Supersymmetric AdS3 supergravity backgrounds and holography

Journal of High Energy Physics, Feb 2018

Lorenz Eberhardt

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Supersymmetric AdS3 supergravity backgrounds and holography

HJE AdS3 supergravity backgrounds and Lorenz Eberhardt 0 1 2 3 Zurich 0 1 2 3 Switzerland 0 1 2 3 0 Princeton , NJ 08540 , U.S.A 1 School of Natural Sciences, Institute for Advanced Study 2 Institut fur Theoretische Physik, ETH Zurich 3 coordinates such that K 4 = @ . We also de ne B We analyse the conditions for AdS3 ux to be supersymmetric. We classify all N = (2; 2) solutions where M7 satis es the stronger condition of being a U(1)- bration over a Kahler manifold. We compute the BPS spectrum of all the backgrounds in this classi cation. We assign a natural dual CFT to the backgrounds and con rm that the BPS spectra agree, thus providing evidence in favour of the proposal. Supergravity Models; AdS-CFT Correspondence; Superstring Vacua - Supersymmetric 1 Introduction 3 The Kahler case 2.1 2.2 3.1 3.2 3.3 3.4 2 Review of the conditions imposed by supersymmetry Killing spinor equations G-structure Further conditions imposed by H Reducing to a local product of Kahler-Einstein spaces All Kahler possibilities CY2 3.5 Induced action on S3 4.1 The Enriques surface 4.2 The hyperelliptic surface 4 The BPS spectra of the Kahler backgrounds 5 Dual CFTs for the Kahler backgrounds 5.1 The DMVV-formula 5.2 The chiral-(anti)chiral spectrum 6 Conclusions A Notations and conventions B Some properties of complex surfaces B.1 T 4 B.2 K3 B.3 HS B.4 ES C Modifying N = 4 multiplets AdS3 supergravity backgrounds provide an interesting playground to explore the AdS=CFT other hand, this makes it hard to classify AdS3-backgrounds. However, one can make some progress when imposing a su cient amount of supersymmetry. There are only three type IIB NS-NS ux N = (4; 4) backgrounds: AdS3 S gebra [2], whereas the latter supports the large N = (4; 4) algebra [3{7]. These backgrounds have very simple properties: they can be supported exclusively by NS-NS elds, exclusively by R-R elds or mixed ux. Those possibilities are related by the SL(2; R)-symmetry of type IIB supergravity. NS-NS backgrounds allow for a simple string world-sheet descrip3 tion [8], while R-R backgrounds are believed to have the simplest dual CFT's [1]. Moving on to less supersymmetry, there is one known N = (4; 2) background [9]. However this background is much more involved, in particular it requires all form- elds to be turned on. For a smaller amount of supersymmetry, classi cations are more di cult. When allowing ve-form ux only, the geometry is very restricted. For constant axio-dilaton and N = (2; 0), the internal manifold is a U( 1 )- bration over a Kahler manifold [10], which satis es some additional curvature constraints. This was generalized in [11] to varying axio-dilaton using an F-theory language. In particular, it was found that for N = (4; 0)supersymmetry, the most general geometry in this case is AdS3 S3=ZM CY3. Here CY3 is an elliptically bred Calabi-Yau three-fold, where the complex structure of the ber is given by the axio-dilaton. One other direction was recently explored in [12]. There, all symmetric space solutions of type IIB supergravity were analysed. Interestingly, it was found that all AdS3 symmetric space solutions are related via T-duality to one of the aforementioned backgrounds AdS3 S 3 M4 with N = (4; 4) supersymmetry. Furthermore, all these backgrounds have (4; 4) or (4; 0) supersymmetry. Thus, both the symmetric space N = (4; 0) solutions and the N = (4; 0) solutions with by either T-duality or are quotients thereof.1 ve-form ux only are related to the known N = (4; 4) solutions In this note we enlarge the classi cation result to incorporate N = (2; 2) supersymmetry. This amount of supersymmetry is particularly attractive, since one still has good control over protected quantities like the BPS spectrum and the elliptic genus. This allows one in particular to determine the dual CFT. While previous structural results in this direction have been obtained [13, 14], they are quite indirect: a background AdS3 enjoys N = (2; 2) supersymmetry when M7 is a U( 1 )- bration over a six-dimensional space which is the target of an N = (2; 2) sigma-model. This result was obtained from string theory. This result is conceptually nice, but provides little intuition on the geometry of M7 M7=U( 1 ) or on the dual CFT. Recently, some N = (2; 2) backgrounds were discussed [15], mostly from a string perspective. They involved taking speci c orbifolds of AdS3 S 3 T4. Most of the orbifold singularities cannot be resolved, which renders the backgrounds non-geometric. However, 1Note that this is strictly speaking only true for M4 = T4, for K3 we cannot perform T-dualities to relate the D3-brane system of [11] to the D1-D5 system considered here. { 2 { HJEP02(18)7 these backgrounds.2 to all of the string models, a dual CFT can be associated. Comparing BPS spectra and elliptic genera yields very non-trivial evidence for the proposal. To our knowledge, very few N = (2; 2)-backgrounds prior to [15] were known, which demonstrates the scarcity of In this paper, we revisit the problem from the point of view of supergravity to understand the scarcity of the N = (2; 2) backgrounds. Motivated by the string computation of [13{15] and to keep the calculation manageable, we consider the case of pure NS-NS ux and constant dilaton. This subsector of IIB supergravity is also known as heterotic supergravity (with trivial gauge group). Via SL(2; R)-symmetry, this also nds the pure R-R solutions. The full U-duality group can typically also generate solutions with ve-form manifold M7 to be a local U( 1 )- bration over a conformally balanced KT manifold [17, 18] M6 fails to be Kahler, as the fundamental (1; 1)-form J of the manifold is not necessarily closed, but J ^ J is. This result was already obtained in [19] using heterotic supergravity. There are known backgrounds which happen to be Kahler, namely the small N = (4; 4) backgrounds we mentioned above. In contrast, the large N = (4; 4) background corresponds to a conformally balanced manifold. main result will be that all are quotients of S3 It is interesting that the dual CFTs of the small N = (4; 4) cases have been known for such a long time, whereas progress in the large N = (4; 4) case was made only very recently [7]. The failure of M6 to be Kahler is just another incarnation of the di culty. To make progress in a classi cation, we will consider the case when M6 is a Kahler manifold. In that case, we succeed in completely classifying the possible internal manifolds M7. The We will subsequently classify all possible quotients of S3 T4 and S3 K3 leading to N = (2; 2) supersymmetry. There is a unique such quotient for S3 seven for S3 T4. This result has a similar avour as the N = (4; 0) classi cation results we mentioned above. Also in this case, the backgrounds are all related to N = (4; 4) backgrounds by quotients. Note that non-abelian T-dualities typically break supersymmetry of K3, whereas there are one chirality completely and thus do not yield N = (2; 2) backgrounds. We will then identify the dual CFTs of these backgrounds. The main claim is that AdS3 (S3 ifold of M4=G. Here, M4 is T 4 or K3 and M4=G is either the Enriques surface or a hyperelliptic surface. To support this claim, we compute the chiral-chiral spectrum and M4)=G is dual to a marginal deformation of the symmetric product orbthe chiral-antichiral spectrum in string theory and match it to the CFT calculation. The methods and ideas follow our earlier paper [15], in particular one of the solutions we present appeared already in this earlier discussion. This paper is organized as follows. In section 2, we review the conditions imposed on the compacti cation manifold imposed by the existence of Killing spinors. Starting from 2In [16], N = (2; 2) backgrounds were constructed by compactifying N = 4 SYM on a Riemann surface backgrounds, which will be matched to the proposed CFTs in section 5. Some more technical material and conventions are summarized in various appendices. 2 Review of the conditions imposed by supersymmetry We will assume throughout this note that the background geometry is of the form AdS3 M7 : (2.1) HJEP02(18)7 We will consider the NS-NS sector of type IIB supergravity on this background. This is no loss of generality, since it was shown in [19] from the point of view of heterotic supergravity that the warp factor is trivial in this case. M7 will be assumed to be compact. We will set all elds, except for the metric and the 3-form, to zero (or constant in the case of the dilaton). In particular, no R-R elds are turned on. We will review the geometry of M7, as determined in [19]. We rederived the results using the constraints imposed by the existence of spinor bilinears [10, 19{27]. 2.1 Killing spinor equations In the following, big latin indices M; N; : : : denote ten-dimensional indices and small latin indices a; b; : : : denote M7 indices. The relevant part of the type IIB action in string frame is then S NS NS = Z d10xp ge 2 R 1 12 HMNP HMNP + 4rM r M ; (2.2) where R is the scalar curvature of the metric gMN and H is the eld strength corresponding to the two-form B. As advertised above, we set the dilaton to a constant. When inserting the prescribed AdS3-part of the elds, we end up with the Einstein equation on M7: It is implied by the existence of a single Killing spinor on M7. On top of the equations of motions, Killing spinor equations should be satis ed to guarantee supersymmetry of the background. The dilatino Killing spinor equation on M7 reads Here, ` is the AdS3-radius. Similarly, we have a gravitino Killing spinor equation: 1 4 Rab = HacdHbcd : Habc abc + = 0 : ra = Habc bc : 12i ` 1 8 { 4 { (2.3) (2.4) (2.5) Out of a Killing spinor , we can form the following real forms: C Ka Jab Zabc y ; y a ; i y ab ; i y abc : T ; abc i T abc : dJ = K ? H ; d! = 2i ` 1B ^ ! : d(J ^ J ) = 0 : C is constant and we normalize such that C 1. We may also de ne some complex forms: 0. These forms satisfy Fierz identities. In particular, we have Z = K ^ Y , so Z is not an independent form. It follows now from the Killing spinor equations that Ka is a Killing K d . After rede ning ! vector and exhibits M7 locally as a U( 1 )- bration over a manifold M6. We can choose e2i ` 1 it turns out that the forms Jab and !abc live entirely on M6. J de nes a complex structure, whereas ! de nes a compatible SU(3)-structure. We have (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) We have also in particular 3 The Kahler case Hence M7 is locally a circle bration over a conformally balanced KT manifold [19]. We will now consider the case where M6 is Kahler, which amounts to the condition K ^H = 0, since then J is closed. We will refer to these backgrounds in the following as `Kahler backgrounds'. We will be able to give a complete classi cation of these backgrounds. 3.1 Further conditions imposed by H The gravitino Killing equation for K reads dK = K H, which together with K ^ H = 0 implies H = K ^ K H = K ^ dK = K ^ dB : (3.1) Furthermore, the form dB may be identi ed with the Ricci-form of the Kahler manifold [10, 28]. Thus, B is xed by , so in particular the U( 1 )- bration is uniquely xed up { 5 { is a (1; 1)-form, it has the standard form to the addition of a parallel vector eld.3 Thus, we have for some choice of coordinates z1, z2 and z3. The condition ^ = 0 implies then that only one of the three eigenvalues 1 , 2 and 3 can be non-zero, say 1 . Moreover, since the norm of has to be constant by (3.5), we conclude that is constant. Thus, the Ricci-tensor has only constant non-negative eigenvalues. Now we can use the theorem proved in [29] to conclude that M6 is locally the product of two Kahler-Einstein manifolds. One manifold is of dimension 2 with positive curvature, the other of dimension 4 with vanishing curvature, i.e. a Calabi-Yau manifold. In other words, we have demonstrated that M6 is of the form M6 = (S2 this does not follow from the conditions we have imposed so far. Since dK = , we need to require Finally, we remark that the norm of the Ricci form of M6 is constant. Indeed, the norm equals the norm of H, which in turn is constant: where G is some group of isometries preserving all the relevant structures. Furthermore, for M6 to be smooth, G has to act freely. Here we used that S2 is the only two-dimensional Kahler-Einstein manifold with positive Kahler-Einstein constant. Finally, we can change our viewpoint again to M7. Namely, since the U( 1 )- bration is parametrized by , it is actually a bration over the two-sphere S2, up to the aforementioned ambiguity of adding 3The argument goes as follows. Since dB = , there is locally a remaining gauge freedom of B ! B + d for some function . Switching our view to M7, K is xed up to K ! K + d . However, K is required to be Killing, so the function must also satisfy r(arb) = rarb = 0 : Thus, d is a parallel vector eld. This is quite restrictive, and we easily see that there are no more conditions. H = K ^ : ^ = 0 : = 3 X i=1 idzi ^ dzi ; (3.3) (3.4) (3.5) (3.6) (3.7) (3.2) a parallel vector eld. To eliminate this possibility, we make a case-by-case analysis. There are two possible choices for CY2: 1. CY2 = T4. Choosing the coordinates in the appropriate fashion, we can assume that we have a U( 1 )- bration over S2 S1. However, through the canonical isomorphism H2(S2 S1; Z) = H2(S2; Z) and using the fact that the second cohomology group classi es U( 1 )-bundles, we see that all U( 1 )- brations are actually only over S2. Thus, in this case the freedom of adding a parallel vector eld is trivial. 2. CY2 = K3. Since K3 has no parallel vector elds, this question does not arise. As we have demonstrated above, the bration cannot be trivial, thus it must be the Hopfbration over S2. This can also be seen explicitly, since we have now uniquely xed K. Thus, we nally conclude that M7 is a nite quotient of S3 CY2 leads to N = (4; 4) supersymmetry, so the group action has to be non-trivial. This also nally demonstrates that the requirements we imposed were su cient, since S3 CY2 satis es the supergravity equations. 3.3 To continue, it is advantageous to have a good understanding of M7 = S review here the background following [15]. We have H / volS3 in this case. Thus the above gravitino Killing spinor reduces to the standard one on S3, while on T 4 we are searching for parallel Killing spinors. Habc abc commutes with all gamma-matrices on S3, but anticommutes with all on T4. Thus, the dilatino spinor equation imposes a de nite 3 T4, so we chirality on T4. It is a mathematical fact that Killing spinors with non-vanishing Killing constant are in one-to-one correspondence with parallel Killing spinors on the Riemannian cone [30]. The chirality of the spinor on the cone translates into the sign of the Killing constant. For the case of S3, its Riemannian cone is R4, so the problem simply reduces to nding parallel Killing spinors on R 4 T4. In addition, they have to satisfy the dilatino Killing spinor equation. Now we can count the number of Killing spinors of type IIB on this background. We are considering rst one ten-dimensional Majorana-Weyl Killing spinor. Standard counting tells us that R 4 T4 possesses 2 24 = 32 parallel Dirac spinors. Half of them have the correct chirality and hence the correct sign of the Killing constant on S3 T4.4 Furthermore, only half of those satisfy the chirality constraint on T4 and hence obey the dilatino Killingspinor equation. Similarly, there are 12 2 22 = 4 Dirac Killing spinors on AdS3 with the correct sign of the Killing constant. Putting these Killing spinors together gives 4 8 = 32 ten-dimensional Dirac Killing spinors. See e.g. [20] on how to combine the two spinors into a ten-dimensional spinor. Now we have to impose also that the ten-dimensional Killing spinor is Majorana-Weyl, which gives then only 12 1 2 32 = 8 Majorana-Weyl Killing spinors. The same holds true for the other ten-dimensional Killing spinor. Thus, we have 4This sign depends on the ten-dimensional Killing spinor we are considering. { 7 { in total 8 left-moving supersymmetries and 8 right-moving supersymmetries. Hence this leads to N = (4; 4) supersymmetry. The case of M7 = S 3 K3 works in essentially the same way. Here the Killing spinors already have a de nite chirality on K3, since the holonomy group is SU(2). Thus, the dilatino equation is super uous, and the same argument as before yields again N = (4; 4) supersymmetry. We will now systematically explore all possibilities of quotients of S3 CY2 which preserve N = (2; 2) supersymmetry. For this, let G which we want to quotient. Obviously, Isom(S3 CY2) some group of isometries by Isom(S3 CY2) = O(4) Isom(CY2) ; (3.8) so we may look on the action on S3 and CY2 separately. To keep things simple, we only consider actions which are orientable on both S3 and on CY2, since otherwise supersymmetry turns out to be completely broken. The spin double cover of SO(4) is SU(2) SU(2), where the two factors correspond to the two di erent chiralities. Clearly the group action has to be non-trivial in both factors, since otherwise we would not obtain N = (2; 2) supersymmetry. Let us rst consider a cyclic subgroup of the group of isometries. We can choose the coordinates in such a way that the S3-part (or rather its lift to the spin-bundle) lies in the standard Cartan-subalgebra U( 1 ) U( 1 ) SU(2) SU(2). Now we claim that the S3-action actually lies in the diagonal or anti-diagonal combination of U( 1 ) U( 1 ), or it lies entirely in one of the U( 1 )'s. If this would not be the case, the group element had four di erent eigenvalues on the spinor representation (2; 1) (1; 2). There is an additional phase which can be produced by the action of the group element on CY2, but it is the same for all four states in the representation. Thus, at least three states get projected out, and we remain at most with (2; 0)-supersymmetry. The case where the isometries lie completely in one of the U( 1 )'s can be discarded, since it destroys one chirality of spinors completely and leaves the other untouched. Thus, it is associated with (4; 0)-supersymmetry, as discussed in [31]. Without loss of generality, we may assume that the cyclic subgroup lies in the diagonal U( 1 ) U( 1 ) U( 1 ). This however means in the SO(4)-language that the action on S3 is given by a rotation. In particular it has a xed point. Each group element has to act x-point free on S3 or on CY2 for the quotient space to be smooth. We have however seen above that each group element has to act with xed points on S3. Consequently the action on CY2 must be free. This in turn implies that the quotient space CY2=G is a Calabi-Yau manifold in the weak sense. This means that it is only required to have a vanishing rst Chern-class in real cohomology, but not in integer cohomology. As a consequence, these manifolds are actually not spin manifolds | only the complete M7 will be a spin manifold. This is extremely restrictive and these quotients are all classi ed by mathematicians. A standard reference is [32]. There are two classes, belonging to T4 and K3, respectively. { 8 { Type 4 has a family with seven members of such quotients, which go by the name of (irregular) hyperelliptic surface.5 They are also called bi-elliptic surfaces, since they admit an elliptic bration over an elliptic curve. Thus, they are best viewed as a of a product of an elliptic curve E = C= with an elliptic curve C = S 1 nite quotient S1. We will denote them generically by HS and write e.g. HSb 2) to indicate a speci c one. They have all the Hodge-diamond For the convenience of the reader, we have listed the di erent possibilities for the group actions in table 1. Further properties of surfaces are presented in appendix B. As a last step, we have to determine the corresponding actions of the quotient groups on S3. For the Enriques surface, this is immediate, since we argued before that every group element acting on S3 has to lie in the diagonal or anti-diagonal U( 1 ). As there is only one non-trivial group element, it can either act trivially or by a rotation by around one axis. The former is 5Not to be confused with hyperelliptic Riemann surface. { 9 { Action of G on E with = with ! = with i = Z2 Z2 Z3 Z3 Z4 Z4 Z6 0 0 0 0 1 1 Z2 Z3 Z2 1 10 1 1 2 1 0 0 1 1 7! 7! 7! 7! ! 7! ! 7! 7! i 7! i 7! 7! + + + ! not possible, since then M7 = S S3 is uniquely determined. The eigenvalues of the U( 1 ) U(1) on the spin representation (2; 1) (1; 2) are given by (i; i) for both the left and right chirality. Thus, precisely half of the left and right chirality Killing spinors survive the projection, since the eigenvalues on the Enriques surface are either i or i, depending on the choice of the lift of the group ES, which is not a spin-manifold. Thus, the action on action to the spinor-bundle. We can similarly argue for the hyperelliptic surfaces a 1) and b 1), where the group actions are given by a rotation by and 2 =3, respectively. For the hyperelliptic surfaces c 1) and d 1), the same argument reveals again that the group actions are given by rotations. However the angle is no longer uniquely xed. Looking again at the eigenvalues of the group action on the Killing spinors, we see that the angles must be =2 and =3, respectively, since otherwise no supersymmetry would survive. Finally, we come to the surfaces of type 2). We see that the second generator acts trivially on the Killing spinors, hence to preserve supersymmetry, it also has to act trivially on S3. Thus, the action on S3 for the surfaces of type 2) are precisely the same as those for their type 1) counterparts. This completes the classi cation of backgrounds coming from Kahler geometries. We have seen that the action on S3 is in all cases uniquely xed. Moreover, the example of the hyperelliptic surface which was mentioned in [15] ts into this classi cation. It is given by the hyperelliptic surface of type a 1). 3.5 Induced action on S3 In this subsection, we will see that the action on S3 is actually very natural. For this, we remember that the background AdS3 S 3 CY2 supports small N = (4; 4) superconformal symmetry. Thus, in particular, it has a spacetime SU(2) SU(2)-symmetry, which is simply given by rotations of S3. We now remember the AdS=CFT-correspondence for AdS3 S 3 CY2. It states that supergravity on this background lies on the same moduli space as the in nite symmetric product CFT Sym1(CY2) : (3.11) All CY2 manifolds are actually hyperkahler manifolds and hence also support N = (4; 4) superconformal symmetry. If we now act by some isometries on CY2, we consequently get an induced action on the SU(2) = S3-current algebra the theory supports. By the AdS/CFT correspondence, we expect this action to be precisely the one we determined in the previous section by brute-force. Since the quotients of CY2 we are considering are still Kahler and Ricci- at, they still support an N = (2; 2) superconformal eld theory. Thus, we conclude that the action on the SU(2)-current has to leave invariant an U( 1 ) SU(2). So the remaining group of automorphisms is only a U( 1 ), in other words, the group acts by rotations on S3. 4 The BPS spectra of the Kahler backgrounds We have established that all Kahler N = (2; 2) backgrounds are of the form AdS3 (S3 where a complete list of the possibilities was provided in the last section. It is the next logical step to compute the type IIB supergravity and BPS spectra of these backgrounds. Note that even though the backgrounds are supported purely by NS-NS ux, we are now considering the full IIB supergravity spectrum, including R-R elds and fermions. Since the backgrounds inherit many properties from their N = (4; 4) cousins, we can use the techniques of [33]. For this, we use the fact that the states are still secretly sitting in N = (4; 4) multiplets, but some states of the multiplets are projected out. We have collected some relevant background for this in appendix C. We have already applied a similar technique in [15]. We will denote by (m; n) a modi ed SU(2) SU(2)-multiplet, where is a unit root of the order of the cyclic group action on S3. Furthermore, we denote by (m; n)S a short modi ed N = (4; 4) multiplet. The re nement of the N = (4; 4) multiplets with insertions of helps us to keep track of the transformation properties under G. 4.1 The Enriques surface Let us rst begin with the K3 case and the associated Enriques surface. In this case is a second root of unity, since the group is Z2. In the following we let be a formal variable satisfying 2 = 1. The action of Z2 on the Hodge-diamond of K3 is The invariant part is the constant part in , which gives the Hodge-diamond of the Enriques surface (3.9). Following [33], we rst compactify to six dimensions and perform subsequently the Kaluza-Klein reduction on S3. During this procedure, we keep track of the eigenvalues of the projection and in the end we only keep invariant states. Furthermore, it will su ce to determine the bosonic eld content, since the fermionic elds will be xed by N = (2; 2) supersymmetry. Compactifying type IIB supergravity to six dimensions, we obtain the bosonic eld content indicated in table 2. The elds come about as follows: (i) Compactifying a ten-dimensional scalar yields again a scalar in six dimensions. Type IIB supergravity contains two scalars, hence this contributes two scalars. (ii) Compactifying a ten-dimensional two-form gives the following eld content in six dimensions. We have one six-dimensional two-form, which splits into a self-dual and an anti self-dual two-form. Furthermore, we obtain b1 vectors, where b1 is the rst Betti number of the internal four-dimensional manifold. Finally, we obtain b2 scalars. Type IIB has two ten-dimensional two-forms and K3 has the cohomology (4.2). Thus, this contributes 2 self-dual two-forms, 2 anti self-dual two-forms and 2 (10+12 ) scalars. (iii) We now consider the ten-dimensional self-dual four-form. Compactifying it yields one scalar, b1 vectors and 12 b2 two-forms. The two-forms can be either self-dual or type scalar 23 + 24 + dim(MES) 0 1 surface. We included also the number of odd elds under the projection, they can still contribute to the three-dimensional eld content. anti self-dual, depending on the signature of the internal manifold. For the case of type IIB and K3, this yields one scalar and 5 + 6 two-forms. These split into 1 + 2 self-dual and 9 + 10 anti self-dual forms. As required, the splitting is dictated by the signatures. K3 has signature 16, whereas ES has signature 8, see appendix B. When ignoring the -dependence ( = 1), there are hence 16 more anti self-dual forms (19) than self-dual forms (3). When performing the projection ( = 0), we would compactify on ES and hence have 8 more anti self-dual forms (9) than self-dual forms ( 1 ). (iv) Finally, we compactify the metric. It yields one metric in six dimensions. Furthermore, we obtain a non-abelian gauge eld realizing the isometry group of the compacti cation manifold. Finally, we obtain as many scalars as there are moduli in the compacti cation. For the present case, K3, as well as ES has a discrete isometry group and hence contributes no vectors in six dimensions. We left the number of scalars undetermined and denoted them by dim(MES). It is not necessary to determine the dimension of the moduli space of string compacti cations MES from rst principles | it will also be xed by N = (2; 2) supersymmetry. Summing up yields then table 2. In the next step, we perform the KK-reduction on the sphere S3. The quotient has the e ect of replacing the standard multiplets (m; n) by the twisted multiplets (m; n) , for more details on those consult appendix C. In this case, has order two and hence we have to decide whether we replace the multiplet (m; n) by (m; n) or by (m; n) . The answer is simple: even spin particles are clearly invariant under the group action on S3, whereas odd spin particles are not.6 Thus vectors will be multiplied by an additional in the end. Hence, following [33], the three-dimensional bosonic eld content is M(m; m m 4) (12 + 16 )(m; m 2) (40 + 36 + dim(MES))(m; m) : (4.3) 6This can also be seen in a less hand-wavy manner. The representations we wrote down are SO(4)representations. A rotation by 180 degrees can be represented by the element diag( 1; This is in the Cartan-torus and the sign picked up under this rotation is then ( 1 ) 21 (n 1; 1; 1) in SO(4). m). Hence we conclude again that vectors receive an additional , whereas the other elds are invariant. type scalar vector 4 4( 2 1 In particular, the chiral-chiral BPS spectrum reads We can also extract the chiral-antichiral ring: dim(MES) = 30 + 28 : 1 m=0 M 12(m; m) : (4.4) (4.5) (4.6) (4.7) HJEP02(18)7 The six-dimensional eld content is then determined as before and is collected in table 3. Now we can perform the KK-reduction as before. Fixing the sign is a bit trickier as before, 7We have not included the second Zm which appears in the type 2) surface, since it acts trivial. elliptic surface. with the result pacti cation: This can be uniquely tted into modi ed N = (4; 4) multiplets as described in appendix C M m (m; m 2)S (12 + 10 )(m; m)S : It is clear that there will be some exceptional cases for small values of m, which we have not treated here. This xes also uniquely the dimension of the moduli space of the com1 1 1 M m=0 odd 1 + 1 + 1 1 1 M m=0 even (m; m 2) 10(m; m) 12(m; m) : There is clearly a quite non-trivial structure in these invariants. 4.2 The hyperelliptic surface We now repeat the analysis for the hyperelliptic surface. The Zn-action7 on the Hodgediamond of T4 is now 1 + 1 + 2 + + 1 1 : (4.8) since does not necessarily square to one. However, the argument of footnote 6 still works and the prefactor of the representation (m; n) is 21 (n m). M m 2(m; m 4) (6 1 + 16 + 6 )(m; m 2) (14 1 + 32 + 14 + dim(MHS))(m; m) : Again, we can t this uniquely into modi ed multiplets with the result M Again, there are some exceptional cases at low spin. Furthermore, this tells us dim(MHS) = From the supergravity spectrum we can now straightforwardly extract the chiral-chiral primary spectrum: M 2(m; m m 1) 4(m; m) : The chiral-antichiral primary spectrum is very interesting in this case. It can in particular distinguish di erent hyperelliptic surfaces. It is in general given by the constant part of M m m(m; m 2) 2 m(m; m + 2) 2 m(1 + )(m; m 1) 2(1 + ) 1 m(m; m + 1) m(1 + 4 + 2)(m; m) : (4.13) It has hence a periodicity in m of period equal to the order of the quotient group. 5 Dual CFTs for the Kahler backgrounds There are almost canonical candidates for dual CFTs to the Kahler backgrounds. First note that the Enriques surface and the hyperelliptic surfaces are the only geometric backgrounds besides T4 and K3 which support an N = (2; 2) superconformal algebra at c = 6. It is thus very natural that the dual CFTs should in analogy to the case of T4 and K3 correspond to the symmetric orbifold of the respective seed theories. This should also work, since we have argued in section 3.5 that we have identi ed the same group actions on both sides of the small N = (4; 4) dualities. We hence propose that type IIB supergravity on the supergravity backgrounds we analysed above lies on the same moduli space as the symmetric orbifolds Sym1(ES) ; Sym1(HS) (5.1) (4.9) (4.10) (4.11) (4.12) HJEP02(18)7 of the Enriques surface and the corresponding hyperelliptic surface, respectively.8 The same proposal was made in [15] for the rst of the hyperelliptic surfaces, so this is the natural generalization of the idea presented there. 8We expect that this correspondence continues to hold for a nite number of copies, where the CFT should be dual to a string theory on the respective background. This is in the spirit of what was found in [15] from the point of view of string theory. To support the claim, we will show in this section that the chiral-chiral and chiralantichiral primary spectrum we calculate from these CFTs agree with the ones we computed in the previous section. Denote by Z(zj ) the partition function of the seed theory ES or HS with the insertion of Z(zj ) = tr ( 1 )FyJ0 yJ0 qL0 qL0 : Here, we included a chemical potential for the U( 1 )-charges. As usual, or z is assumed. We add a subscript `NSNS' or `RR' to indicate whether the trace is taken in the NS-NS sector or in the R-R sector. We add a superscript N to refer to the symmetric orbifold theory with N copies. As one can see from the de nition of the partition function, we suppressed ground state energies. We write In [34] and [35], a formula was given for the partition function of the symmetric orbifold: 1 N=0 X pN ZRNR(zj ) = 1 Y Y It is convenient to let this formula ow to the NS-NS sector: 1 N=0 X pN ZNNSNS(zj ) = 1 Y Y n=1 `; m; `; m (1 pnqmqmy`y`)c(n(m `);n(m `);` n2 ;` n2 ) : ZRR(zj ) = X c(m; m; `; `)qmqmy`y` : m;` 1 1 (5.2) (5.3) (5.4) (5.5) (5.6) p) 1. (5.7) (5.8) We note that the right hand side of (5.6) contains exactly one factor of the form (1 Following the argument of [36], we can extract ZN1SNS(zj ) as follows. The right hand side of (5.6) is of the form 1 1 p 1 X xipi = 1 X i X xj pi : i=0 i=0 j=0 Hence, we can extract ZN1SNS(zj ) = Pj1=0 xj by omitting the factor of (1 p) 1 and setting p = 1. We will indicate the fact that this factor is omitted by a prime in the product. Thus, we have ZN1SNS(zj ) = 1 Y Y0 n=1`; m; `; m (1 1 qmqmy`y`)c(n(m `);n(m `);` n2 ;` n2 ) : This is actually not the expression with which we should compare the supergravity answer. The reason is that this partition function also counts multi-particle states, whereas we only X1 1 k=1 Zmulti(zj ) = exp k Zsingle(kzjk ) : (5.9) It is then easy to see that the single-particle version of (5.8) is ZN1SNS; single(zj ) = 1 X X dealt with single-particle states in supergravity. The transition between the two partition functions is simple, they are related by 1 2 1 2 Here, we omitted the prime on the sum, since it simply corresponds to the vacuum in this partition function. We now extract chiral-chiral primary states of (5.10). Clearly, only terms with m = ` and m = ` contribute in the sum. Then the sum localizes onto the Ramond ground states of the seed theory. These in turn correspond via spectral ow to chiral-chiral primary states in the seed theory. We use the same trick as in supergravity to determine the chiral-chiral and the chiral-antichiral primary spectrum in one go. For this, we consider the modi ed supergravity spectrum of K3 and T4 with the insertions of 's. Enriques surface. Using the Hodge-diamond (4.2), we see that c 0; 0; = 1 ; c 0; 0; = ; c(0; 0; 0; 0) = 10(1 + ) ; (5.11) 1 2 ; 1 2 M m 3 1 2 ; 1 2 1)S ! n 2 ; ` n 2 1 ; (5.13) and all other ground state coe cients vanish. Thus, the modi ed K3 supergravity spectrum reads after translating to the supergravity conventions: (1; 1)S (11 + 10 )(2; 2)S (1; 3)S (m; m 2)S (12 + 10 )(m; m)S ; (5.12) which is in perfect agreement with (4.4), up to the aforementioned exceptions at low spin. As a corollary also the chiral-chiral and chiral-antichiral primary spectrum will agree. Hyperelliptic surface. The Hodge-diamond (4.8) tells us this time that c 0; 0; = 1 ; c 0; 0; ; 0 = 1 + 1 2 1 + 1 ; c (0; 0; 0; 0) = 1 + 2 + : c 0; 0; 1 2 This translates into the following supergravity spectrum from the symmetric orbifold: (1; 1)S (1 + )(1; 2)S ( 1 + 1)(2; 1)S 1 + 3 + )(2; 2)S 2(1 + )(2; 3)S (m; m 2)S (1 + )(m; m ( 1 + 4 + )(m; m)S ; (5.14) which is again in agreement with (4.10), up to low-lying exceptions. Consequently, also the chiral-chiral primary and chiral-antichiral primary spectrum will agree. Let us mention that we have also applied the techniques developed in [15, 36] to the present case. We have found that the supergravity elliptic genera of the backgrounds agree with the elliptic genera of the symmetric product orbifold theory. While for the hyperelliptic surfaces, the elliptic genus is vanishing, it equals half of the K3 elliptic genus (B.3) in the case of the Enriques surface. 6 Conclusions In this paper, we have discussed the conditions imposed by supersymmetry on AdS3 backgrounds with pure NS-NS ux. We reviewed that supersymmetry implies that the internal manifold is a U( 1 )- bration over a conformally balanced manifold. Strengthening the condition to Kahler, we were able to give a complete classi cation of these backgrounds. Moreover, it was relatively easy to identify their dual CFTs. Several directions for future work seem promising. First, it would be interesting to understand also non-Kahler backgrounds on the same level as the Kahler backgrounds | maybe also there a classi cation could be possible. This would greatly enhance our understanding of AdS3 backgrounds. Furthermore, one can consider warped products of AdS3 with M7, add a non-trivial dilaton pro le, or turn on RR- elds. For the latter case, [10, 37] gives some classi cation results. Each of these complications adds new interesting ingredients, but the dual CFT will be substantially harder to identify. However, we feel that N = (2; 2) supersymmetry is particularly suited for exploring the landscape of AdS3=CFT2 dualities. Another exciting direction is black hole counting, particularly for the Enriques surface. The background can be viewed as a near horizon limit of a black hole sitting at a boundary of a ve-dimensional space-time. While not a black hole in at space, one can still perform microscopic state counting. Since the black hole is sitting on the boundary of space-time, the surface of its horizon is precisely half of its original value. This is re ected on the CFT side by the fact that the elliptic genus is half of the K3 value (B.3). It would certainly be interesting to explore this in more detail. Furthermore, one should embed the background into string theory. In particular, it would be interesting to nd a suitable D-brane construction, which may provide some insight on how to construct other N = (2; 2) backgrounds. The symmetric orbifold of the hyperelliptic surface supports a higher spin symmetry. However, it is unknown whether the same holds true for the symmetric orbifold of the Enriques surface at least at special points in the moduli space. For K3, this is possible thanks to free eld constructions [38]. In a similar vein, it would be interesting to see whether the corresponding higher spin symmetry can also be realized as (possibly an orbifold of) a coset [39]. Finally, there is still moonshine to be found in the Enriques surface elliptic genus [40]. It seems that the Mathieu group M12 acts on the BPS states of this compacti cation. Hence our construction provides another geometric example of moonshine. Acknowledgments It is a pleasure to thank Matthias Gaberdiel for guidance in this work and for a careful reading of the manuscript. I also would like to thank Elena Asoni, Shouvik Datta, Andrea Dei, Kevin Ferreira, Anna Karlsson, Christoph Keller, Edward Witten and Ida Zadeh for useful discussions. Furthermore, this manuscript has greatly pro ted from correspondences with Dario Martelli, Eoin O Colgain and Sakura Schafer-Nameki. My research is (partly) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation. I gratefully acknowledge the hospitality of the Institute for Advanced Study in the nal stages of this work. A Notations and conventions We use a mostly plus metric for AdS3 M7. Hence M7 has a standard Riemannian metric. We de ne a generalized inner product between forms. Assuming p > q, it reads: HJEP02(18)7 For a p-form , the Hodge-norm is de ned by We have moreover ( )aq+1 qp 2 1 (p q)! a1 aq a1 ap : a1:::ap : where vol is the canonical volume form. For a complex form, we have the natural analog The Hodge dual in n dimensions and Euclidean signature of a k-form is de ned to be (? )a1 an k = k! b1 bkbk+1 bn b1 bk : B Some properties of complex surfaces In this appendix, we collect some interesting and useful properties of the complex surfaces we use in the main text, namely the four-torus T4, K3, the Enriques surface ES and the hyperelliptic surfaces HS. All of these surfaces are projective and therefore Kahler. They are furthermore distinguished among other complex surfaces, since their rst Chern class vanishes in real cohomology, i.e. it is a torsion element in integer cohomology. This su ces for Yau's Theorem [41] to hold and consequently these surfaces support a Ricci- at metric. It is furthermore possible to use them as target spaces for N = (2; 2) sigma-models, since there is no axial anomaly. We should note that there are other complex surfaces with vanishing rst Chern class in real cohomology. These are the primary and secondary Kodaira surfaces. However, these are not algebraic and hence not Kahler, so they are unsuitable for our purposes. (A.1) (A.2) (A.3) (A.4) (A.5) = ^ ? j j 2 1 1 p! a1:::ap = 2 vol ; : 1 2 2 1 4 1 1 0 0 0 1 1 2 2 0 0 1 and the cohomology ring is the exterior algebra over four generators | two of degree (1; 0) and two of the degree (0; 1). To determine the action of group actions on the cohomology, it hence su ces to determine the action on these four generators. The Euler-characteristic, the signature and all other genera of the surface vanish. The canonical bundle is trivial. T4 is a spin manifold. B.2 K3 K3 is the unique simply-connected Calabi-Yau surface. It can be realized as various orbifolds of T4. However away from these orbifold points, the Ricci- at metric is not explicitly known, but exists thanks to Yau's Theorem. The Hodge-diamond reads T4 is certainly the most explicit of the four surfaces. In particular the Ricci- at metric is the canonical metric inherited from C2, when thinking of T4 as a quotient thereof. The Hodge-diamond reads (B.1) HJEP02(18)7 (B.2) (B.3) (B.4) and equals B.3 HS The Euler-characteristic is 24, while the signature is 16 | the intersection lattice is the unimodular lattice II3;19. The holonomy group equals SU(2). The canonical bundle is again trivial. K3 is also a spin manifold. The elliptic genus of string theory is non-vanishing ZK3(zj ) = 8 2(zj ) 2( )2 2 + 3(zj ) 3( )2 2 + 4(zj ) 4( )2 2 : Hyperelliptic surfaces are nite quotients of tori | we gave an overview of the di erent possibilities in table 1. The Hodge-diamond reads for all possibilities Hyperelliptic surfaces are elliptic brations over elliptic curves. For this reason, they are also called bi-elliptic surfaces. The Euler-characteristic vanishes in all cases, as does the signature. The holonomy group is Zn, of which the generator is a rotation by an angle of 2 . Here, n = 2, 3, 4 and 6 for type a, b, c and d, respectively. The canonical bundle is a torsion bundle, i.e. it is not trivial but its n-th power is. Finally, hyperelliptic surfaces are not spin manifolds. B.4 ES Enriques surfaces can be realized as Z2-quotients of K3 surfaces. They have Hodge-diamond 0 0 0 1 10 1 0 0 0 : (B.5) HJEP02(18)7 The Euler-characteristic is 12, the signature is 8. The intersection lattice is the unimodular lattice II1;9. The canonical bundle is a torsion bundle of order two. The holonomy group is a semidirect product SU(2) oZ2. Finally, Enriques surfaces are not spin manifolds. The string theory elliptic genus is half of the K3 elliptic genus. C Modifying N = 4 multiplets To determine the BPS and supergravity spectrum in the main text, we used the underlying N = 4 multiplet structure of the compacti cation. In this appendix, we provide some details of the modi cations of the N = 4 multiplet structure we used. We rst pick an N = 2 subalgebra inside the N = 4 algebra of which the corresponding supercharges will be denoted by Gr+ and Gr . The remaining two supercharges will be denoted by Ger+ and Ger . They are not invariant under the quotient we are performing | they have eigenvalues and 1, respectively. Here, denotes a unit root of the same order as the group by which we are performing the quotient. Similarly, the Cartan-element of the su(2)-current algebra is invariant under the quotient, while the two raising and lowering operators Jm pick up the eigenvalues . Let us denote by `(y) an su(2) character of spin `. We further denote by su(2)-character twisted by : ` (y) = ` j yj : ` X j= ` j+`2Z The corresponding multiplet will be denoted by (m) in the main text, where m = 2` + 1. When combining left- with right-movers, we write (m; n) for the twisted multiplet. One has to pay attention that one has to use for the left-movers and 1 for the right-movers, i.e. we have by : (m; n) = (m) (n) 1 : It is simple to write down a short N = 4 character of the global su(1; 1j2)-algebra twisted ` q ` 1 q ` (y) 1 q 2 (1 + ) ` 12 (y) + q ` 1(y) : ` (y) an (C.1) (C.2) (C.3) Here, we inserted a ( 1 )F in the de nition of the character. An N = 2 character is by de nition invariant under the projection and reads jN;h=2(q; y) = 1 h > where the three cases correspond to chiral primary, anti-chiral primary and long representations, respectively. 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Lorenz Eberhardt. Supersymmetric AdS3 supergravity backgrounds and holography, Journal of High Energy Physics, 2018, 87, DOI: 10.1007/JHEP02(2018)087