Surveying 4d SCFTs twisted on Riemann surfaces

Received: April Surveying 4d SCFTs twisted on Riemann surfaces Antonio Amariti 0 1 2 3 Luca Cassia 0 1 2 Silvia Penati 0 1 2 Open Access 0 1 2 c The Authors. 0 1 2 Supersymmetry, Conformal Field Theory 0 Piazza della Scienza 3 , 20161, Milano , Italy 1 Sidlerstrasse 5 , Bern, ch-3012 , Switzerland 2 Institute for Theoretical Physics, University of Bern 3 Albert Einstein Center for Fundamental Physics Within the framework of four dimensional conformal supergravity we consider N = 1; 2; 3; 4 supersymmetric theories generally twisted along the abelian subgroups of the R-symmetry and possibly other global symmetry groups. Upon compacti cation on constant curvature Riemann surfaces with arbitrary genus we provide an extensive classi cation of the resulting two dimensional theories according to the amount of supersymmetry that is preserved. Exploiting the c-extremization prescription introduced in arXiv:1211.4030 we develop a general procedure to obtain the central charge for 2d N = (0; 2) theories and the expression of the corresponding R-current in terms of the original 4d one and its mixing with the other abelian global currents. Supergravity Models; Anomalies in Field and String Theories; Extended 1 Introduction 2 3 4 Twisted reduction of N = 1 SCFTs Twisting with avors Classi cation of the solutions Twisted reduction of N = 2 SCFTs Twisting with avors Classi cation of the solutions Twisted reduction of N = 3 SCFTs Classi cation of the solutions Twisted reduction of N = 4 SCFTs Classi cation of the solutions Further directions A The anomaly polynomial B Supersymmetry variations of the auxiliary elds in N = 3; 4 SCFTs Introduction Two dimensional (super) conformal eld theories ((S)CFTs) play a central role in the worldsheet description of string theory and in the formulation of the AdS3/CFT2 correspondence. Moreover, being the conformal group in nite dimensional, many exact results can be extracted from their algebraic structure. Classifying 2d CFTs is anyway a di cult nding new examples of conformal theories is not straightforward. A powerful laboratory to build in nite families of 2d CFTs is supersymmetry. SCFTs in 2d can be obtained by compactifying 4d SCFTs on curved compact 2d manifolds. In this process some of the original supersymmetry charges survive whenever Killing spinor equations arising from requiring fermion variations to vanish, admit non-trivial solutions. In general, this does not happen since on curved manifolds there are no covariantly constant Killing spinors. However, as suggested in [1] (see also [2, 3]), this problem can be circumvented by performing a (partial) topological twist, i.e. by turning on background gauge elds for (a subgroup of) the R-symmetry group along the internal manifold in such a way that its contribution to the Killing spinor equations compensates the contribution from the spin connection. More generally, one can also turn on properly quantized backuxes for other non-R avor symmetries. In this case preserving supersymmetry also requires to set to zero the associated gaugino variations. Although this procedure does not allow to extract the matter content of the 2d theory, useful information on its IR behavior is provided by the 2d global anomalies that can be obtained in terms of the 4d ones and of the background Focusing on 2d theories with N = (0; 2) (or equivalently, N = (2; 0)) supersymmetry, the corresponding central charge cL (cR) is proportional to the anomaly of the abelian R-symmetry current inherited from the exact 4d R-current JR, obtained by a-maximization [5]. However, under dimensional reduction 4d abelian global currents can mix with the exact 4d J , hence the exact 2d R-current has to be determined by extremizing the 2d central charge cL (cR) as a function of such a mixing. The program of such c-extremization principle was derived in [4]. An interesting phenomenon regarding the mixing of global currents with J Riemann surfaces. There, it was observed that even though there is a global (baryonic) symmetry, that does not mix with J mixing with J richer structure of baryonic symmetries. xed point [18, 19], it has a non-trivial xed point. This phenomenon is generalizable to cases with a Motivated by the former discussion, in this paper we engineer the partial topological twist in the natural setup of conformal supergravity and study systematically the twisted compacti cation on constant curvature Riemann surfaces of 4d SCFTs with different amount of supersymmetry. In this uni ed framework we investigate the cases of is a genus g Riemann surface. We study the conditions to preserve di erent amounts of supersymmetry in 2d by solving the Killing spinor equations arising from setting to zero the variations of the gravitino and of the auxiliary fermions in the Weyl mulwe also turn on vector multiplets associated to global non-R symmetries. In this case an additional constraining equation for Killing spinors arises from setting to zero the variation of the corresponding gaugino. = 3; 4 theories, where avor symmetries are absent, we also discuss the possibility of twisting in two steps. This consists in a rst twist along an abelian subgroup of SU(3)R U(1)R or SU(4)R, reducing the R-symmetry and leaving some vector multiplets associated to non-R global symmetries. A further twist along such symmetries corresponds vide the (formal) expression for the anomaly coe cients and the central charge in terms of the 4d anomalies and the uxes, as obtained by c-extremization. Concurrently, the explicit expression for the exact 2d R-current is given as a linear combination of the 4d R-current and global non-R symmetries. In section 7 we conclude by commenting on some possible future lines of research. In appendix A few necessary details on the anomaly polynomial are collected. In appendix B we provide further details on the vanishing of the Twisted reduction of N = 1 SCFTs manifold M = R1;1 is a Riemann surface of genus g and constant scalar curvature. Twisted compacti cation of this class of theories has been already discussed x the general scheme that we will use in the N -extended cases. R1;1 coordinates are labelled (x0; x1), while the ones on connection ! on satis es the relation are (x2; x3). The spin where R2(!) is precisely a representative 2-form for the rst Chern class of the tangent bundle class is usually denoted as c1(T ) 2 H2( ; Z). The curvature for a Riemann surface can be written in terms of the volume form d Vol and the Gaussian curvature K as For later convenience we de ne the normalized scalar curvature sgn(K), the normalized volume form R(!) = 2 R(!) = K d Vol 8> jKjd Vol for K 6= 0 for K = 0 and the total normalized volume so that R(!) = 2 In general, compacti cation on breaks supersymmetry completely, since on arbitrarily curved manifolds there are no covariantly constant Killing spinors. Along the lines of [21], in order to put a 4d theory on a curved manifold and preserve some supersymmetry we couple the theory to a conformal supergravity background that reproduces the desired spacetime geometry. The whole superconformal group is gauged and the corresponding gauge elds are organized into the Weyl multiplet as follows (we use notations and conare indicated in table 1. Here Pa,Ka are vector generators of translations and special conformal transformations, Mab and are generators of Lorentz rotations and dilatations, Q and S are the spinorial supercharges. The U(1)R R-symmetry generator TR assigns charge 1 to the positive chirality supercharges Q and charge +1 to their conjugates we will often write TR = 5 with 5 = i 0123. The supersymmetry transformation laws of the independent gauge elds read ea = b = = D " where "; are the Majorana spinors associated to Q and S transformations, respectively. The covariant derivative is de ned as D " iA TR)". Since we are only interested in theories on curved manifolds with rigid supersymmetry, we x the Weyl multiplet to be a collection of background elds describing the geometry of spacetime. In order to preserve Lorentz invariance on R1;1 we set all the spinor elds to zero and assign possibly non-vanishing components to bosonic forms only in the (x2; x3) directions. As follows from (2.8), in general this choice breaks superconformal invariance. However, some Q-supersymmetry survives if the geometry admits non-trivial covariantly may have non-trivial solutions if we turn on a non-zero background also for the R-symmetry gauge connection A [1] such that the two contributions coming from A and !ab in the = 0).1 This equation covariant derivative cancel each other. More precisely, focusing on constant solutions, we rst apply the exterior derivative to ] = iR (A) 5 " = 0 where R (!23) and R (A) are the curvatures of the connections !23 and A , respectively. Given the particular form of the curvature R (!) = , we choose A such that its curvature is also proportional to the normalized volume form R (A) = where the parameter a is constrained by the Dirac quantization condition Substituting (2.10) in (2.9), we then obtain R(A) = a 5 " = 0 We postpone the search and classi cation of non-vanishing solutions to section 2.2.2. 1To begin with one could solve the equation = 0 for non-vanishing , by setting = 14D=" [23]. The We now consider the case in which the original 4d theory also admits a global abelian nonR symmetry that can be either avor or baryonic symmetry. With an abuse of notation, we call it U(1) avor. This symmetry can be weakly gauged by turning on a background connection.2 However, in order to preserve the original superconformal symmetry one has to turn on a whole gauge vector potential B , the gaugino and the auxiliary scalar Y , all in the adjoint representation of the avor symmetry. The corresponding supersymmetry transformations are where R (B) is the curvature 2-form of the gauge connection B and the covariant derivative on spinors is de ned as in eq. (2.8). connection with constant curvature Similarly to the case of the R-symmetry background in (2.10), we can choose a U(1) avor Y = i" 5 R (B) = b Y i 5 " = 0 together with vanishing background gaugino. In order to preserve some supersymmetry we have to require where jej = e22e33 e23e32 is the vielbein determinant on . 23. Therefore, setting Y = bjej 23 we nally obtain the condition 01) " = 0 We then see that in principle, turning on a background for an abelian non-R global symmetry, introduces additional constraints on the supersymmetry generators. More generally, we can consider 4d theories with rank{n avor symmetry group, i.e. with n generators Ti in the Cartan subalgebra. In this case we can gauge one vector multiplet (Bi ; i; Y i) for each Cartan generator. If the corresponding auxiliary scalars are bijej 23) we are led to the same constraints (2.16). 2Similar discussions appeared in [6, 8, 24]. =2 = 0 a + =2 = 0 are automatically irreducible representations of the R-symmetry group corresponding to charge 1. Classi cation of the solutions We are now ready to discuss the most general solutions of the two supersymmetry preserving conditions a 5 " = 0 ; 01) " = 0 as Spin(3; 1) ! Spin(1; 1) also splits as where the constant a signals the presence of a non-trivial U(1)R background, eq. (2.10), while bi are associated to Bi connections for U(1) avor symmetries, eq. (2.14). We note that the second equation is nothing but a 2d (anti)chirality condition. In order to nd solutions to these equations, we write the Majorana spinor " in terms of its Weyl components, " = ( to the positive chiral spinor _ ), and with no loss of generality we restrict the discussion transforming in the 2 of SL(2; C). On the product manifold R1;1 the original Lorentz group of 4d Minkowski is reduced Spin(2) , and consequently the spinorial representation of Here the representations on the right hand side are labelled by the eigenvalues of the hermitian generators 01 and i 23 of Spin(1; 1) and Spin(2) , respectively. The generator 01 corresponds also to the chirality operator on R1;1, hence we refer to 11;1 and 1 1; 1 as the 2d positive (left) and negative (right) chirality representations respectively, and denote the corresponding spinors as + and As summarized in table 2, for to + for a = 2 and for a = . The second equation in (2.17) does not restrict the Killing spinors any further, since we can always choose bi such that (2.16) projects on the same chirality as that of the Killing spinor. Therefore, independently of the presence of for a = . These solutions are compatible with the quantization condition a 2 Z, being an even number. In the special case of compacti cation on a torus, = 0, when no avor symmetry to zero, the Killing spinor equation reduces to @ R-symmetry U(1)left U(1)right generated by the two combinations T = 12 TR M23, where M23 is the Lorentz generator on . Supersymmetry can be reduced by gauging some avor symmetry. In this case, in fact, the second equation in (2.17) constrains the supercharges b = 0 b 6= 0 a = 2 N = (2; 0) N = (2; 0) a = N = (0; 2) N = (0; 2) b = 0 b 6= 0 N = (2; 2) N = (2; 0) or (0; 2) 1; 0 in terms of the surviving amount of supersymmetry in 2d. We include the possibility of a twist along the avor symmetries, with ux b. N = (0; 2) for Yi = in table 3, where the resulting 2d theories are classi ed in terms of the surviving amount of supersymmetry. Twisted reduction of N = 2 SCFTs We now consider a N U(1)R. The Lie Pauli matrices. The four-dimensional chiral supercharges Q I are in the (2; 2) 1 representation of the group Spin(3; 1) form in the (2; 2)+1 representation. In particular, the U(1)R generator TR acts on the supercharges as group U(1)RN =1 generated by the combination TRN =1 = manifold M = R1;1 Here we give a systematic derivation within the superconformal gravity setup. and then gauge xing the background Weyl multiplet as to reproduce the desired geometry with possibly non-trivial uxes turned on in order to preserve some supersymmetry. group ea ; f a; b ; !ab, the superconnections I associated to supersymmetries QI and SI , the connections A and V A for the R-symmetry groups U(1)R and SU(2)R and the auxiliary elds Tab; D (bosonic) and I (fermionic), needed to close the algebra o -shell. Under supersymmetry transformations the fermionic elds of the gravity multiplet transform as I = @ + 1 b + 1 !ab I = 4 D= Tab"IJ "J V A(i A)IJ "J Rab(A)i 5" In order to preserve Lorentz invariance on R1;1 the background fermions must be set to zero. This choice automatically sets to zero the Q-supersymmetry variation of all bosonic elds, which can then be chosen such that the Q-variation of the fermions vanish as well. From (3.2) and (3.3) we deduce that we can safely set the background elds b and Tab to zero and simplify these expressions to I = I = V A(i A)IJ "J 1 ab Rab(A)i 5"I + Rab(V A)(i A)IJ "J 2 Z, and The remaining background connections A topological twist as we now describe. Turning on a background ux for V A breaks explicitly the SU(2)R invariance of the theory down to a U(1) subgroup of it. Without loss of generality we choose this subgroup to be the one generated by i 3. Namely, we parametrize the R-symmetry gauging as follows and V A can then be used to perform partial R (A) = R (V A=1;2) = 0; R (V 3) = is the normalized volume form of . This choice is actually equivalent to gauging the 1-parameter subgroup of SU(2)R U(1)R generated by a1TR + a2 3. Looking for constant spinor solutions of (3.4) and (3.5) we can apply the exterior covariant derivative to thus turning the Killing spinor equation into an equation for the curvatures. Substituting the background (3.6) we nd I] = = i I = where (3.8) is obtained by substituting (3.7) in (3.5) and therefore it is only valid on the components of I that are actual solutions of the Killing spinor equation. are then left with a single de ning equation for Killing spinors. Twisting with Before solving the Killing spinor equation (3.7) we generalize the discussion to the case of 4d SCFTs admitting some global abelian non-R symmetry U(1) avor. Weakly gauging (B ; X; I ; Y A). Such a multiplet contains one gauge with curvature R one complex scalar X, two gaugini I forming a SU(2) doublet, and one auxiliary eld the supersymmetry variations of the bosonic components of the multiplet are identically vanishing, and they can be chosen to satisfy I = Rab(B) ab JI + Y A(i A)IJ " Gauging the global symmetry along with R (B) = b , and setting for instance Y 1;2 = 0, Y 3 = 2b for the positive chirality component of "J we obtain J = 0 ( 01 + 1) 1 = 0 1) 2 = 0 where we have used i 23 = 01 5 and 5 J = J The previous condition is equivalent to requiring that the two components of the I Another possibility to perform the avor twist would be via a two-step procedure. We section 2. Observe that we could engineer such a reduction also in the absence of avor symmetries. In that case we should rst perform a R-symmetry twist that preserves four supercharges. This twist would break R-symmetry and leave an unbroken U(1) that could be treated as avor symmetry useful for further twisting. Classi cation of the solutions chirality components I in the (2; 2) representation as In order to nd solutions to eq. (3.7) we observe that the selected background breaks SU(2)R ! Spin(1; 1) U(1) 3 , and correspondingly the positive where on the r.h.s. indices denote the 2d chirality of the reduced spinors I = I = We can nd solutions to (3.7) by appropriately choosing the values of the twisting parameters ai as summarized in table 4. A further constraint comes from eq. (3.10) when a global non-R symmetry is also gauged. = 0, separately. amount of supersymmetry preserved in 2d. All the other choices are related by a trivial change of basis of the symmetries or a di erent choice of sign for the auxiliary elds. For a1 = 2 which form a SU(2)R doublet. The 4d R-symmetry is left unbroken and the 2d theory is a chiral =2 = 0 a1 + a2 + =2 = 0 =2 = 0 a2 + =2 = 0 are preserved when the twisting parameters ai satisfy the corresponding equations in the column on the right. imply that only one of the two components of the doublet can be preserved according to the particular choice of the auxiliary For a1 = 0 and a2 = broken to U(1)2 with generators T 2 the preserved supersymmetries are 1 M23 and the preserved supersymmetry eld Y A in the vector multiplet, hence M23 + 12 3 becomes a avor symmetry in two dimensions since, by de nition, the preserved supercharges transform trivially under it. In this case, gauging a global non-R symmetry with the corresponding connection B together with the choice of auxiliary Y 3 = 2b , does not constrain the Killing spinors any further (see For a1 + a2 = (0; 2) with U(1) R-symmetry. In this case there are two new abelian avor symmetries that were not present in the original 4d theory, generated by the two combinations 2 TR + M23 Turning on a avor ux B does not constrain this solution any further. case where there is no twist, since the dimensional reduction on at space preserves all supersymmetry. The compacti ed theory ows to N = (4; 4) in 2d with global symmetry SU(2) U(1)2 where the two abelian groups are generated by the combi M23. Both sectors (4; 0) and (0; 4) provide a four dimensional real representation of the SU(2) R-symmetry group. Another possible choice of supersymmetry preserving background on the torus corre1 that transform trivially with respect to the background symmetry b = 0 b 6= 0 N = (4; 4) N = (2; 2) N = (2; 2) N = (0; 2) or (2; 0) 6= 0 b = 0 b 6= 0 a1 = 2 , a2 = 0 a1 = 0, a2 = a1 + a2 = N = (0; 4) N = (0; 2) N = (2; 2) N = (2; 2) N = (0; 2) N = (0; 2) 1; 0 in terms of the surviving amount of supersymmetry in 2d. We include the possibility of a twist along the avor symmetries, with ux b. with the auxiliary Y 3 = 2b further breaks supersymmetry to 1 , as can be seen avor symmetries which correspond precisely to the T background (3.14) and the left R-symmetry T+ (under which the right sector is invariant). Alternatively, choosing The results of this section are summarized in the table 5. V A, A = 1; transformations, respectively. Twisted reduction of N = 3 SCFTs can exist at strong coupling. These theories have SU(3)R U(1)R R-symmetry and their non-R global symmetries. , a partial topological twist can be performed on using an abelian subgroup of the R-symmetry group. In this section we study all possible solutions of the Killing spinor equations for such a twist, classifying all di erent con gurations of preserved supercharges in two dimensions in terms of the di erent choices of the uxes for the R-symmetry group. a curved manifold is N the corresponding non-linear supersymmetry transformations have been recently derived ; 8 are the gauge elds associated to the R-symmetry U(1)R and SU(3)R # of real d.o.f. 1; : : : ; 8. We choose a basis in which the SU(3) can be embedded into the top left 3 of SU(4), so that the rst 8 generators of SU(4) reduce straightforwardly to the generators of SU(3). The U(1)R group is obtained by mixing the U(1) from the decomposition of SU(4)R into SU(3)R U(1) and the chiral U(1) that enhances the superalgebra from PSU(2; 2j4) to SU(2; 2j4) [31, 35]. We observe that these two U(1) groups act proportionally to each N = 3 Weyl multiplet. As in the previous cases, we are interested in preserving supersymmetry while coupling the SCFT to a curved background describing the geometry of the manifold M . We choose a background Weyl multiplet where, together with the fermions, all the bosonic elds are set to zero except for ea , A , V A and DJI . Consequently, the conditions for the fermion variations to vanish read [34] IJ = I = = 0 I = @ + 1 DKI "K V A(i A)IJ "J = 0 abRab(V A)(i A)JK)"L = 0 1 abRab(A)i 5"I = 0 These provide the set of constraints that select the surviving Killing spinors in two dimensions. In order to nd non-trivial solutions, we choose the R-symmetry V A and A background elds such that R (V 8) = R (V A) = 0 for A 6= 3; 8 R (V 3) = R (A) = I] = = i and subject to appropriate quantization conditions (see the remark at the end of section 5). The non-trivial Killing spinor equations then reduce to R (V 3)(i 3)IJ + R (V 8)(i 8)IJ " together with the two auxiliary conditions (4.2), (4.3). a1 + a2 + a3 a1 + a2 + a3 =2 = 0 =2 = 0 =2 = 0 Classi cation of the solutions In order to nd non-trivial solutions to equation (4.7) we restrict the discussion to the positive chirality components of the "I spinors. We observe that under the breaking U(1)R ! Spin(1; 1) U(1)R realized still indicate the 2d chirality as de ned in (3.12). The spinors are charged under U(1) 3 U(1)R according to 1 p23 ; 1). Supersymmetry preserving equations are then given in table 7. Once the need to be satis ed. In appendix B we prove that solutions to exist if we appropriately choose the value of the components of the auxiliary eld DJI . IJ = 0 and I = 0 always In table 8 we list all possible solutions to the conditions in table 7 together with the corresponding preserved supersymmetries and the remaining 2d R-symmetry. We focus on the cases with mostly right supersymmetry and for each possibility we pick up just a choice of uxes. All the other possibilities can be obtained through a change of basis for the SU(3)R it associated to the diagonal generator ( 2 i 23 case, one extra U(1) symmetry emerges from the topological twist, which is generated by T itself (or any linear combination of T with the avor symmetry generator). Although under T the supercharges are charged, this symmetry cannot be a R-symmetry of the low energy SCFT. It might be that this symmetry is not a symmetry of the 2d theory, or appears as an outer automorphism of the 2d supersymmetry algebra.3 However, in order to get more insight on it one should know the actual SCFT algebra that emerges from the twisted reduction and the relation of T with the rest of the superalgebra generators. From table 8 we note that, while for to further reduce supersymmetry. 3We are grateful to Nikolay Bobev for raising this interesting point. N = (6; 6) N = (4; 4) N = (2; 2) 6= 0 N = (2; 4) N = (0; 6) N = (2; 2) N = (0; 4) N = (0; 2) a1 = 0, a2 = 0, a3 = 0 a1 = 0, a2 + a3 = 0 a1 + a2 + a3 = 0 a1 = 0, a2 = 3 , a3 = a1 = 0, a2 = 0, a3 = a1 = 2 , a2 + a3 = 0 a1 = 0, a2 + a3 = a1 + a2 + a3 = surfaces in terms of the surviving amount of supersymmetry in 2d. In the last column we indicate the subgroup of 4d R-symmetry that is compatible with the twisted compacti cation. However, also in the without avor symmetries. This works as follows. First we perform a R-symmetry twist that preserves either four or eight supercharges. This twist breaks R-symmetry as well, leaving some avor symmetries with the associated vector multiplets. The second step of multiplet that preserves only half of the supercharges. For example, if we use this procedure the original SU(3)R U(1)R survives as avor symmetry. In the second step we can gauge an abelian subgroup of this avor symmetry. The corresponding gaugino background then Twisted reduction of N = 4 SCFTs This case has been extensively discussed in the literature [2{4, 6, 36]. For completeness, here we brie y review the main results in the language of conformal supergravity. The supercharges are in the antifundamental representation of the SU(4)R R-symmetry in which the Cartan subalgebra is spanned by 3 = 6 8 = p 15 = p # of real d.o.f. and an auxiliary parameters "I satisfying the theory on the curved manifold4 M = R1;1 I , the bosonic auxiliary elds C, EIJ , TaIbJ DKIJL and the fermionic auxiliaries I , IKJ . In table 9 we list the corresponding SU(4)R representations. For a complete , by freezing the Weyl multiplet to contain as only non-vanishing components the vielbein, a R-symmetry background V A eld DKIJL. Supersymmetry is (partially) preserved if there exist spinor I = @ "I + IKJ = 1 DKIJL"L V A(i A)IJ "J = 0 1 ab K[I Rab(V A)(i A)JL]"L = 0 I is identically zero in the selected background. In order to nd non-trivial solutions we choose the R-symmetry gauge eld such that R (V 3) = R (V A) = 0 for R (V 8) = A 6= 3; 8; 15 R (V 15) = subject to appropriate quantization conditions (see the remark at the end of this section). Equations (5.2) and (5.3) then reduce to I] = = i Classi cation of the solutions The selected background induces the breaking Spin(3; 1) SU(4)R ! Spin(1; 1) Spin(2) U(1) 15 under which the chiral supersymmetry parameters split as where, once again, the indices indicate chirality as de ned in (3.12). The spinors are charged under U(1) 3 U(1) 15 according to 1 Therefore, equation (5.6) translates into the set of supersymmetry preserving equations listed in table 10. For any set of ai parameters satisfying one of the conditions in the = 0 N = (8; 8) N = (4; 4) N = (2; 2) 6= 0 N = (4; 4) N = (0; 6) N = (2; 2) N = (0; 4) N = (0; 2) a1 = 0, a2 = 0, a3 = 0 a1 = 0, a2 + a3 = 0 a1 + a2 + a3 = 0 a1 = 0, a2 = 3 , a3 = a1 = 0, a2 = 0, a3 = a1 = 2 , a2 + a3 = 0 a1 = 0, a2 + a3 = a1 + a2 + a3 = a1 + a2 + a3 a1 + a2 + a3 =2 = 0 =2 = 0 =2 = 0 =2 = 0 1; 0 in terms of the surviving amount of supersymmetry in 2d. In the last column we indicate the subgroup of 4d R-symmetry that is compatible with the twisted compacti cation. previous table, equation (5.3) can be satis ed by a suitable choice of the background auxiliary elds DKIJL without further constraining the I parameters. In table 11 we list explicit solutions for the ai parameters and the corresponding 2d surviving supersymmetry with its R-symmetry group. We focus on the cases with mostly right-handed supersymmetry and for each possibility we pick up just one particular solution with emerges from the topological twist. Although T acts non-trivially on the supercharges, this cannot be a R-symmetry of the low energy SCFT, but it could be identi ed as an outer automorphism of the 2d superconformal algebra. there are no avor symmetries, we can further reduce supersymmetry by performing a two step reduction. The rst step consists of turning on an R-symmetry twist, breaking half of the supercharges are preserved. avor symmetry, such that only if we look at the N = (4; 4), 6= 0 case in table 11 the solutions a2 = =3 and a3 = symmetry bundle would be ill-de ned. However, in this case the quantization condition that one has to actually impose is that the combination T background symmetry that has been gauged by the twist) assigns integer charges to every eld/representation of the theory. Substituting the explicit values of a2 and a3 we can see that the background symmetry T corresponds precisely to the U(1) R-symmetry of the a2p3 8 + a3p6 15 (i.e., the N = (4; 4) theory T = ( 3 8) + ( 6 15) = The quantization condition then becomes 2 2 Z, which is satis ed for any choice of genus g. A similar analysis applies to the other cases, leading to the same conclusion. by twisted compacti cation of N -extended supersymmetric theories in four dimensions, as described in the previous sections. In particular, we determine a general expression for the central charge and the other 2d global anomalies. 4d anomaly polynomial I6 for the U(1) global symmetries, including the abelian symmetry coupled to the twisting supergravity background, and integrate it along the The resulting expression is a 4-form that can be identi ed with the anomaly polynomial I4 of the 2d theory. From this expression we can then infer the 2d anomalies as functions of the 4d anomalies and of the background uxes. In this procedure we have to take into account that, even if the R-symmetry we start with is the exact R-symmetry in 4d, along the dimensional ow the U(1)R can mix with other abelian avor symmetries. The exact 2d central charge is then reconstructed by extremizing a trial central charge as a function of the mixing coe cients [4]. Because of this potential mixing, in the reduction procedure we can start with any trial U(1) R-symmetry TR in four dimensions, as di erent choices will simply shift the mixing parameters of the 2d theory without a ecting the nal result of the extremization procedure. We consider a generic SCFT in four dimensions with di erent amount of supersymmetries will have di erent matrix forms, but they can be identi ed up to a mixing with the abelian avor symmetries where ti are the generators of the abelian avor symmetries U(1)i in the 2d representation, while i are the mixing coe cients. The relation (6.1) represents the most general trial 2d R-current, involving abelian currents that do not necessarily mix with the R-current in the 4d SCFT, as the baryonic symmetries in toric quiver gauge theories [18, 19]. Our discussion can be applied also to the case of extended supersymmetry. In that be treated as avor symmetries that can potentially mix with the 2d R-symmetry. In the In order to compute the anomaly polynomial I6, which encodes all the global and gravitational anomalies of the twisted theory,5 we rst couple each global symmetry to a background connection on R1;1, which being topologically trivial can be compacti ed into a torus T = S1 S1. The topological twist introduces additional background components for U(1)R and U(1)i also along the Following the notations of appendix A, if we denote fR the rst Chern class of the R-symmetry bundle and fi the class associated to the gauging of the abelian U(1)i avor symmetries, then we can write where, the components in the direction of are de ned by (2.10) and (2.14) as fR = f fi = fiT + fi R = so that the total Chern class of the global symmetry bundle E (see appendix A for the de nition) restricted to the Riemann surface where TR and Ti are the 4d generators and the trace means summing over positive (negative) chirality fermions with plus (minus) sign. Here the twisting parameter a is xed by the Killing spinor equation (2.12) to the value . We can then interpret the combination 2 TR + Pi biTi to be the abelian symmetry which generates the topological twist on . 5The gauge theory is assumed to be free of local gauge anomalies, i.e., anomalies for symmetries coupled to dynamical gauge vectors. According to formula (A.14), the anomaly polynomial is given by the six-form I6 = ch3(E) Tr[TR3 ]fR3 + 1 X Tr[TR2 Tj ]fR2 fi 1 X Tr[TRTiTj ]fRfifj + 1 X Tr[TiTj Tk]fifj fk where Tr[TA1 Having compacti ed the theory on it is natural to identify the anomaly polynomial of the corresponding two-dimensional theory with the expression obtained by integrating I6 on the Riemann surface. The result of the integration is kA1:::Al are the l-degree 't Hooft anomaly coe cients of the I6 = Tr[TR2 T ] fR2 + X Tr[TRTiT ]fRfi + X Tr[TiTj T ] which can be compared to the general formula for the anomaly polynomial in 2d I4 = ch2(E) kRR fR2 + X kRifRfi + X kij fifj kRR = kRi = kij = Tr[TRTiT ] Tr[TiTj T ] k = leading to the following identities is de ned in (2.4). We note that (6.8) relates 4d 't Hooft anomaly coe cients on the right hand side with 2d anomaly coe cients, kAB Tr[tAtB], on the left hand side. As already mentioned, when we ow to two dimensions the generator TR corresponding to a trial four dimensional R-symmetry can mix with the other global U(1)'s to give rise to the exact two dimensional R-symmetry. Therefore, reinterpreting equation (6.8) in a two-dimensional language, requires substituting the generator TR with (6.1). Explicitly, kRtrRial = kRtriial = kij = k = The mixing parameters i are now determined by extremizing the trial central charge 3kRtrRial (a sign appears, due to our choice of 2d chirality matrix 01, see table 2) which implies 0 = = 0 Equation (6.14) can be solved by inverting the matrix kij , provided that it has nonvanishing determinant. The expression for the extremized central charge is nally given by cr = 3 2 in terms of the anomaly coe cients of the original four dimensional SCFT. We note that eq. (6.14) determines the coe cients i at the (possible) 2d superconformal xed point, giving raise to the exact 2d R-current once plugged in (6.1). These coe cients may di er from the ones appearing in the 4d exact R-current. There are abelian currents that do not mix in 4d but their mixing in 2d is in general non-vanishing. This is for example the case of the baryonic symmetries in the Y pq models discussed in [13]. Further directions We conclude our analysis by discussing some open questions and future lines of research. A rst generalization of the program of constructing 2d SCFTs from four dimensions consists of decorating the Riemann surfaces discussed here with punctures. A possible way to study such a problem consists of exploiting the doubling trick discussed in [38, 39]. In this case one can gain information on the e ective number of 2d chiral fermions by gluing a Riemann surface with a copy of itself (with the opposite orientation), thus obtaining a closed surface. One can apply our results to classes of 4d SCFTs with a gravitational dual. For example one can consider theories associated to D3 branes probing the tip of three dimensional Calabi-Yau cones. The analysis of such models was initiated in [13], for the in nite Y pq family of [40]. Such theories are characterized by the presence of a SU(2) U(1) mesonic avor symmetry and a U(1) baryonic symmetry. The baryonic symmetry does not mix with the 4d R-current, but it has been observed that this mixing is non-trivial once the theory is reduced to 2d. For more general quivers the gauge group is a product of U(N )i factors. In the IR the U(1)i U(N )i are free and decouple. The non anomalous combinations of these U(1)s are the baryonic symmetries. While in the Y pq case there is just a single baryonic symmetry, in other cases one can have a richer structure. The formalism developed in section 6 is necessary for extending the analysis to such families. One can also study the problem from the AdS dual setup along the lines of [13], reconstructing the central charge from the gravitational perspective. The solution in this case should correspond to D1 branes probing a type IIB warped AdS3 where M7 represents (locally) a U(1) bundle over a 6d Kahler manifold. It should be ! M7 geometry, possible to formulate the central charge and its extremization in terms of the volumes of M7, in the spirit of [41]. It should be also possible to study models arising from the compacti cation of 6d theories, such as class S theories [42] or theories with lower supersymmetry, as the Sk problem, especially because the central charges a and c can be computed along the lines of [29]. The analysis of the gravitational dual mechanism of the topological twist in this case can be performed by studying the consistent truncation of [35] in gauged supergravity. uxes are turned on. In such a case it might be possible to compare the eld theory and the supergravity results. Acknowledgments S(M ) = S the bundle S operator D= = Mazzucchelli for useful discussions. The work of A.A. is supported by the Swiss National Science Foundation (snf) under grant number pp00p2 157571/1. This work has been supported in part by Italian Ministero dell'Istruzione, Universita e Ricerca (MIUR), Istituto Nazionale di Fisica Nucleare (INFN) through the \Gauge Theories, Strings, Supergravity" (GSS) research project and MPNS-COST Action MP1210 \The String Theory Universe". The anomaly polynomial In this section we brie y review the general formalism of the anomaly polynomial that has been used in section 6. We refer the reader to the original paper [45] for further details (see also [46] for a review). We begin by recalling the Atiyah-Singer index theorem for the Dirac operator on a compact manifold. Let M be a compact closed manifold of even dimension 2l and E a smooth complex vector bundle over it, associated to some representation of a Lie group G. If M is a spin manifold, we can de ne fermionic elds as sections of the spinor bundle are the two chiral irreducible spinor representations of the spin group of M . A chiral fermion eld charged under G is then described by a section of The gamma matrices a act on spinors S exchanging their chirality, hence the Dirac can be represented by the o -diagonal 2-by-2 matrix where the operators D= : S Singer index theorem then states that E are the adjoints of each others. The Atiyahindex(D= +) dim ker D= + dim ker D= A^(M ) ch(E) D= = = + D0 where A^(M ) is the so called A-roof genus of the tangent bundle of M A^(M ) = 1 4p2(M )] + : : : expressed in terms of the Pontryagin classes pi(M ) 2 H4i(M; Z), while ch(E) is the Chern character of the bundle E. dimensional and the bundle E can be decomposed as a Whitney sum of line bundles E = L (r) is de ned, up to isomorphisms, by its rst Chern class (n), one for each representation (particle species). It follows that each c1(L(r)) = where R(A(r)) is the cohomology class of the curvature of the associated abelian connection A(r). In this case the Chern character is de ned additively as ch(E) = X chk(E) = Since each line bundle L (r) is associated to a unitary one-dimensional representation of integer charge q(r), we can equivalently describe the bundle E as follows. If we de ne fG to be the rst Chern class of the line bundle of unit charge,6 then for each L (r) we can write Qim=1 U(1)i, we have to consider the bundle E = (r) = L1 where each L (r) is a tensor product representation for the group G, labelled by the set of charges (q1(r); : : : ; qm(r)). If, as before, we de ne fi to be the rst Chern class of the line 6Note that a principal U(1) bundle and the associated line bundle of charge 1 have the same rst from which we obtain where we assembled all the charges q(r) into the diagonal matrix TG, which now represents the Lie algebra part of the connection on E. Using this rede nitions, the Chern character (A.5) is c1(L(r)) = q(r)fG c1(E) = Tr[TG]fG ch(E) = bundle of unit charge for the U(1)i symmetry, we can write chk(E) = k! r=1 i=1 i1 ik r=1 e ective action of a massless chiral fermion where we used the property of the rst Chern class, c1(Li(r) L(jr)) = c1(Li(r))+c1(L(jr)). The diagonal matrices Ti can be taken to be the hermitian generators of the U(1)i symmetries, written in the representation associated to E. In fact, under a chiral rotation the fermionic path integral picks up a non-zero phase eiW = R iD= + W = 2 index(D= +) proportional to the index of the Dirac operator. Since the anomaly for a negative chirality fermion is minus that of a positive chirality index(D= ) dim ker D= dim ker D= + = index(D= +) in a theory with many fermions of both chiralities, the total chiral anomaly is given by the sum of the anomalies of the positive-chirality fermions minus the sum of the anomalies of the negative-chirality ones. Finally, in [45] it was shown that one can construct a Dirac operator in 2l+2 dimensions in such a way that its index reproduces the gauge anomaly for a charged chiral fermion in 2l dimensions. The corresponding index density is a (2l + 2)-form A^(M ) ch(E) which is called the anomaly polynomial. Supersymmetry variations of the auxiliary = 3; 4 SCFTs In this section we show that it is always possible to satisfy the conditions IJ = 0, I = 0 in (4.2), (4.3) and auxiliary eld DJI and DKIJL, respectively. as usual to the positive chirality transformation, the IIJ variation reads that can be trivially solved by setting the corresponding DKIJL components to zero. 1 abRab(V )JJ J = 0 After twisted compacti cation the J spinors decompose as i 23 eigenvectors and we 1 according to the 2d chirality of the spinor. where we chose DIILJ to be diagonal in the J; L indices. write i 23 J = sJ J with eigenvalue sJ = Using equation (5.6) we eventually nd J = R (V )JJ J R23(!23) = "J = 0 Note to the reader: in this section we do not assume Einstein summation notation for repeated R-symmetry indices. along a subgroup of the Cartan of SU(4), the curvature R(V )IJ R(V A)(i A)IJ is diagonal in the adjoint (I; J ) indices. As a consequence, the Killing spinor equations (5.6) split into to the preserved supersymmetries, whereas the rest of the components are set to zero. Having this in mind, we now discuss the condition IJ = 0, where the variation is generated by the preserved supercharges. Three possible cases can arise. If K 6= I; J from (5.3) we immediately nd IKJ = 1 X DKIJL"L = 0 I vanishing. always possible to choose a non-vanishing DNM background that makes the supersymmetry 6= 0, from table 11 it turns out that for each xed J solution only one chirality is present and (B.4) can be always satis ed by an appropriate choice of the DIIJJ components. Finally, if "J is a Killing spinor but "I is not, the IJ variations do not vanish in general. However it is possible to show with a case by case analysis that these components always decouple from the representation of the supersymmetry algebra and they are not relevant for the counting of the supersymmetries. Open Access. 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