Shortening anomalies in supersymmetric theories

Journal of High Energy Physics, Jan 2017

We present new anomalies in two-dimensional \( \mathcal{N}=\left(2,2\right) \) superconformal theories. They obstruct the shortening conditions of chiral and twisted chiral multiplets at coincident points. This implies that marginal couplings cannot be promoted to background superfields in short representations. Therefore, standard results that follow from \( \mathcal{N}=\left(2,2\right) \) spurion analysis are invalidated. These anomalies appear only if supersymmetry is enhanced beyond \( \mathcal{N}=\left(2,2\right) \). These anomalies explain why the conformal manifolds of the K3 and T 4 sigma models are not Kähler and do not factorize into chiral and twisted chiral moduli spaces and why there are no \( \mathcal{N}=\left(2,2\right) \) gauged linear sigma models that cover these conformal manifolds. We also present these results from the point of view of the Riemann curvature of conformal manifolds.

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Shortening anomalies in supersymmetric theories

Received: December Shortening anomalies in supersymmetric theories Jaume Gomis 1 2 4 8 9 10 Zohar Komargodski 1 2 4 6 9 10 Hirosi Ooguri 1 2 3 4 7 9 10 Nathan Seiberg 1 2 4 5 9 10 Yifan Wang 0 1 2 4 9 10 Princeton 1 2 4 9 10 NJ 1 2 4 9 10 U.S.A. 1 2 4 9 10 Open Access 1 2 4 9 10 c The Authors. 1 2 4 9 10 0 Joseph Henry Laboratories, Princeton University 1 Caltech , Pasadena, CA 91125 , U.S.A 2 Rehovot 76100 , Israel 3 Kavli IPMU, WPI, University of Tokyo 4 Waterloo , Ontario, N2L 2Y5 , Canada 5 School of Natural Sciences, Institute for Advanced Study 6 Weizmann Institute of Science 7 Walter Burke Institute for Theoretical Physics 8 Perimeter Institute for Theoretical Physics 9 Princeton , NJ 08540 , U.S.A 10 Kashiwa , 77-8583 , Japan We present new anomalies in two-dimensional N theories. They obstruct the shortening conditions of chiral and twisted chiral multiplets at coincident points. This implies that marginal couplings cannot be promoted to background super elds in short representations. Therefore, standard results that follow from N = (2; 2) spurion analysis are invalidated. These anomalies appear only if supersymmetry is enhanced beyond N = (2; 2). These anomalies explain why the conformal manifolds of the K3 and T 4 sigma models are not Kahler and do not factorize into chiral and twisted chiral moduli spaces and why there are no N = (2; 2) gauged linear sigma models that cover these conformal manifolds. We also present these results from the point of view of the Riemann curvature of conformal manifolds. Anomalies in Field and String Theories; Conformal Field Theory; Supersym- - Dedicated to John Schwarz on his 75th birthday 1 Introduction 2 3 4 N = (4; 4) conformal manifolds Riemannian curvature Supercurrent bundle A The Wess-Zumino perspective B Special geometry relation C Four-point function involving supercurrents D SU(2)out selection rules Supersymmetric contact terms and the shortening anomaly Curvature of conformal manifold and factorization Conformal Field Theories (CFT's) often come in a continuous family labeled by their exactly marginal couplings. This family, known as the conformal manifold M, is endowed with a canonical metric, the Zamolodchikov metric [1]. The Zamolodchikov metric is determined by the two-point functions of the exactly marginal operators. The general Riemannian structure of conformal manifolds was rst discussed in [2, 3]. Such conformal manifolds appear in theories with extended symmetries, such as suThese conformal manifolds also play an important role in the AdS/CFT correspondence and on string worldsheets. In the former case, the conformal manifold of the boundary CFT maps to the space of vacua in the bulk Anti-de Sitter space (AdS). In the latter case, the conformal manifold of the worldsheet theory maps to the space of solutions of the equations of motion in spacetime. One of the fundamental properties of the conformal manifold of two-dimensional (2; 2) superconformal eld theories (SCFT's) is that it factorizes locally2 into the product N = 1Some general arguments about when such families may exist can be found in [4]. 2There are examples where the conformal manifold is modded out by a discrete symmetry, which prevents it from being a product globally [5, 6]. We thank D. Morrison for a useful discussion about this point. of two Kahler manifolds, Coordinates of Mc and Mtc are the coupling constants of the exactly marginal operators constructed from the dimension ( 1 ; 12 ) operators in the chiral and twisted chiral rings [7] of 2 linear sigma model with a Calabi-Yau threefold as its target space, the factorization (1.1) was proven in [8] by combining string theory worldsheet SCFT and target space arguments.3 worldsheet view points, see [9, 10] and [8, 11{13], respectively. More recently, in [14] the Weyl anomaly on the conformal manifold of N = (2; 2) theories was used to rederive the factorization (1.1). The argument in [14] applies to tion made in [14] was that the coupling constants parameterizing Mc and Mtc could be promoted to supersymmetric dimension (0; 0) background chiral and twisted chiral multiplets, in the spirit of [15]. Factorization of the conformal manifold (1.1) then followed from the classi cation of anomalies of the partition function under super-Weyl transformations. Alternatively, one can also easily provide an argument for factorization (1.1) in the spirit of [16]. torus T n SCFT is locally [17], On the other hand, it is well-known that the conformal manifold of the n-dimensional while the conformal manifold of the K3 SCFT is locally [2] , (see also [19]), for some n.4 These examples appear to be at odds with the factorization (1.1) of the conforcase in (1.2), these conformal manifolds do not factorize locally into a product of Kahler manifolds. In fact, they are not even Kahler manifolds. This is in spite of the fact that the Another way of presenting this puzzle is the following. Normally, when a particular global symmetry of a theory implies some special properties, extending that symmetry does not ruin those properties. Here, we see a counter-example to that. Speci cally, if 3For SCFT's realizing Calabi-Yau compacti cations, Mc and Mtc parameterize the moduli space of complex structure and complexi ed Kahler class. that it should factorize as in (1.1) . This conclusion turns out to be wrong. Later we will extended supersymmetry. factorization mentioned above, the OPE has a pole5 expansion (OPE) between a chiral multiplet O and a twisted chiral multiplet O~ of scaling dimensions ( 12 ; 12 ) does not have poles. On the other hand, in all the counter-examples to J++(x2) + where the symbol j on the left-hand side picks up the bottom component of each multiplet. The operator J++ appearing on the right-hand side must be both chiral and twisted chiral, and with scaling dimensions (1; 0) and R-charges (2; 0). J++ is the bottom component of a multiplet implies that the R-symmetry is enhanced and that the SCFT enjoys enlarged SCFT, the current J++ enlarges the R-symmetry from U(1) to SU(2).6 Let us imagine that we deform the SCFT with exactly marginal operators A powerful idea [15] is to promote and ~ to background chiral and twisted chiral superelds. We will see that whenever the OPE between O and O~ is singular as in (1.5), then the couplings cannot be promoted to such background super elds due to an anomaly! We can either promote the s to background chiral super elds or the ~s to background twisted chiral super elds, but we cannot do both simultaneously. To our knowledge, this type of obstruction has not been discussed before. Before we explain how this comes about let us explain the physics of promoting couplings to background elds. The e ective action as a function of the background satis es the required symmetries when the operators do not have contact terms that spoil those symmetries. For example, if a conserved current is not conserved at coincident points then the background e ective action will not be gauge invariant. Another example is if in a CFT the operator equation T e ective action fails to depend just on the conformal class of the background metric. Similarly, in supersymmetric theories, for the e ective action to depend on and twisted chiral super elds, the operators O and O~ have to obey their de ning equations and ~ as chiral O = 0 ; D+O~ = D O~ = 0 , as if we are in Lorentzian signature. state in the SCFT. 5Throughout this note we study the theories in Euclidean space. However, we use Lorentzian signature notation with coordinates x etc., which are complex conjugates of each other. The reason for using this notation is that we use complex conjugation notation on the charged chiral objects like the odd coordinates 6It is important that the current J++ which enlarges the R-symmetry corresponds to a normalizable not only at separated points (which holds by de nition) but also at coincident points. Loosely speaking, one could say that the equations (1.7) have to be obeyed o -shell. As with all anomalies, our shortening anomaly can be understood as the failure of the partition function to be invariant under certain background eld transformations. The operator equations (1.7) at coincident points would be a consequence of the partition function being invariant under certain background super eld transformations, akin to the standard way conservation laws follow from the invariance of the partition function under background eld transformations. When the partition function is not invariant under these back eld transformations, we encounter an anomaly. This point of view is elaborated in appendix A. We will show that the OPE (1.5) induces contact terms of the form D+(1)O(Z1) O~(Z2) D+(2)O(Z1) O~(Z2) and that it is impossible to tune both of these contact terms away (the notation will be explained in section 3.) Therefore, the e ective action does not depend on the background coupling constants as if they were chiral and twisted chiral super elds. Some of the background couplings have to sit in long multiplets. We call this phenomenon a shortening anomaly.7 The discussion above is reminiscent of the clash between conservation of the vector and axial current in theories with fermions, where by adjusting counter-terms either symmetry can be preserved, but not both simultaneously. As is standard in such situations, where some operator equations are violated at coincident points, when we turn on nontrivial backgrounds for and ~ then the contact terms (1.8) lead to nontrivial operator equations. Depending on which counter-terms we choose, one of the equations below has to be true: For constant couplings, where only the bottom components of operators O and O~ remain short, as they should. The obstruction to promoting both chiral and twisted chiral couplings to short multiplets (which exists only if supersymmetry is enhanced) invalidates the conclusions found in [14] for such theories.8 We therefore conclude that factorization (1.1) breaks down only 7We would like to emphasize that the operators O, O~ remain short in the standard situation where the couplings are constant. Indeed, ; ~ have vanishing beta functions, since the operators O; O~ have no operators to combine with (see [20] and also earlier literature, e.g. [11]). In particular, (1.5) does not induce a beta function. The shortening anomaly is in the background elds, not in the operators. The operator equations are modi ed only in nontrivial con gurations for the background super elds (see equation (1.9)). 8The analysis in the spirit of [16] is also invalidated by this anomaly since an implicit assumption in such an analysis is that it is possible to preserve the shortening conditions of all the dimension ( 12 ; 12 ) chiral and twisted chiral operators both at separate and coincident points. and ~ are turned on, the does not factorize.10 ifolds (1.2) (with n > 2 even), (1.3) and (1.4) are indeed endowed with such an operator of R-charge (2; 0) and have enhanced supersymmetries, thus resolving the paradox.9 This phenomenon is similar to a familiar situation in supergravity. The target space = 2 supergravity with hypermultiplets has a quaternionic target space, which is not Kahler (see [22] for a review of the scalar manifolds of supergravities in various dimensions). supergravity theory because it has a multiplet including a graviphoton and a gravitino, that resolves our puzzle, which includes a conserved spin- 32 current, plays a similar role to the graviphoton multiplet in this supergravity analog. For a related discussion see [23]. We can relate our discussion to supergravity more directly if we view our 2d models as worldsheet theories for strings and we study their spacetime description. The low energy scalar manifold indeed takes the factorized form (1.1). On the other hand, compacti cations on manifolds leading to enhanced spacetime supersymmetry, such as T 6 or K3 As with all 't Hooft-like anomalies, our analysis leads to theorems about the nonsupersymmetric RG ow in which all the infrared marginal couplings are realized as chiral or twisted chiral couplings along the ow, then it is guaranteed that the corresponding coupling constants in the infrared are in short representations and hence there is no shortening anomaly. Therefore, the conformal manifold would have to factorize into a chiral Kahler manifold and a twisted chiral Kahler manifold. Therefore, one can immediately conclude that there cannot exist an N = (2; 2) RG ow that realizes the full conformal manifold of the K3 SCFT. Indeed, constructions of gauged linear sigma models (GLSM's) [24] which lead to subspaces of the K3 conformal manifold are known (see for example [18, 25]), but it has never been possible to embed the K3 SCFT in a UV completion that covers the full conformal manifold. The same holds for T 4 but not for T 2, which does not have a the complete T 2 SCFT conformal manifold.11 We now see that the obstruction for K3 and T 4 is due to an anomaly in the infrared that must be matched in the ultraviolet.12 9Such an operator does not exist for the T 2 SCFT. And indeed, the T 2 conformal manifold (1.2) 10The conformal manifold of the worldsheet SCFT is a subspace of the supergravity scalar manifold. (1; 1; 1; 3) with a superpotential W 3. The twisted chiral coupling is realized by the complexi ed FI parameter and the chiral coupling by the single complex parameter in G3(X). conformal manifold of sigma models on K3 or T 4. It is interesting to relate this discussion to recent developments concerning the S2 compactify such theories on S2 and they compute, respectively, the Kahler potentials of the chiral and twisted chiral deformations [28, 30] (see also [14, 31]). However, this statement is not meaningful, if the total space does not factorize as in (1.1)! Therefore, when (One can understand it again as being due to the failure of spurion analysis.) Because of manifold, therefore, what one can extract from the sphere partition function is at best the Kahler potential on some Kahler submanifolds of the conformal manifold. An alternative sphere compacti cation that utilizes the extended supersymmetry may exist and it may probe the full conformal manifold. shortening anomaly. We also provide a complementary perspective on the anomalies by studying the Riework [8, 19, 32]. The study of the mixed chiral and twisted chiral exactly marginal curvature components leads us to establish a factorization theorem: the conformal manifold conserved current of R-charges (2; 0), precisely the same operator J++ responsible for the The outline of the paper is as follows. In section 2 we set the stage by discussing the OPE of chiral and twisted chiral super elds. In section 3 we show that if a particular short representation J++ appears in the OPE, then one inevitably nds an obstruction to imposing the shortening conditions simultaneously on both the chiral and twisted chiral super elds. In section 4 the same result is established by analyzing the Riemann curvaperconformal algebra and prove that the conformal manifold of such theories indeed takes the form (1.4). In this section we further show that the extended supercharges acquire non-trivial holonomies when transported around the conformal manifold. In appendix A, we discuss the shortening anomaly from the Wess-Zumino point of view. Some of the calculations of the various curvatures are presented in appendices B, C, OPE of chiral and twisted chiral operators in N = (2; 2) SCFT's operator can be realized as the top component of a chiral multiplet O with U(1)+ R-charges (1; 1) or as the top component of a twisted chiral multiplet O~ with R-charges (1; 1). These multiplets obey the shortening conditions D+O = 0 D+O~ = 0 O = 0 ; O~ = 0 : Our aim is to determine to what extent these shortening conditions can be maintained as we explore the conformal manifold of an N = (2; 2) SCFT. Monitoring the shortening conditions (2.1) under an exactly marginal perturbation leads us to analyze the contact terms in ; I ) are points in superspace. In this section we determine the operator product expansion (OPE) of a chiral and a twisted chiral multiplets, whose top components yield marginal operators, leaving the analysis of contact terms to the follo In supersymmetry, it is often useful to employ spurion analysis. E.g., the coupling constants of chiral operators O are promoted to background chiral multiplets [15] . This procedure makes sense only if the operator equation D O = 0 is respected also at coincident points (loosely speaking, we can say that O is chiral o -shell). This is because when we write the partition function depending on some background elds, by taking derivatives with respect to the background elds, we can probe the correlation functions of the corresponding operators both at separated and at coincident points. By de nition, operators equations are always obeyed at separated points, but they may fail at coincident points. The famous (continuous) 't Hooft anomalies arise when a conservation equation is not obeyed at coincident points. As a consequence, the partition function depending on the associated background elds does not obey the naively expected equations (in the famous case of the chiral anomaly, the partition function is not gauge invariant). Similarly, we can couple twisted chiral background elds to twisted chiral operators as long as the corresponding shortening conditions are valid o -shell. A general method that guarantees that some conservation equation is obeyed also at coincident points is to construct a regularization obeying the conservation equation. This automatically tunes the contact terms in the infrared to zero. Such a regularization may not exist, if there is a genuine anomaly in the conservation equation. In the next section we will establish the existence of some contact terms that violate the shortening conditions (2.1) at coincident points. Therefore, the standard arguments relying on the selection rules of ows that contain all the infrared chiral and twisted chiral couplings cannot exist. SCFT in two dimensions when it has an operator of dimensions (1; 0) and R-charges (2; 0). Unitarity implies that this operator, which we denote by J++, is a conserved current, that the supersymmetry is enhanced. A typical (but not the only) example of supersymmetry enhanced by such an extra encode the contributions of non-chiral operators and descendants. For simplicity, we are suppressing some coe cients that we will make more explicit later. In particular, a hyper-Kahler manifold M , there is a unique holomorphic 2-form 2 H2;0(M ), and the extra R-current is expressed as J++ = ab +a +b. Chiral and twisted chiral operators are Oj = kab + O~j = kabgbc cd are fermions. The OPE (2.5) where k 2 H1;1(M ), gab is the Kahler metric on M , and then follows from More generally, one can show that the existence of J++ of dimensions (1; 0) and Rcharges (2; 0) alone is su cient for the pairing (2.4) and therefore the OPE (2.5) follows commute with the U(1)+ group must be larger than just U(1)+ R-symmetry in an N = (2; 2) SCFT, the R-symmetry U(1) and the dimension ( 1 ; 1 ) chiral and twisted chiral operators must furnish a representation of it. Therefore, J++ cannot act trivially and hence some pairing as in (2.4) and (2.5) must be present. Unitarity and supersymmetry further imply that the extra R-current J++ is the bottom component of a very short multiplet, which we denote by J++. It obeys extra R-current as where Oj denotes the bottom component of the operator O. By taking OPE's of both sides of the equation with another twisted chiral operator O~(x3) (or, in other words, using the symmetry of the OPE coe cients), we nd J++(x3) + U(1)+ U(1) D+ J++ = D J++ = D J++ = 0 : For convenience, we summarize the R-charges of various objects used in the following: Our nal goal is to monitor the shortening conditions (2.1) as we explore the conforsupersymmetry manifest. Introducing coordinates (x; ; ) (suppressing the +; The chiral coordinates y D = D = fD; Dg = 2i@ : D y = 0 D y = 0 (c; c), O~ is (c; c) and J++ is simultaneously (c; c) and (c; c). We want to determine the dependence of three-point correlators on the superspace position of the operators. We start with the supertranslation Ward identities. two points in superspace (x1; 1; 1) and (x2; 2; 2), we can de ne two independent even linear invariants:13 i 12 12 = y1 2i 2 1 = y1 Supertranslation invariance implies that correlators depend on the position of operators through zi j and z i j . The coordinates we have de ned are rather convenient. Indeed, if the i-th operator is chiral, the correlators depend on zi j only, while if it is anti-chiral they depend on z i j . Our correlator of interest is: ; I ). The shortening conditions, hO(Z1) O~(Z2) J ++(Z3)i ; D+O = D D+O~ = D O = 0 ; O~ = 0 ; D+ J ++ = D J ++ = D J ++ = 0 ; together with supertranslational and rotational invariance imply that the correlator (2.13) depends on z1+3+, z2+3+ and z1 2 . A subtlety in SCFT's that must be taken into account is the existence of a superconformal invariant X(Z1; Z2; Z3) constructed out of three points in superspace [33]. Superconformal invariance and nilpotency of X(Z1; Z2; Z3) imply that the most general correlator consistent with superconformal invariance is given by hO(Z1) O~(Z2) J ++(Z3)i = [1 + a X(Z1; Z2; Z3)] : id ; d ; d ), where g = exp i(xP + Q + Q) . This yields the supertranslation invariants ( 12 2). Note that z1 2 = 12 i 12 12 and z1 2 = hO(Z1) O~(Z2) J ++(Z3)i = Moreover, using that we obtain our desired OPE: hJ++(Z2) J ++(Z3)i = The superspace correlators we have constructed obey, by construction, the shortening conditions (2.14) at separated points. Our next task is to study the shortening conditions at coincident points (2.3) and determine whether counter-terms can be adjusted so that the shortening conditions for chirals and twisted chirals (2.1) can be both simultatherefore conclude that the correlator at separated points is Supersymmetric contact terms and the shortening anomaly The OPE (2.18) may lead to some contact terms (2.3).14 To understand these we need to study the superspace derivatives (the derivatives with respect to the second argument follow from these) = @++@ log(jx12j2) = log(jx12j2) = Our strategy to compute (3.1) is to extend the well-known formula, @++ to Green's function in superspace. This line of inquiry makes manifest an inherent ambiguity in de ning the derivatives (3.1) in superspace. In order to de ne these derivatives we need to specify the behavior of Green's function in superspace for the left-movers, and di erent choices yield di erent answers. This ambiguity is a manifestation of the fact that bution.15 In fact, we will nd an ambiguity of 1+2 12 + (2)(z12) in de ning 1=z1 2 . This is 14A related discussion appeared in [34], where it clari ed the need for a contact term, which had been found earlier in [35, 36]. A more modern discussion of that problem appeared in [37]. 15On the other hand, the pole 1=x on the 2-plane of (x++; x ) can be extended to an unambiguous distribution because R d2xf (x++; x superspace, however, is akin to 1=(x distribution de ned via 1=(x )2 and cannot be extended to an unambiguous distribution. The is well-de ned for any smooth function f . The pole 1=z1 2 in @ (1=x ) corresponds to a particular choice of regularization. the addition of (x).16 As a result, the derivatives (3.1) su er from certain ambiguities. However, we will nd that, whichever choice we make for these ambiguities, we cannot remove the contact terms for both chiral and twisted chiral operators simultaneously. We can compute (3.1) starting from the Green's functions in superspace D+(1)D(1) log(z1+2+z1 2 ) = D+(1)D(1) log(z1+2+z1 2 ) = 4 Physically, (3.3) and (3.4) can be interpreted as the Schwinger-Dyson equations or the Ward identities for the shift symmetry of a chiral multiplet and a twisted chiral multiplet respectively. But these are not the only choices. Consider the following identity D(1)z1 2 = D(1) a log(z1+2+z1 2 ) + b log(z1+2+z1 2 ) ; = 2 ib (2)(z12) 1+2 : fD+; D+g = 2i@++ : Amongst these, there are distinguished canonical choices Preserves chiral Ward identity Preserves twisted chiral Ward identity Symmetric violation of Ward identities We note that there is no choice of a that preserves simultaneously the chiral and twisted chiral Ward identity. Even though the derivatives are subject to some ambiguities, we now proceed to unambiguously establish our shortening anomaly. and D+(1) 1=z1 2 . is proportional to i ) di er by distributions proportional to (x). 17Note that equations (3.3) and (3.4) only x D+(1) 1=z1 2 and D+(1) 1=z1 2 up to terms propor Anomalies arise when we have an operator that should satisfy an operator equation, such as (2.1), and one nds that such an equation is only correct at separated points while at coincident points there are various contact terms.18 Establishing the anomaly amounts to showing that these contact terms cannot be removed by rede ning the scheme since scheme rede nitions change the theory by contact terms. the most general OPE between a chiral and a twisted chiral super eld in a theory with an enhanced R-symmetry, now allowing for supersymmetric contact terms, is given by J++(Z2) + r (2)(z12) 1+2 1+2 J++(Z2) ; where r is a scheme dependent constant, which can be shifted by changing the scheme. In order to establish our shortening anomaly we must show that it is not possible to tune the contact term r such that the shortening conditions for a chiral and a twisted chiral multiplet can be maintained simultaneously. Since the contact term in (3.9) is annihilated by D(1) and D(2), these shortening equations are automatically preserved. Acting with D+(1) and D+(2) we get using (3.6)19 D+(1)O(Z1) O~(Z2) D+(2)O(Z1) O~(Z2) 2 iC a (2)(z12) 1+2 J++(Z1) + r (2)(z12) 1+2 J++(Z1) : Preserving the chiral and twisted chiral shortening conditions along the conformal manifold requires tuning the coe cient of the supersymmetric counter-term to obey twisted chiral : r = 2 iCb ; r = cannot simultaneously preserve the chiral and twisted chiral shortening conditions along the conformal manifold. By tuning the contact term, we can either preserve the chiral or the twisted chiral constraint, but not both. This is our shortening anomaly. The fact that we cannot preserve both constraints simultaneously is analogous to the situation in two dimensions with vector and axial anomalies [39], where contact terms cannot remove both anomalies simultaneously. For a complementary derivation of our shortening anomaly in the cohomological approach based on the Wess-Zumino consistency conditions, see appendix A. The contact terms we encountered lead to an operatorial violation of the shortening equations upon deforming the theory by background super eld sources for the exactly 18By turning on suitable background elds the ambiguities in the contact terms can be described as an ambiguity in adding local counter-terms constructed out of the operators in the theory and the background elds. When such background elds are present, the problem with contact terms can be uplifted to problems at separated points. marginal operators (as in (2.2)). The shortening conditions become ators to full- edged background chiral and twisted chiral super elds is impossible whenever the theory includes the operator J++. We emphasize that the violations in (3.12) depend on the fermionic components of the multiplet to which the couplings have been promoted. Curvature of conformal manifold and factorization A CFT with exactly marginal operators comes equipped with additional structure beyond the metric on the conformal manifold M. Operators in the CFT are sections of vector bundles over M. A canonical example of this is the set of exactly marginal operators, which are sections of the tangent bundle T M. Transporting operators in the vector bundle along M leads to operator mixing, which is governed by a connection on the vector bundle [3, 40, 41]. The curvature of these connections captures geometrical and topological data about the vector bundle of operators in the CFT. The curvature of the tangent bundle T M can be used to prove theorems about the conformal manifold M. The computation in [8] of the curvature of T M in two-dimensional guments, was used to prove that the conformal manifold of such SCFT's factorizes (1.1). The result in [8] follows from the vanishing of the mixed chiral and twisted chiral curvature components, which implies that the holonomy group is the direct product of two commuting subgroups, which in turn implies factorization (1.1). In this section we establish the and only if the SCFT has a conserved current with R-charges (2; 0). We now proceed to compute the mixed components of the Riemann tensor of M using CFT techniques, in the spirit of [3, 19, 41, 42]. The tangent bundle T M comes equipped with a metric compatible connection, whose curvature we would like to nd. The curvature of this connection is determined by a certain four-point function of the exactly marginal operators, which we denote by Ui. The formula for the curvature can be written as [42]20 Rijkl = ) hUi(0) Uk(y) Ul(1) Uj (1)ic : The symbol `RV' stands for the prescription where we cut out small discs around the xed operators and remove the power-law divergent terms.21 The subscript c stands for the 20Note that our de nition of the curvature includes an overall normalization of 4 compared to that of [42]. This is the convention in which the special geometry relation takes the standard form. In addition, we de ne as usual O(1) = limx!1 x 2 O O(x) while keeping all the other insertions angular dependence and thus this singularity vanishes upon integrating over the angles. We recall that dimension (1; 1) operators do not appear in the OPE since a nonzero OPE coe cient would lead to a connected correlator, which is de ned by subtracting the three possible disconnected contributions to the four-point function. As shown in [41], an explicit formula for the curvature can be given in terms of the CFT data: spectrum of operators and OPE coe cients. further exploited in [19] we can show that the only non-trivial mixed chiral and twisted chiral moduli four-point function that needs to be studied is = @y++ @y = @y++ @y where we have denoted the exactly marginal operators constructed from operators in the chiral and twisted chiral ring by Fi = 1 Z F~~i = Using the superconformal Ward identities introduced in [8], the four-point function of interest (4.2) can be expressed in terms of the four-point function of chiral and twisted chiral operators of dimension ( 1 ; 1 ) We are now ready to compute the curvature using (4.1). It follows from (4.4) that the independent components of the Riemann curvature tensor are Rik~`~j and Ri`~k~j . The connected component prescription in (4.1) can be extended to both sides of (4.4), and therefore, by pulling the operator located at x to in nity we arrive at Integrating by parts in (4.1) and remembering that we are integrating over the complex plane with disks around the punctures removed, the answer reduces to contour integrals around the punctures. In order to get a non-zero contribution to the curvature, the function c z=0;w=1 22More singular terms are removed by the prescription in (4.1). One can verify that there is no contribution from in nity. where J++ has R-charges ( 2; 0). Using J++(w) O~k~(z) Oi(1) = around y = 0. demonstrates that there is no contribution to the curvature from the contour integral In conclusion, we have shown that there is a non-trivial component of the mixed only if the SCFT has a current with R-charges (2; 0). In such a SCFT, the Riemann curvature is given by and the conformal manifold M no longer factorizes. = (4; 4) conformal manifolds by S+A++, and the SU(2)R currents by J++ ), where ( is an important example of extended supersymmetry. In this section, we use the forgive a purely eld-theoretic derivation that the local geometry of the coset is (1.4). We This gives a geometric perspective on our shortening anomaly. and A are doublet indices for the algebra, respectively. We use the convention 12 = 21 = 1 for the invariant tensors ; AB and their inverses. For the right moving sector, we use dotted indices. We denote the weight 12 ; 12 are exactly marginal operators which preserve the N = (4; 4) superconformal symmeTheir two-point functions are F AA_ = hOi _ (x)Oj _ (y)i = hFiAA_ (x)FjBB_ (y)i = where ij is the Zamolodchikov metric. Riemannian curvature RiAA_;jBB_ ;kCC_ ;`DD_ = log jy j FiAA_ (0) FjBB_ (y) FkCC_ (1) F`DD_ (1)E : (5.4) 2 D (anti-)chiral multiples are Fi11_ (Fi22_ ), while those for (anti-)twisted chiral multiples are As shown in the previous section, there are non-zero curvature components in mixed Ri11_;j12_;k21_;`22_ = Ri11_;j21_;k12_;`22_ = using the Bianchi identity, we can also determine23 Ri21_;j12_;k11_;`22_ = On the other hand, the curvatures in the purely chiral directions are controlled by the Ri11_;j22_;k11_;`22_ = ij k` + i` jk where CiJk are the chiral ring coe cients. between the curvature components (5.5), (5.6), and (5.7) Ri11_;j22_;k11_;`22_ + Ri21_;j12;k11_;`22_ + Ri21_;j22_;k11_;`12_ = 0 : This allows us to determine Ri11_;j22_;k11_;`22_ = Comparing this with (5.7), we obtain as a by-product the following constraint on the chiral ring of any N = (4; 4) SCFT CiIkCjJ`gIJ = would be nice to verify that the cohomology ring of M satis es the constraint (5.10).24 Combining these results, the full Riemannian curvature is RiAA_;jBB_ ;kCC_ ;`DD_ = AB CD + ( AC BD + AD BC ) A_B_ C_ D_ : This implies that the conformal manifold of an N = (4; 4) SCFT is locally the coset M = Supercurrent bundle RiCC_ jDD_ A B = identities to be RiCC_ ;jDD_ ; _ A_; _B_ = The nontrivial SU(2)out SU(2)out holonomies shown in these curvatures means that it is Since the tangent bundle T M is a tensor product of the left and right supercurrent bundles and the bundle of weight chiral primaries, the curvature tensor for the supercurrents computed here can be combined with the curvatures for the chiral primaries computed in [19] to reproduce the Riemann curvature (5.11) on M. Although M is not Kahler, it has Kahler sub-manifolds. In fact, the maximal Kahler sub-manifold of a quaternionic-Kahler manifold is middle-dimensional [44]. In our case, the maximal Kahler submanifold of M is locally S = 24A curious observation is that (5.10) implies that there is a uniform bound on chiral ring coe cients (squared) associated with the R-charge (1; 1) chiral primaries. The bound takes the schematic form C2 < up, say, as at the conifold point. ) DS A+++(0) FiCC_ (y) FjDD_ (1)S B+++(1)E : hS+A++(0) FiCC_ (y) FjDD_ (1)S+B++(1)ic = The y-integral in (5.13) can then be performed AC BD RiCC_ ;jDD_ ; A; B = If we only turn on the marginal couplings associated to chiral multiplets and explore that sub-manifold of M, then there is no shortening anomaly, and the argument of [14] leading associated to chiral or twisted chiral operators corresponds to the sub-manifold (5.17) . We thank Kevin Costello, David Morrison, Kyriakos Papadodimas, Ronen Plesser, Adam Schwimmer, Stefan Theisen, and Edward Witten for useful discussions. J.G.'s research was supported in part by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. Z.K. is supported in part by an Israel Science Foundation center for excellence grant and by the I-CORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number 1937/12). Z.K. is also supported by the ERC STG grant 335182 and by the United States-Israel BSF grant 2010/629. H.O. is supported in part by U.S. Department of Energy grant DE-SC0011632, by the Simons Investigator Award, by the World Premier International Research Center Initiative, MEXT, Japan, by JSPS Grantin-Aid for Scienti c Research C-26400240, and by JSPS Grant-in-Aid for Scienti c Research on Innovative Areas 15H05895. N.S. was supported in part by DOE grant DE-SC0009988. Y.W. was supported by the NSF grant PHY-1620059 and by the Simons Foundation Grant #488653. H.O. thanks the hospitality of the Institute for Advanced Study and Harvard University, where he spent his sabbatical in 2015 - 2016, and of the Aspen Center for Physics, which is supported by the National Science Foundation grant PHY-1066293. N.S. thanks the hospitality of the Weizmann Institute of Science during the completion of this The Wess-Zumino perspective Anomalies arise when we have some operator that should satisfy an operator equation, e.g. @ j = 0 or T points while at coincident points there are various contact terms. The essence is to show that these contact terms cannot be removed by rede ning the scheme. Indeed, scheme rede nitions change the theory by various contact terms and so we need to demonstrate that the anomaly is invariant under scheme rede nitions. A convenient way to establish it is to introduce background elds for the various operators. Then scheme rede nitions correspond to adding new local terms to the action, which depend on these background elds and also, possibly, on the operators in the theory. We would like to examine the operator [Q+; Oj], which is normally zero if O is chiral. To this end we couple a background eld to this operator. A standard procedure is to couple the super eld O to a background eld in the superpotential but then we do not have a source for the redundant operator. Therefore we will couple O to a source in the Kahler potential. We add a corresponding term for a twisted chiral super eld O: Now we imagine computing the partition function Z[A; B] (with A; B super elds). It is useful to tabulate their charges 0.26 So the standard expectation is that The standard expectation is that the partition function would not actually depend on Z[A; B] = Z[A + D+ Z[A; B] = Z[A; B + D+ . This should be viewed, for example, in analogy with Z[g ] = 0 for the conformal anomaly case. What we would like to test is whether we can respect (A.3) whether the shortening of the background multiplets is consistent with supersymmetry. The general principles that we reviewed above tell us that for in nitesimal Z[A; B] should be a local functional of the sources and operators in the theory i.e. log Z[A; B] = 1 ( 1=2; 1=2) ( 1=2; 1=2) with Llocal some local function of the couplings and operators. The right hand side in (A.5) is restricted by demanding that it is supersymmetric and also by demanding that it obeys the Wess-Zumino consistency conditions [45]. Let us assume that the partition function is invariant under the namely, O~ obeys the twisted chiral shortening conditions at both separated and coincident points. We can then write the variation under as follows (the formula for the variation under + is analogous) log Z[A; B] = some constant. Equation (A.6) respects supersymmetry (because it is a R d4 integral), and it is consistent with the R-symmetry (D+D B carries R-charges (0; 0) and carries R-charges ( 2; 1) and therefore D carries ( 2; 0) and thus it exactly cancels the R-charge of J++). Furthermore, (A.6) obeys the Wess-Zumino consistency 26This standard expectation follows from the fact that these conditions hold \o shell," namely there is a regularization where this is true. Technically, it means that there are no cohomologically nontrivial contact terms in correlation functions of these redundant operators. condition since D+D B is invariant under B ! B + D+ +. Therefore, (A.6) does not violate the fact that the partition function is postulated to be invariant under of the local term We now proceed to prove that (A.6) is cohomologically non-trivial. If (A.6) were cohomologically trivial then one could add a local term to log Z[A; B] such that the right hand side of (A.6) would vanish while retaining supersymmetry and invariance under transformations. It is clear (by integration by parts and using (2.7)) that the could cancel the right hand side of (A.6). However, (A.7) spoils the invariance of the partition function under transformations. One can easily verify that indeed the right hand side of (A.6) is physical as long as we insist on supersymmetry and invariance under To summarize let us make some comments in (A.6) is nonzero, then it turns out that we may not be able to respect both (A.3) and (A.4). In other words, we cannot embed the coupling constants of chiral and twisted chiral operators into short multiplets. At least some of the couplings have to be in longer multiplets. 2. We can view the and transformations as analogous to U(1)A and U(1)V transformations in 2d electrodynamics. If we preserve one we must give up on the other, but choosing which one to preserve is at our discretion. Therefore, the situation is very similar to the way the usual chiral 't Hooft anomalies arise [39]. Note that from equation (A.6) we can immediately write the anomaly in operatorial formalism. This is because couples to D+O and so we nd But since our partition function is invariant under transformations and hence depends Hence in a \fermionic background" for the twisted chiral coupling, the operator O ceases to be chiral. It is now straightforward to make contact with the analysis in the bulk of the paper. Our discussion in this appendix has shown that there may be an anomaly with coe cient and that it would manifest itself as (A.9). Comparing with (3.12) we thus see that this coe cient is nonzero whenever the OPE coe cient in (3.9) is nonzero. Furthermore, the analysis in this appendix sheds light on the choices we could make in (3.11). Indeed, we could have chosen whether to postulate that the partition function preserves (A.3), (A.4), or none of the two. As in all cases with anomalies, these various choices are related to each other by adding counter-terms to the action, e.g. the one we discussed in (A.7). By choosing the coe cient of this counter-term carefully, we can change the scheme from the transformations are obeyed to the one where transformations are obeyed. Special geometry relation formations generated by chiral primary elds is determined in terms of the Zamolodchikov geometry relation. Below we will give a simple derivation of this relation using (4.1) (our derivation is valid for any central charge c). Fi(1)Fj (y)Fk(1)F`(0) = @y++ @y )hOi(1)Oj (y)Ok(1)O`(w)ic The curvature is then computed by integration by parts, with non-vanishing contribuin the limit limy!0 @w++ @w )hOi(1)Oj (y)Ok(1)O`(w)ic We can act with the derivatives on the prefactor jy wj2 and thus reduce the problem to studying the y ! 0 limit of hOi(1)Oj (y)Ok(1)O`(0)ic. There is a contribution from the unit operator in the t and u channel as well as a contribution from the (2; 2) chiral primaries, Rk`ij = yli!m0hOi(1)Oj (y)Ok(1)O`(0)ic = gij gk` + gi`gkj where gIJ is the metric associated with the R-charge (2; 2) chiral primaries. Four-point function involving supercurrents Let us focus on the connected four-point function hS+A++(x) FiCC_ (y) FjDD_ (z)S+B++(w)ic : Since the four-point function is holomorphic in z and w, it is determined completely by the poles in the OPE between the supercurrents and other insertions. We rst consider the singularities in x, and denote the polar terms in (x z) by I1; I2; I3 From the OPE of the N = 4 supercurrents, S+A++(x)S+B++(w) = AB ful analysis of the integrand shows that such contributions are absent for this particular four-point function. Using again (C.5) and also where we have dropped the disconnected pieces and also the terms involving the SU(2)R currents since Fi are SU(2)R singlets. It is then easy to obtain using the OPE between T I1 = I1 = Finally from the OPE between S+A++ and FjDD_ we get and following the same steps as above we obtain I3 = Putting together (C.4), (C.8) and (C.10) while taking the limit w ! 1, we arrive at S+A++(z)F BB_ (w) = 2 @w++ @ I2 = 2 h@y++ S+A++(z)O _ (w) = Similarly looking at the OPE between S+A++ and FiCC_ I2 = I3 = = 2 hS+A++(0) FiCC_ (y) FjDD_ (1)S+B++(1)ic AC BD Fi11_ (0) Fk11_ (y) F`22_ (1) Fj22_ (1) + Fi21_ (0) Fk11_ (y) F`12_ (1) Fj22_ (1) Fi21_ (0) Fk11_ (y) F`22_(1) Fj12_ (1) = 0; function of the exactly marginal operators FiAA_ FjBB_ FkCC_ F`DD_ respects SU(2)out Ward 2@x++ Oi1(x)Q2+2Ok1(y)O`2(z)Q1+1Oj2(1) ; Q2+1Oi1(x)Q2+2Ok1(y)O`2(z)Oj1(1) ; SU(2)out selection rules In this appendix we prove that identities.28 Q2+2Oi1(x)Q2+2Ok1(y)Q1+1O`2(z)Q1+1Oj2(1) Q2+1Oi1(x)Q2+2Ok1(y)Q1+2O`2(z)Q1+1Oj2(1) Q2+1Oi1(x)Q2+2Ok1(y)Q1+1O`2(z)Q1+2Oj2(1) Moreover, from (C.5) we can derive Q2+1Oi1(x)Ok1(y)O`2(z)Q1+2Oj2(1) = Q2+2Oi1(x)Ok1(y)O`2(z)Q1+1Oj2(1) : Putting together (D.2), (D.3), (D.4) and (D.5), we obtain Q2+2Oi1(x)Q2+2Ok1(y)Q1+1O`2(z)Q1+1Oj2(1) + Q2+1Oi1(x)Q2+2Ok1(y)Q1+2O`2(z)Q1+1Oj2(1) + Q2+1Oi1(x)Q2+2Ok1(y)Q1+1O`2(z)Q1+2Oj2(1) = 0; which leads to (D.1). 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Jaume Gomis, Zohar Komargodski, Hirosi Ooguri. Shortening anomalies in supersymmetric theories, Journal of High Energy Physics, 2017, 67, DOI: 10.1007/JHEP01(2017)067