Anomalies, Conformal Manifolds, and Spheres Jaume Gomis, Nathan Seiberg JHEP03(2016)022 3 ) Is Our (1.2) K 25 to Be

Home , Conformal anomaly

JHEP03(2016)022 Springer March 4, 2016 , c : February 3, 2016 February 12, 2016 : : is the space of con- follows immediately M , Published M 10.1007/JHEP03(2016)022 Received Accepted 2) supersymmetric theories doi: , = (0 Adam Schwimmer, = 4 we also show that the relation c = 4. This reasoning leads to new N d d Published for SISSA by 2) and , [email protected] , [email protected] = (2 , e N = 2 theories in N Zohar Komargodski, b = 2 and . d 3 [email protected] , = 2 supersymmetric theories in and Stefan Theisen 1509.08511 Po-Shen Hsin, d The Authors. N a c Supersymmetric gauge theory, Anomalies in Field and String Theories

The two-point function of exactly marginal operators leads to a universal con- , [email protected] 2) theories in , = 2 and = (2 d [email protected] [email protected] Rehovot 76100, Israel School of Natural Sciences, InstitutePrinceton, NJ for 08540, Advanced Study, U.S.A. Max-Planck-Institut fürGravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany E-mail: Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada Department of Physics, PrincetonPrinceton, University, NJ 08544, U.S.A. Weizmann Institute of Science, b c e d a Open Access Article funded by SCOAP can be ruled out by a more detailedKeywords: analysis. ArXiv ePrint: information about the conformalthe manifolds manifold is of and Kähler-Hodge we these furtherN theories, argue for that it example, hasbetween we vanishing the show Kählerclass. sphere For that partition functionfrom the and appropriate the sigma Kählerpotential models of thatples we of construct. potential Along trace the way anomalies we that find several obey exam- the Wess-Zumino consistency conditions, but emphasizing its interpretation as aformal sigma model, field whose theories target (a.k.a. space theory the is conformal supersymmetric, manifold). this sigma Whenexamples, model the we has underlying to consider quantum be in field appropriatelyin some supersymmetrized. detail As Abstract: tribution to the trace anomaly in even dimensions. We study aspects of this trace anomaly, Anomalies, conformal manifolds, and spheres Jaume Gomis, Nathan Seiberg JHEP03(2016)022 3 ) is Our (1.2) K 25 to be . This λ 1 . ( } I I λ IJ M g {O 22 , I λ ]. By allowing 5 is interpreted as the space of vacua in ) (1.1) x d d ) ( 2 I ]. It is the metric on the conformal ) K M 1 y O λ ( I − IJ parameterize the space of conformal field xλ x g ( d 21 I d λ = Z i 7 ) – 1 – y . The two-point functions I ( X = 2 J 17 M 2 O 14 d 21 1 d/ ) π x = 4 ( I d = ]. For example, the Ricci scalar associated to hO 3 can be interpreted as the space of classical vacua of the theory. In the δS [ I M λ 2) supersymmetric backgrounds in superconformal gauge 1 , ]). These correspondences allow to connect our results to various other topics. 4 22 24 correspondence, the conformal manifold of the CFT 2) supersymmetry in , 2) 2) ]. It carries nontrivial information that cannot be removed by redefinitions d , , 2 -dimensional conformal field theories have exactly marginal operators (see e.g. [ 2) and (0 = (2 = 2 supersymmetry in /CFT d , +1 +1 The purpose of this note is to explore the geometry and the topology of d d In worldsheet string theory, C.1 (2 C.2 (0 N N 1 main tool will be the conformalAdS anomaly firstAdS discussed in [ define a metric, known asmanifold the [ Zamolodchikov metric [ of the coupling constants invariant under all such redefinitions. the theory remains conformal. Thetheories, coefficients a.k.a. the conformal manifold 1 Introduction Some means that when we add them to the action with coupling constants D (2 A Normalization of the anomaly B The FTPR operator and itsC properties Review (and conventions) of two-dimensional supersymmetry 3 4 (0,2) supersymmetric theories 5 Contents 1 Introduction 2 The anomaly associated with the metric on JHEP03(2016)022 or = 2 c (1.3) (1.4) ) and K λ N λ, ]. ( c 17 ] as well as – K 9 ) and a space ]. λ 15 6 , leads to a new λ 2) chiral and twisted 2) theories in two , 3 . , ) e λ . = (2 e λ, ]. ( IJ = (0 N 7 g tc K N − = 2. Our analysis leads to e M ] and proven in [ c 3 8 d 2) supersymmetry there are two 0 , r r 2) and . , 12 is Hodge and suggests that its Kähler cannot be a smooth compact manifold. = (2 = K/ 2) theories in section e = (2 M V , ] and was further used in [ = 4. Again, the sphere partition function and will further argue that its Kählerclass M N ). Their Kählerpotentials are a Z 3 d 4 14 e λ N − , = (2 2) theories in e λ a (scheme dependent) scale. The dependence , – 2 – 0 depending on chiral couplings ( r ; 0 ] derived a contribution to the trace of the energy- r N r ) c 5 λ λ, M = (0 ( = c ) were conjectured in [ ]. K Z N − and shows that 13 1.3 e must be Hodge – ] as well as in [ c = 2 theories in 3 11 M 10 M N 0 r r , to place the theory on the sphere and the partition function has and it also leads to global restrictions. Second, the anomaly forces = V in terms of the Riemann tensor of the Zamolodchikov metric. As in IJ A g = 2 theories in four dimensions. Z and M we will discuss N we review the analysis of these conformal anomalies (without supersymme- factorizes into a space A 4 2 is devoted to M 5 ). We will show that is the radius of the sphere and that e λ r ] based on the work of [ reflects the ordinary conformal anomaly. As we will show, the appearance of 2 depending on twisted chiral couplings ( e λ, In section Section We will also study the sphere partition function. Without supersymmetry, there are In section In the remaining sections we will study Our discussion of two-dimensional In supersymmetric theories the anomaly above must be supersymmetrized. This in- reflects another contribution to the conformal anomaly depending on exactly marginal In this paper we assume that the couplingA constants manifold Kähler-Hodge can is be a promoted Kählermanifold to for which the flux of the Kählertwo-form through any ( 10 r tc 2 3 tc tc supersymmetry fixes an additionala contribution four-tensor to in the conformal anomaly depending on chiral superfields. This assumption is non-trivial as ittwo-cycle can is fail an in integer. some cases [ has universal content and computes the potential Kähler on This relation was proven in [ in [ restrictions on the metric on class is trivial. But we will argue that the two-sphere partition function is not universal. Here on K couplings. The identifications ( counterterms that render itways, ambiguous. denoted With universal content. It is given by proof M K should be trivial. This, in particular, shows that try). Here we spell outleads the to conditions new they constraints have beyond to the satisfy Wess-Zumino and consistency show conditions howdimensions a [ and careful analysis momentum tensor, which depends on the Zamolodchikov metric troduces a few newform elements of into the the metric analysis.us to First, introduce it some leads contact to terms. restrictions We will on study the both local aspects in detail. spacetime dependent background fields, [ JHEP03(2016)022 re- (2.1) (2.2) C = 1 theories in 5 ]: N , which is a contact = 4. Appendix 2 23 ] suggested a construction , p d 5 18 + 1 n . n -dependent. Then, the trace = 2 x concerns with the normaliza- = 2 d M , d I A ) therefore remains intact under λ 2 2 1.2 ]. In both cases nontrivial contact . p Λ J d λ 20 p and spacetime dependent coupling con- , 2 d ) take the following form log 19 n µν I 1.2 2 is Hodge and suggests that its Kählerclass ]. γ λ p IJ – 3 – 22 M , g IJ g 21 ⊃ we collect some properties of the Fradkin-Tseytlin- µ µ T i ∼ B ) p = 1 theories. − = 1 theories in four dimensions. In fact, [ ( = 3 see [ N J N d O ) p = 4. 2) Poincarésupergravity in superconformal gauge (which always ( , I d hO 2) supersymmetry in two dimensions and their linearized supergravities. , = 2 and = 2 theories in d considers (2 N D . We assume that the theory can be regulated in a diffeomorphism-invariant 2) and (0 4 I , λ We study CFTs with spacetime metric For a related supersymmetric analysis of conformalFour anomalies appendices in contain technical results. Appendix Our discussion is reminiscent of that of [ We do not make this claim for This term is in addition to the ordinary conformal anomalies, which depend only on the spacetime 4 5 = 4 and fashion. Specifically, we assume thatincident the points energy-momentum (apart tensor from is the conserved even ordinary at Ward identity co- relations). We willof be compact interested conformal in manifolds in metric. The precise action ofthis the later Laplacian for could be to the left andstants to the right. We will specify the non-conservation of the dilatationfields). charge in One the way presence toanomaly of detect in non-vanishing background it even dimensions is includes to a make term, the roughly couplings, of the form [ The explicit scale (orreason cutoff) is that Λ rescaling interm. Λ the changes The logarithm the correlation doesrescaling answer function not Λ. by at violate Such a separated logarithms scale polynomialthough appear points invariance. in abundantly they ( in The do conformal field not theories violate (CFTs). the Even conformal Ward identities, they lead to anomalies (i.e. In momentum space the two-point functions ( views (2 Appendix exists locally). We classify the allowed rigid supersymmetric backgrounds in2 this gauge. The anomaly associated with the metric on theory is placed on the sphere. d tion of the anomalies.Paneitz-Riegert (FTPR) In operator, appendix which appears in the anomaly in is trivial. terms are identified. Theytherefore cannot correspond be to absorbed anomalies. by Theyin supersymmetric reflect the local short flat distance counterterms space physics and theory. and Then, can be these contact analyzed terms have interesting consequences when the two dimensions, our analysis shows that JHEP03(2016)022 is Z (2.5) (2.6) (2.3) (2.4) log σ δ . It mani- M . . J A λ ν ) as a type-A and ∂ responsible for the I ) is cohomologically 2.5 λ Z µ 2.5 ∂ µν γ . IJ ] 6 under infinitesimal Weyl trans- = 0 γ g √ Z Z . ). Therefore, ( log log ) x δσ 1 I 2 σ σ µν 2.5 d λ δ δ ( 2 Z σ when the background fields have trivial topology. δ δσ γ π 1 4 − . Below we will argue that it should also be δσ γ RF = 2 – 4 – √ Z − M . 6 x µν 2 log γ δσ γR d σ 2 √ δ σ Z δ 1 and σ ]. Therefore, even though the anomaly arises due to a ) includes a sigma model with target space ] one could refer to the first term in ( . -independent I x δσ δ ] and its variation 2 x 24 λ I 25 d M 2.5 , λ ; Z µν γ µν π γ that is obtained by a Weyl variation of a local term is considered c [ 24 Z = 2. The infinitesimal Weyl variation of log Z of compact support. Naively, conformal invariance means that the = d log ) and the ordinary trace anomaly is given by δσ Z σ δ 2.2 log ] is a nonlocal functional of its arguments. However, its variation must be coordinate invariant in spacetime. I σ functional of λ δ Z ; µν log . The normalization of the second term is worked out in appendix γ is the Ricci scalar and the first term is the universal contribution due to the central local [ c σ short distance regulator, it is universal — it does not depend on the regularization. trivial. An anomalyby is changing a “cohomologically a nontrivial” counterterm.renormalization term. scheme Equivalently, [ It it cannot cannot be be removed removed by changing the globally well defined on Z a δ R A term in It must be coordinate invariant in It must obey the Wess-Zumino consistency condition [ An important part of our discussion will be the analysis of the allowed local countert- The anomaly functional ( Let us start in A type-A anomaly vanishes for 5. 3. 4. 1. 2. 6 A type-B anomaly doescertain not correlation vanish functions. for constant sigma even for trivial topology and reflects logarithms in no local counterterm, whose Weyl variationnontrivial. yields ( In the language ofto [ the second term as a type-B anomaly. erms (related to item 5 in the list above). In two dimensions, an important counterterm is Here charge festly obeys the Wess-Zumino consistency condition because it is Weyl invariant. There is trace anomaly ( formations with infinitesimal variation vanishes. Butnumber because of important of properties: the anomaly, it does not. This variation satisfies a the partition function JHEP03(2016)022 ). ∼ for n and ] I 2.5 (2.7) (2.8) (2.9) J I (2.10) (2.11) ··· 2 V B I I 1 [ I ∂ A = ) is invariant , IJ B 2.10 ··· ) transforms under + . Also, we have two 2.9 . is determined entirely n I ) M n I ··· I , 2 λ I ··· ( J 1 2 , . I I λ I 1 I ν A I λ λ ) obey the Wess-Zumino con- ∂ δσF = 0 it is clear that ν A ν I r ∂ ∂ λ p µ γ ··· µν 2.10 µν to be quadratic polynomials in the . We note that ( ∂ is locally given by ). This allows us to identify γ √ I = I n I X µν I e M x ), ( V λ 4 IJ 2 ( by picking specific simple combinations ··· p γ δ γ V 2 d B f IJ I n 2.9 on 1 √ I √ = . I Z ∼ I ··· 3 2 e . On the other hand, ( A δσ 2 ), ( δσ V γ B I M p e F µ f µ 1 − √ I I 2.8 ∂ – 5 – = log Λ A x ∂ x ∂ and 2 and 2 i + ) = d ) x δσ d I I I 1 on the conformal manifold 2 n e ) with V λ p V d p Z ( Z ( − IJ n 2.6 = 0 and hence → Z ⊃ I ⊃ B ] = I O ⊃ Z e Z 2 V γ RF JK p Z √ ··· B log log ) I x [ 2 σ are exactly marginal, the coefficient of the logarithm must be σ 2 log ∂ δ p δ d ). We will discuss it below. I ( σ 2 δ = O I Z ), there are other potential trace anomalies that we need to consider. O σ 2.10 ) δ 1 IJK 2.5 . Similar other specific cases show that p . By integration by parts we find that this anomaly is now identical to ( 2 H I 1 J 1 I ). Scale invariance constrains I B I , but the change is cohomologically trivial. It can be absorbed in the Weyl O g . Consider the momentum space correlation function of the exactly marginal λ h n f ) controlled by a two-form cannot be present. More precisely, the argument I µν I . Therefore, we can determine ∂ δ ∂ r 2.8 p = + ··· I 4 I µν V ∂ γ First, a simple argument excludes all type-B anomalies that are beyond that in ( We will now show that even though ( In addition to ( 3 I as connections on the conformal manifold → ∂ I I 2 1 e anomaly ( above shows that some one-form the type-A anomaly ( ultra-local, i.e. a polynomialthe in couplings momentum (otherwise, there wouldmomenta be a beta functionof for momenta. For example,p for by derivatives of the Zamolodchikov metric. This means that the additional parity odd operators where the ellipses onΛ. the Since right-hand the side operators represent terms independent of the UV cutoff analysis can further restrict anomalies. We recall that afunction. type-B anomaly Without loss is of associatedwith generality to we a can logarithm study appearing the in theory a in correlation flat Euclidean spacetime V variation of the local countertermV ( sistency conditions, a moreanomalies. detailed analysis This leads demonstrates that to constraints further that restrictions, go ruling beyond out the standard these cohomological and They are characterized byunder one-forms the gauge transformation with an anti-symmetric two-form type-A anomalies We will see various consequences of this counterterm inFirst, what we follows. have the parity odd type-B anomaly Its Weyl variation is JHEP03(2016)022 . ) ]. ). K I , or and 2.5 31 2.4 , , the y M ]. In a I (2.13) (2.12) 30 q µν , which 23 6= . ] γ − are pure J x e I V e I J V [ = λ ∂ ν ) must satisfy z ∂ or and ] I J 2.10 R , or V , then it is natural V z a µ I ), of the type already µν [ j ), are a priori allowed ), ( γ ∂ 6= 1 3 . In addition, one could 2.12 y 2.9 J 2.10 − λ (0) with some matrix = 0 some of the symmetries = a µ K µν I ), ( x A . There could also be ’t Hooft λ O R a J a µν 2.9 K I iq F I are taken to carry charges M . λ − ), ( I ) µ i I x ) ∂ O is proportional to the Zamolodchikov metric λ ( 2.5 y µ ( IJ and examine the anomaly as a function µν ∂ (2) J g δ R a µ 2 O I = ) . Away from µν λ A a − x I I δ µ ( q ∂ λ J I J ∼ µ – 6 – λ . 8 λ O σ ν b ) and covariantize this expression both in spacetime V ∂ z I (0) σ ( I λ . in the type-A anomalies ( IJ → ∇ I = 6 unfortunately appears to be more complicated [ ) that are associated to separated points physics are b µν ··· O µνρ . The operators e I I V ) T a µ d R e λ V + h IJ x µ A ( γ δσ g g J = 2.12 ∂ λ √ ρ and µν . Thus, at least locally, the connections V x T and ] I 4 I J ν d V I λ ∇ γ δσ R V , thus carry charges µ √ ). If the CFT also has conserved currents and I ∇ IJ x λ 4 I 1.1 , we should now also demand gauge invariance. The equation ( d γ g . They cannot lead to nonzero “field strengths” √ M Z one finds the expression may be thus explicitly broken. Related expressions appear in [ IJ ]. The situation in 2 ]. 7 g π = 0, i.e. these connections are flat. These anomalies can be extracted from a 29 µ as in ( 32 x δσ 1 – j ] 4 I J d , and the gauge fields 192 e λ V 26 I , R I [ 5 λ . In the first case the only contact term allowed by dimensional analysis contains . One also requires that it satisfies the Wess-Zumino consistency condition ( = 4 the local functional that reproduces the logarithm in the two-point func- ⊃ ∂ x ) is a four-derivative local term. One can construct it by starting with the ansatz ⊃ d Z M = . There could also be contributions with support at 6= Z 2.1 ] In Summarizing, we have seen that even though ( Thus far we limited ourselves to deformations by exactly marginal operators with Therefore, all the terms in ( Next, we argue that y J IJ log We use the convention [ The fact that the anomaly g , which has zero separated-points correlation functions. In the second and third case, V 7 8 log σ = I µ µ [ δ -theorem [ σ at the fixeda point playsFor a a review see very [ important role in perturbative proofs of the strong version of the tion ( δ and in After some work spacetime and on is modified by simplyencounter replacing new anomalies that containanomalies the under field gauge strength transformations. to couple them toof classical these background fields. fields Thecouplings anomaly sigma modelsthe now coupling depend constants on thegenerated spacetime by metric addition to the previous requirements of conformal invariance, coordinate invariance in anomalies that obey theThis will Wess-Zumino have consistency important conditions, consequences they in what can follows. becoefficients all excluded. contribute to the anomalies proportional to are anti-symmetric in gauge and the associated anomalies vanish. z T we can have the contactThis term would lead to a logarithmicanalyzed term in above, the and three-point function hence, ( it is proportional to the Zamolodchikov metric and does not ∂ the following energy-momentum correlator Using the conformal Ward identity atto separated points, the correlator must be proportional JHEP03(2016)022 . ) ∼ M 2.13 (2.15) (2.14) IJKL ]. If the c with com- 38 ). We will or M 2.13 . KL g IJ g ··· ) satisfies the Wess- . ) and ( L ∼ λ and the Zamolodchikov The Wess-Zumino con- ) + 2.13 ν 2.5 I ∂ λ 9 . ( ]. K 3 IJKL λ IJKL 37 F c M ν c ∂ J . could be an arbitrary connection, λ 2 µνρσ = 2 supersymmetric theories, such ] in related contexts. The combina- µ B R ∂ 31 I JK I N λ reside in various superfields [ µ ) + I ∂ is the usual Christoffel connection on I λ λ ( ] and [ 2 so that the anomaly is coordinate invariant I JK . See for instance [ F 28 IJKL = 2 b – 7 – +1 d γc 2 µν d to be the Levi-Civita connection. Note that ( ) in the renormalization scheme in two dimensions √ R I JK 2.6 ) + , where Γ x δσ I 4 K is proportional to the Riemann curvature tensor of the λ d is the Riemann tensor on λ ( 1 µ Z ∂ F we will show that in J 2 IJKL ⊃ 5 λ is enriched to c IKJL R µ Z ∂ R , respectively. may be either an independent rank-four tensor on the manifold γ A ) does not imply a relation between log I JK √ e λ σ x δ 2.4 4 + Γ IJKL d I and c , where λ ], which we discuss further in appendix ]). I that we would like to mention: Z 2) supersymmetry in λ 36 , 14 2) supersymmetric theories in two dimensions. Here the exactly marginal – ) to the question of locality in AdS , = JKIL , I 33 R 10 ) can be viewed as an interesting variant of the Fradkin-Tseytlin-Paneitz-Riegert IJKL ) in four dimensions is restricted to have holomorphic dependence on the coupling λ 2.14 c = (2 + b = (2 , as we demand in general. At this juncture Γ 2.13 2.15 We will be particularly interested in the case where the underlying theory is supersym- For future references, let us also list some of the allowed counterterms in four dimen- While we do not present an exhaustive classification of anomalies in four dimensions, N We thank Y. Nakayama for a discussion on the topic and for stressing the potential relevance of the M N , or it may be fixed by the Zamolodchikov metric, e.g. 9 IKJL in parameters belong either to background chiralwe multiplets or denote twisted by chiral multiplets, which anomaly ( partition function meaningful (up tofunction a [ transformation Kähler generated by a holomorphic 3 Our goal in this section is to determine the conformal anomaly and analyze its consequences we must further requireWe that will the study local someshow anomaly of that the functionals the consequences remaining above ambiguity of beand ( ( supersymmetrizing supersymmetrized. ( constants. This fact has several important consequences. In particular, it makes the sphere sions metric. Then, the exactlysuperconformal field marginal theory couplings (SCFT) can be regularized in a supersymmetric manner, then sistency condition ( metric. However, in section a relation must exist,Zamolodchikov and metric. The four-tensor M R tion ( operator [ there is an additionalponents conformal anomaly that depends on a four-tensor on The ordinary Laplacian on not necessarily the Levi-CivitaZumino consistency one. condition forces However, Γ demandingcoincides with that expressions ( that appeared in [ Above JHEP03(2016)022 ]. A , 39 (3.2) (3.1) ). For 2.6 and U(1) ) + c.c. e λ V 2) SCFT has e λ, , . Although this . λ, to a background ) + c.c. is a chiral superfield A e λ V Z λ, A R e ( λ, Clearly, these expres- K λ, supergravity it is in a Σ λ, 11 ( V δ K or U(1) θ Σ ) is then straightforward. In 4 δ V 2.5 x d 2 ]. (We repeat this analysis in the θ E ) and the counterterm ( d 4 44 x d 2.5 Z 2 d e π Σ respectively. In what follows we will 1 4 δ Z ] and, in particular, the possibilities for π 1 10 44 4 – Σ) + + 40 Σ and – 8 – ). The corresponding superconformal variations, δ C is the Berezinian superfield, and here .) Σ)(Σ + ) to find the Σ dependence of log Σ + c.c. E δ , the anomaly is given by δ D A 3.1 component. The first term represents the ordinary anomaly. R-symmetry but not both. These contact terms are de- ER 2 Σ + θ θ A δ 2 ( . It is straightforward to repeat it for U(1) θ A 4 x d 2 d xd e 2) SCFT this distinction is the following. The (2 Σ (see appendix 2 , Z or U(1) d . In particular, we assume that there are no gravitational anomalies A V π Z c — the Liouville field. This statement is easily supersymmetrized. Every R-symmetry. We can couple either U(1) 24 ]. These are labeled by whether the U(1) symmetry preserved in the π σ c − A . 40 24 R = − or U(1) c A U(1) V = Z = supergravity it is in a chiral superfield Σ and in U(1) is a complex function of the exactly marginal couplings. × 2) supergravity was discussed in [ A L log , c A is the chiral superspace measure, Z V K Σ δ E One might try to integrate ( The supersymmetrization of the conformal anomaly ( We find it convenient to use a simplification specific to two dimensions. Since locally In terms of the (2 We should discuss two distinct supergravity formulations known as U(1) First, we should supersymmetrize the anomaly ( log In the full supergravity without using the conformal gauge the anomaly takes the form To follow the discussion below (in our analysis of two-dimensional theories) no familiarity with super- = (2 Σ 11 10 δ gravity is necessary. where that contains the Ricci scalar in its can be done asit a is local valid expression and in localproperty terms makes in of it Σ, the particularly the superconformal interesting, answer gauge, as is but it nonlocal. it cannot is The be point nonlocal absorbed is in in that local other counterterms. gauges. This where sions obey the Wess-Zumino consistency condition. twisted chiral superfield whose anomalies we are interestedconcentrate in, mainly are on U(1) the regularization preserving U(1) every two-dimensional metric isconformal factor conformally flat,supergravity we background can can be described describe locallyIn the by a U(1) superconformal metric factor using in a the superfield. preserve either U(1) scribed by the correspondingbe supergravity. regularized Equivalently, while we preservingeither assume diffeomorphism U(1) that invariance the and theory supersymmetryso can as that well as supergravities [ Poincarésupergravity theory is vector or axial. a U(1) gauge field, but an anomalyCorrespondingly, prevents the us coincident from points coupling divergences both and of them the to associated background contact fields. terms can that we need toN place the theory notrigid only supersymmetry in in curved curved space spacesuperconformal were but gauge analyzed in in in [ curved appendix superspace [ JHEP03(2016)022 . . 12 tc )) e λ = 2 (3.4) (3.7) (3.8) (3.3) (3.5) (3.6) e λ, ×M ( N c tc K M − = ) , λ M λ, ( c K ) + c.c. λ . ( Σ) ( F δ ) λ ) are absent. Note that this Σ ( θ Σ+ ) in components and requiring 4 F δ . ( 2.10 Σ θ 3.1 x d , δ 4 2 . ) ), ( d ) under Kählertransformations. It ) (3.9) ) e λ λ e λ x d ) + c.c. ) + 2.9 ( ( 2 Z e λ, λ 3.7 λ and the dependence is holomorphic. ) of the supersymmetric local counter- ( ( d ( 2) worldsheet theories lead to F G π , 1 F tc ), ( λ F 4 3.5 Z K Σ ) + ) + 2.8 π ER δ 1 ] λ e λ − 4 θ ( ( 2 ) 10 F G λ θ – 9 – x d 4 2 + Σ)+ + λ, d ( ) + c.c. = x d c 2 Z λ K ( d π and it depends only on the chiral parameters and the 1 4 F = Z c Σ)(Σ+ K → K K → K = δ R K π K 1 and it depends only on the twisted chiral parameters. This θ A 4 is a product metric of manifolds two Kähler 2 S is can be found by expanding ( Σ+ tc = δ c x d K M ( K 2 A θ M d 4 is S Σ tc xd Z δ 2 d π M 1 4 the local counterterm is [ Z = A π c Σ is the chiral curvature superfield in superconformal gauge. The countert- A 2 24 S D − is real and = ). = ) depends only on the chiral parameters A K R ). In U(1) 3.3 Z 3.3 Next, we would like to check the invariance of ( We conclude that the anomaly is Further restrictions on In order to proceed we must find the most general supersymmetric expression, which is 2.6 In the full supergravity without using the conformal gauge the counterterm is log 12 Σ term ( In addition, under the Kählertransformation the anomaly shifts by the super-Weyl variation ( δ is trivially invariant under in the context ofsupersymmetry type in II spacetime. stringfactorized theory, The as where a hypermultiplet consequence. (2 andthe the worldsheet. Here we vector see multiplet that metrics it are follows from properties of anomalies on and therefore the metric on The Kählerpotential on Kählerpotential on splitting between the chiral and the twisted chiral parameters is well known and is natural that the forbidden two-dimensional anomalies ( goes beyond the Wess-Zumino consistencyis conditions. that After some algebra, the conclusion where erm ( Under a super-Weyl transformation local in any gauge, andof can ( serve as a local counterterm. This is the supersymmetrization JHEP03(2016)022 M with (3.11) M ) we obtain and therefore 3.7 M ]). In that context 2) theories. It is also 49 , – ). Here different parts . and since the transition 46 2 M M K c 6 − Σ ), there is no single Lagrangian in all ) (3.10) λ ) can be interpreted in several different Σ + 2.6 ( is a holomorphic section of a line bundle, θ F 3.9 4 Σ c 6 c 6 e x d ) and we thus have a global description of the 2 d Σ + – 10 – where Σ is the spacetime dilaton superfield, or 3.9 Z → K π must have vanishing Kählerclass. This argument is Σ c explicitly and is therefore not invariant under Kähler 48 M K ). Σ) + ⊃ − 2.6 ) to achieve full Kählerinvariance of the anomaly (here we A Z (Σ + 3.7 c 6 ], with the difference being that we are considering the properties − is trivial, there is no immediate problem since we are not forced log ] (see a refinement of this statement in [ is nontrivial, there is a difficulty. In that case we must cover 50 45 M M 6= 0). This perspective means that c More generally, we should always require that the scale variation of the partition Now, consider the couplings changing in spacetime in such a way that we must use the Finally, we can try to use this analysis to suggest a stronger result. It is well known, The lack of strict Kählerinvariance under ( must be Hodge. This result, which we have now derived using the anomaly, is known of the space of theories rather than the usualfunction target is space a of a local,the specific anomaly globally-defined sigma functional functional model. contains oftransformations. the The background anomaly fields. functional is well In defined our only context, if the Kählerclass vanishes. of spacetime have couplingfunctions constants between them in need different aof patches counterterm, spacetime! in e.g. ( We suggestand that the such various fields a on situation itwe are is such arrive inconsistent. that at no This such the transition wouldanalogous conclusion functions mean to are that needed. that that Therefore, of [ the Kählerclass of patches and transition functions thata involve change Kählertransformations in and the correspondingly counterterm ( transition functions (e.g. spacetime wraps a nontrivial cycle in The lack of strictthe Kählerinvariance means Kählerclass that of our anomalyto is perform not the quite Kählertransformations ( theory. well defined. Different presentations If ofbut the this theory can might be be related absorbed by in Kählertransformations, a local counterterm or in a redefinition of Σ. However, when and we have used it extensively, that the anomaly variation is a well defined local term. supergravity theory [ the action dependsequivalently, on it is theis spacetime the conformal two-dimensional compensator. conformal factor.the This anomalous is Indeed, piece integrating similar of the to the anomaly our effective ( action Σ, in which superconformal gauge assume whose first Chern classM is the cohomologyfor class sigma of models the with Kählerformnatural Calabi-Yau on target in spaces the and context for of general string (2 compactification as a property of the four-dimensional ways with interesting consequences.mation First, should we be can accompanied bya simply a transformation state change law that in to the a Σ Kählertransfor- local counterterm. Second, we can assign and use the first term in ( JHEP03(2016)022 ) as tc 3.13 ). It (3.14) (3.15) (3.12) (3.13) for the M 3.7 c K , c and ! c a δσK M . We can absorb 1 2 c on δa Σ K δ − B + , A σ e g ). This is due to the fact )) B in the first line of ( e λ e λ µ c and of 3.13 e λ, δσ and ∂ ( A A J tc e λ e λ c I in ( 6 , whose exterior derivatives give δσK µ g K µ ∂ e + δσ A − . B and ) A ν I λ e g W e λ A and λ, µ + iδa , ( µ ∂ D . c J + A is non-vanishing, upon letting the coupling µν λ K , ( µ A 2 = δσ ∂ e λ + M I I – 11 – D µ = µ λ λ ∂ π ∓∓ 1

µ µ A 4 tc J , we would not be able to define a single energy- ∂ ∂ µ Σ c J K δ . We note that only the Kählerpotential ∂ + I M A K tc D transformation. We find g I ∂ R K ∂ V δa π − satisfying c − δσ + A by improvements of the energy-momentum tensor multiplet. 24 I e λ and W

c λ − µ c x µ K ∂ , 2 = 0. Our anomaly is ∂ = K tc c , d tc c , but not under Kählertransformations of K K W K W Z I tc K A B ∂ J ∂ π K 1 ∂ ∂ vanishes. 2 I A c i i 2 2 − ∂ ∂ and a chiral K Σ is the chiral curvature superfield. As above, it is invariant under Kähler , the relevant axial supercurrent multiplet consists of real (in Lorentzian = = = = = 2 µ µ J A B C D I e A Z A A J g e g parameterizes the U(1) = log R δa Σ is a chiral multiplet. This point will be important below. Σ δ δ In the absence of supersymmetry, the last term Another way of stating our equations uses the supercurrent multiplet. As we review Let us now extract some useful physical information from our anomalies ( constants wrap some two-cyclemomentum in tensor throughout ourKählerclass of two-dimensional space. This again suggests thatwould the be cohomologically trivial and could be tuned away by an appropriate choice of where transformations of Kählertransformations of Alternatively, we can makeit it thus invariant follows by that also if shifting the Σ. Kählerclass of From the first point of view In a conformal theory chiral multiplets appears in thethat term proportional to in appendix signature) Here we have integratedwell by as parts the and pull-back usedthe of the two-forms Kähler the of metrics Kählerone-forms suffices for our purposesthe to multiplets evaluate of the the anomaly exactly keeping marginal only parameters the bottom components of where JHEP03(2016)022 . We is not (3.17) (3.16) (3.20) (3.18) (3.19) )) λ A ( F Z )+ λ ( and vanishing F . ( A ). We can thus c 3 e λ ) re e λ , 3.13 and c e λ, → I K ( B c r in ( λ − and for constant sources e λ e K c 2 µ c 3 vanish identically since all ∂ S ) ). This part of the anomaly − A . A )). Therefore, modulo Kähler ) 0 c e λ δσK r . λ 2.5 Z r µ 3.13 c 3.3 2) SCFT regularized preserving ∂ supergravity. In this case the λ, , ( δσK B = tc ). This is the reason the coefficient V A c γRK e g K γ = (2 √ + ) (see ( 3.16 √ x γRK λ J 2 x e N ( Σ) √ 2 d λ δ x d F µ 2 ∂ Z d Z 2 I , which arises from the ordinary central charge π e S Σ + 1 λ r – 12 – R π 8 δ 1 µ . First we covariantize it 4 ( π c 1 ∂ 8 θ J 4 ⊃ − I ) is a genuine new contribution to the trace anomaly ⊃ − e g A δσK c 3 A x d Z 2 Z 3.13 partition function is independent of squashing. d 0 2 r r xδσ log supergravity invariants. The most general such term is log Z S 2 Σ d we learn that the partition function contains π A δ 1 4 Z )) with holomorphic ] = δσ ⊃ 2 π λ partition function of a 1 ( 2 S . However, since we are, by assumption, defining the partition V [ 2 c F γ A Z − S √ Z ). Supersymmetry relates this nonlocal universal term to local terms 2 ) + log ) is correct for any compact manifold with the topology of the two-sphere, − γRK λ e Σ ( δ √ 3.16 = F 3.19 x ) becomes ( 2 d 2) SCFTs. γR 3.7 , γR R √ √ ) is physical. The super-Weyl invariant terms in σ σ ], who argued that the is x δ δ 9 2 = (2 A d The analysis extends almost verbatim in U(1) Upon evaluating the partition function on the two-sphere In the evaluation of the partition function for constant sources Even in the absence of supersymmetry, the terms in ( 3.18 N R , it suffices to focus on the term σ with [ anomaly ( where we exhibit the radiusanomaly. of the Note sphere thatKählerinvariant, in or we agreement accompany with Kählertransformationsremark the with that picture ( above, wethe either prefactor say being that reexpressed in terms of the area of the manifold. This is consistent two-dimensional supergravity backgrounds are superconformally flat. we obtain that the U(1) We repeat that this issome not nonlocal a supersymmetric terms local thatof term. generate ( It the is anomaly related ( by supersymmetry to a Using local term. These areis precisely captured the by terms we athe discussed nonlocal anomaly in term ( ( inthat the upon effective a action, Weylreconstruct whose these transformation Weyl terms give variation in reproduces rise the to effective action the by term integrating the anomaly sigma model. in are cohomology nontrivial since they cannot be generated by the Weyl variation of any term, function using a supersymmetricthe regulator, cohomologically Weyl trivial variation termsδ of must arise U(1) from transformations, the anomaly in ( regularization scheme. Indeed, this term is proportional to the variation of the local JHEP03(2016)022 ), w 3.13 (3.21) (3.22) in the and 3 σ ]). If one in ( c/ . a − 53 transforma- ! – tc with various w vanishes. V 51 a e w 2 xδa δσK θ 2) SCFTs. It has 2 M ) does not vanish a e , d partition functions δ + . 1 2 R 2 3.3 tc + 6 S ia − K = 0 but with σ c/ − + appears explicitly in the a e σ B c 3 tc e λ δσ µ ], and it would be interesting K 0 ∂ r c 6 r 56 A e λ + µ ∂ = ν B tc A A µ e g ∂ γRK + ]. Another interesting open question con- µν √ J x 2 λ 55 d + , µ 2 ]). Our derivation shows that this phenomenon ∂ µ S – 13 – 54 e R I A 10 = 0). Due to the anomaly λ π 1 µ 8 µ ∂ ∂ R − e J I a e c 3 g δ + 0 r r δσ ] (see also [ , then this phase can be absorbed by including

9 , w x 8 2 ] = 2 d S (such that Σ = [ Z partition function on the twisted sphere vanishes (see [ a e V i iσ 2 we discuss a classification of supersymmetric backgrounds using our π 1 Z 2 S + = D − e σ Σ is a twisted chiral multiplet. Integrating the anomaly, as above, we a = . The round sphere discussed above corresponds to 2) SCFTs are expressed in terms of the Kählerpotential on the appropriate V e Σ = , w Z log = (2 e Σ In appendix It is important that the anomalies we discussed reflect UV physics. They are indepen- In summary, we have re-derived the result that supersymmetric , and δ a N , to see if our methods shed light on it. of the coupling constants.(the anti-holomorphic But one since vanishes), only theconstants the holomorphic is singular counterterm part physical. ( ofmean. the It dependence They would on were coupling be recentlycerns interesting Calabi’s computed to diastasis, in which understand [ isan a what elegant nice these interpretation observable Kählerinvariant in in singular (2 terms pieces of conformal interfaces [ the two-sphere the partition function needstions. to Hence, transform the with aintroduces nonzero the phase parameter under U(1) partition function. The partition function can be argued to be holomorphic as a function superconformal gauge formalism. Specifically,σ we consider Σ = non-vanishing. We would likesymmetric now backgrounds, to and make in a particular, few about comments the on topologically the twisted background other on possible super- metric. Our methods also led to the suggestiondent that of the Kählerclass the of backgroundpartition spacetime function and was can used besubstituted as Σ explored of a locally a tool in sphere. to flat space. extract this The anomaly. sphere We note that we simply of moduli space of theories [ follows directly from a newanomaly trace anomaly is in tied supersymmetric field by theories. supersymmetry The to new trace the anomaly associated with the Zamolodchikov Now the Kählerpotential foranomaly the since twisted chiralarrive multiplets at And using JHEP03(2016)022 R c ] (see 45 2) models , . Similarly, R c . In the linearized = ++ 2) theory. Then this L , A c + and therefore the sphere θ + couples to the right-moving . This function is not fixed = 1 supergravity theories in θ H 2) models and in particular, , H + N −− 2) models). The corresponding + A , ]. Ψ + 59 θ , i 2 ; (4.1) 57 tc + K + − Ψ = 0, where c ). But there is an additional term in the trace + θ K i – 14 – 2 2) theory viewed as a (0 −− 2.5 , ∼ A + . We will determine their contribution to the con- H σ I λ = ] for background on (0 V 58 2) models are used to construct , ] (see [ 57 2) SCFTs and study their trace anomaly. Our conventions are such , in the superconformal algebra. In this case the remaining degrees of 2) theory, we find it convenient to use the superconformal gauge. But , 2) models contains a term depending on the Kählerpotential, supersym- ++ , j 2) theory there are two natural “conformal gauges.” The difference between , is given as ]). For a related stringy discussion see [ 2) theory. It depends on precisely how we couple the theory to the background 2) supergravity above) and then the vector U(1) symmetry is a global symmetry, , H , 49 – (2 46 A One possibility is to use the gauge As in the (2 Before delving into a technical discussion, let us summarize what we find. The trace Under these assumptions, the supergravity transformations (which include gauge trans- We plan to place the theory in a nontrivial supergravity background. This is simpler unlike the (2 them is in the gauge condition imposed on theU(1) U(1) current gauge field. freedom are in a real superfield the moduli space ofheterotic SCFTs string, cannot where be (0 compact.spacetime. This In is such in cases accordalso it with [ is intuition known from that the the vacuum manifold is [ Kähler-Hodge i.e. in this caseit the is function ambiguous. ispartition physical The function and sphere in unambiguous. partition suchthe function But theories conclusion depends in is that on general the not (0 Kählerclass universal. vanishes do However, hold the in arguments (0 leading to metrizing the ordinary bosonic anomalyanomaly ( that depends onby a the new (0 function offields. the couplings, It isfunction instructive to consider a (2 grounds. We willU(1) refer to thesuffering gauged from an U(1) anomaly. symmetry as axial (asanomaly in in our (0 discussion of linear combination of these two currents.free Note setup, that we even can if we imagine do adding not decoupled have such fields an to anomaly formations achieve for it. a U(1)theory gauge field) to are the non-anomalous. corresponding One can supergravity then and naturally study couple it the in nontrivial supergravity back- when the supergravity theorymust is relate anomaly the free. left-moving First,determines and the to the anomaly avoid in right-moving gravitational the centrala anomalies right-moving charges left-moving U(1) we current U(1) and current we with assume that the there same is anomaly. also Then we can gauge an anomaly free Here we consider (0 that the supersymmetry isin right-moving. Fermi The multiplets exactly [ couplings marginal are operators in are chiral necessarily formal anomaly. superfields 4 (0,2) supersymmetric theories JHEP03(2016)022 − H ++ I I ∂ A (4.5) (4.3) (4.4) (4.2) A J = . ∂ Σ). But I ( B . µν ! + c.c. V. (Σ + δσ 1 2 I is Hermitian. I −− λ = λ ∂ ]) this corresponds M −− V = 0, which is solved Σ) −− 60 δ ∂ µ I K∂ − A I B µ ∂ Σ ∂ δ Σ) , this is consistent only if ( − δ Σ and simply shift Σ. Their 2 + δ I H − θ λ . (see also [ 2 + Σ Σ C −− δ xd V + 2 c d 3 D 2 + ( K∂ I I Z . Conversely, a constant mode of ∂ −− λ is nonlocal. The lack of locality affects ∂ π V c −− 6 V by a constant represents the action of the −− = i ∂ Σ) ∂ + a I δ V Σ) D A δ + + . – 15 – Σ) = D − Σ) D Σ + V ). δ ) leads to a term proportional to δ Σ i −− ia ++ δ ∂ ∂ has to be real in order to eliminate type-A anomalies ( 4.3 + ( . + c.c. (Σ + + Σ + −

H δ K − . σ = 2 + ( 1 2 −− ( a θ R ∂ − 2 = Σ + ++ R δ θ Σ) Σ. The shift of ∂ xd δ d V δ + 2 + + d θ . − + xdθ H Z Σ Σ + 2 . Then the remaining degrees of freedom are in a chiral multiplet xdθ iθ δ δ d 2 a π ( i represent the anomalies that arise in the presence of exactly marginal ν d 4 − + = ∂ I Z θ . Furthermore, + = 0 with i Z 2 = µν Ψ ,B K π δV + c.c.. However as we discussed in section i Z I xd I + π 4 2 ∂ c are arbitrary functions of the couplings. J A = d iθ 12 λ is 2 + I log −−−− = µ µ + Z Σ ∂ I H = A V ,B δ I π I A ia λ Z = c , the chiral superfield Σ = A ν 12 + . Its zero mode is present in Σ but not in V ∂ i ) a log ++ σ Therefore, the anomaly must be of the form The components expansion of ( The first term is the ordinary central charge anomaly. It can also be written as follows Then, the most general expression for the anomaly action is Super-Weyl transformations are associated with a chiral The gauge invariant chiral curvature superfield can be expressed using either of Alternatively, we impose Lorentz gauge on that gauge field These two multiplets are almost identical. Given Σ we can write J Σ H A δ I ∂ (locally) that are present upon expanding thewith first some term. real Similar function considerations show that The functions coupling constants. Already from this we can infer that the metric on So far, in our two slightly different versions of conformal gauge: action on global vector U(1) symmetryimaginary that constant and is does not not gauged. act on This global symmetry shifts Σ by an given only correspond to a linearly growing these fields to locally by Σ = supergravity approximation which we review in appendix JHEP03(2016)022 . H (4.8) (4.9) (4.6) (4.7) con- ] (see c.c. 2 60 − I . In other λ I 2) theories. , λ . , −− ) I λ H ). We can either K∂ I satisfying [ −− ∂ 2.6 ∂ = 0 to + Σ) I δ −− is the left-moving current −−−− λ must be Kähler(in accord = 1 supergravities). Below R T + −− Σ + M N −− δ D , . ) is cohomologically nontrivial. ). This modifies the couplings of the ( R and I K∂ + , I λ 4.5 θ , ∂ = 0 − 2 I λ ) = 0 R − ( = 0. xd ) R −− I 2 I H −− seems like a nontrivial anomaly, which d λ λ R −− ( R . There is no a priori principle that fixes h Z R H −− ++ ]), or simply use linearized supergravity as . + −− + H ∂ Physically, this means that we redefine the π i 61 4 i∂ D K∂ M + I xdθ + 13 − ∂ 2 . – 16 – ( d ++ V + R Z by the function D c.c. π −−−− π −− i i Σ by a local transformation. However, this redefinition is indeed H − 4 8 T ∂ ( − + − + R 2) theories. − D , in Σ + 2) theory in which case this freedom does not exist and Σ 2) supergravity. It can be understood in linearized supergravity before , R δ , 2)-supersymmetric local counterterms). This will allow us to H + π , c 12 ) satisfies the usual consistency conditions including the Wess- but most of the information in ( i xdθ we conclude that the anomaly can be written as to show that the local counterterm is 2 ), we can express the anomaly as an operator statement. The theories ). K 4.3 d = C H 4.1 Z 3.15 −− and R π )–( + H c 12 D ) i ) is holomorphic. This counterterm allows us to absorb some holomorphic trans- 3.14 I C = λ ( Z h 2) theories and it leads to the anomaly ] and appendix , After removing Even though the anomaly associated with Next we should identify the ambiguity (i.e. the cohomologically trivial terms that As in ( The expression ( Note that we cannot absorb log 60 , unless the theory is a (2 13 Σ δ a truly local transformation inpicking any (0 gauge. Theretheory we to simply shift curved spaceFor by conformal additional theories, termsambiguities this in do modification the not play Lagrangian, only a which depends role depend in on on (2 derivatives the of coupling the constants. coupling constants. Such becomes physical ( metric and its superpartnerswords, by when some we function allow the ofLagrangian couplings the that to coupling vanish be constants upon generalin functions, setting (0 we the can couplings addH new to terms constants. to the Such a freedom exists where formations on cannot be absorbedIt in can a be local absorbed counterterm, in in a fact, redefinition it of is not physical in (0 determine the actual anomaly.use For the that full we nonlinear shouldin supergravity supersymmetrize [ (see ( e.g. [ that we assumed exists in the CFT. Our anomaly modifies arise from variations of (0 we study have aappendix supercurrent multiplet with real When the theory is conformal we also have Zumino conditions. We thereforecerning see which anomalies that are our allowed show additionalwith that intuition considerations from the heterotic in metric compactifications, on section whichwe lead to will also find some global restrictions on JHEP03(2016)022 I ). 2) λ , and (5.1) ] and 2.13 (4.10) ia and 59 + I , λ σ 57 ) associated ]. , ) is the Kähler 2.5 62 λ ) J ν λ, . λ ( A ) µ I µ K ∂ λ ∂ I . , , the partition function λ I I µν µ λ λ ∂ ( is the target space of some J + δσK I K µ and should be Kähler[ M A Σ) 2) theories. Kählertransforma- I δ µ , δσG λ ∂ = 2 supergravity [ M is a holomorphic section of a line ( ) + Σ + Σ N δa a c 6 δ 2) SCFTs and argue that the Kähler 1 4 ( , e . + is in fact trivial. δa θ E I 4 + λ M µ θ d σ δσK 4 – 17 – K∂ 1 4 I x d ) contains the term δσ . 4 ∂ ( = 4 d − Σ are chiral and anti-chiral superfields, respectively. c 6 − M δ d 4.10 Z I x λ 2 2 is not only Kähler,but it is also Hodge. Also, as in (2 µ π d 1 M Σ and Z K∂ K, 192 (which we have set to zero for simplicity) and therefore it is δ I J π ∂ 1 ∂ 2 ⊃ H I ∂ i − Z . = = = 2) theory is used as the worldsheet of a compactified heterotic string , log µ J Z M I Σ A δ G log and therefore Σ = 1 supersymmetry in four dimensions and δ M should be trivial. N ]. We extend these conclusions to all (0 M 45 = 2 supersymmetry in = 2 supersymmetry the appropriate superspace expression is In addition to the anomaly that contains the moduli, we also have the usual Weyl While the anomaly functional ( These results are consistent with the expectation from the string application of these The situation with Kählertransformations is as in (2 Expanding it in components with the only nonzero background fields Σ = N N we find | I are chiral and anti-chiral superfieldsthe with exactly Weyl weight marginal zero, couplings, whose whichpotential lowest we on components also the are denote conformal as manifold anomaly, which depends only on the supergravity multiplet. Its superspace expression is The super-Weyl parameters They can be viewed as a conformal compensator in 5 We now proceed toFor the supersymmetric generalization of the conformal anomaly ( class of depends on the choicenot of universal. models. When the (0 it leads to of its chiral superfields.Hodge [ In this case it is known that with the metric on tions can be absorbedbundle over in a shifttheories, of we suggest Σ. that the As Kählerclass of there, The first term is the ordinary anomaly. The second term is the anomaly ( λ JHEP03(2016)022 = 0, (5.7) (5.5) (5.6) (5.2) (5.3) (5.4) w ) can be ) for each A C λ 5.5 , λ , A = 0 is a chiral . λ ( B w λ K R µ K for details) B ν + c.c. ∇ ]. We start with the B 2 3 ∇ A 64 λ − , αβ µ . 4 ) µν N ∇ W B E I C , M λ and we can use ( λ αβ , g ) , a R γ i I J AB A 1 3 B W ∂ λ A − λ K ) ( B λ , , and the metric background (i.e. L − a λ ), we need to know the component A + K K B ν ) we follow [ λ − g µν ] and appendix µνρσ B ( D I ∇ 5.1 c Σ) = ) we expand around a reference point R A C λ ∂ 63 K 5.3 δ ν A K N λ µν B| ∇ , A + µνρσ M θ E , Ξ + ( µ C A Σ + 4 g K, is the Weyl superfield, while Ξ is constructed R γ A C ) and the following definitions for the metric, δ a λ λ ∇ L ( ( c C ν λ 1 3 − θ d ∂ 2 and anti-chiral 2 αβ = 4 K 5.4 ∇ Σ – 18 – L i π K − δ − are chiral and anti-chiral multiplets respectively. 1 W ∂ B K A E x d B A| g K, J λ µν 4 B 192 λ θ B J J ∂ µ γ δσ 2 d 4 R µ ∂ ∂ L ∇ √ I I I ∇ ∇ Z ∂ g ∂ ) we will then specify to A x x d A ) = A and A 4 4 1 4 λ λ A ). Using ( 2 = = = d d λ µ 5.2 µ I λ µ ∇ A = J λ , L ∇ ∇ I Z Z , ∇ A I JK with chiral B g S 2 2 Σ λ JK γ Γ i D A BC ( I π π = 2 chiral and anti-chiral multiplets with Weyl weight 1 1 C = 2 superspace (see e.g. [ B A √ K i K R = ( x 16 16 K 2 A 4 AB N N A , where i d + − ⊃ ⊃ K λ B P Z Z are = 0 and similarly for anti-chiral multiplets. Then )( A γ I = A √ w log = λ x , λ S 4 Σ δ d K and Ξ are chiral superfields. and Z = (Σ αβ A A = λ λ W S For calculating the component expansion of ( To work out the component field expansion of ( connection and curvature on a Kählermanifold with Kählerpotential where multiplet with expressed as a sum term in the sum. Doing this we arrive at In order to findand the then answer use the for fact a that generic product of chiral multiplets with Weyl weight special case Keeping only the bottomdropping components the bosonic auxiliary fields in the supergravity multiplet), we get where respectively. For our anomaly ( from curvature superfields thatin appear curved in superspace. the commutators of super-covariant derivatives expansion of an action of the general form where an integral over chiral JHEP03(2016)022 ) ). B 5.8 (5.9) (5.8) (5.12) (5.10) (5.11) δa ). We also δa and its Weyl µ 5.8 ). I δσ λ 4 ∇ R 5.8 b ∆ I 1 2 2 3 λ K ν I γ − ) to ( ∇ √ 4 J − ∇ E 5.2 = 2 supersymmetric ver- = λ I δσ ν . ν c λ N b . 2 ∇ L . b ∇ µ π λ . δσ µ ν b ν ) + c ∇ K ∇ ∇ I ∇ δa − , µ = 64 ν ∇ K αβ J J ∇ λ R ∇ ) in the second line of ( λ λ + + c.c.) ν W is a sum of a holomorphic and anti- the FTPR operator (see appendix ν 2 ν 2 µν J 4 σ ∇ I γ ∇ αβ ∇ R 4 λ F ]). The supersymmetric Gauss-Bonnet λ J 2.13 c C 1 3 I I µ c C µ ∆ λ µν W λ λ + 65 ∇ µ + γ ν − ∇ F µ + I − ( 1 3 F ∇ b ). ∇ λ R ∇ R K µν γ I Ξ µ µ I − = λ R b √ – 19 – ∇ ∇ 2 3 ) µ µν 5.10 1 x µν λ 12 one realizes that the terms in the third line of ( K 4 ∇ K ( − − ∇ J R d L J I K R γ B 4 I F − b 1 3 λ ∇ E 4 E i g Z + KJ I µ + θ I E 2 − b 4 + ∇ ∇ 1 8 contains the Euler combination π R δσ γ ) in the first line. It appears with the Riemann tensor of δσ 1 µ µν − x d K µ δσ √ δσ 4 γ I αβ R ∇ ν J 192 4 ∇ d ( λ √ ∇ S 2.14 W ∇ R γ 2 µ R Z µ µ Z µ partition function we simply need to integrate the combination αβ √ ∇ − . Using σ ∇ ∇ . Note that this action is completely covariant under target space I x 4 δ ∇ 4 I J J 4 W λ µν S 1 6 λ S K λ d λ µ = ν 1 6 − b + ∇ Z R γ ∇ I + 1 3 2 ) on λ I K δσ π J λ b 2 1 δσ − µ 5.9 b ∇ 2 96 I ∇ 1 2 µν J b ), the hats denote covariant derivatives with respect to target space diffeomor- I J ⊃ ∇ R I K ) is cohomologically trivial if g γ Z is an arbitrary function of the moduli and ∆ i 2 2 1 i 2 δσg 2.13 √ c 5.8 = 0 this is precisely the combination that appears in the + + − − + log To arrive at the Using the expressions in appendix c Σ . To take into account the complete anomaly we have to add ( δ variation is the supersymmetrization of ( appearing in ( Indeed, consider the following local superspace counterterm The combination Ξ sion of theterm Gauss-Bonnet may invariant include (see as afore, e.g. prefactor ( [ an arbitraryholomorphic holomorphic function function of of the the moduli, moduli. in There- which case it reduces to where For identify the new anomalyM ( can be written as a variation of a local term. Specifically, As in ( phisms acting on the diffeomorphisms and we nicely identify the term ( we arrive, after several integrations by parts, at JHEP03(2016)022 , M = 4 (5.14) (5.15) (5.17) (5.13) N 2 (the full − vanish. + c.c. (5.16) Z F ∆ Σ δ . E θ 4 transforms homogeneously x d 4 αβ d e Σ + c.c. it transforms as δ W Z , F ∆ , 2 transforms with weight 12 ) under an infinitesimal Weyl trans- π e Σ Σ + 1 δ δ E K/ F 5.2 E e 192 ∆ ) is true for any superconformally flat θ + a 2 = 2 supersymmetric generalization of the 4 F, 4 a − is not only Kähler,but it is also Hodge. In − K ) = is special-Kähler.It would be interesting to N 5.13 x d 1 = 2, we will now see that the trace anomaly 4 F 24 0 → d of the partition function is taken care of by the d r M r e Σ M ΣΞ – 20 – δ δ K F Z e Σ = + 2 + ). δ must be trivial. (For certain cases with an a π ] = F F 8 ] that 4 ). The Weyl superfield Ξ = 2 Σ + − M S ). As in 5.13 e [ Σ 66 δ B ( δ K Z = that vary in spacetime and wrap a nontrivial cycle in 5.11 Z θ E I → 8 ]) λ ]. We note that ( K log 63 x d 4 Σ 14 d δ e Σ is invariant). We then find δ Z 2 E π ] and [ 1 10 192 background is superconformally flat, the super-Weyl invariant terms in 4 = S e Σ, we use (cf. [ δ Z 14 ∆, the chiral projection operator, is the log dual, it has been argued in [ Σ As in two dimensions, this means that To find the change of the anomaly polynomial ( The Kählerambiguity 5 δ Since the F 14 δ S. Kuzenko, D. Morrison, Y.mas, Nakayama, V. K. Niarchos, Skenderis, H. E. Ooguri, Witten,the H. and Perimeter Osborn, J. Institute K. Zhou for Papadodi- for its usefulwork very discussions. of kind Z.K. NS hospitality would was during like the to supported course thank in of part this by project. DOE The grant DE-SC0009988. Z.K. is supported by understand when this happens in general.) Acknowledgments We would like to thank C. Bachas, C. Closset, S. Cremonesi, L. Di Pietro, N. Ishtiaque, tion and therefore also underto a hold finite for transformation. the The partition invariance can function be ( explicitly seen addition, using background we argue that theAdS Kählerclass of Therefore, choosing the anomaly polynomial is invariant under an infinitesimal joint transforma- Kähler-Weyl On the other hand, under a Kählershift where FTPR operator (see also appendix with weight one, while thesuperspace chiral density superspace density ambiguous local counterterm ( is invariant under a correlated Kählershift and Weyl transformation. formation as claimed incompact [ four-manifold if we express the prefactor in terms of its volume. we find JHEP03(2016)022 x λ e ]. It (B.1) (A.1) (A.2) (A.3) (A.4) → 36 – x 33 ). 2.5 , . , . ) ) ) x x ( ) 2 ( 2 µ (4) µ 2 2 λ , (2) 2 δ 4 x ) is fixed as follows. We compute the x δ 2 x ( ∆ 2 2 log( 2.13 2 λπ 96 log λπ 3 γ δσλ 2 = = √ – 21 – 4 | x 8 1 1 | ) and ( 1 4 x 1 32 768 | x d ). | 2.5 − = Z = 4 2.13 | 1 = 2 under constant rescaling of the coordinates (anom) λ 8 x | (anom) | λ δ 1 d δ x | ) by parts. One obtains, up to cohomologically trivial terms, that the 2.13 = 4 we write d In We collect some propertiesarises of in the our Fradkin-Tseytlin-Paneitz-Riegertone context operator integrates in [ ( theanomaly case is that there is only one exactly marginal modulus and This is matched by the anomaly ( B The FTPR operator and its properties whose anomalous Weyl variation is whose anomalous Weyl variation is This is the contact term that is reproduced by the anomaly functional ( change of the contactby term writing in the two-point function as necessarily reflect the views of the funding agencies. A Normalization of theThe anomaly normalization of the anomaly ( and Development. This research wasical supported Physics. in part Research by at Perimeterthrough Institute Perimeter Industry for Institute Canada Theoret- and is by supported theand Province by Innovation. of the J.G. Ontario Government through also of the acknowledgesand Ministry Canada further from of support Research an from ERA ansions grant NSERC or by Discovery recommendations Grant the expressed Province of in Ontario. this material Any are opinions, those findings, of and the conclu- authors and do not United States-Israel Bi- nationalas Science by Foundation (BSF) the under Israel grantand Science 2010/629 Foundation Z.K. as center are for well supported excellencemittee. grant by P.H. (grant the is no. supported I-COREacknowledge 1989/14). by Program support Physics A.S. of from Department GIF the of — Planning Princeton the University. and German-Israeli A.S. Budgeting Foundation and Com- for S.T. Scientific Research the ERC STG grant 335182, by the Israel Science Foundation under grant 884/11, by the JHEP03(2016)022 2 = ∇ and γ Σ δ (B.7) (B.2) (B.3) (B.4) (B.5) (B.6) 2 √ e R ∆ Σ (with δ → δσ 2 2 − ∇ e is → 5 and in Lorentzian 2 δσ , 2 −− δσ . x ], where one also finds 4 6 63 R − 2 2 3 R + 4 ∆ δσ 2 3 − µ ν which transforms as + ∇ R ). . It satisfies ∆ . ∇ 2 2 2 . R µ µν ∇ µ π ∇ π 2 3 . ∇ R ∆ under a super-Weyl transformation ≡ ∇ 4 µν − 2 µν Σ µν which is used in section γ = 8 2 ∆ δ 4 R R 4 = 64 2 − δσ E 2 4 4 δσ , Σ. Its precise definition in terms of super- − γE δσ + 2 δ − e = det γ E √ µ ∆ = 2 δσ µν – 22 – x γ g √ Σ → ∇ R 4 2 δσ , δ µνρσ d x 2 4 R − − we have ∆ 4 2 C µ is the complex conjugate of ∆ d 6 − S = 4 ∇ 4 δσ R − R ν S 1 3 ++ µνρσ Z ∇ x R + C µ 2 with for the coordinates, which makes it easier to compare with δσ δσ R ∇ = 2 3 4 2 2 4 δσ − − − − = E 2 x 4 4 = = = E ∆ 2 ∆ R R µν σ σ − = 4 the analog of the FTPR operator is the chiral projection operator R δ δ σ σ d δ δ δσR 2) this is the chiral projection operator 2 , − is the Weyl tensor. = 2 in in the conformal gauge, where = 2) = (2 , γ N µνρσ δR N C log = For In two dimensions instead of ∆ The conformally covariant operators have generalizations in chiral superspace. In 2 1 2 = 2, This can be derived using C.1 (2 We will use the notation spinors. In Euclideansignature signature they are two real− independent coordinates. The Ricci scalar is given by covariant derivatives and curvaturereferences superfields to is the original reviewed literature. in [ C Review (and conventions) of two-dimensional supersymmetry under super-Weyl transformations, which are parameterizeda by similar a transformation chiral superfield for the anti-chiral projector ∆ with the infinitesimal transformation parameterized by a chiral scalar superfield Here δE d such that and the expression for the Euler density which we normalize to when it acts on a scalar. Another property of ∆ is the FTPR-operator with the defining property that under Weyl rescaling of the metric, where JHEP03(2016)022 V (C.5) (C.6) (C.7) (C.8) (C.9) (C.1) (C.2) (C.3) (C.4) (C.11) (C.10) . 2) theory with + , χ , − , D = 0 is a conserved vector = ∓ . ∂ Y − V ∓∓ e T. χ −− j + + D i θ i∂ ≡ D +

− e T = 2 − are defined by , θ ∂ } = . . ∓∓ ∂ ,D − D e λ R . . − − D ++ = 0 = 0 = 0 , W = 0 R = = − ∓ = 0 − Y V A ++ ++ D − W D j j e λ χ D { such that D − − D e −− −− T D D ∂ ∂ = = – 23 – , = 0 = = + − , e ∓∓ ,D e T, e λ λ ) are automatically satisfied and the first becomes T , χ Y J ++ − ∓ + V A −− −− χ + j j ∂ i∂ + . D D D C.4 D D Y D − ++ ++ θ = ∂ ∂ = 2 i = = 0 = + } − + χ + ∓∓ , Y −− D J − ∂ ∓ , R such that ∂θ χ D + + is a conserved axial current and twisted chiral superfields D

W D D + = ) λ { = − ∓∓ ]. They are related by mirror symmetry. First, there is the U(1) Y C.4 D + ∓∓ 60 ≡ J supermultiplet is obtained formally by acting with a mirror symmetry D R ) is replaced by A A ∓∓ D j C.8 The U(1) There are two interesting energy-momentum supermultiplets in a (2 Chiral superfields The supercovariant derivatives, which can be obtained from the four-dimensional ones Often, there is a chiral and then ( In this case transformation on ( and then the last two equations in ( It immediately follows thatcurrent the bottom component, Often, there exists a twisted chiral operator an R-symmetry [ supermultiplet of Wess and Bagger by dimensional reduction, are The algebra is JHEP03(2016)022 . ] ... 60 + (C.14) (C.15) (C.16) (C.17) (C.12) (C.13) ].) We ia 44 + Σ), where σ , gauge field in . H A −− . j ) . ++ , ∂ ) is consistent with the ) by using the defining −− −− −−−− U ∂ C.13 C.13 H + , , + c.c. , θ ++ , . ++ ) ∂ + θ + R W −− + L Σ − = 0 + θ , − ++++ ++ global symmetry transformation. , ) = 0 2 represents the U(1) e T T Σ. H D d = 0. In components we have ++ V −− −− + + . a −− Λ + θ θ L − R ++ −− Z R S + + ++ − −− − Λ R θ θ j ++ D U L −− −− + ∂ + + − − + ++ −− ∂ − i∂ −− ∂ + + Λ D ∂ ++ ( ++ ∓∓ θ −− − – 24 – + 2 is a general multiplet. L ) is invariant under ( Λ J +++ + H + ++ + D S S -symmetry the supercurrent multiplet consists of −− U R , and the linearized coupling takes the form −− = 0 and the only degree of freedom is Σ = + + − D + R j i∂ θ θ −− C.12 H i i −−−− −− = + −−−− + T ∂ ++ ++ −−−− S ( T − − ∂ Σ = ( −−−− X H δ + −− θ + H −− , which are real in Lorentzian signature. They satisfy [ H 4 D ∂ θ , → H → H supergravity. The supergravity multiplet is ( by a constant is a U(1) +++ + δ d d + supergravity is analogous. In that case the only mode in the H S S θ + . Under the super-Weyl transformations Σ is shifted by a chiral a H → H A , Z + + ++ a V − dθ ν −−−− iθ iθ H ++ = ∂ −−−− T Λ is anti-chiral. , Z H − − L H µν δ = −−−− ++ −− = R L j j T µ and that the action ( = = = A ). ++ −− Σ. The shift of δ are arbitrary superfields. Note that the first line of ( C.4 2) R R H −−−− , is the conformal factor in the metric and T 2) theories with a conserved L , is real (in Lorentzian signature) and Σ is chiral. The linearized coupling to matter σ We couple this theory to linearized supergravity in a standard fashion. We introduce The situation in U(1) In the superconformal gauge Now we discuss linearized coupling to supergravity (see also the analysis of [ Above Λ is chiral and three real superfields, The complete super-diffeomorphism group is generated by the following transformations For (0 three superfields, Hence, the R-current is vectorial superfield superconformal gauge is the twisted chiral superfield C.2 (0 reality of relations ( Here Lorentz gauge This is invariant under the linearized transformations where start from the caseH of U(1) then takes the form JHEP03(2016)022 . , = w 2 + with θ ζ (D.1) + −−−− (C.18) α charge. ζ ]. These H ia . , A w − 44 α σ ) and . ζ and − are invariant. ζ , H Σ = 0 = 0 Σ = e + , ζ + + . ζ ζ w 2 are neutral under it. ) ia ia and θ + − w iF + σ σ Σ 2 e ia . we we − α and + i i 2 2 ζ γ σ , ia √ − − α + ζ σ R ( e −−−− − − + ζ ζ H θ ia ia i 4 − + ++ σ σ ∂ − e e is not the complex conjugate of H − −− −− w −−− −− Ψ R-charge, and ∂ – 25 – A ++ + , ∂ , ∂ ∂ − ++ = 0 = 0 = 0 + H ∓ − − Ψ ζ ζ ζ 2 −− .) ia ia −− i∂ V + − ∂ ∂ σ σ + i = we we D − i i can be complex and 2 2 ∓ ζ = = are the Ricci scalar and the field strength of the U(1) R-gauge field, a , which is the conformal factor, is real. But we will allow non-unitary + + have the same U(1) 2) supersymmetric backgrounds in superconformal gauge are components of the “gravitino” and originate from σ − F , − + ∂ R + ζ 2) supersymmetric background Σ can be classified as follows according ζ ζ , ia ia and −−− − + and σ σ R supergravity where the conformal factor is in a chiral multiplet. It is trivial e = (2 e + A ζ and Ψ N 2) and (0 ++ ++ , + ∂ , and will view them as four independent complex variables (no particular reality). ∂ The The first class of backgrounds preserves one supercharge of a given U(1) We will set the fermionic components of Σ to zero; i.e. Σ = Most of our analysis will be local. The global considerations are easily implemented As we said, the advantage of using this presentation is that there is no need to use The curvature is in the invariant chiral superfield 2) background in two dimensions can be brought locally to a superconformal gauge and = , Note that to the preserved supercharges given by the Killing spinors Without loss of generality we can take the Killing spinors as nonzero ( The first equation isThe the remaining standard equations restriction state due that to the flat fermionic space components superconformal of symmetry. α The conditions for supersymmetry are We will assume that backgrounds in which later. We will denote the Killing spinors for the supersymmetry variation as then all the informationusing is U(1) contained in ato chiral repeat superfield the Σ. analysis in (More U(1) precisely, we will be supergravity. We simply use flat space ordinary superconformal symmetry. In this appendix we repeatauthors the used classification linearized of supergravity supersymmetric to backgroundsand find of the then [ equations covariantized them. for supersymmetric Instead,will backgrounds we not will have to use rely the on(2 superconformal linearized or gauge. the This full way nonlinear we supergravity. The point is that every respectively. respectively. D (2 Here Ψ JHEP03(2016)022 a R- ) = A − 1 (D.6) (D.4) (D.5) (D.2) (D.3) and (up to ζ respec- , σ + 1 ) we find −− ζ −− − 1 ∂ x ζ − 1 , ζ . + 1 − 2 ζ ζ and ) reproduces the = 0 + ++ ) becomes the topo- D.5 . . ) x ++ ) ∂ 6= 0. They lead to the ). Therefore, D.5 −− + 2 −− x ++ v ζ − ( vanishes. Without loss of −− x + x ζ ∂ + ζ − 1 1 v ζ ζ ) we find another expression, ) . The supersymmetry algebra −− − 2 ) gives ≡ , x + log + log ++ −− , ζ ) + log v , equation ( + log = 0 or )). σ x x − + 2 D.2 ia . Consistency of the two solutions ( ia ia ζ v ia (2 a + 2 −− σ, ++ − + ++ ζ + x − x ( ∂ σ σ −− σ σ ( − 1 0, the solution ( ∂ → − ζ ++ ) Σ = 2 x a + log −− ++ −− → ( is a constant describing the Ω-deformation. −− ++ ∂ must have an isometry given by the vector ∂ , x ++ ∂ ∂ 2 ia w x a θ − + + − ( 1 ++ w, ++ + = – 26 – ζ ζ ζ ζ ). Imposing invariance under ( i with nonzero 2 x + 2 x ↔ 2 2 ) are arbitrary functions of θ 2 σ ζ − 2 2 . This is the topological twist. The anti-topological v . The solution ( θ θ θ and iθ i iσ w w + 2 − 1 , ζ i −− i 2 ] ζ , 2 + 2 σ x σ ). Here Σ = 2 ( ζ = + 2 − 44 ζ −− − and − a −− ∂ a ζ ) ) + 2 ia ia σ − 1 → x ia ia Σ = 2 ζ − + = . ζ − − and 2 σ σ + − , − ζ ) and ( iσ , ζ σ 6= 0 and σ σ 6= 0. And the background is − 1 are invariant under the vector + − ζ v ). For . Imposing invariance under ( v ) by a , Σ = Σ = 1 L = ζ + i 1 ++ ) = (1 ζ Σ = 0 D.4 a Σ = ( Σ = ( ∂ D.2 ) and − 2 = and → + 2 = 0 with nonzero 1 ζ , ζ ζ ++ a σ δ + + + 2 1 1 2 x ζ ζ ζ ( δ + = 0 with arbitrary and + ζ 2 w σ = 0 it suffices to assume that one component of ) with δ = , 1 v δ + iσ D.2 1) and ( ζ , = Two limiting cases are interesting. For In the second case There are two such cases depending onFor whether The second class of backgrounds preserves two supercharges with opposite U(1) a = 0. We find for every ++ . Up to a conformal transformation we can take the Killing spinors to be ( − x logical background ( two-dimensional Ω-background in [ are arbitrary function of thesatisfies invariant combination twist corresponds to v ( The corresponding isometry vector is generality take i.e. This means that a superconformal transformation by log topological twist and Ω-deformation respectively. again ( which is related toconstrains ( where tively. (In Euclidean spacethat they our are analysis holomorphic is and local; anti-holomorphic global functions.) considerations Recall restrict thesecharge, functions. denoted as ( ζ JHEP03(2016)022 ]. , (D.7) (D.8) Field Phys. , Σ. The D 2 B 733 B 37 SPIRE 6= Nucl. Phys. IN , ∗ ]. (2014) 1139 SPIRE Phys. Lett. Nucl. Phys. ]. IN , , [ (1986) 565] [ . Two-Sphere Partition 325 12 , 43 = 0 11 , gauge field has zero curvature 2 2 + x x A (1991) 486 ζ i i/ 2 ia 2 − 1 + 1 + ) is not unitary since Σ σ 2 2 e ]. θ θ B 363 D.7 ]. + + ++ 58 2 2 SPIRE x x , , ∂ Commun. Math. 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