S-folds and 4d \( \mathcal{N} \) = 3 superconformal field theories

Journal of High Energy Physics, Jun 2016

S-folds are generalizations of orientifolds in type IIB string theory, such that the geometric identifications are accompanied by non-trivial S-duality transformations. They were recently used by García-Etxebarria and Regalado to provide the first construction of four dimensional \( \mathcal{N} \) =3 superconformal theories. In this note, we classify the different variants of these \( \mathcal{N} \) =3-preserving S-folds, distinguished by an analog of discrete torsion, using both a direct analysis of the different torsion classes and the compactification of the S-folds to three dimensional M-theory backgrounds. Upon adding D3-branes, these variants lead to different classes of \( \mathcal{N} \) =3 superconformal field theories. We also analyze the holographic duals of these theories, and in particular clarify the role of discrete gauge and global symmetries in holography.

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S-folds and 4d \( \mathcal{N} \) = 3 superconformal field theories

Accepted: May 0 University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 1 University of Tokyo , Bunkyo-ku, Tokyo 113-0033 , Japan 2 Weizmann Institute of Science , Rehovot 7610001 , Israel 3 Department of Physics, Faculty of Science S-folds are generalizations of orientifolds in type IIB string theory, such that the geometric identi cations are accompanied by non-trivial S-duality transformations. They were recently used by Garc a-Etxebarria and Regalado to provide the of four dimensional N =3 superconformal theories. In this note, we classify the di erent variants of these N =3-preserving S-folds, distinguished by an analog of discrete torsion, using both a direct analysis of the di erent torsion classes and the compacti cation of the S-folds to three dimensional M-theory backgrounds. Upon adding D3-branes, these variants lead to di erent classes of N =3 superconformal eld theories. We also analyze the holographic duals of these theories, and in particular clarify the role of discrete gauge and global symmetries in holography. Extended Supersymmetry; Brane Dynamics in Gauge Theories; F-Theory - 3 superconformal eld theories D-branes 2.1 2.2 3.1 3.2 3.3 3.4 4.1 4.2 4.2.1 M2-brane and D3-brane charges 5 Special cases and N = 4 enhancement 1 Introduction and summary of results 2 Preliminary analysis 3 Holographic dual D3-branes on S-folds and their discrete symmetries 1 Introduction and summary of results Field theories with superconformal symmetry are useful laboratories for learning about the behavior of quantum eld theories in general, and strongly coupled eld theories in particular. This is because the superconformal symmetry allows many computations to be performed in these theories, using methods such as localization, integrability, and the superconformal bootstrap. The eld theories (above two dimensions) about which the most is known are N = 4 superconformal eld theories (SCFTs) in four dimensions, that have been called the `harmonic oscillator of quantum eld theories'. These theories have an exactly marginal deformation, and it is believed that they are all gauge theories with some gauge group G, such that the exactly marginal deformation is the gauge coupling constant. In these theories many observables have already been computed as functions (trivial or non-trivial) of the coupling constant, and there is a hope that they can be completely solved. Four dimensional theories with N = 2 superconformal symmetry have also been extensively studied. Some of these theories have exactly marginal deformations and correobservables can be computed also in N = 2 SCFTs, at least if they have a weak coupling limit, they are connected by renormalization group ows to theories that have such limits, or they are one of the theories of class S. Four dimensional N = 3 theories should naively provide an intermediate class of theories, that is more general than N = 4, but such that more computations can be made than in general N = 2 theories. N = 3 SCFTs (that are not also N = 4 SCFTs) have no exactly marginal deformations [5, 6] and thus no weak coupling limits that would aid in classifying and performing computations in these theories. Until recently no N = 3 SCFTs were known, but recently a class of such theories was constructed by Garc a-Etxebarria and Regalado in [7]. Their construction uses a generalization of orientifolds in string theory. Orientifold 3-planes (the generalization to other dimensions is straightforward) are dened in type IIB string theory as planes in space-time, such that in the transverse space y 2 R6 to these orientifolds, there is an identi cation between the points y and ( y), but with opposite orientations for strings (or, equivalently, with an opposite value for the B2 and C2 2-form potentials of type IIB). This means that we identify con gurations related by a Z2 symmetry that involves a spatial re ection in the transverse SO(6), and also a transformation ( I) in the SL(2; Z) S-duality group of type IIB string theory. This breaks half of the supersymmetry, preserving a four dimensional N = 4 supersymmetry. In particular, putting N D3-branes on the orientifold (these do not break any extra supersymmetries) gives at low energies four dimensional N = 4 SCFTs. In [7] this was generalized to identifying con gurations related by a Zk symmetry, that acts both by a (2 =k) rotation in the three transverse coordinates in C 3 = R6, and by an element of SL(2; Z) whose k'th power is the identity. We will call the xed planes of such transformations S-folds;1 for k = 2 they are the same as the usual orientifolds. Viewing SL(2; Z) as the modular group of a torus, the Zk S-duality transformation may be viewed as a rotation of the torus by an angle (2 =k); such a rotation maps the torus to itself if and only if k = 3; 4; 6 and its modular parameter is = e2i =k, so S-folds of this type exist only for these values of k and . It is natural to de ne such an identi cation using F-theory [17], in which the SL(2; Z) S-duality group is described as adding an extra zero-size torus whose modular parameter is the coupling constant of type IIB string theory; in this language the S-folds of [7] are the same as F-theory on (C3 T 2)=Zk. This speci c Zk identi cation preserves a four dimensional N = 3 supersymmetry. So, putting N D3-branes on the S-fold gives at low energies theories with N = 3 superconformal symmetry. Note that D3-branes sitting on the S-fold are invariant under all the transformations discussed above, and, in particular, the D3-brane charge of the S-fold is well-de ned. In this note we analyze further the theories constructed in [7]. We focus on asking what are the extra parameters associated with S-folds, analogous to discrete torsion for 1This term, rst coined in [8], generalizes the term T-folds that is used to describe identi cations by elements of the T-duality group. S-folds of string theory involving S-duality twists together with shifts along a circle were studied, for instance, in [9{13], and similar S-folds were studied in the 4d N = 4 SYM theory in [14{16]. { 2 { orbifolds and orientifolds (namely, to non-trivial con gurations of the various p-forms of type IIB string theory in the presence of the S-fold). S-folds with di erent parameters can carry di erent D3-brane charges. Upon putting N D3-branes on the S-fold, these discrete parameters label di erent N = 3 SCFTs. Our main result is a classi cation of these extra parameters (following a preliminary discussion in [7]); we show that there are two variants of S-folds with k = 3; 4 and just a single variant with k = 6. Each variant leads to di erent N = 3 SCFTs, with di erent central charges and chiral operators. In some cases discrete global symmetries play an interesting role. In a speci c type of orientifold (the O3 plane), the theory of N D3-branes on the orientifold is an O(2N ) gauge theory, and this may be viewed as an SO(2N ) gauge theory in which a discrete the e ects of discrete symmetries. In section 3 we discuss the holographic duals of the N = 3 SCFTs, and the realization of the discrete symmetries there. We show that holography suggests that only some of the possibilities found in the naive analysis are consistent. The compacti cation of F-theory on (C3 T 2)=Zk on a circle gives M-theory on (C3 T 2)=Zk, with each S-fold splitting into several C4=Zl singularities in M-theory. In section 4 we show that the consistency of this reduction implies that indeed only the possibilities found in section 3 are consistent. In section 5 we discuss the fact that for speci c values of k and N some variants of the N = 3 theories have enhanced N = 4 supersymmetry, with gauge groups SU(3), SO(5) and G2, and check the consistency of this. This provides new brane constructions and new (strongly coupled) AdS duals for these speci c N = 4 SCFTs. In this paper we discuss only the S-folds which give rise to four dimensional N = 3 supersymmetric theories. It would be interesting to study S-folds that preserve di erent amounts of supersymmetry, and that have di erent dimensions. A particularly interesting case, which should have many similarities to our discussion, is S-folds preserving four dimensional N = 2 supersymmetry. We leave the study of these theories to the future. 2 Preliminary analysis In this paper we study the S-folds introduced by Garc a-Etxebarria and Regalado in [7], which are equivalent to F-theory on (C3 T 2)=Zk for k = 3; 4; 6, generalizing the standard orientifold 3-planes that arise for k = 2. Just as the orientifold 3-planes have four variants, O3 , Of3 , O3+ and Of3+, that di er by discrete uxes, we expect that these new S-folds could also have a few variants. In this section we perform a preliminary analysis of the possible variants, from the viewpoint of the moduli space of the N = 3 superconformal eld theories realized on N D3-branes probing these S-folds. In the following sections we will discuss additional constraints on the possible variants. { 3 { We start from the following three properties of theories of this type, that are generally believed to hold: The conformal anomalies (central charges) a and c are equal in any N = 3 SCFT [5]: a = c: The value of a = c determines also all anomalies of the SU(3)R U(1)R symmetries in these theories. HJEP06(21)4 The geometry of the gravitational dual of N D3-branes sitting on a singularity involving a Zk identi cation of their transverse space, and carrying units of D3-brane charge, is AdS5 S5=Zk, with (N + ) units of 5-form large N central charge of the corresponding CFTs is ux. This implies that the a c k(N + )2=4 + O(N 0): In any N = 2 theory, the Coulomb branch operators (chiral operators whose expectation values label the Coulomb branch of the moduli space, namely the space of vacuum expectation values of the scalars in the vector multiplets) form a ring generated by n operators, whose expectation values are all independent without any relation. Furthermore, up to a caveat mentioned below, there is a relation n i=1 2a c = X(2 i 1)=4; where operators. i are the scaling dimensions of the generators of the Coulomb branch The property (2.2) can be derived easily following the analysis of [18{20]. Essentially, the curvature of the AdS space is supported by the total energy of the ve-form eld, and the volume of the Zk quotient is 1=k of the volume before the quotient, changing the value of Newton's constant on AdS5 by a factor of k. The equation (2.3), originally conjectured in [21], was given a derivation that applies to a large subclass of N = 2 theories in [22] (though it is not clear that this subclass includes the theories we discuss here). More precisely, this relation applies only to gauge theories whose gauge group has no disconnected parts. To illustrate this, consider the N = 4 super Yang-Mills theories with gauge groups U(1) and O(2). They have the same central charges 2a c = 1=4, coming from a single free vector multiplet, but the gauge-invariant Coulomb branch operator has dimension 1 for the former and 2 for the latter, so that property 3 only holds for the former. This is because the scalars in the vector multiplet change sign under the disconnected component of the O(2) gauge group. The O(2) theory is obtained by gauging a Z2 global symmetry of the U(1) = SO(2) theory, where the Z2 symmetry is generated by the quotient O(2)=SO(2); this operation does not change the central charges. { 4 { (2.1) (2.2) (2.3) This means that the O(2) theory includes a Z2 gauge theory, implying that it has a nontrivial Z2 2-form global symmetry in the sense of [23], as we further discuss below. We assume in the following that the relation (2.3) holds whenever the theory does not have any non-trivial 2-form global symmetry, even if the theory does not have a gauge theory description. 2.2 D3-branes on S-folds and their discrete symmetries Let us now see what the three properties recalled above tell us about the properties of S-folds. Let us probe the Zk S-fold by N D3-branes. The moduli space of N = 3 theories is described by the expectation values of scalars in N = 3 vector multiplets, which are identical to N = 4 vector multiplets; viewing an N = 3 theory as an N = 2 theory, these decompose into a vector multiplet and a hypermultiplet. Let us choose a speci c N = 2 subgroup of the N = 3 symmetry, and consider the component of the moduli space which is a Coulomb branch from the point of view of this N = 2 subgroup. This implies that all the D3-branes lie on a particular C=Zk within C3=Zk. Denote by zi (i = 1; : : : ; N ) the positions on C of these D3-branes. Since the D3-branes are identical objects, the identi cations that are imposed on the moduli space by the geometry are erators are the symmetric polynomials of zik, and their generators are (PiN=1 zijk) for j = 1; ; N , with dimensions From (2.1) and (2.3), this naively implies that k; 2k; ; N k: 4a = 4c = kN 2 + (k 1)N: Applying (2.2), we nd that the D3-brane charge of the Zk S-fold is = (k 1)=(2k). As discussed above, we know that even for k = 2 this is not the whole story, due to the possibility of discrete torsion and discrete symmetries. Let us review in detail this k = 2 case, in a language that will be useful for our later analysis. Among the known O3-planes, the above analysis only applies to Of3 , O3+ and Of3+, for which = (k 1)=(2k) = +1=4 is the correct value. When the orientifold is O3 , the O(2N ) gauge group arising on N D3branes can be viewed as a Z2 gauging of the SO(2N ) gauge theory, and thus, to compute a and c, we need to use the SO(2N ) theory instead of the O(2N ) theory. In the SO(2N ) theory the residual gauge symmetries on the Coulomb branch, acting on the eigenvalues of a matrix in the adjoint representation of SO(2N ), are generated by matrix of SO(2N ), z1z2 : : : zN . From this list we can compute a and c again using (2.1) and (2.3) and nd (2.8) (2.9) from which we nd = 1=4 from (2.2). This is indeed the correct value for O3 . The SO(2N ) theory has a discrete Z2 global symmetry, corresponding to gauge transspace one of these transformations acts as z1 ! z1, and with this extra identi cation the group generated by (2.7) becomes the same as the group (2.4) acting on the D3-branes. In particular, this identi cation projects out the Pfa an operator of dimension N , such that after it we obtain the Coulomb branch operators with dimensions (2.5). So, for this speci c orientifold plane, the theory on the D3-branes is described by a Z2 quotient of some `parent' theory which has a di erent group of identi cations (2.7), and correspondingly di erent Coulomb branch operators. Only after gauging a Z2 global symmetry of this `parent' theory do we get the theory of the D3-branes on the O3 orientifold. We stress here that this gauging of Z2 symmetry does not change the central charges. It is natural to ask how we could know directly from the theory of the D3-branes on the O3 plane that its central charges are not the same as the naive ones, because it arises as a Z2 gauging of some other theory. As discussed in [23, 24], when we have a discrete Zp global symmetry, we have local operators that transform under this symmetry, as well as 3-plane operators that describe domain walls separating vacua that di er by a Zp transformation. When we gauge the Zp global symmetry, these local and 3-plane operators disappear from the spectrum. Instead we obtain new 2-plane operators (that may be viewed as worldvolumes of strings), characterized by having a Zp gauge transformation when we go around them. These 2-plane operators are charged under a 2-form Zp global symmetry in the language of [23]. So whenever we have a theory with a 2-form Zp global symmetry, it is natural to expect that it arises by gauging a Zp global symmetry of some `parent' theory. And indeed, the analysis of [23] implies that this `parent' theory can be obtained by gauging the 2-form Zp global symmetry. Thus, whenever we have a theory with a Zp 2-form global symmetry, we expect that its central charges would not be given by (2.3), but rather by those of its `parent' theory.2 Suggested by this analysis for k = 2, we expect that also for k = 3; 4; 6, di erent versions of S-folds will be characterized by di erent 2-form Zp Zk global symmetries for the corresponding theories on the D3-branes, that will imply that these theories arise from `parent' theories with Zp global symmetries. The analysis above suggests that the identi cations on the moduli spaces of these `parent' theories should be subgroups of the 2Gauging a discrete Zp global symmetry does not change the dynamics on R4, but it does change the spectrum of local and non-local operators. { 6 { group generated by (2.4), such that the ring of invariants is polynomial without any relation, and such that adding an extra Zp generator produces the group (2.4). Luckily such groups are already classi ed and are known as Shephard-Todd complex re ection groups; the fact that the ring of invariants is polynomial without any relation if and only if the group acting on CN is a complex re ection group is known as the theorem of Chevalley, Shephard and Todd.3 For more on complex re ection groups, see [26]. In our case, the available groups are known as G(N; p; k), and are generated by the following elements: (zi; zj ) 7! ( zi; zi 7! pzi; 1zj ); where p is a divisor of k. By adding another Zp generator acting as zi 7! zi, all these groups become the groups (2.4). Denoting ` = k=p, we see that if there is a `parent' theory with the identi cations (2.10) on its moduli space, then its gauge-invariant Coulomb branch operators would be generated by the symmetric polynomials of zik and by (z1z2 and therefore the dimensions of the Coulomb branch generators would be given by k; 2k; : : : ; (N 1)k; N `: From (2.1) and (2.3) we then nd that the central charges of the `parent' theory, and also of its Zp quotient that would describe the theory of N D3-branes on the corresponding S-fold, are given by 4a = 4c = kN 2 + (2` k 1)N: Note that even though the Zp-gaugings of the theories with di erent values of ` all have the same moduli space, they are distinct theories with di erent central charges. When p is not prime, one can also gauge subgroups of Zp, giving rise to additional theories, which again are not equivalent to the theories for other values of ` (even though they may have the same moduli space). Therefore, it seems at this stage, that each Zk S-fold with k = 2; 3; 4; 6 can have variants labeled by ` which is an integer dividing k. The D3-brane charges k;` of these S-folds can be easily found using (2.2) and (2.12): (2.10) zN )`, (2.11) (2.12) (2.13) ` = 1 ` = 2 ` = 3 ` = 4 ` = 6 Note that just from the analysis above it is far from clear that S-folds and N = 3 theories corresponding to such identi cations actually exist. However, we expect any Sfold to fall into one of these categories. In the next two sections, we will see that the 3See [25] for a recent use of Shephard-Todd complex re ection groups in four-dimensional N = 2 theories. { 7 { S-fold variants that really exist are only those shaded in the table (2.13) above, rst by carefully studying the holographic dual and then by comparing with M-theory. Note also that there could be more than one theory with the same identi cations. For k = 2 there are three orientifold-types with the same ` = 2 identi cations, though they all give rise to the same theory on D3-branes because they are all related by S-duality. We do not know the corresponding situation in our case. We also cannot rule out the existence of N = 3 theories having Coulomb branch operators (2.11) also for the values of ` and k that do not come from S-folds. It would be interesting to shed further light on these questions, perhaps by a conformal bootstrap analysis of N = 3 SCFTs, or by analyzing the two dimensional chiral rings of the corresponding theories [27, 28]. 3 3.1 Holographic dual AdS duals of CFTs with discrete symmetries Theories of quantum gravity are not expected to have any global symmetries (see, for instance, [24]). There are very strong arguments that this is the case for continuous symmetries, and it is believed to be true also for discrete symmetries. In the AdS/CFT correspondence, this implies that any global symmetry in the CFT should come from a gauge symmetry in anti-de Sitter (AdS) space. For continuous global symmetries it is known that indeed they come from gauge elds in the bulk, with a boundary condition that sets their eld strength to zero on the boundary. In some cases (like AdS4 [29]) there is also another consistent boundary condition in which the eld strength does not go to zero on the boundary, and its boundary value becomes a gauge eld in the CFT. Changing the boundary condition may be interpreted as gauging the global symmetry in the CFT (in cases where this gauging still gives a CFT). In this section we discuss the analogous statements for discrete symmetries. We focus on the AdS5 $ CFT4 case, but the generalization to other dimensions is straightforward. The discussion here is a special case of the discussion in appendix B of [23], generalized to ve dimensions, but as far as we know its implications for the AdS/CFT correspondence were not explicitly written down before (see also [30]). Consider a Zk gauge symmetry on AdS5 (thought of as a sector of a full theory of quantum gravity on AdS5, that is dual to a 4d CFT). A universal way to describe such a gauge symmetry in ve space-time dimensions is by a topological theory of a 1-form A and a 3-form C, with an action A can be thought of as the gauge eld for a Zk symmetry; for example, the action above arises from a U(1) gauge symmetry that is spontaneously broken to Zk. The forms A and C are both gauge elds, whose eld strengths dA and dC vanish by the equations of motion. There are gauge transformations that shift the integrals of A and C over closed cycles by one. A gauge symmetry is just a redundancy in our description, but an invariant property of this theory is that it has line and 3-surface operators, given by ei H A and ei H C , L = ik 2 A ^ dC: { 8 { (3.1) multiplies the line operators ei H A by e2 i=k, and a similar 3-form global symmetry. with A and C integrated over closed cycles. And, it has a 1-form Zk global symmetry that When we put such a theory on AdS5, we need to choose boundary conditions; the possible boundary conditions for such topological gauge theories were discussed in [23]. The variational principle implies that we need to set to zero A ^ C along the boundary. If we set to zero C along the boundary, then the boundary value of A gives a gauge eld in the dual CFT, corresponding to a Zk gauge symmetry in this CFT. With this boundary condition line operators in the bulk are allowed to approach the boundary and to become line operators in the CFT, while 3-surface operators cannot approach the boundary, but can end on the boundary, giving 2-surface operators in the CFT. This is as expected for a Zk gauge symmetry in four space-time dimensions. On the other hand, if we set A to zero on the boundary, we obtain a Zk global symmetry in the dual CFT. The line operators ending on the boundary now give local operators in the CFT, and the 1-form global symmetry in the bulk becomes a standard Zk global symmetry in the CFT, under which these local operators are charged. The 3-surface operators going to the boundary give 3-surface operators in the CFT, which are domain walls between di erent vacua related by Zk. Thus, whenever we have a Zk gauge symmetry on AdS5, there are two natural boundary conditions. One of them gives a conformal eld theory with a Zk global symmetry, and the other gives a conformal eld theory with a Zk gauge symmetry (and a Zk 2-form global symmetry). The second theory is related to the rst one by gauging its Zk global symmetry, and similarly the rst one arises from the second by gauging its Zk 2-form global symmetry. When k is not prime, there are also additional possible boundary conditions, corresponding to gauging subgroups of Zk. In the context of our discussion in the previous section, this implies that the `parent' theories and their Zk-gaugings should arise from the same holographic dual, just with di erent boundary conditions for the Zk gauge elds. We will see in section 3.4 how the Zk gauge elds of (3.1) arise in F-theory on AdS5 (S5 T 2)=Zk from integrals of the type IIB ve-form eld F5 on discrete cycles. Note that all this applies already to the AdS5 S5=Z2 case discussed in [31]; typically in that case only the option of having a global Z2 symmetry, leading to the SO(2N ) gauge theory, is discussed. Finally, note that a very similar story occurs already in type IIB string theory on AdS5 S5, which includes a topological sector in the bulk corresponding to a 1-form ZN gauge symmetry, with an action i2N B2 ^ dC2 (where B2 and C2 are the 2-form elds of type IIB string theory) [32, 33]; this topological theory was discussed in section 6 of [23]. In this case both simple boundary conditions give rise to a 1-form ZN global symmetry in the dual CFT, and the resulting theories are SU(N ) and SU(N )=ZN gauge theories [23, 34, 35]. In this speci c case there is also an option of coupling these theories to a continuous U(1) gauge symmetry, leading to a U(N ) theory [36, 37]. One can also quantize the topological theory in ways that do not lead to local eld theories [38]. { 9 { In [31], Witten showed how to characterize the variants of O3-planes by studying the discrete torsion on S5=Z2. In this section we generalize this to S5=Zk S-folds, for k = 3; 4; 6. For k = 2, the discrete torsion of the NSNS and of the RR three-form eld strengths takes values in H3(S5=Z2; Z~), where the tilde over Z means that the coe cient system is multiplied by ( 1) when we go around the Z2 torsion 3-cycle of S5=Z2. For k = 3; 4; 6, we have a Zk torsion 3-cycle in S5=Zk, and we have an action of an element NS-NS and R-R eld strengths in Z Z when we go around the Zk torsion cycle of S5=Zk. 2 SL(2; Z) on the We can choose a speci c form for this , given, say, by = for k = 3, = these matrices are and 1. The discrete torsion is then given by H3(S5=Zk; (Z Z) ), and its computation is standard in mathematics.4 In general, H (S2n 1=Zk; A) where A is a Zk-module is given by the cohomology of the complex C0 1 t ! C1 1+t+ +tk 1 ! C2 1 t ! 1+t+ +tk 1 ! C2n 2 1 t ! C2n 1 (3.2) where all Ci ' A, t is the generator of Zk, and the di erential d is alternately given by the multiplication by 1 t or by 1 + t + (1 t)(1 + t + + tk 1) = 1 + tk 1. It is easy to see that d When k = 2, t is just the multiplication by 1. Then 1 + t = 0 and 1 t = 2, from which we conclude H3(S5=Z2; (Z Z) ) = Z2 Z2, reproducing four types of O3-planes. When k = 3; 4; 6, the action of the generator of Zk on Z Z obeys 1+ + + k 1 = 0, and det(1 ) = 3; 2; 1 for k = 3; 4; 6, respectively, and therefore 8 >>Z3 (k = 3); Z2 (k = 4); > >:Z1 (k = 6): This gives the discrete torsion groups arising from the 3-form elds of type IIB string theory on these S-folds. For k = 3, we have three di erent possibilities, but two non-trivial elements of Z3 are related by conjugation in SL(2; Z), so up to S-duality transformations there are just two types of S-folds with k = 3 (in the same sense that up to S-duality there are just two types of O3-planes that give di erent theories for the D3-branes on them). For k = 4, the cohomology is Z2, and therefore we expect two types of S-folds. Similarly, for k = 6, there is only one type of S-fold. 3.3 Generalized Pfa ans In the holographic duals of the S-folds, the discrete torsion described above corresponds to discrete 3-form uxes on the S3=Zk discrete 3-cycle in S5=Zk, and it a ects the spectrum 4The method to compute cohomologies with twisted coe cients is explained in e.g. Hatcher [39] chapter example 2.43. Section 5.2.1 of Davis-Kirk [40] was also quite helpful. of wrapped branes on this 3-cycle. We can use this to match the discussion of the previous subsection to our analysis of section 2. For k = 2 this was analyzed in [31]. There is a Z2-torsion three-cycle of the form S3=Z2 within S5=Z2, and only when the discrete torsion is zero, i.e. when the 3-plane is O3 , we can wrap a D3-brane on this cycle. By analyzing the properties of this wrapped D3-brane we nd that it corresponds to a dimension N operator in the dual CFT, which can be naturally identi ed with the Pfa an operator, that only exists in the SO(2N ) theory but not in SO(2N + 1) or Sp(N ) theories. Note that, according to the discussions above, this operator exists in the `parent' SO(2N ) theory, but not after we gauge the Z2 to get the O(2N ) theory. So in the AdS dual it exists when we choose the boundary condition for the Z2 gauge theory in the bulk that gives a Z2 global symmetry, corresponding to the `parent' theory, but not for the other boundary condition, that corresponds to the theory on the D3-branes. The obstruction to wrap D3-branes on S3=Z2 can be understood as follows. The NSNS 3-form ux G can in general be in a non-trivial cohomological class. But when pulled-back to the worldvolume of a single D3-brane, G is the exterior derivative of a gauge-invariant object B F , where F is the gauge eld on the D3-brane, and therefore the cohomology class [G] should be trivial. The argument for the RR ux is the S-dual of this. For all k = 3, 4, 6, there is a Zk-torsion three-cycle in S5=Zk of the form S3=Zk. When the discrete torsion is zero, there is no obstruction to wrapping a D3-brane on this cycle. The scaling dimension of this wrapped D3-brane can be easily found to be kN=k = N . This matches the scaling dimensions we found for the ` = 1 variants of section 2. So we identify the S-fold with no discrete torsion with the ` = 1 case (either the `parent' theory, or its Zk gauging that gives the theory on the D3-branes, depending on the boundary conditions). For k = 6, there is nothing more to discuss, since H3(S5=Zk; (Z Z) ) itself is trivial, so we do not nd any variant except ` = 1. For k = 3, when the discrete torsion in H3(S5=Zk; (Z Z) ) = Z3 is non-trivial, we cannot wrap a D3-brane on this cycle. So this should correspond to the ` = 3 variant (with no discrete symmetries). The most subtle is the k = 4 case, when the discrete torsion is given by an element in Z) ) = Z2. We claim that the non-trivial element of this discrete torsion gives the ` = 4 variant of section 2, not the ` = 2 variant. To see this, it is instructive to recall why in the case k = 2 we can wrap two D3-branes on S3=Z2 even with the discrete torsion. Note that it is not just that for N D3-branes wrapping on the same cycle, the triviality of N [G] su ces. For one thing, if we have two D3-branes wrapping on the same locus with at least U(1) U(1) unbroken, then B F1 and B F2 are both gauge-invariant (where F1;2 are the U(1) gauge elds coming from the rst and the second Chan-Paton factor), and therefore [G] still needs to be zero. To wrap two D3-branes on S3=Z2 with discrete torsion consistently, one needs to require that the Chan-Paton indices 1,2 are interchanged when we go around the Z2 cycle. This means that in fact there is a single connected D3-brane of the shape S3, which is wrapped on S3=Z2 using a 2:1 quotient map. In this particular setting, we know how to judge the consistency of wrapping: on S3, (B F ) is a gauge-invariant object, so [G] should vanish there. What needs to be checked is then to pull back the spacetime G using the map S 3 ! S3=Z2 ! S5=Z2; where the rst is the 2:1 projection and the second is the embedding. The result is zero, from the trivial fact that H3(S3; Z) = Z does not have non-zero two-torsion elements. Thus we see that we can wrap two D3-branes in this way. Now, let us come back to the k = 4 case, and try to wrap two D3-branes on S3=Z4 with discrete torsion, which should naively be possible since the discrete torsion lies in Z2. Again, if the Chan-Paton structure is trivial, we cannot wrap them. If we try to do the analogue of the k = 2 case above, we can try wrapping two D3-branes such that the Chan-Paton indices 1 and 2 are exchanged. Just as above, this is equivalent to wrapping one D3-brane on S3=Z2, covering S3=Z4 with a 2:1 quotient map. To test the consistency of the wrapping, one needs to pull back the spacetime obstruction on S5=Z4 via S3=Z2 ! S3=Z4 ! S5=Z4: (3.4) (3.5) One nds that the pull-back is non-trivial, showing that this embedding is inconsistent.5 It is clear that we can wrap four D3-branes on the discrete cycle, so this implies that the theory with this discrete torsion should be the ` = 4 variant discussed in section 2, and that we cannot have S-folds with the ` = 2 variant. This does not prove, we admit, that there are no other non-Abelian con gurations on two D3-branes on S3=Z4 that still allow the wrapping. But we will see that the choice ` = 4 for this discrete torsion variant matches with the M-theory computation below. 3.4 Realization of the Zk gauge theory in F-theory After these discussions we can understand how the Zk gauge theory of section 3.1 arises in F-theory on S5=Zk, when the discrete torsion in H3(S5=Zk; (Z Z) ) is zero. Recall that Hn(S5=Zk; Z) is Zk for n = 1 and 3, and its generator is S1=Zk and S3=Zk, respectively. We can wrap a D3-brane on these cycles to obtain a world 3-cycle and a worldline on AdS5. (Note that the latter is the generalized Pfa an particle discussed above.) Clearly, they both carry Zk charges. We are then naturally led to identify the worldline as coupling to the 1-form A and the world 3-cycle to the 3-form C in (3.1). For this identi cation to make sense, it should be the case that if we rotate the generalized Pfa an around an S1 that has a unit linking number with the world 3-cycle of the D3-brane wrapped on S1=Zk, there should be a non-trivial Zk holonomy of unit strength. This can be seen as follows. A D3-brane wrapped on S1=Zk creates an F5 ux given by the Poincare dual of its worldvolume. In this case this is given by s c, where s is the generator of H1(S1; Z) where this S1 is linked in AdS5 around the worldvolume of the D3-brane, and c is the generator of H4(S5=Zk; Z) = Zk. Now, we use the cohomology long exact sequence associated to the short exact sequence 0 ! Z ! R ! U(1) ! 0 to conclude that there is a natural isomorphism H3(S5=Zk; U(1)) ' H4(S5=Zk; Z) = Zk: (3.6) 5In general, the pull-back map from Sn=Zab to Sn=Zb is a multiplication map by 1 + t + t2 + : : : + ta 1, where t is the generator of the Za action that divides Sn=Zb to Sn=Zab. In physics terms, this means that the discrete Zk eld strength (which is Z-valued) with four legs along S5=Zk can be naturally identi ed with the discrete Zk holonomy (which is U(1)-valued) with three legs along S5=Zk. From this we see that we have a holonomy of the type IIB C4 eld given by the element s c 2 H1(S1; Z) H3(S5=Zk; U(1)); (3.7) which can be naturally integrated on the cycle S1 S3=Zk to give exp(2 i=k). This means that the generalized Pfa an particle wrapped on S3=Zk, when carried around the S1 linking the worldvolume of the D3-brane wrapped on S1=Zk, experiences this holonomy. This is indeed the behavior we expect for the objects charged under the Zk 1-form A and In this section, we consider M-theory con gurations on (C3 T 2)=Zk. We know that we obtain such con gurations from any of our S-folds on a circle, using the standard relation between type IIB theory on a circle and M-theory on a torus. However, the opposite is not true, since when we translate some con guration of discrete uxes in M-theory on (C3 T 2)=Zk to F-theory on a circle, we could also have some non-trivial action of the shift around the F-theory circle, corresponding to a `shift-S-fold' where there is a rotation on the transverse C 3 when we go around the compacti ed S1. By analyzing all possible discrete charges in M-theory and translating them to F-theory, we learn about all possible variants of S-folds. 4.1 Discrete uxes in M-theory on (C3 Let us start by analyzing the possible discrete uxes in M-theory, which come from 4-form uxes. For a given k, (C3 T 2)=Zk has several xed points of the form C4=Z`i , each of which has an associated H4(S7=Z`i ; Z) = Z`i in M-theory [41]. From this viewpoint, the possible discrete charges are given by the orbifold actions on the xed points, M H3(C4=Z`i ; Z) = <>Z3 8 >>Z2 > > > > >>Z4 >:Z6 Z2 Z3 Z4 Z3 Z2 Z3 Z2 Z2 We can alternatively measure the same charges by considering H4 of the `asymptotic in nity' of (C3 T 2)=Zk, which has the form (S5 T 2)=Zk. This is a T 2 bundle over S5=Zk, and as such, one can apply the Leray-Serre spectral sequence,6 that says that it has the ltration H4((S5 T 2)=Zk; Z) = F 2;2 F 3;1 F 4;0; 6For an introduction on the Leray-Serre spectral sequence, see e.g. [42]. In our case, the computation goes as follows. The second page E2p;q of the spectral sequence is given by E2p;q = Hp(S5=Zk; Hq(T 2; Z)), with the di erentials d2p;q : E2p;q p+2;q 1. The third page is given by E3p;q = Ker d3p;q=Im dp 2;q+1, and has the di erentials d3p;q : E3p;q !!EE3p+23;q 2. The second page and the part relevant for us of the third page (4.1) (4.2) where F 2;2=F 3;1 = H2(S5=Zk; H2(T 2; Z)) = H2(S5=Zk; Z); F 3;1=F 4;0 = H3(S5=Zk; H1(T 2; Z)) = H3(S5=Zk; (Z Z) ); F 4;0 = H4(S5=Zk; H0(T 2; Z)) = H4(S5=Zk; Z): (4.3) (4.4) (4.5) The di erent subgroups here correspond in a sense to the `number of legs along the base S5=Zk and the ber T 2'. As we are taking the Z-valued cohomology, we only have a ltration and it is not guaranteed that the group (4.2) is a direct sum of (4.3){(4.5). We can easily compute (4.3){(4.5) to obtain HJEP06(21)4 F 2;2=F 3;1 = Zk; F 3;1=F 4;0 = <>Z3 ; F 4;0 = Zk: (4.6) 8 >>Z2 > > > > >>Z2 >:Z1 Z2 (k = 2) (k = 3) (k = 4) (k = 6) From the standard F-theory / M-theory mapping, we see that each piece (4.3){(4.5) has the following e ect in the F-theory language: The piece F 2;2=F 3;1 becomes a component of the metric of F-theory on S1. Concretely, it speci es the amount of the rotation on the transverse space while we go around S1. Therefore, the system is a plain S1 compacti cation if the ux in this piece is zero, whereas it is a shift S-fold if it is non-zero. The piece F 3;1=F 4;0 becomes the discrete RR and NSNS 3-form uxes around the Sfold. Indeed, the coe cient system H1(T 2; Z) of the ber is exactly the one (Z Z) that we discussed in the previous section, and there is a natural isomorphism of F 3;1=F 4;0 with the discrete torsion (3.3). The piece F 4;0 becomes an F5 ux having one leg along the S1. As discussed in section 3.4, we have a natural identi cation H4(S5=Zk; Z) = H3(S5=Zk; U(1)). So, this piece can also be regarded as specifying a C4 holonomy having one leg along the S1. Note that are then given by (F 2;2=F 3;1) (F 3;1=F 4;0) F 4;0 = <>Z3 2 Z 0 Zk 0 Zk Z 1 0 X 0 X 0 X 0 Z 0 Zk 0 Zk Z 0 1 2 3 4 5 ; 8 >>Z2 > > > > >>Z4 >:Z6 2 1 0 Z2 Z3 Z4 Z3 0 Zk X Z2 Z3 Z2 Z2 Zk Z Z2; Z3; Z2; Z1 for k = 2; 3; 4; 6, respectively. From this we conclude that in fact F p;q=F p+1;q 1 = E1p;q ' E2p;q for p + q = 4. this case the map that is the `sum' and that The cases we use are (M2-brane charge) = 1 24 1 k + `(k `) : 2k ` = 0 ` = 1 ` = 2 ` = 3 ` = 4 ` = 5 is the diagonal embedding. Note that a b is indeed a zero map. We assume that a similar relation holds also in the other cases. Namely, we assume a : M H3(C4=Z`i ) ! F 2;2=F 3;1 = Zk b : F 4;0 = Zk ! M H3(C4=Z`i ) is the `diagonal embedding'. For k = 4 and k = 6, we need to use natural maps such as Z2 ! Z6 and Z6 ! Z2. We just choose a multiplication by 3 in the former, and the mod-2 map in the latter, and similarly for k = 4. We see that the composition a b is indeed a zero map, which gives a small check of our assumptions. 4.2 M2-brane and D3-brane charges Using the discussions above, let us test our identi cation by working out the M2-brane and D3-brane charges of our various con gurations. We use the formula of [43] for the charge of C4=Zk orbifolds with discrete torsion, and we see that (4.7) and (4.1) are the same as abstract groups. What remains is to gure out the precise mapping between the two. This is a purely geometrical question since it is just the relation of the H4 of (C3 T 2)=Zk computed at the origin and at the asymptotic in nity. As we have not yet done this computation, we will use guesswork, and then verify that it leads to consistent results for the M2-brane and D3-brane charges of the various The k = 2 case can be worked out using the known properties of the orientifolds. In is given by the sum of the four Z2's, and the map a : M H3(C4=Z`i ) = (Z2)4 ! F 2;2=F 3;1 = Z2 b : F 4;0 = Z2 ! M H3(C4=Z`i ) = (Z2)4 1 4 1 3 3 8 5 12 A long list of tables analyzing all possible cases follows. The end result is that every possible M-theory con guration can be interpreted as a plain S1 compacti cation or a k = 2 : k = 3 : k = 4 : k = 6 : shift-S-fold on S1 of precisely the S-folds that we discussed in the previous section (the types shaded in (2.13)), namely the S-folds with (k; `) = (2; 1); (2; 2); (3; 1); (3; 3); (4; 1); (4; 4); and (6; 1): (4.14) In all cases we compute the D3-brane charge using the sum of the M2-brane charges of the orbifold singularities (4.12) and successfully compare it with our expectation (2.13). Let us denote the discrete charges of the four xed points as elements in piece F4;0 = Z2 is generated by ( 00 00 ) and ( 11 11 ). The piece F 2;2=F 3;1 is given by the sum of the four entries. In the interpretations below, we consider the columns of the matrices to correspond to O2-planes, when we interpret our M-theory con guration in type IIA by shrinking one cycle of the torus (as we can do for k = 2). Of course everything should work out correctly in this k = 2 case, and has been already worked out in [44, 45]. We reproduce the analysis here since it is a useful warm-up for k = 3; 4; 6. ZZ22 ZZ22 . The Shift = 0=2 2 F 2;2=F 3;1: O2+ + O2+ + 1 4 4 label IIA #D3 label IIA #D3 label IIA #D3 label IIA #D3 label IIA #D3 label IIA #D3 4 ( 10 01 ) 4 ( 10 00 ) f O2+ + O2 0 O2 + Of2 The rst two come from O3 , but the latter of the two has one additional mobile D3-brane stuck at the origin, due to a non-trivial Wilson line around S1 in the component of O(2N ) disconnected from the identity. The rest all come from the other three O3-planes wrapped on S1. Shift = 1=2 2 F 2;2=F 3;1: O2+ + Of2 f ( 10 11 ) bration over S1 such that when we go around S1, we have a multiplication by ( 1) on C3. The ones with D3-brane charge 0 are an empty shift-orientifold, and the ones with charge 0 + 1=2 have one D3-brane wrapped around S1. Note that the charge 1=2 we are seeing here re ects the fact that the T 2 of M-theory is bered over C3=Z2. In the type IIB frame, a ber over a particular point on S1 is C3, and therefore the D3-brane charge is 1 if we integrate over the asymptotic in nity of this C3. Let us denote the discrete charges of the three xed points as elements in (Z3; Z3; Z3). The piece F4;0 = Z3 consists of (0; 0; 0), (1; 1; 1), (2; 2; 2) = (1; 1; 1). The piece F 2;2=F 3;1 = Z3 corresponding to the shift is given by the sum of the entries. There are 27 choices in total. HJEP06(21)4 Shift = 0=3 2 F 2;2=F 3;1: They are the Z3 shift-S-folds, with the rotation angle 2 =3, of the at background on S1, with 0, 1, 2 D3-brane(s) wrapped around at the origin. label (0; 0; 0) (1; 1; 1) (2; 2; 2) label (0; 1; 2) (1; 2; 0) (2; 0; 1) label (0; 2; 1) (1; 0; 2) (2; 1; 0) 13 + 1 13 + 1 label (1; 0; 0) (2; 1; 1) (0; 2; 2) label (1; 1; 2) (2; 2; 0) (0; 0; 1) label (1; 2; 1) (2; 0; 2) (0; 1; 0) Among the rst three, the rst entry has the right D3-brane charge to be the (k = 3; ` = 1) S-fold, see (2.13). The other two have one more mobile D3-brane, stuck at the origin through a non-trivial Z3 background holonomy around S1. The others all have the right D3-charge to be the (k = 3; ` = 3) S-fold, see (2.13) again. The second three and the third three have opposite Z3 charge characterizing the type of the S-fold. Shift = 1=3 2 F 2;2=F 3;1: #D3 #D3 #D3 #D3 #D3 #D3 1 3 3 3 0 2 3 2 3 3 3 2 3 1 3 1 3 They are again the Z3 shift-S-folds, but with the rotation angle 4 =3, of the at HJEP06(21)4 background on S1, with 0, 1, 2 mobile D3-brane(s) wrapped around at the origin. We denote the charges at the xed points as elements in (Z4; Z4; Z2). The piece F4;0 = Z4 consists of (0; 0; 0), (1; 1; 1), (2; 2; 0), (3; 3; 1). Shift = 0=4 2 F 2;2=F 3;1: label (0; 0; 0) (1; 1; 1) (2; 2; 0) (3; 3; 1) label (0; 2; 1) (1; 3; 0) (2; 0; 1) (3; 1; 0) 38 + 1 38 + 1 + 3 8 4 + 1 2 These are Z4 shift-S-folds of at space, with rotation angle =2, and with 0, 1, 2 or 3 mobile D3-brane(s) around S1. Shift = 2=4 2 F 2;2=F 3;1: #D3 #D3 #D3 #D3 3 8 38 + 1 18 + 12 + 18 + 12 Among the rst four, the rst has the right D3-brane charge to be the (k = 4; ` = 1) S-fold, see (2.13). The other three have one more mobile D3-brane, stuck at the origin through non-trivial Z4 background holonomy around S1. All the second four have the right D3-charge to be the (k = 4; ` = 4) S-fold, see (2.13). Shift = 1=4 2 F 2;2=F 3;1: These are the Z2 shift-S-folds on S1 of the O3-planes. #D3 0 #D3 label (0; 3; 1) (1; 0; 0) (2; 1; 1) (3; 2; 0) label (0; 0; 1) (1; 1; 0) (2; 2; 1) (3; 3; 0) label (0; 2; 0) (1; 3; 1) (2; 0; 0) (3; 1; 1) 18 + 1 + 18 + 12 charge of O3 is is 1=2 ( 1=4) = 1=8. trapped at the origin. by this Z2 operation. Shift = 3=4 2 F 2;2=F 3;1: Among the rst four, the rst has the right D3-brane charge to be the Z2 shift-Sfold of O3 . Recall that the base of the M-theory con guration is C3=Z4, but the in the type IIB description, the ber at a particular point on S1 is C3=Z2. The D3-brane 1=4, and therefore the charge as seen from the M-theory con guration The others have one or two additional mobile D3-brane(s) trapped at the origin. Among the second four, the rst and the third have the right D3-brane charge to be the Z2 shift-S-fold of Of3+. The second and the fourth have one additional mobile D3-brane Note that we cannot take the Z2 shift S-fold of Of3 or O3+, since they are exchanged 4 0 2 + 1 4 4 + 1 2 0 4 These are Z4 shift-S-folds of at space, with rotation angle 3 =2, and with 0, 1, 2 or 3 mobile D3-brane(s) around S1. 4.2.4 k = 6 We denote the charges at the xed points as elements in (Z6; Z3; Z2). The piece F4;0 = Z6 consists of (0; 0; 0), (1; 1; 1), (2; 2; 0), (3; 0; 1), (4; 1; 0), (5; 2; 1). Shift = 0=6 2 F 2;2=F 3;1: label (0; 0; 0) (1; 1; 1) (2; 2; 0) (3; 0; 1) (4; 1; 0) (5; 2; 1) #D3 5 12 152 + 1 152 + 1 152 + 1 152 + 1 152 + 1 The rst has the right D3-brane charge to be the (k = 6; ` = 1) S-fold, see (2.13). The other ve have one more mobile D3-brane, stuck at the origin through a non-trivial Z6 background holonomy around S1. Shift = 1=6 2 F 2;2=F 3;1: label (1; 0; 0) (2; 1; 1) (3; 2; 0) (4; 0; 1) (5; 1; 0) (0; 2; 1) #D3 0 5 6 2 3 1 2 1 3 1 6 This is the Z6 shift-S-fold of at space, with zero to ve D3-branes stuck at the origin. The rotation angle is 2 =6. label #D3 (2; 0; 0) 112 + 13 (3; 1; 1) 112 + 1 (4; 2; 0) 112 + 23 (5; 0; 1) 112 + 13 (0; 1; 0) 1 12 (1; 2; 1) 112 + 23 The fth has the correct charge to be the Z3 shift-S-fold of the standard O3 -plane. Since the O3 -plane itself has an identi cation by the angle , its Z3 quotient involves the rotation by =3 of the transverse space. Note also that the O3 -plane has the D3-brane charge 1=4, therefore we see 1=3 ( 1=4) = 1=12 in M-theory. The others have one or two additional D3-brane(s) at the origin. Shift = 3=6 2 F 2;2=F 3;1: label (3; 0; 0) (4; 1; 1) (5; 2; 0) (0; 0; 1) (1; 1; 0) (2; 2; 1) #D3 16 + 12 16 + 1 16 + 12 1 6 16 + 12 16 + 1 The fourth has the correct charge to be the Z2 shift-S-fold of the (k = 3; ` = 1) S-fold. The others have one or two additional mobile D3-brane(s) on top. Shift = 4=6 2 F 2;2=F 3;1: label #D3 (4; 0; 0) 112 + 13 (5; 1; 1) 112 + 23 (0; 2; 0) 1 12 (1; 0; 1) 112 + 13 (2; 1; 0) 112 + 23 (3; 2; 1) 112 + 1 The third has the correct charge to be the Z3 shift-S-fold, of rotation angle 2 =3, of the standard O3 -plane. The others have one or two additional D3-brane(s) at the origin. Shift = 5=6 2 F 2;2=F 3;1: label (5; 0; 0) (0; 1; 1) (1; 2; 0) (2; 0; 1) (3; 1; 0) (4; 2; 1) #D3 0 1 6 1 3 1 2 2 3 5 6 This is the Z6 shift-S-fold of at space, with zero to ve D3-branes stuck at the origin. The rotation angle is 2 5=6. Comments: note that in the cases with shift 2; 4 2 Z6, we only nd Z3 shift-S-folds of O3 , but we do not have Z3 shift-S-folds of Of3 , O3+ and Of3+. This is as it should be, because these three types of O3-planes are permuted by the Z3 action. Similarly, there is no Z2 shift-orientifold of the (k = 3; ` = 3) orientifold, since the Z2 action exchanges the two subtly di erent versions that we denoted by (1; 1; 1) and (2,2,2) in section 4.2.2. So far, we saw that N D3-branes probing various variants of the S-folds give rise to N = 3 superconformal eld theories characterized by (k; `) = (3; 1); (3; 3); (4; 1); (4; 4); (6; 1): (5.1) When ` < k we expect to have both `parent' theories and their discrete gaugings. In this section we discuss some interesting special cases, including cases where the N = 3 supersymmetry is enhanced to N = 4. As discussed in [5], an enhancement of supersymmetry to N = 4 occurs if and only if there is a Coulomb branch operator of dimension 1 or 2, since N = 3 supersymmetry then dictates the presence of extra supercharges. The dimensions of the Coulomb branch operators of our `parent' N = 3 theories were given in (2.11). For k = 2 we always have such an enhancement, but for k = 3; 4; 6 we see that it happens just for ` = 1 and N = 1; 2. Note that the theory with the lowest central charge which does not have any enhancement is the N = 1, k = ` = 3 theory, whose only Coulomb branch operator has dimension 3. The central charges of this theory are the same as those of ve vector multiplets. Since the Coulomb branch operators of N = 3 theories must be integers as shown in section 3.1 of [28], then this must be the `minimal' N = 3 SCFT, assuming the general validity of the formula (2.3). It would be interesting to test this by a superconformal bootstrap analysis, generalizing the ones in [46, 47]. Going back to theories with N = 4 supersymmetry, the case N = 1 is rather trivial: we just have a Coulomb branch operator of dimension one, so the moduli space is just C3, and we get the N = 4 super Yang-Mills theory with gauge group U(1). So let us discuss the N = 2 cases. The spectrum of the Coulomb branch operators of the `parent' theory is given by > < 8>2; 3 (k = 3); 2; 4 (k = 4); >>:2; 6 (k = 6): (5.2) These spectra agree with those of an N = 4 super Yang-Mills theory with gauge group SU(3), SO(5) and G2, respectively. Below we give evidence that indeed, the `parents' of these S-fold con gurations give rise to these N = 4 super Yang-Mills theories, realized in a somewhat unusual manner. Note that N = 4 theories always have an exactly marginal deformation, sitting in the same multiplet as the dimension two Coulomb branch operator, and our conjectured relation implies that for our N = 2 theories this is the gauge coupling of these N = 4 gauge theories. Our discussion in the previous sections implies that in the AdS dual of these `parent' theories, this marginal deformation corresponds to a scalar eld coming from a D3-brane wrapped on the torsion 3-cycle; for this speci c case this wrapped D3-brane gives rise to a massless eld. Again let us limit ourselves to the points on the moduli space corresponding to a Coulomb branch from the point of view of an N = 2 description of our SCFTs. In the N = 2; ` = 1 theories, this subspace is parameterized by z1;2 2 C, with the gauge symmetry (2.10) (z1; z2) 7! ( nz2; nz1); where = e2 i=k and n is any integer. The charges in a basis that is natural from this point of view can be written as (e1; m1; e2; m2). Naively, one would expect the electric charges e1; e2 to correspond to electric charges of the corresponding N = 4 theory, but this cannot be the case because of the non-trivial Sp(4; Z) action on these charges, induced by the SL(2; Z) transformation that accompanies the identi cation (5.3). So instead we consider the rank-2 sublattice containing charges of the form (Q; Q) where we regard (e; m) 2 Z Z as a complex number Q = e + m . Two charges from this sublattice are local with respect to each other. To see this, note that given Q = e + m and Q0 = e0 + m0 , their Dirac pairing is em0 e0m = (QQ0 Q0Q)=( ): Then the Dirac pairing between the two charges (Q; Q) and (Q0; Q0) is clearly zero. This means that we can take these charges to be the \electric charges" in the N = 4 description. We provide some consistency checks for this below. Now, note that the SL(2; Z) element associated to the Zk orbifold then acts on Q just by multiplication by . Then the gauge transformation (5.3) acts on this variable Q as Q 7! nQ; For k = 3, Q = !n and Q = (1 + !)!n = !n 1 where ! = e2 i=3, For k = 4, Q = in and Q = (1 + i)in, For k = 6, Q = n and Q = (1 + ) n. Clearly they can be identi ed with the roots of SU(3), SO(5), and G2, respectively. (5.3) (5.4) (5.5) (5.6) and nz2 for n = 0; : : : ; k of half-BPS particles are given by which is a re ection of the complex plane along the line e in=kR. This makes it clear that the group generated by (5.3) for k = 3; 4; 6 is the Weyl group of SU(3), SO(5) and G2, respectively. Let us test our identi cation by looking at half-BPS particles. There's no string connecting z1 and nz1, since we know nothing happens when z1 = 0 for ` = 1. So there are only strings connecting nz1 and mz2. Using (5.3) we can always restrict z1 to have a phase between 0 and 2 =k. Then we just have to consider all (p; q)-strings connecting z1 1. Since the IIB coupling constant is = , the central charges (p + q )(z1 nz2): We conjectured that \electric" states have the charge (Q; Q), and then their central charges are given by Qz1 Qz2. Comparing with (5.6), we see that \electric" objects have the following Q: The metric on the moduli space is also correctly mapped to that on the Cartan subalgebra of these groups. The original metric is dz1dz1 + dz2dz2 on C2, and we choose a real subspace R2 of the form Qz1 Qz2. Then, two vectors Qz1 Qz2 and Q0z1 0 Q z2 in R2 have the induced inner product (2Re(QQ0)). Using this, we can easily check that the vectors listed above have the same inner products as the root vectors of SU(3), SO(5), and G2, in the normalization that the short roots have length squared 2. Finally, recall that the dyons of N = 4 SYM have central charges of the form (p + q Y M )( s p + q Y M r ( l ); (5.7) where s;l are short and long roots, and r is the length squared of the long roots divided by that of the short roots. We can check that the spectrum (5.6) can be matched with (5.7) with the identi cation of the roots given above, if we take Y M = 1=(1 + ), uniformly for k = 3; 4; 6. As a further check, note that for k = 6 this is exactly the value of Y M for which the G2 N = 4 theory has a discrete Z6 symmetry [48]. Acknowledgments The authors would like to thank I. Garc a-Etxebarria, D. Harlow, Z. Komargodski, T. Nishinaka, H. Ooguri and N. Seiberg for useful discussions. The work of OA is supported in part by an Israel Science Foundation center for excellence grant, by the I-CORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number 1937/12), by the Minerva foundation with funding from the Federal German Ministry for Education and Research, by a Henri Gutwirth award from the Henri Gutwirth Fund for the Promotion of Research, and by the ISF within the ISF-UGC joint research program framework (grant no. 1200/14). OA is the Samuel Sebba Professorial Chair of Pure and Applied Physics. The work of YT is partially supported in part by JSPS Grant-in-Aid for Scienti c Research No. 25870159, and by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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Ofer Aharony, Yuji Tachikawa. S-folds and 4d \( \mathcal{N} \) = 3 superconformal field theories, Journal of High Energy Physics, 2016, 44, DOI: 10.1007/JHEP06(2016)044