γ γ µ ρ ρ γ , γ ( 76 γ γ = 1 conformal supergravity. Only a subset of the algebra relations need ) and the anomalies ( νκ = 0 γ N 4.3 (2 W ] ) of ε , δ ξ 2.12 δ [ In particular, the following identities apply specifically to which hold trivially. All computationsmanifold in this or appendix that assume eitherdropped. the a compact Moreover, fields spacetime we go only keep to the zero leading at non trivial infinity terms so in the that gravitino total derivative terms can be are trivially satisfied. Moreover, the Wess-Zuminoare conditions for known purely to bosonic hold symmetries commutation [ relations involving atfor least the one four fermionic commutators symmetry transformation, except In this appendix we provide thesatisfy details of the the Wess-Zumino proof consistency that thealgebra conditions superconformal ( ( anomalies ( be checked explicitly since all commutators between any two non-anomalous symmetries B Solving the Wess-Zumino consistency conditions Finally, the following three identitiesWard in identities ( four dimensions help compare the superconformal extensively: JHEP04(2019)040 , . ) ) ), ν σ the ψ ψ c , σ (B.6) (B.7) (B.8) (B.2) (B.3) (B.4) (B.5) λ 2.11 ( ( ) σν and σ ρν εγ εγ g
ε
Γ ρµν a δ ε σ ρ κλ µν δ µ R ∇ ρσ κλ ∇ R κλ σκλ θ R ρ ρσ σκλ ρ R ρ ∂ R κλµν ( ε R κ δ ]. For generic θ  8 -supersymmetry trans- µ ∇ κλµν ∂ ) determine S κλµν κλµν θ  ρ µ 4.5 µνρσ ∇ 6 . - or . x e θ  Q x e θ  4 x e ∂ x e x e  . 4 4 4 4 d ) ensure that the Weyl anomaly = 0 d d d d . ) = 0 Z = 0 4.5 Z W W Z Z Z W -supersymmetry anomalies are Weyl ) ] ] 2 ) ) ) ) a η η A S 2 2 2 2 W a π a a a , η π − e π π π , δ , δ δ − ( − − − c 12 σ σ ( c σ η c c c 48 12 12 12 δ δ = 0 ( [ [ ( ( ( + - and − + + σ W )+ Q σ σ σ ] A θ φ φ φ ε ρσ – 17 – , , δ , δ , 5 5 5 δ -supersymmetry, i.e. , δ F ρ ε S W εγ εγ εγ ∂ δ ] Let us first consider the two commutation relations µν = 0 = 0 [ ε F µν µν µν µν ) = 0 ( , δ F F F F ε W W W σ ] δ ε - and ε δ δ [ A σ Q , δ = 0. µνρσ µνρσ µνρσ µνρσ e δ σ ( µνρσ ε δ θ  θ  θ  [ W δ ρ ρ ρ ] ) terms in the Weyl anomaly can be found in [ -supersymmetry, the anomalies in ( 2 η x e θ  Q ψ 4 x e θ  ( , δ R Q x e ∂ x e ∂ x e ∂ 4 d O 4 4 4 σ d A A d d d δ Z ε ) terms in the Weyl anomaly in ( Z the ) 2 Z Z Z c ) c x e x e θ ψ i i i The only remaining commutation relations involving a bosonic symmetry 3 c 2 ( 4 4 ) ) ) 2 3 = π c c c d − d 2 2 2 π O 3 3 3 a − a π π π 27 a Z Z ) can be used to derive the fermionic terms in the Weyl anomaly. 54 − − − θ ε 18 18 18 a a a 2(5 (5 δ δ = 0 and [ = 0. B.4 (5 (5 (5 − − − − is invariant under both W W ======] ] ε θ W W For the case , δ ε θ , δ 6 δ δ ε σ θ ε δ δ δ δ Hence, as required by the Wess-Zumino conditions. conditions ( [ are those between localformations. gauge Staring transformations with and either Combining these results we concludeconsistency that these conditions two compatible commutators with satisfy the the local Wess-Zumino symmetry algebra, namely Moreover, the density Taking into account the Weylit transformation is of straightforward the to supersymmetry checkinvariant, parameters i.e. that in both ( the [ JHEP04(2019)040 ) σ ψ 0 ε (B.9) δ (B.14) (B.13) (B.10) (B.11) (B.12) λ ( εγ
0 ε µν λ κλ γ ) R )  ρ ε ε . λ τ ) κτ A , , ε γ γ F 0 0 κ σ σ ε κ ε ε σ γ λ κτ ( ψ ψ ) ( 0 ∇ 5 5 A σ γ ε 0 ρσ σλ η ( ε ∇  δ λ ηγ ηγ F κ ρ κλστ µνρσ σλ 1 ∇ ∂ 12 µν µν µν R ∇ F F F F µν ρ µν − ( µν µν xe 5 F η A -supersymmetry transformations κλ 4 δ γ κλ κλ d µνρσ µνρσ R Q µν R R σλ Z F ρ ρ µνρσ ρσ ) ) F θ  θ  µνρσ 0 2 a F A A ρ ρ i 3 ε . π ) − µνρσ σ + There remain only the four commutators c 12 , ( λ 0 µνρσ µνρσ µνρσ D = 0 x e θ  σλ x e ∂ x e ∂ ε λ ψ 4 x e θ  4 4 . ( + 0 λ P R x e  4 d ε d d λ W σ 4 1 2 d δ D ] εγ S A x e  x e  x e  φ d θ κ κ ( A 0 Z 4 4 4  Z Z – 18 – ε A κ ) Z d d d 5 ) , δ δ i i εγ εγ η ε Z c η ) ) ) ∇ 5 λ ) δ c 3 c c x e θ εγ µν µν Z Z Z [ 2 c 2 2 4 3 3 3 γ µν ρ 2 εγ 0 ) ) ) x e 3 π 2 − π π d ρ κλ κλ π A c 4 ε 2 2 2 a a κλ − − − π a ( d A R R − 27 π π π 18 18 Z 54 µν R a a a − 1 2 − − 27 a ρ µν ρσ ρσ η F 48 24 48 Z c c a (5 2(5 (5 (5 δ − F ( ( ( F A F (5 gives θ − − − δ − − − − − + = µνρσ W ======µνρσ µνρσ µνρσ µνρσ = θ Q xe W W -supersymmetry we have similarly 4 A W θ xe η xe xe xe ] ε d δ 4 4 4 4 δ S 0 ε θ η d d d d δ δ Z Z , δ x e , with i i Z Z Z ε 4 For ) ) δ ) ) ) d [ c c W 2 2 2 c c a 2 2 3 3 θ π π π Z π π − − δ − − − 0 a c a ε 48 12 48 a a 18 18 ( ( ( δ = = 0. (5 (5 − + − + W = = = W ] ] 0 θ ε W ε , δ , δ δ 0 ε η ε δ δ δ and hence [ among fermionic symmetries. Applyingon two the successive generating function and hence, [ JHEP04(2019)040 σ ψ 0 0 η η (B.20) (B.18) (B.19) (B.15) (B.16) (B.17) ρ δ ρ D σ D ] γ σ ] ρσ ψ 0 ρσ η 0 ηγ δ η ν ηγ , ρ
[ ρ ) . ν ) µ [ D ) ε ε D ) µ g ) ε η τ λ σ ε g ε τ γ γ τ µν τ 0 0 µν γ 0 µν ηγ γ ε ε γ ε . 0 0 ( ) F ε ( λ τ ε ε ( ε µν κ ( γ τ ρσ λ xeP A 0 F ∇ 4 xeP γ ) ε A 0 κλστ , µν 4 d κλ ( ε ρσ ε κλστ µν λ d λ σλ κσλτ R = 0 in four dimensions leads λ ρσ R Z γ A µν ] F κλ 0 R A Z µν xe ρ ρ ) R ε κλρσ ( µν 4 R ρσ c ) ( µν A A . σ ρ κλ d c R κ 0 F 2 ρσ xe κλ − ψ η κλ A 2 4 R στ µν π 0 − F ∇ a µνκλ µν Z µν ρ R ρ d η π 4 R F a , ρ ) κλ δ F R 4 µν D ρ µν A κλ λ c ρ A 3(2 Z σ R µνρσ F 2 3 A R A κλ 3(2  γ D ) π ) µνρσ + c µν − µνρσ µνρσ R [   ε µνρσ − 2 σ σ 3 ] ]   a 0  12 µνρσ λ µνρσ π µνρσ κλ ψ ρσ µνρσ 1 4 1 4 η ∇ − γ 1 2 ρσ ρσ 0  (5 0 ρ F η x e  R a γ γ 12 − − ε δ 4 ν ν ( + + D , + [ [ x e  ρ x e  (5 d x e  ) + σ σ 1 2 R = = µ µ 4

– 19 – 4 4 ε γ D ) ) µνρσ g g d ψ d − ρ ρ d λ A 5 5 − Z  0 ε ε 0 η γ λ λ γ γ µν µν A A η 1 2 ) 0 Z δ Z

Z = γ γ -supersymmetry transformations on the generating c σ ε ] ] 0 0 5 ) κλ ) σλ 3 ) ρ ρ γ ν ν 2 S − ε ε x e θ = 0 and θ c Rg Rg ]( 2 2 2 5 a δ δ ( ( a F 4 π 3 ) ] σ ηγ 1 2 1 2 στ π π − κ κ σ σ π d ρ ε [ [ ρ µ µ − − µν − ηγ ∇ R 27 λ δ δ a − − c ∇ ∇ A 96 A ρ 24 , a c F γ Z ( 0 108 ( κ − − (5 A µν µν µν µν µν ] ] ε µν (5 − + ρ ρ γ γ ν ν ( ∇ F µνκλ − F − [ µν κλ κλ δ δ σ µνρσ = = R σ σ F  λσ [ [ R R = [ ∇ ρ ρ µ µ λµν µνρσ µνρσ κ F W µνρσ γ γ κ ] A A R R 0 W ∇ µνρσ µνρσ ] ε R  0 η η   ρ ε µν xe ,δ η η Two successive µν µν ε 4 A µνρσ µνρσ , δ xe κλ   δ d ε [ 4 xe xe δ R d 4 4 [ σ σ ρ Z µνρσ d d xeF xeF i ∇ ∇  A Z 4 4 ) d d i Z Z c give = 0. 2 = = = ) 3 ) ) c π c c 2 Z Z µνρσ 2 2 3 − W  π 2 2 π π W − − 18 a − ] 8 8 π π a a ic ic 0 18 a W 6 ( 6 ( (5 η η (5 δ − − − + + 0 , δ η η = = δ δ as required by the Wess-Zumino consistency conditions. [ function with Therefore, we finally get Moreover, the fact that to the two identities so that The last two terms can be rearranged as JHEP04(2019)040 . =0
µν η (B.26) (B.24) (B.25) (B.21) (B.22) (B.23) ) F λ λ ρ g ∂ ν 0 γ η ρ , µνρσ −  we note that D  0 ν λ ) ρ
g η η µν η ρ λ κν ν P γ γ γ D 0 ( 5 . ↔ 5 0 η
0 γ η ( η ηγ
η ) ν µκ  µ νκ λ γ − g g ρ κ 0 µν F ν
= 0 we find that the second g 0 + η 1 2 η γ 2 η κ . Rg η ν ρ µν ) ) 0 D 0 + − 1 2 γ − ν 0 λ F D λ η
ν λ ρ g 5 η ¨* ¨ γ σ − η g νκ κ ∂ γ ) 5 g µ µκ ¨ γ λν σ η ηγ ρ γ µν g γ γ ( λ γ 3 − η.
ρ interchanged we obtain ηγ γ R νκ µνρσ + ρ κ σ 0  ρ κ η λ ∇ + − g γ µκ µν η ) vanishes trivially since λ ρ g γ  5 σ 0 ν ρ κ g γ 0 η ν = 0 D g γ η g ) µν µ η γ 0 5 η 2 ν + ρ ( γ η η ( 0 γ ρ ∇ µν γ ( 0 and B.20 ρ 0 κ η D η − g 1 3 η ∇ η κ νκ = D σ η
κ − κ σ κ ρ µλ 0 σ γ η γ D – 20 – νκ g κ κ D γ 5 g η γ κ λ 0 − ∇ ρ ρ 0 λ . ) = σ γ F − γ 0 0 ν η ν η D γ
+ η γ 1 2
η γ η ] γ λ ν 5 ρ µν νκ ρ − ∇ κ η η − µκ νκ γ γ ) ] κ 0 γ ν − − + 0 , 0 g F D 0 ηγ δ κ λ η 0
γ η ν η 5 ρ ρ κ ηγ κ }D η κλ 1 2 0 σ g κ ( ρ η η [ γ g g γ λ η γ λ 5 ρ µ ( ν ν µλ σ D − D 0 ν + γ δ γ [ νκ ν γ g , γ λ 0 D σ µκ ∇ η 0 µ ρ 5 3 γ η g η − νκ − g µν µν + γ µν κ 0 + ηγ ρ ρ ] 0 λ η ηγ η λ γ η − ρ γ R D η D νκ g η 0 { ηγ κ λ 5 κ η νκ νκ νκ ρ 1 3 κ κ η η δ − γ κ γ γ γ 0 σ F F F 0 D [ ( γ ρ ηγ = = 2 η 0 0 η λ µν 1 2 1 2 1 2 ν νκ κ ) + η η γ ∇ ρ γ νκ µν η − − − D ηγ µν µν g µ νκ F κλ η λ ηγ g 3 g = = = γ 1 2 γ F 0 ] νκ νκ ηγ η 1 2 κ κ κ − ( F κλ F µν µ ∇ ∇ ∇ γ 1 2 = = g ν ∇ [ µν µν µν µ − P P P R µν 1 3 2 3 1 6 1 3 interchanged is subtracted. P ηg 0 1 3 − − − − η µν = = = = = In order to evaluate the term proportional to the Schouten tensor The third term can be simplified as The first term in the second equality in ( P and Hence, which again vanishes up to a total derivative term due to the Bianchi identity Hence, subtracting the same quantity with Integrating by parts and usingterm the Bianchi vanishes identity as well. η For the second term we have We will now show that each of these terms vanishes once the corresponding expression with JHEP04(2019)040 ) η σ ψ
η η δ (B.32) (B.31) (B.27) (B.28) (B.29) (B.30) κ λ ( D λ 0 εγ η. γ εη
]
: κλ , µν κλ  0 γ η εγ σµν κλ  ν ρ κ [ ) also vanishes. µν , R µ  D R ρ g  µ ρ 0 κλ )  A 0 η A R ηγ B.20 η  κ The final commutator ν − κ ρσ µρ . 0  ∇ ) F ∇ in ( η ηγ 0 R . (  κ η η µ µν D  5  µνρσ = 2 µνρσ µνρσ λ P ηγ  0 µν  γ ( εγ ] η i ν xe ρ 4 xe 3 xe κλ Rg 4 4 ∇ 4 D . γ d 1 2 d ) d − ν [ µν Z Z µ − Z µσ ) = 0 ) in the last equality. R ) = g ) 2 c 2 -supersymmetry transformations. We , c 2 c ν ηg  µν π = π θ π − S γ − − R a µν η 12 B.27 a 48 = 0 a µν ( 12 − ν ( ( λ  Rg − D − and Rg νσ ψ + W µ µ 1 2 σ ] η η g η 1 2 η 0 - and γ φ δ µ ∇ 0 σ η σ – 21 – η − κ κλ Q γ η εη − D δ D , δ 1 2 0 5 5 5 η εγ εγ η µν + µν δ ρ anomaly to the commutator, namely ) we have [ ) = 2 : R εγ εγ R µν µν εγ ) vanish upon subtracting the same quantities with = D ρ ρ η ρ Q   µνσ σ − µ κλ κλ µ A A A σ γ A γ 0  γ ( R R B.20 0 ∇ η µν µν B.20 µν η µν η ν ρσ ρσ ( F F γ F 0 D µ F F η µ ∇ µνρσ µνρσ µνρσ , with µνρσ ηγ R R µνρσ µνρσ µνρσ µν W Q = R xe xe xe θ 0 ) + R 4 4 4 A 0 δ xe xe η ε d d d − 4 4 η ρ 0 d d µ + Z Z Z D η xe i i i Z Z ρ σ ηγ 4 ) ) ) ) ) ( γ W c c c d D 2 2 c c 2 2 2 ν 3 3 3 σ σ π π µν π π π − − δ Z γ ∇ − − − a a η 48 48 18 18 18 ηγ a a a ( ( µν µν δ = (5 (5 (5 R − − + ηγ µνρσ W = = = = interchanged, leading to the commutator ] = 2 R 0 η µνρσ η W To summarize, all terms in ( Finally, for the last term in ( ε , δ R δ ε η δ δ we need to considerstart with is the the contribution one of the between in agreement with the Wess-Zumino consistency conditions. [ where we have again used the Bianchi identity ( and Hence, with It follows that the term proportional to the Schouten tensor where in the last equality we have used the Bianchi identity JHEP04(2019)040 . ) σ ε ε 5 ψ σ ε . γ A.6
δ σλ
D ρ στ σφ g ε τφ (B.36) (B.35) (B.33) (B.34) ρ στ g g 5 D g ρφ ] D γ ηγ ] ρτ ρκ g ρφ ρσ g g µν ρσ g νκ νκ νλ g ρσκλ ηγ σ ηγ ηγ νλ g g ν ε [ ν R , ε ψ g µτ R [ σ µ ε 5
µλ µφ g µ ε δ g ε µν γ µκ g g µν D g 5 ρ 5 g ρ F + F γ µν ρµ γ D + µν , D − + η µν +
στ µν η ηγ
σκ g ε µν σκ σλ g ηγ µν 5 x e P ρν ηγ x e P σκ F ρλ g g 4 γ F 4 R ρτ R g g R d d ρτ ν ρφ g µρ µν ρλ g νκ µ g µν Z µνρσ g νφ g F Z µνρσ F F ηγ ) ντ νφ g ) 4 c ντ 4 g g µφ c νρ + g µλ 2 g + x e  + − 2 ε µκ µλ x e  − R g 4 π ε a 5 µφ ε 4 π g g ν a σ 4 d 5 ε. 5 ε g 4 d γ µ + − ψ γ σ 5 γ + + F ε 3(2
Z µρ γ + 3(2 Z D δ 4 ρµ σλ µρ ) ρ ρ + ) σκ σλ σφ g c τφ + ηγ c σλ − g g g σ 2 ηγ D D 3 ε ηγ 2 3 ρτ g ε π νρ ψ ηγ σ   ρλ ρλ ρφ π g − ρν 5 ε νρ ] ] − ρκ g g g R δ D a 12 γ R a g νκ R ν 12 ρ ρ ρσ ρσ µ ντ ντ νφ ν g µ (5 ρσκλ ρσ γ γ (5 ν D D νφ g g g µ ε ν ν F 5 5 [ [ R g µφ + – 22 – 5 F F 4 + ηγ µ µ γ γ µκ µκ µτ g σ 4 4 γ ε

µν µτ g g g g g ] ] − ψ σ g + F ρ ρ τφ − ε − µν µν ε + + + ν ν δ D µνρσ 5 ε ε + δ δ 5 5 ηγ σφ 5 5 R γ σ σ Rg Rg στ στ σκ [ [ g κλτφ γ γ στ g g ηγ ηγ g 1 2 1 2 µ µ  ρσ µν g ρ ρ ρκ δ δ anomaly to the commutator is ρσ µρ ρσκλ ρτ ρκ ρλ F g − − A A ηγ − − ρφ g g g S R ] ] ηγ ηγ i g ντ µνρσ µν µν ρ ρ µν µν A 2 νλ  νφ νφ g γ γ ν ν µν νρ νκ F F g g g i 2 δ δ − µνρσ F g µλ R σ σ µνρσ [ [ µφ µκ µλ R g µ = = µ µ µλ R µνρσ µνρσ g g g ν µνρσ µνρσ γ γ g µν ε + κλτφ R R F µν + + +  F η η κλ   4 F σφ + η η x e  x e  σκ σλ σφ g + µν µν 4 4 4 ηγ g g g µνρσ d d σφ ρλ = = 4  x e x e µν S ρκ g ρκ ρφ g 4 4 g Z Z g g A − d d κλ ρτ x e F x e F i i νκ η 4 4 g ντ νλ νλ ) ) g R κλτφ d d Z Z c c g g g  2 2 νλ 3 3 µτ ) ) ρσ x e g c c µτ µφ µτ π π Z Z g 4 2 2 F − − g g g d 2 2 π π − − µκ a a 18 18 µνρσ 8 8 π π g a a ic ic + + +  + Z 6 ( 6 ( (5 (5 The contribution of the µνρσ W ε  = η − and terms of a given color give the same result when contracted with − δ − + + + + δ 4 ε = = = δ S where we have grouped terms according to the conjugacy classes of the symmetric group and so In particular, and the product of two Levi-Civita tensors The last term in this expression can be simplified further using the last identity in ( JHEP04(2019)040  , λ
µ (B.40) (B.39) (B.37) (B.38) ε ε 5  5 εF ε γ 0 5 γ

γ µν : νρ µκ νλ g  ηγ g λσ R ηγ  σλ i g g µν  ε + − 5 F 0 ε − γ λρ ε + : ρσ g 5 κλ σκ γ ε  γ g 5 νσ ηγ νλ g γ  ρσ 0 µλ + g ε µρ  ηγ ε, ε ηγ : ηε.  ) νρ µνκλ  +  ηγ  g  λρσ R κλ νσ 2 ρσ νρ µ µν  µκ ε. λσ F F R µν R 5 g ). F 5 R R γ 5  5 1 3 γ ν F 2 1 4 µρ ε σλ iγ 0 µ iγ i g 8 iγ ρσ − γ −   2 F + 2 B.35 : + 4 + + µν λρ µν κλ ηγ − ρσ  ) − g ε P F R − ε κλ γ 5 ρσ 5  3 1 νσ γ ε νσ γ γ µν σκ κλ 5 iγ γ ε R  g − µρνσ R γ 5 κλ κλρσ ηγ +  R µρ γ γ  µλ ρσκλ ρσ +2 µνρσ R g γ ρσ 5 – 23 – 0 κλ R κλ 1 1 4 µσ R 12 γ γ ηγ + 2 − 1 4 ρσκλ ε 1 4  0 ¨* εF ηγ σ ηγ + + 2 
R 5  ε 5 ¨ 5 D : γ  5 κλρσ νλρσ µνρσ γ λν µν ρ γ µνρσ µνρσ  = ¨ γ
γ µνκλ R i F κλ R νσκλ  D ρ R R ε 1 4 5 νλ ν ] ( ( R R − µκ µν  ν ηγ δ γ ν g µσκλ  i
µν µν µν σ F  ηγ µ µνρσ ) η ] D  i µ +  ρ λ + µ F F F F δ [ i νλ ν µ 1 8 4 ε i i i i 8 4 8 8 − νκ δ λρσ γ D − σ µ ρσ γ 2 × iF [ + − + + + + ρ µκ γ η µ R ε ε   µλ ν g δ 5 5 1 4 δ g κλ 2 ηε ηε ηε ηε µν µν σ − + + − ηγ ηγ νσκλ µ µν µν µν µν ηγ ] ε ε ε γ ρ Rg Rg R λ 5 κλ ρσ ρσ F F F F κλ ν σ ν 1 3 1 3 γ γ δ γ D F F η µ D µν µν µν µν κλρσ κ σ − − [ ρ νλ F µν µν µν µν µν F F F F µ R D g i g 8 D ] 1 2 1 2 1 2 1 2 F F µν µν γ F ( 1 4 ηγ η i 2 η η λ κλ R R − − − − +  η µ µν   µν µν µνρσ µνρσ R = = = = = ηγ F µν 6 1 6 1   P P ν iF [ 1 i i 2 2 1 6 36 1 6 − − µ  iF g µνρσ ======Next we evaluate the term We next consider the term  µν P . The easiest term to evaluate is where in the last line we have utilized the identity ( we now evaluate the four termsε involving two covariant derivatives on the spinor parameter Using the identity, JHEP04(2019)040 , (1971) (B.42) (B.43) (B.41) ]. ηε , 37B ) we finally
µν SPIRE ηε F 2 IN B.36 µν (1988) 353 R  − ε ηε ) in ( 5 2 117 x e F Phys. Lett. γ Adv. Theor. Math. R 4 , , d 1 3 µν ]. B.41 (1982) 661 [ Z − ηγ 2 µν ηε 17 π c SPIRE ε ) and ( IN 12  iRF ηε µνρσ ][ 2 − εη, ρσ  R 2 ε 1 2 B.40 F
5 1 5 12 R = 1 3 ), ( ηε µνρσ iγ ηγ − 2 ε R + ρσ − 5 R ε Commun. Math. Phys. ]. F
γ J. Diff. Geom. + , κλ B.39 − µν κ ν , γ ε ε µν µν g 5 R 5 , γ F λ ), ( µ ) γ η, σ g µν SPIRE 5 ηγ – 24 – ρσ ρσκλ W µν 0 R IN arXiv:0712.2824 F P µνρσ R εγ +2 i A B.38 [  i ][ ηγ κ ν 1 4 ¨* σ ε  ¨ 3 4 g σ µν µνρσ λ + ¨ − x e x e  D µ  R 4 4 x e ] ρ γ R ¨ i 4 d 2 = d iRF D ), which permits any use, distribution and reproduction in ρσ ), and ( d A θ 2 + γ θ is the Pontryagin density. +4 (2012) 71  Z Z ] ε ν 5 Z [ − i ) γ 1 µ ( ) c ρσ κλ ) 12 B.32 Consequences of anomalous Ward identities 2 c g c 2 γ 313 2 κλ 3 π ρσ − µνρσ ν − π [ µν π − x e ε R µ µν a R − 16 4 hep-th/0206161 5 32 g a ) in ( 72 d i [ Rg γ a (2 ( CC-BY 4.0 µν 1 2 − (5 µν µνκλ µνρσ Z Seiberg-Witten prepotential from instanton counting + − γ − B.35 This article is distributed under the terms of the Creative Commons R R Rg = = 1 4 ρσ 1 2 µν ]. Topological Quantum Field Theory Supersymmetry and Morse theory Localization of gauge theory on a four-sphere and supersymmetric Wilson loops F κλρσ γ − − W µνρσ ] R ε  ]. (2003) 831 η 5 µν 1 2 γ µνρσ µνρσ γ 7 , δ SPIRE ε = R P µνρσ R δ IN i i [ 2 4 R  SPIRE [ P µνρσ 8 1 2 1 − + IN R Phys. Commun. Math. Phys. 95 [ Substituting ( Finally, the last term gives V. Pestun, J. Wess and B. Zumino, E. Witten, E. Witten, N.A. Nekrasov, = = =  [4] [5] [1] [2] [3] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. with as required by the Wess-Zumino consistency conditions. where obtain JHEP04(2019)040 , , JHEP (1969) , 08 JHEP JHEP , ]. , (2017) (2013) 072 ]. A 60 (2016) 2617 JHEP 01 , SPIRE A 50 IN SPIRE (1969) 2426 B 47 ]. ][ IN ]. (2014) 577 ][ JHEP ]. ]. 177 , ]. Nuovo Cim. Supersymmetry in J. Phys. (2017) 038 327 SPIRE , , SPIRE Supersymmetric Field IN SPIRE IN 07 SPIRE [ ][ SPIRE IN IN Anomalous Supersymmetry [ IN model ][ ][ σ Phys. Rev. , JHEP Acta Phys. Polon. , arXiv:1207.2785 , [ arXiv:1212.3388 supersymmetry and localization in the [ ]. 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Killing spinors and SYM on curved spaces ]. ]. ]. ]. ]. ]. ]. ]. , SPIRE SPIRE SPIRE SPIRE SPIRE SPIRE SPIRE SPIRE IN IN hep-th/9708042 IN hep-th/0002230 IN hep-th/9806087 IN arXiv:1405.5144 IN IN IN Quant. Grav. [ supersymmetry [ [ multiple supercurrent insertions central functions of supersymmetric gauge[ theories superYang-Mills theory in the Wess-Zumino gauge Functions, Effective Action and Superconformal[ Anomalies renormalization in the AdS/CFT[ correspondence Theories [ to Twisted Holomorphic Theories [ [ [ Supersymmetric Partition Functions (2014) 032 Curved Spaces and Black Hole[ Holography [ J. Erdmenger and C. Rupp, J. Erdmenger and C. Rupp, D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, O. Piguet and S. Wolf, I.L. Buchbinder and S.M. Kuzenko, F. Brandt, I.N. McArthur, L. Bonora, P. Pasti and M. Tonin, P. Benetti Genolini, D. Cassani, D. Martelli and J. Sparks, M. Henningson and K. Skenderis, S. de Haro, S.N. Solodukhin and K. Skenderis, C. Closset, T.T. 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