4d \( \mathcal{N}=1 \) from 6d (1, 0)

Journal of High Energy Physics, Apr 2017

We study the geometry of 4d \( \mathcal{N}=1 \) SCFT’s arising from compactification of 6d (1, 0) SCFT’s on a Riemann surface. We show that the conformal manifold of the resulting theory is characterized, in addition to moduli of complex structure of the Riemann surface, by the choice of a connection for a vector bundle on the surface arising from flavor symmetries in 6d. We exemplify this by considering the case of 4d \( \mathcal{N}=1 \) SCFT’s arising from M5 branes probing ℤ k singularity compactified on a Riemann surface. In particular, we study in detail the four dimensional theories arising in the case of two M5 branes on ℤ 2 singularity. We compute the conformal anomalies and indices of such theories in 4d and find that they are consistent with expectations based on anomaly and the moduli structure derived from the 6 dimensional perspective.

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4d \( \mathcal{N}=1 \) from 6d (1, 0)

Received: December Published for SISSA by Springer Shlomo S. Razamat 0 1 2 5 Cumrun Vafa 0 1 2 3 Gabi Zafrir 0 1 2 4 5 0 Kashiwa , Chiba 277-8583 , Japan 1 Cambridge , MA 02138 , U.S.A 2 Haifa , 32000 , Israel 3 Je erson Physical Laboratory, Harvard University 4 Kavli IPMU (WPI), UTIAS, the University of Tokyo 5 Physics Department , Technion We study the geometry of 4d N = 1 SCFT's arising from compacti cation of 6d (1; 0) SCFT's on a Riemann surface. We show that the conformal manifold of the resulting theory is characterized, in addition to moduli of complex structure of the Riemann surface, by the choice of a connection for a vector bundle on the surface arising from symmetries in 6d. We exemplify this by considering the case of 4d N = 1 SCFT's arising from M5 branes probing Z k singularity compacti ed on a Riemann surface. In particular, we study in detail the four dimensional theories arising in the case of two M5 branes on Z 2 singularity. We compute the conformal anomalies and indices of such theories in 4d and nd that they are consistent with expectations based on anomaly and the moduli structure derived from the 6 dimensional perspective. Duality in Gauge Field Theories; Field Theories in Higher Dimensions; Su- 1 Introduction Adding punctures M5 branes probing ADE compacti ed on a surface N = 2 sub case and the resolution of a puzzle Four dimensional perspective: preliminaries Linear and circular quivers Higher genus theories Marginal directions Symmetries and group theory Punctures and gluings The Gmax = so(7) models The Gmax = so(5)u(1) models Two M5 branes on A1 singularity: preliminaries and summary The Z2 orbifold of the N = 2 SYM 5.2 IR dual descriptions A 5.3 IR dual descriptions B Supersymmetric index and anomalies The Gmax = su(2)diagu(1)2 models Theories of type Gmax = u(1)3 Relations between the models The Gmax = so(5)u(1) models from Gmax = su(2)diagu(1)2 ones u(1)t models with general u(1)t ux. Models with Gmax = su(2) Models with Gmax = su(2)u(1)2 Model with Gmax = su(3)u(1) Model with Gmax = sgo(5)u(1) Anomalies from six dimensions Models with Gmax = so(5)u(1) Models for Gmax = su(3)u(1) Models for Gmax = su(2)u(1)2 A Free trinion from TB B Trinion TA from trinion TB C Calculating the 6d anomaly polynomial C.1 The anomaly polynomial for the Z2 orbifold of the A1 (2; 0) theory D Computation of anomalies and indices from Conformal manifold of the orbifold theory Anomalies for more general case Maximal punctures as two minimal punctures and integrable models Introduction eld theories in a given number of dimensions can exhibit rather surprising properties. Important examples of such properties are the dualities relating seemingly di erent quantum eld theories in four and lower dimensions. In many cases such surprising properties can be given a geometric interpretation once the theories of interest are realized as a dimensional reduction on a compact manifold of a higher dimensional theory. Di erent duality frames correspond to di erent ways to construct the same compact manifold. A paradigmatic example of this is the compacti cations of (2; 0) theories on a Riemann surface by product of this construction is the derivation of existence of new strongly coupled SCFT's with no tunable parameters. A generic SCFT in four dimensions has then a description in terms of such SCFT's and only in very special cases a weakly coupled Lagrangian can be constructed. A natural question to ask is whether by starting from more generic six dimensional theories, in particular considering less supersymmetric starting points, new four dimensional phenomena can be derived. In particular a variety of six dimensional setups can be considered having (1; 0) supersymmetry [2] and a classi cation of them based on F-theory constructions has been proposed [3{5], leading to a vast number of 6d SCFT's. Taking such will be labeled by the choice of the six dimensional theory and by the choices made during the compacti cation. The latter include the choice of the Riemann surface and choices of bundles for di erent background vector elds associated to avor symmetries of the six dimensional theory. Di erent compacti cations, thus di erent labeling, might produce same theories in four dimensions [6{8], but in a generic scenario the four dimensional theories coming from di erent compacti cations are distinct. This then possibly produces a very wide variety of theories and relations between those. The purpose of this paper is to establish a direct link between choices made in six dimensions and theories conjecturally obtained in four dimensions [9] (see also [10{12]). In particular the choices made during compacti cation determine in a rather tractable way the dimension of the conformal manifold and the anomalies of the four dimensional theories. The conformal manifold is determined by the moduli space of complex structures of the Riemann surface and by the moduli space of gauge connections for the symmetries of the six dimensional setup. These can lead to distinct 4d theories coming from choices of in general non-abelian at bundles1 leading to a large number of moduli [13]. More precisely, let G denote the 6d avor symmetry. We pick an abelian subgroup of it L G and denote the commutant by G0 which we take to be non-abelian. In particular we have and de ne Gmax = L G0. Then we pick a at gauge eld for Gmax on the Riemann surface modulo the possibility of picking a non-trivial ux for the abelian part c1(L) 2 Zdim(L). The dimension of the conformal manifold for compacti cations with no punctures for each non-trivial choice of c1(L) is given by, Note that the 1 in the (g 1) term is there only for the non-abelian part of Gmax, which comes from the fundamental group relation of the Riemann surface as well as gauge transformations which only impact the non-abelian part of Gmax. The anomalies of the four dimensional theories can be obtained on the other hand by integrating the anomaly polynomial of the six dimensional theory on the Riemann surface taking into account the particular choice made for the uxes, c1(L). After making some general remarks and predictions we will discuss in detail the case of two M5 branes probing Z2 singularity. Following [9] we will derive a set of theories in four dimensions which we will then map to the six dimensional compacti cations on Riemann surfaces with genus g > 1 and with di erent choices of uxes for the avor symmetry. In particular we will compare the anomalies of the four dimensional construction with those obtained from six dimensions and the dimensions of the conformal manifolds. The theories in four dimensions will have a description in terms of \strongly coupled" Lagrangians. The construction will start from weakly coupled Lagrangians and then take us on a parameter space of these models to special loci where certain symmetries are enhanced. Coupling these symmetries to dynamical gauge eld the theories which correspond to the di erent compacti cations can then be constructed. Since the enhancement of symmetry used for the construction happens only at strong coupling and we do not have a precise road map to the enhancement locus, we refer to these models as having \strongly coupled" Lagrangians. Such Lagrangians are nevertheless su cient to derive a plethora of protected information about the models. In particular all the anomalies and supersymmetric indices are easily extractable. We will compute here the supersymmetric index and the central charges from these Lagrangians. 1This can be viewed as the choice of a connection for a holomorphic vector bundle. The organization of this paper is as follows. In section 2 we study the general structure of 6d (1,0) SCFT's compacti ed on a Riemann surface and the resulting 4d theories. We exemplify it using 6d SCFT's arising from M5 branes on G singularity compacti ed on a Riemann surface and in particular predict the dimension of the conformal manifold of the four dimensional eld theory arising in low energy. In section 3 we review a four dimensional construction conjectured to give the eld theories arising in the compacti cation, and discuss the dimension of the conformal manifold from this perspective. In section 4 we of M5 branes is also two. In section 5 we derive a description of some trinions, theories in terms of \strongly coupled" Lagrangians. We also discuss superconformal indices and anomalies of theories constructed from these building blocks. In section 6 we derive a variety of examples of theories corresponding to di erent choices of uxes upon compacti cations. In section 7 we compute anomalies of the putative four dimensional theories arising in compacti cations from six dimensional anomaly polynomial. In section 8 we summarize our results. Several appendices complement the text. Basic setup In this section we discuss the basic setup we have and use that to make predictions about preserve half the supersymmetry we need to partially twist the theory along the Riemann surface [14]. What this means is that we modify the spin connection by mixing with it a u(1) in the Cartan of su(2) R-symmetry of the (1; 0) theory: u(1) su(2)R. In this way, on We expect, as in the case of its (2; 0) cousin, that in the infrared, which corresponds to the choice of the complex structure of the Riemann surface. In other words, the conformal the Riemann surface. However with lower supersymmetry we will have more options as we go down to 4d. The reason for this is that the (1; 0) theories often (but not always) have additional global symmetries. We can turn on backgrounds [15] corresponding to gauge eld con gurations for this avor symmetry along the Riemann surface in a way that supersymmetry be preserved is that the supersymmetry variation of the gaugino elds in the multiplet be zero. This variation of the gauginos represented by is given by = F where the D-term in the variation breaks the symmetry to an abelian subgroup. For the try. Thus for non-abelian avor symmetry we can turn on at non-trivial bundles on the Riemann surface and still preserve the supersymmetry. For a genus g surface this gives rise to an (g 1) dim G complex dimensional moduli space where G is the non-abelian avor group. To see this note that as usual one can open up the Riemann surface given the Ai; Bj cycles. We can assign arbitrary G holonomies for each Ai; Bj subject to Y[Ai; Bi] = 1 ; and the overall gauge transformation. This gives (2g 2) dim G real dimensions, or a 1) dim G complex dimensional manifold. For the abelian u(1) case, we have a more general possibility. Let denote the spinor which is covariantly constant after the topological twist. Then to preserve the variation we need to satisfy the equation This can be solved by zz + D = Fzz zz + D = 0: ; Fzz = const: zz : We can now have F be non-zero. In such a case the allowed constants (related to the F representing a given class in c1(F ) has been xed, we can still solve the above equation by the addition of at u(1) gauge elds to it. So for each u(1) avor symmetry and for any integer n representing its c1 we will get an additional g complex moduli. However, there is a more general possibility which allows combining the abelian and non-abelian cases. Let G denote the avor symmetry group. We can choose an abelian subgroup of it L whose non-abelian commutant inside G is given by G0: L we can choose a non-trivial ux in L which can be deformed by addition of at bundle in in 4d whose conformal manifold Mg is expected to have complex dimension where Gmax = L G0. Note that Gmax is the maximal symmetry group we expect to have for the 4 dimensional theory corresponding to submoduli where the holonomies of G0 are turned o , but for which there is a background ux in L. It is useful to consider a familiar example to illustrate these ideas. Consider the 6d (2; 0) in 4d. In such a case the R-symmetry group is so(5). To twist we need to pick a u(1) in it. Consider the su(2)L su(2)R = so(4) We choose so(2) R-symmetry in the Cartan of su(2)R (which can be viewed as the sum of the two canonical Cartans so(4)). After this twist, we still have an extra avor symmetry group mentioned above. This will give us a theory whose moduli space has dimension dimMg = (3g 1) 3 = 6g which has been studied in [16]. This corresponds to the geometry of the normal bundle to the Riemann surface being R Lg 1, where Lr denotes a line bundles of degree r. to our discussion above Here the su(2) at bundle mixes the two line bundles of degree g 1. Instead we can also consider choosing an abelian subgroup L G which in this case is simply u(1) theories studied in [17] and corresponds to the normal bundle to the Riemann surface being Lg n 1. The dimension of the conformal manifold for this case is according dimMg = (3g 3) + g = 4g Let us just consider two examples to illustrate the diversity of choices available for 6d (1; 0) theories. Consider the exceptional 6d SCFT, with F-theory base given by O( 1) as part of the conformal manifold. Or we can choose any abelian subgroup L dimension can vary from 1; : : : ; 8. For each dimension there are numerous possibilities for how it embeds in E8 leading to di erent non-abelian commutants G0. For each such choice, the 6d SCFT corresponding to F-theory geometry O( 12) bundle over P1. This theory has no global symmetries and so for each genus g we get a unique choice whose conformal manifold is simply the complex structure moduli space of the Riemann surface.2 Adding punctures So far we have discussed compacti cations on Riemann surfaces without punctures. If we add punctures to the Riemann surface, there would be additional moduli for the resulting conformal eld theory. The most obvious has to do with the choice of the position of the punctures. If we have s punctures this will add s complex moduli. Moreover, depending on the type of the puncture, the choice of the holonomy of G will have to be restricted to a special one compatible with preserving conformal symmetry. One way to think about this is that a puncture on the Riemann surface can be viewed as attaching a semi-in nite cylinder to the surface. So we e ectively obtain the reduction of the 6d theory on a circle to 5d. We will have to choose a holonomy for G along the circle. Such holonomies typically play the role of mass parameters for 5d theories. Only at special holonomies can we expect the 5d theory to be gapless. Thus the choice of such holonomies will typically break the G symmetry to a subgroup P G, which preserves the element. The inequivalent choices of G holonomy in a particular conjugacy class, which are preserved by P are given by of such moduli spaces). If the holonomy in the bulk is broken to Gmax due to picking an with no extra moduli. All of the well-studied cases have extra moduli (even those coming from (2; 0) theory as already discussed) when viewed as an N = 1 theory in 4d. by P max = Gmax conformal manifold: dimMg = (3g Abelian subgroup L to turn on ux in, then the allowed holonomies will be smaller given \ P . In this way we get the following formula for the dimension of the 4d 3 + s) + (g 1)dim(Gmax) + dim(L) + = (3g 3 + s) + g dim(Gmax) + dim(L) As we have discussed, the choice of punctures of a given type corresponds to a conjugacy class of G which preserves a subgroup P of G. There could be various di erent though physically equivalent ways to embed P in G. These are usually related to one another by automorphisms of G, which may be both inner or outer automorphisms. For di erent choices related by an inner automorphism, if G is a simple non-abelian group then there is no need to have an extra designation of the actual element of the conjugacy class of the puncture, because one can obtain all choices of the conjugacy element by the action of G which can be absorbed into the moduli of at connections on the punctured Riemann surface. However, even if G is simple, when we choose an abliean subgroup L to put ux in, it could happen that not all conjugacy elements compatible G0 which is the left-over symmetry. In such a case, inequivalent choices of embedding of the conjugacy classes of the punctures in G should be viewed as leading to distinct classes of theories. The same is true also for embeddings di ering by an outer automorphism. These Gmax inequivalent embeddings of the conjugacy element of the puncture in G is an extra designation for the puncture and explains the appearance of the `color' (and `sign') of the puncture in [9]. We shall call such extra choices for a puncture for a general theory as the `color' of the puncture. M5 branes probing ADE compacti ed on a surface The main class of examples studied in this paper involve compacti cations of the 6d SCFT obtained by probing ADE singularities with N M5 branes. In fact we will concentrate on the Ak 1 case and our main example will involve 2 M5 branes probing the A1 singularity. As we will see, already this example is highly non-trivial and interesting. Consider M-theory in the presence of K-type singularity where K can stand for any of the ADE groups. In other words, we consider the background given by K is the discrete subgroup of su(2) associated to the group K. As is well known this gives rise to a 7 dimensional singularity on which a K-type gauge symmetry emerges. Now consider probing this singularity with N parallel and coincident M5 branes. Since M5 branes have a 6 dimensional worldvolume, i.e. 1 lower dimension than the singularity locus, they will appear as points on the line of the singularity. The resulting theory will have a G = KL KR symmetry [8]. The KL;R arise from the bulk gauge symmetries 1 X dim 1 X dim(Pimax) case, which we will be mainly interested in, there is also a global u(1) symmetry coming from the fact that K enhances to u(k). Generally these theories are non-trivial 6d (1; 0) they simply give rise to a hypermultiplet which transforms in the (k; k)+1 representation of u(1). For low k; N the avor symmetry can get accidental enhancements. For example for k = N = 2 it turns out that su(2)L u(1) enhances to G = so(7). Now we wish to compactify this 6d theory on a Riemann surface and as already discussed we partially twist the theory by adding to the spin connection the Cartan of su(2)R symmetry. In the case of T 2 we can preserve all the supersymmetries, leading to an N = 2 supersymmetric theory in 4d. However, as already discussed we can do more: since the theory in the bulk has G gauge symmetry we can turn on at G-bundles on the Riemann surface. In fact, because the M5 branes split the K-symmetry in the bulk to two parts, we can turn on independent at K bundles for KL and KR. This still preserves supersymmetry. A simple example case the supersymmetry charge is given by T r( @ ) where are bi-fundamental hypermultiplet fermions and bosons transforming as, which leaves the supercharge invariant as the undergo GL GR monodromies around non-trivial cycles of the Riemann surface. In addition we can turn on a at bundle for the u(1) global symmetry (which can also have a non-trivial rst Chern-class). Moreover we can also choose other abelian subgroups of su(k) u(1) as already discussed, and turn on ux in them. Similarly we can add punctures. There are various punctures allowed. To characterize them we compactify the theory on a circle to 5d, leading to a ne su(N )k quiver theory. Punctures which preserve the full symmetry or a simple puncture with just a u(1) symmetry have been studied in [9]. Moreover the structure of allowed punctures is rather intricate and has been studied in [19]. In this paper we will mainly focus on the full and simple punctures. For full punctures, we have evidence that the conjugacy class corresponding to corresponding to leaving the holonomy of the full puncture invariant is su(2) N = 2 sub case and the resolution of a puzzle at su(k)diagonal It is instructive to consider the sub-case which leads to higher supersymmetry. To obtain complex dimensional moduli. Or we can consider the Ak 1 singularity case and consider compactifying the theory on T 2 with the u(1) at connection turned o . This gives rise su(k) connections on T 2. This is the case where we obtain a quiver gauge theory using the duality between M-theory compacti ed on T 2 and type IIB on S1. Wrapped M 5 branes on T 2 become dual to D3 branes and so in the type IIB theory we have N D3 branes probing Ak 1 singularity which leads to the a ne su(k) N = 2 supersymmetric quiver theory with su(N ) gauge groups on each node of the quiver. It is known [21] that the moduli of this theory is that of at su(k) connections on T 2 which is k dimensional. For a discussion of this in the context of 6d theory compacti ed on T 2 see [8]. Now we come to a puzzle: from the discussion of the previous section, we expected the moduli space to be given by two copies of at K bundles. However, here we are only getting one copy. This is because we are in a special situation where the generic argument discussed in the previous section does not apply. In particular naive application of the formulas of the previous section would have given a factor of (g 1) = 0 for the multiplicative factor for the dimension of at bundles at genus 1. Of course in that case the moduli would have to be abelian because we are on a torus. Nevertheless we are still getting only one copy of part of at bundles lead to mass parameters for the bifundamentals of the quiver theory. One may wonder whether this is a generic feature, and that somehow the o -diagonal part of K-bundlles give mass terms. We will now argue that this is not the case, and the case of torus with no punctures is misleading us. If we compactify this theory on T 2, and turn on at su(k)L su(k)R bundle, indeed if the L and R connections are not equal we will mass up the hypermultiplets because the light modes will come from the zero modes of the theory on T 2 and if the left and right connections are not equal there are no zero modes left. However, if left and right connections are equal, since the connections are abelian on the torus, this leads for generic connections to k zero modes leading to k massless bifundamentals, as is expected for a ne Ak 1 quiver with rank 1 on each quiver node. Having con rmed the expectation based on expected again that if the left and right connections are not equal the theory in 4d will become trivial. But this is not the case, as we will now argue. To see this, recall that the theory on a curved Riemann surface is twisted, which means that the elds acquire non-traditional spins. In particular let us consider the fermions in the hypermultiplet. They will transform as 1-forms as well as 0-forms, coupled to the KR bundle. To nd if there are any low energy 4d modes left, we need to count the zero modes of the internal theory. By Hirzebruch-Riemann-Roch theorem we deduce that n0 = (dimV )(2g where n1 denotes the number of 1-form zero modes and n0 the number of 0-form zero dimV = k2. We thus learn that for 2g 2 + s > 0 we will typically get 1-form zero modes we set left and right K-connections equal, then we get equal number of n1 and n0 modes and strengthens the picture that for arbitrary at KL KR connections on the Riemann Four dimensional perspective: preliminaries We now turn our attention to eld theories in four dimensions conjectured to be obtained once N M5 branes probing Zk singularity get compacti ed on a Riemann surface. The in this way are quiver theories with standard descriptions in terms of Lagrangians, but the vast majority are built from strongly coupled ingredients. The theories possessing standard Lagrangians correspond, for general N , to Riemann surfaces with genus zero and punctures with low enough symmetry, linear superconformal quivers, or to genus one with a number of u(1) punctures, circular/toric quivers. For higher genus and (N > 2 when review the essentials of the linear and toric quiver cases and then discuss the higher genus generic theories. Linear and circular quivers We start with reviewing some aspects of the linear and circular quivers, for details we refer the reader to [9]. The building block for constructing linear and circular quivers is the free trinion. This is a collection of free chiral elds which we organize into k su(N )lb, and Qel are in the bifundamental representation of su(N )lb+1 elds Qei, gure 1. The chiral elds Ql are in the bifundamental representation of We also have 2(k 1) + 1 + 1 abelian symmetries u(1)t convenient to encode the charges of the elds in fugacities. For Ql we have t 2 u(1) . It is maximal punctures with . We think of this theory as associated to a three punctured sphere with two avor symmetry su(N )ka and su(N )bk, and a minimal puncture with symmetry u(1) . The other symmetries ( ; ; t) are not associated to the punctures but have more general geometric origin. We will refer to the symmetries which are not associated to punctures as internal symmetries. These symmetries are conjectured to come from the Cartan subgroup of u(1) su(k)R discussed in the second section. We have natural operators which are charged only under the symmetry of one of symmetries, the baryons QlN and QelN . There are k mesonic operators for every maximal puncture and 2k baryonic operators for every minimal puncture. The k mesons for the two maximal punctures of the free trinion have di erent charges under the 2k symmetries. They have charge +1 under u(1)t and are charged under certain diagonal combination of the u(1)k 1 and u(1)k 1 and have charge zero under the complementary combination. In fact there is a choice of the diagonal subgroup corresponding to mapping u(1)k 1 to u(1)k 1. We could have in principle maximal punctures with mesons charged punctures. The mesons are the bilinear combinations of Q and Qe denoted in brown in the picture. In the example here the meson is charged under the maximal puncture symmetry corresponding to the groups on the left side of the quiver and singlet under other puncture symmetries. Note that the meson operators in the free trinion are \mesons" of the maximal symmetry on the right in the usual sense of the word, that is are bilinears and singlets under that symmetry, but are operators we associate to the symmetry on the left under which they are charged in the bifundamental representation. The baryons are operators of the form QN and are singlets of the maximal puncture symmetries while being charge under the minimal puncture abelian symmetry. under any one of these choices. We associate a label to the maximal punctures we call color de ning the choice of the diagonal abelian group, that is a map between the u(1) to u(1) symmetries. In linear quivers we discuss here only a Z k valued index will appear, [9]. which the mesons are charged enhances to su(2) u(1). In this case we have two mesons in the bifundamental representations of the same groups and having opposite charges under a diagonal combination of u(1) and u(1) . We can think of the punctures as breaking the u(1) symmetry to the Cartan in general, and to su(2)diag: when k and N are both two. From the point of view of six dimensions the internal group is enhanced to so(7) with the su(2)diag: a particular subgroup, which we will discuss in detail soon. We denote the symmetries that the punctures preserve as P . This will become important in what follows. We will momentarily discuss a six dimensional interpretation of the color label. We can combine two theories together by gauging a diagonal combination of symmetries associated to maximal punctures, one from each theory. There are two di erent ways to perform the gluing. The gluing we refer to as -gluing, see gure 2, also introduces k bifundamental chiral elds l which couple to the mesonic operators through a superpodi erent theories. This gluing is the Z lMl0), where Ml and Ml0 are the mesons coming from the two k orbifold of gluing with N = 2 vector eld in class Gluing two trinions together the resulting theory will have two maximal punctures, two minimal punctures, and also have all the additional abelian symmetries discussed above. It is important to note that the maximal punctures have a natural cyclic ordering of the k su(N ) groups and when gluing we keep that ordering. The gluing together of free trinions into a linear quiver theory breaks no global symmetries speci ed above. The linear quivers have a conformal manifold. A sub-manifold of this preserves all the symmetries and has dimension equal to the number of minimal punctures minus one. The linear quivers enjoy dualities which geometrically are associated to exchanging the minimal theories with 3N We can also glue two maximal punctures of a linear quiver together. The theory so obtained is associated to torus with a number of minimal punctures. If the number of minimal punctures is a multiple of k no symmetries are broken by the gauging, however if the number of minimal punctures is not a multiple of k, a u(1)k 1 subgroup of the u(1)2k 1 internal symmetry group is broken. As a basic example, gluing two maximal punctures of k singlet chiral elds. This theory corresponds to torus with one minimal puncture. Removing the singlet chiral elds one interprets the a ne quiver itself as associated to torus with no punctures [8]. This theory has u(1) u(1)k 1 symmetry. It has k + 1 dimensional manifold of conformal deformations. This manifold can be thought of as 1 for complex structure moduli, k language the a ne quiver is a torus with k punctures. We have k complex structure moduli and one N = 1 deformation. The breaking of the symmetry when one glues a torus can be understood by considering k valued label, color, of maximal punctures introduced above. The basic trinion has two maximal punctures with color di ering by one unit. Cyclic shift of the color label has no physical meaning so gluing several free trinions we can always think of them as glued along maximal punctures of same color. The colors of two maximal punctures of a sphere with some number of minimal punctures di er by that number. When we glue two ends together however if the number of minimal punctures is not a multiple of k we glue two punctures of di erent colors together and this breaks some symmetries. We can obtain di erent theories by starting from one of the theories above and closing di erent punctures. This corresponds to vacuum expectation values for baryonic operators, which closes minimal punctures, or mesonic ones which closes maximal punctures down to minimal ones. In general closing minimal punctures does not correspond to a theory with one minimal puncture less. This leads to the need to introduce discrete charges for the We can also discuss a choice of such a bundle for the additional u(1) symmetry [9{ 11, 17]. This corresponds to another type of gluing which we call S-gluing ( gure 2) . This k orbifold of gluing with an N = 1 vector in class S . Since there are no adjoint chiral elds, no additional elds are introduced by orbifolding. Therefore in S-gluing, we do not introduce the bi-fundamental elds when two models are glued but only turn on a superpotential coupling the mesons of the two maximal punctures. For this gauging not to break the symmetries we need to map appropriately the internal symmetries of the glued components. In particular, we couple the mesons of the two copies and these are charged under u(1)t. For this symmetry to be present after the gluing the u(1)t symmetry of one theory is identi ed with u(1) 1 of the other theory. This leads to the notion of a sign of maximal punctures depending on the pattern of charges under u(1)t. This idea was introduced in [17] and further developed in [9{11, 23{25]. One can dwell on the question of what happens when a maximal puncture is closed. In eld theory closing a puncture corresponds to turning on vacuum expectation values to operators which are charged under the puncture symmetries. For general N and k giving a vacuum expectation value to a meson operator for example will change the maximal puncture having su(N )k symmetry to a non maximal one having a smaller symmetry. Closing a puncture, one needs to give vacuum expectation values to several mesons completely breaking the symmetry. It was observed in [9] that doing so one does not just obtain a theory without a puncture but rather one has a theory corresponding to a Riemann surface with one less puncture and with certain additional uxes/discrete charges turned on. To summarize, the theories with Lagrangians, corresponding naturally to certain sphere and torus compacti cations of M5 branes on Ak 1 singularity, possess 2k 1 abelian symmetries which are not associated to punctures. These symmetries can be thought of as corresponding to the Cartan of su(k)L u(1) of the basic geometric setup of section 2. The discrete charges are possible choices of bundles/ uxes for these abelian symmetries. Higher genus theories We wish to construct theories associated to general surfaces. This requires introducing theories associated to a genus zero surface and having three maximal punctures. Such known mostly by indirect arguments involving dualities and cross dimensional relations. tured sphere, trinion, here is denoted by TN . These theories for example can be seen at strong coupling cusps of a linear quiver with N 1 minimal punctures. For su(2) this quiver itself is the T2 theory, but for higher N the resulting theory is the TN theory with certain superconformal tail attached to it by gauging a subgroup of it's symmetry [1]. The construction of the interacting trinions in the k > 1 case is more involved [9]. We will The addition of handles brings several complications. Gluing two punctures of the same theory together might break some symmetries as we mentioned. We will not consider such situations in general. We can also have additional moduli associated to cycles of the surface and not just to the number of tubes as we will soon discuss at length. Here we just where the extra g break supersymmetry down to minimal one and come from holonomies for the Cartan of the enhanced non abelian R symmetry [17, 23]. Tuning the degrees of the line bundles this number can be enhanced to 3g + s 3 + s where the term coming in addition to the complex structure is given by the number of at su(2) holonomies [22]. Marginal directions Let us constrain the number of marginal deformations of a theory corresponding to M 5 branes on Z k singularity. To derive a constraint we will have to assume something about the theories. The assumptions we choose to make are motivated by the linear quivers we have discussed. We assume the following, of an su(N )i maximal puncture. For each puncture we have k relevant operators Mi in a bifundamental representation su(N )i+1 subgroup of the su(N )k avor group associated to the Gluing two maximal punctures together can be achieved in two di erent ways, where in both we gauge a diagonal combination of the su(N )k symmetry of the punctures. In the rst method of gluing, the -gluing, we introduce k elds i charged u(1)t and in bifundamental representation of of su(N )i+1 su(N )i. We also turn on the superpotential, In the second way of gluing,the S-gluing, no bifundamental elds are introduced and we turn on a superpotential coupling the meson elds of the punctures, The only new marginal operators after we perform the gluing are the ones appearing in the superpotential and the ones one can build from -gluing, and the eld strength W (i) corresponding to the gauged vector elds. We also assume that operators which are marginal before gluing remain marginal. The number of marginal operators which are singlets of the symmetries associated to punctures minus the conserved currents for the internal symmetries will be assumed to be given by a linear expression, m(g; s) = Ag + Bs + C : Here g denotes the genus of the surface and s the number of maximal punctures. We do not consider here more general types of punctures. We are considering marginal directions minus the currents as this is a robust and easily computable quantity parametrizing a model. For example, it is an invariant of the Higgs mechanism for avor symmetry [26]. Using these assumptions we can quickly derive a constraint on m(g; s). When we glue two theories together we combine the genera and the number of punctures is the sum of punctures minus two. The number of marginal operators minus currents is the sum of these numbers of the two components plus the following new operators. We analyze rst -gluing. We have k operators Mi i, k Mi0 i, k gaugino bilinears W (i)W (i), and we have k fermionic opertaors i i. Here i and i are the fermionic and bosonic components of i. The latter operators correspond to the k u(1) symmetries rotating the i which are broken by the superpotential/anomalies. All in all we deduce, m(ga + gb; sa + sb 2) = m(ga; sa) + m(gb; sb) + 2k : The same computation applied to increasing the genus by gluing two punctures of the same m(g + 1; s 2) = m(g; s) + 2k : In the S-gluing we have k gaugino bilinears W (i)W (i), and we have k operators MiMi0. Thus again every S-gauging introduces 2k marginal operators and the relations (3.4) and (3.5) -gauging still hold. From relations (3.5) and (3.4) we deduce, C = B = We deduce two observations. First, if we do not have any punctures the number of marginal deformations minus the currents is proportional to g 1. We also can write a general solution to the above constraints in a informative way, m(g; s) = 3g 3 + s + is a parameter which should be integer. Note that from our discussion in six dimensions both 1 should be dimensions of groups. The former is the dimension of the group preserved on the Riemann surface, = dim Gmax, whence the latter is the dimension of the intersection of the group xed by the puncture and the group preserved by the surface, 2k 1 = dim(Gmax \ P ). The rank of Gmax is 2k 1 and thus we see that in a generic compacti cation our result is consistent with xing the conjugacy classes of the holonomies around punctures. There are a variety of values can have depending on the choice of R symmetry. Taking This is the symmetry we obtained from our Lagrangian constructions of linear and toric quivers, and we will argue that in a general compacti cation it gives the correct dimension (minus the currents) of the conformal manifold. Another natural choice we could consider 1. This gives m(g; s) = 3g + s 3 + g which implies that the Gmax symmetry is su(k) u(1) and the expression gives the number of complex structure moduli with at connections for these groups. The general arguments presented here should be modi ed in certain cases. We give In excess to the operators we counted also T rM 2 and T rM 02 here are gauge invariant since the mesons here are the moment map operators in the adjoint representation of the gauge group. These operators can be also marginal in certain cases and we will assume they are in what follows. The mesons squared are associated to the each glued component and thus do not change the number of marginal directions associated to gluing, but do change the we should consider. Each gauging adds then 2 operators. Number of marginal operators minus the conserved currents is, m(g; s) = 3g From six dimensions the group preserved by the compacti cation for which the additional operators are marginal is so(3). We choose a u(1) sub-group of the so(5) R symmetry to perform the compacti cation and the commutant is the global symmetry. We thus expect to be equal to dim so(3) and we can write the above as, m(g; s) = 3g This is the known result of [16, 17]. general N . The operators M1M2 and M10 M20 in this particular case become gauge invariant. In some situations these become also marginal. Since these operators are from only one of the glued copies they do not change the counting of operators one has to add but this does a ect the number of operators. From the six dimensional analysis the symmetry here enhances to su(2)3. We thus would expect there to be marginal directions for at holonomies of this symmetry. In this case then we can deduce from six dimensions that then can be written as, m(g; s) = 3g dim su (2)3 rank su(2)3 : which come from one of the two glued theories. Also here in certain situations these are marginal. Here the dimension of the symmetry group enhances to su(2k). This gives A = 3 + dim su(2k) = 4k2 + 2 which sets B = 2k2 k + 1. Then the dimension of the conformal manifold minus the conserved currents becomes, m(g; s) = 3g Finally in case both k and the number of branes are equal to two we have the gauge invariant deformations, which are also marginal in some cases, derived above and two additional deformations constructed from ingredients of the two copies, M10 M2 and M1M20 . The latter operators are added when we glue theories together and thus they change the number of operators added for each gluing to six. The group here enhances to so(7) m (g; s) = 3g = 3g (dim su(2) + 1 + 1) : We will analyze this case in detail in what follows and will see this result emerging from com Next we consider the -gauging. When both k and number of M5 branes are two we have additional gauge invariant deformations becoming marginal in certain compactications. These are M1 2, M2 1 . We also have more fermionic operators, 1 2. Gluing thus changes the number of marginal directions minus currents. The six dimensional symmetries here are broken to so(5) u(1). Thus we expect to have m(g; s) = 3g (dim su(2) + 1) + g We will recover this expression in explicit computations in what follows. Here too in general compacti cation the additional marginal operators are not marginal and only for special choices of R symmetry they are exactly marginal. In all the cases above we assumed that the additional operators we add are exactly marginal. Although they are marginal for any setup they are only exactly marginal for compacti cations with particular choices of R symmetry. Some of the operators which are not marginal before gluing can become marginal in certain cases which will violate our in the rest of the sections of this paper. There can be additional cases when operators become marginal, for example operators made from s, for some special choices of the R symmetry. It will be interesting to explore this further. We have derived here constraints on the structure of the conformal manifolds of the four dimensional theories based on simple assumptions of what the punctures are and how you glue them together. This structure ts well with the geometric expectation of section 2 and we will see in what follows that even ner details, like values of , agree with the expectations. Two M5 branes on A1 singularity: preliminaries and summary We will next study in detail the case of two M5 branes probing Z symmetry of the six dimensional setup, which for general Z 2 singularity. The here enhances to so(7) [7, 8]. This fact makes this case in certain respects richer than the general setup. However, we will be able to completely determine supersymmetric properties of the four dimensional theories corresponding to such compacti cations. In this section we discuss the general properties of this case and state the main results to be derived more rigorously in the following sections. To avoid confusion since both k, the order of the orbifold, and N , the number of branes, are two we will try to keep the variable N where the confusion between the two might arise. Everywhere in the following sections unless explicitly said otherwise N stands for the number two. Symmetries and group theory When we compactify the six dimensional model, we can turn on a ux on the Riemann surface for an abelian subgroup L = u(1)r of the avor symmetry, so(7) in this case.3 The four dimensional theories obtained when compacti ed on a Riemann surface with no punctures will have a conformal manifold with maximal symmetry on a sub locus of it being the commutant of the chosen u(1)r in so(7). We denote this symmetry by Gmax. On a general point of the conformal manifold the symmetry will be given by the abelian theory are summarized in the table below and in table (4.7). Gmax s]o(5)u(1) su(2)u(1)2 (a; b; c) (a; 0; b)=(0; a; b) su(2)diagu(1)2 (a; 0; 0)=(0; a; 0) (a; 0; a)=(0; a; a) We will refer to compacti cation leading to Gmax maximal symmetry on a locus of conformal manifold as being of type Gmax. In the table F denotes a triplet of uxes for the Cartan of so(7) one turns on. The three Cartans we will denote as u(1) and will now de ne. It is convenient to parametrize the symmetries with fugacities. The character of the adjoint representation of so(7) we will parametrize as, where we have de ned so(5) characters, 21so(7) = 1 + 10so(5) + 10so(5) = 3su(2)1 + 3su(2)2 + 2su(2)1 2su(2)2 ; 5so(5) = 1 + 2su(2)1 2su(2)2 ; 3In most parts of this paper we are not cautious with the global structure of the groups. In particular by quoting groups we are referring to their Lie algebras. u(1)3 = u(1)t su(2)u(1)2 = u(1)t su(2)diagu(1)2 = u(1)t su(2)u(1) = u(1) so(5)u(1) = so(5) 1=tsu(2) su(3)u(1) = (su(2)1 or (su(2)2 so(7) = so(7) : su(2)2 ; or u(1)t su(2) = ; or u(1)t u(1) = su(2) ; or u(1) It is informative to decompose so(7) into its so(6) maximal subgroup. The so(6) maximal subgroup is given in terms of an so(3) so(3) = su(2) su(2) decomposition as su(2) = su(2) .4 In particular the adjoint of so(7) decomposes as 15 + 6 of so(6) where, 15 = 6 = and su(2) characters are de ned as, 3su(2)1 = 1+ 3su(2)2 = 1+ 4 2su(2)1 = 2su(2)2 = We choose here a somewhat unusual normalization for the su(2) fugaicities which is done to be consistent with [9] and avoid proliferation of square roots in the equations. In terms of these fugacities the groups appearing in the table above are as follows, we see here. Note that so(5) and s]o(5) from both the six dimensional point of view and group theory wise are equivalent and related by a choice of u(1) in so(7). However, we treat them di erently here as in the four dimensional constructions the eld theoretic description of the two cases is di erent. Because the six dimensional origin of the two is the same the are more choices of uxes which give same groups as the ones appearing in table (4.1) having di erent four dimensional constructions. We list these for completeness but will not discuss these in detail in what follows as an interested reader can easily generate eld theories corresponding to them from the constructions of other models. Gmax su(2)u(1)2 su(2)diagu(1)2 su(2)su(2)u(1) 4One can also decompose so(6) into su(2)t su(2) or su(2) = su(2)t with similar expressions. These are equivalent decompositions of so(7) di ering by a Weyl transformation. T r(RFs2u(3)) = T r(Q3a) = T r(RQ2a) = T r(R2Qa) = T r(RFs2u(3)) = The anomalies here will be compared to four dimensional computation in appendix D. Models for Gmax = su(2)u(1)2 In this section we consider turning on two uxes each under a di erent u(1) inside so(7). We choose the u(1)'s so as to preserve the maximal possible symmetry, which is u(1)2 There are in fact two di erent ways to do this. In both we start with the previous u(1) so(5) case and turn on a ux in a u(1) within so(5). There are two di erent u(1) subgroups inside so(5). One is the maximal u(1) su(2) subgroup, under which the 4 dimensional representation of so(5) decomposes as: 4 ! 21 + 2 1. The other is taking the su(2) maximal subgroup and using the Cartan of one of the su(2)'s. Under this subgroup the 4 dimensional representation of so(5) decomposes as: 4 ! 11 + 1 1 + 20. We shall rst deal with the rst case and latter deal with the second. Models with Gmax = su(2)diagu(1)2. First we must decompose the so(5) Chern classes to those of the subgroup. For this we again utilize the splitting principle: C2(usp(4))4 + C22(usp(4))4 2C4(usp(4))4 = ch(usp(4)4) = ch (u(1)1 1 + C1(u(1)b) + C1(u(1)b) + C2(su(2))2 + C2(su(2))2 + C22(su(2))2 C22(su(2))2 where we have denoted this u(1) as u(1)b. 1, and for so(7) it is 2. over the Riemann surface nding: xt + 2C1(F ) + C10(u(1)b). Like previously, the rst term is the ux on the Riemann surface which strength is measured by x. The second term takes into account possible mixing with the R-symmetry, while the third is the curvature for the 4d u(1)b global symmetry. Again x needs to be quantized according to (7.12). Now the minimal charge of the u(1) for Spin(7) in our convention is Next we insert these decompositions in the anomaly polynomial (7.11), and integrate Z Isu(2) = (2 31z +32 32x 6 1z +12 1 22z +12 2 12x 3 21 12 22 24 2x+11)(g 1) C13(F ) 8 +2(g 1)( 12z +2 22z +4 1 2x z 1)C12(F )C10(u(1)a)+ +4(g 1)( 12x+8 22x 2x 2 2 +2z 1 2)C12(F )C10(u(1)b) +(g 1)(2z 1 +4x 2 1)C1(F )C102(u(1)a) (g 1)zC10(u(1)a)p1(T )0 2(g 1)zC103(u(1)a) 2(g 1)xC10(u(1)b)p1(T )0 4(g 1)zC10(u(1)a)C2(su(2))2 +4x(g 1)C102(u(1)a)C10(u(1)b) +4z(g 1)C102(u(1)b)C10(u(1)a)+8(g 1)( 1x+ 2z)C1(F )C10(u(1)b)C10(u(1)a) 32(g 1)xC103(u(1)b) +4(g 1)(8 2x+ 1z 1)C1(F )C102(u(1)b) 16x(g 1)C10(u(1)b)C2(su(2))2 : From these we see that: T r(R3) = 2(g 1)(2 13z +32 23x 6 1z +12 1 22z +12 2 1x 3 12 12 22 24 2x+11); 2 T r(R) = 2(g 1)(1 2z 1 8x 2) : This gives the a central charge: a = 3 (g 1) 34 36 22 80 2x+96 32x 9 21 +36 21 2x+6 31z 20 1z +36 1 2z 2 Next we need to perform a-maximization with respect to both 1 and 2. In general the solution is quite involved and we won't write it here. However for the speci c case of z = 1; x = 14 , we nd: a = 7:99177 (g c = 8:30369(g This matches the result in four dimensions (5.20). Also note that for g even the ux for x is consistent only with so(7) and not Spin(7). There are a considerable number of other anomalies, which similarly to the previous cases can be calculated from (7.33), though we shall not consider this here. of the 4 dimensional representation and thus in the relation between the usp(4) and u(1) su(2) Chern classes. Recall that under this subgroup the 4 dimensional representation of so(5) decomposes as: 4 ! 11 + 1 1 + 20. Thus using the splitting principle for the Chern classes we nd: C1(u(1)b))(1 + C2(su(2))2) : (7.37) C2(su(2))2; C4(usp(4))4 = C12(u(1)b)C2(su(2))2. xt+ 2C1(F )+ C10(u(1)b). Again x is quantized according to (7.12), where now the minimal charge in our C12(u(1)b) + convention is 1 for both cases. Riemann surface nding: Next we insert all these in the anomaly polynomial (7.11) and integrate over the Z Isu(2) = (2 31z +10 32x 6 1z +6 1 22z +6 2 12x 3 21 6 22 12 2x+11)(g 1) C13(F ) 8 1)C12(F )C10(u(1)a)+ +2(g 1)( 12x+5 22x 2x 2 2 +2z 1 2)C12(F )C10(u(1)b) +(g 1)(2z 1 +2x 2 1)C1(F )C102(u(1)a) (g 1)zC10(u(1)a)p1(T )0 2(g 1)zC103(u(1)a) (g 1)xC10(u(1)b)p1(T )0 2x(g 1)C10(u(1)b)C2(su(2))2 : From these we see that: This gives the a central charge: 2(g 1)zC10(u(1)a)C2(su(2))2 +2x(g 1)C102(u(1)a)C10(u(1)b) +2z(g 1)C102(u(1)b)C10(u(1)a)+4(g 1)( 1x+ 2z)C1(F )C10(u(1)b)C10(u(1)a) 10(g 1)xC103(u(1)b) +2(g 1)(5 2x+ 1z 1)C1(F )C102(u(1)b) 3 = 2 (g 1) 2 13z +10 23x 6 1z +6 1 22z +6 2 1x 3 12 6 22 12 2x+11 ; 2 T r (R) = 2 (g 1) (1 2z 1 4x 2) : a = 3 (g 1) 34 18 22 40 2x+30 32x 9 21 +18 21 2x+6 31z 20 1z +18 1 2z 2 Next we need to perform a-maximization with respect to both 1 and 2. In general the solution is quite involved and we won't write it here. However we can write the result for speci c cases. For instance for z = 1; x = 14 , we nd: a = 7:8911 (g c = 8:19276(g one of the uxes and so we expect symmetry enhancement. However the ux forms are di erent than before leading now to the breaking of so(7) ! u(1) su(2)2 global symmetry. In this case we can solve for 1 and 2 exactly nding: su(2)2 so we expect a 4d theory with Inserting these values to a and c we nd: a = (g c = (g 9 + 100x2 + 25x2 117 + 8 9 + 100x2 9 + 100x2 + 10x2 147 + 11p9 + 100x2 For the case of x = 14 , we get: a = This matches (6.15) taking g = 3. c = from 6d (1; 0) SCFT's. We have seen that for each such theory, the resulting conformal manifold is enriched, in addition to complex moduli of the Riemann surface, by the structure of the bundle associated with avor symmetries of the 6d theory. The diversity of the theories one obtains is further enhanced by the choice of an abelian subgroup of the avor group where in addition to at holonomies one can turn on discrete uxes for them on the Riemann surface. We checked the general expectation based on 6d reasoning in detail for compacti cations of two M5 branes probing Z 2 singularity on a genus g > 1 Riemann surface. In particular we have detailed a eld theoretic construction of a large set of models from which we read o the anomalies and the conformal manifolds. These objects then were matched to their six dimensional counterparts. This is a special case of a general story but nevertheless it is rather rich. The richness is due to the fact that the compacti cation outcome is determined, in addition to the number of branes and the type of the singularity, by a choice of ux. We have obtained eld theoretic descriptions of the theories in four dimensions in terms of \strongly coupled" Lagrangians. The subtlety here is that in order to build theories corresponding to general Riemann surfaces one has to tune couplings of a theory with standard Lagrangian to a very speci c, presumably strongly coupled, point. There are several directions for further research. First, we have studied the di erent models by de ning building blocks and gluing them together in di erent ways. A way to generalize the construction is to study more systematically di erent ows triggered by vacuum expectation values. Such ows geometrically correspond to shifting uxes and closing punctures. At the very least these should provide non trivial checks of the statements we are making here. The six dimensional computations detailed here can be easily generalized to arbitrary (1; 0) SCFT's. In particular we can apply it to M5 branes probing arbitrary singularities, numbers of branes, and choices of ux. For example, the conformal and avor anomalies for N M5 branes on Zk singularity with no uxes for any abelian subgroup of the su(k) su(k) u(1)t symmetry can be computed, and the a; c values are (see appendix F for derivation), a = c = 1)(12 + k2(9N 2 + 9N 1)(8 + k2(9N 2 + 9N work. On the eld theory side however the construction we used in this work is very tuned to the special case of two branes and Z 2 singularity. There are several ways in which one can attempt to address the question from four dimension nevertheless. One is straightforward but technically involved. The indices of the four dimensional theories were shown in [9] to be written in terms of eigenfunctions of certain integrable models for any number of branes and k of Z k (there should be also a generalization for any ADE type of singularity). See [38, 39] for explicit expressions for the Hamiltonians of these a theory one can in principle extract its anomalies [43{46]. However, technically case. The di erent limits in which the problem can be solved more easily [9, 42] are blind to exactly marginal deformations though capture the relevant deformations. Another road to generalizations is to understand better the dualities leading to relations between theories with Lagrangians and strongly coupled building blocks. This is the study of the superconformal tails. Such tails were studied for the Z k orbifold case in [9]. In particular, if relevant dualities can be harnessed to isolate the strongly coupled building blocks one would be able to write \strongly coupled" Lagrangians of the type discussed in [28] and in this work. Technically this requires analyzing integral kernels arising from supersymmetric tails and nding procedures to invert them extending the results of [31]. The theories with more branes and general singularities will be classi ed in addition by a choice of an abelian symmetry where we turn on ux embedded in G G( u(1)t) symmetry of the M5 brane setup. In addition we will have a rich choice of punctures, recently addressed in [19], which will further come in di erent \colors" (corresponding to choices of abelian uxes) classi ed by the subgroups of G G( u(1)t) which they preserve. Other challenges include making contact with the di erent limits one can consider. Examples are the holographic limits [47, 48] and the compacti cation limits to lower dimensions. One can also consider studying general partition functions. On general grounds we would expect that various indices (lens index [49], T 2 S2 index [50, 51]) should be derivable in terms of a corresponding TQFT structure (see [52, 53] for the lens index for work we have concentrated on genus larger than one though studying genus one should be also feasible and it would be interesting to make contact to the results of [6{8]. Acknowledgments We would like to thank Thomas Dumitrescu, Davide Gaiotto, Sergei Gukov, Jonathan Heckman, Patrick Je erson, Zohar Komargodski, Tom Rudelius, and Amos Yarom for useful discussions. The research of CV is supported in part by NSF grant PHY-1067976. GZ is supported in part by the Israel Science Foundation under grant no. 352/13, by the German-Israeli Foundation for Scienti c Research and Development under grant no. 1156-124.7/2011, and by World Premier International Research Center Initiative (WPI), MEXT, Japan. SSR is a Jacques Lewiner Career Advancement Chair fellow. The research of SSR was also supported by Israel Science Foundation under grant no. 1696/15 and by I-CORE Program of the Planning and Budgeting Committee. Free trinion from TB Starting from the TB trinion we can obtain the free trinion by partially closing the maximal puncture, which appears twice, to a minimal one. The puncture is closed by giving a vacuum expectation value to one of the M operators associated to the puncture. Let us choose the component of M with charges t z 1 1 z2 1. This means that a combination of the puncture su(N )2 symmetries and u(1)t is broken. In terms of fugacities we have t = z1z2 which can be solved by taking z1 = t 2 1 and z2 = t 21 1 . We use the Lagrangian we discussed for TB to understand the IR xed point of this ow. It is easiest to perform the analysis at the level of the index since it captures all the relevant details of the physics. Iuzv = e(pq 2 ) e(t( u2) 1u1 1)(p; p)(q; q) e( z 1u1 1)Iz;v;pzu2;pu2=z : are using here the choice of R symmetry giving the bi-fundamental chirals charge two in this and next appendix. The function e(z) is the elliptic Gamma function and we refer the reader to [9] for all the de nitions used here. We need to compute the residue in z which appears only in the orbifold theory. This index gives us the index of an su(2) SYM with four avors and a bunch of singlet elds. Let us see how this comes about. First we write the index of this theory in detail, Iz;v;a;b = (p; p)2(q; q)2 According to the prescription to close maximal puncture one also has to introduce ve singlet elds and couple them through superpotential. In the index this amounts to multiplying the above by e(pq 2) . Although it is not obvious from the expression after this multiplication it is symmetric under the exchange of the u(1) fugacities a, b, and . This follows from Seiberg duality. Let us denote the residue above by Iz;a;b; and use the symmetry to write the residue of (A.1) as, We see now that the z su(2) theory has only two avors and thus is described by the quantum deformed moduli space which in the index implies a delta function identifying u1 with w1. We thus get the index for the xed point to be, This is the index of a free trinion. We thus deduce that the theory TB under an RG triggered by a vacuum expectation value for one of the mesonic operators associated to the maximal puncture ows to free trinion. Trinion TA from trinion TB We can also obtain trinion TA starting from a four punctured sphere built from TB trinions. As we saw when one considers RG ow triggered by vacuum expectation values for puncture appearing twice on TB the xed point is given by a free trinion. Let us glue together two TB along the puncture appearing once and then close one of the remaining punctures to (p; p) (q; q) =u2) 1 u1 1 Ic;w;pzu2;pu2=z : We close the u(1) puncture completely by giving a vacuum expectation value to baryonic operator implying superpotential. The index becomes, Ic;w;pzv2;pv2=z : Ic;w;pzv2;pv2=z : (q; q) (p; p) (p; p) (q; q) (p; p) (q; q) The u1 su(2) gauge part has three avors and thus is described in the IR by quadratic gauge invariant composits, This is exactly the index of TA. The trinion TA can be obtained thus as a xed point of ow starting from theories built from TB trinions, and the same of course holds for any theory built from TA trinions. The equation (B.2) implies also that TA trinion is obtainable from TB trinion by gauging an su(2) subgroup of the latter. There is also an inverse relation giving TB from TA. The operation of gauging an su(2) subgroup is a color-changing operation for a single puncture. Note that if for some reason the u(1) is broken the kernel of (B.2) is trivial and the two theories are the same. Calculating the 6d anomaly polynomial In this appendix we brie y review and collect the various formulas that we use in this article to calculate and manipulate anomaly polynomials. For a more comprehensive discussion about the calculation of the anomaly polynomial for 6d SCFT's see [37] and references within. The anomaly polynomial receives contributions from the various elds in the theory. For 6d SCFT's the relevant possible elds are the vector, hyper and tensor multiplets. Their contributions can be evaluated using the known contribution of a Weyl fermion [54]: ch(E)A^(T ) ; where in 6d we must project to the 8 form part. A^(T ) is the Dirac A-roof genus conveniently given by: A^(T ) = 1 where p1(T ); p2(T ) are the rst and second Pontryagin classes of the tangent bundle re ch(E) is the Chern character of the total bundle for any additional local or global symmetries. It is convenient to expand it in terms of the Chern classes of the bundle: ch(E) = rank(E) + C1(E) + C14(E) + 4C1(E)C3(E) 4C12(E)C2(E) + 2C22(E) 3C1(E)C2(E) + 3C3(E) where Ci(E) stands for the i'th Chern class. and product of vector bundles: A useful property of the Chern character is its decomposition under the direct sum For the theories we consider the additional symmetries we encounter are the su(2)R Rch(R)ch(G)ch(F ). We next evaluate the contribution to the anomaly polynomial for each multiplet in turn. V ) = ch(U ) + ch(v); ch(U V ) = ch(U )ch(v) : There are similar formulas also for the Chern and Pontryagin classes. These are collectively known as the splitting principle. For the Chern classes, de ning the total Chern class as: C(E) = P Ci(E), it obeys: V ) = C(U )C(v) : For the Pontryagin classes, given a decomposition of the bundle as a sum of complex line bundles with rst Chern classes ei, then: p1(T ) = X ei2; p2(T ) = We consider a single hypermultiplet in a representation rG of the gauge symmetry and rF of the avor symmetry. It contains a single Weyl fermion which is an su(2)R singlet. Thus its addition to the anomaly polynomial receives contributions from the tangent, gauge and avor symmetry bundles. Using the formulas presented in this section, it is given by: p1(T )C2(G)rG 2 C2(G)rG (C12(F )rF C14(F )rF +4C1(F )rF C3(F )rF 4C12(F )rF C2(F )rF +2C22(F )rF where we use Ci(G)rG for the i'th Chern class of the G-bundle with representation rG we consider are simple. We consider a single vector multiplet. It contains a single Weyl fermion which is an su(2)R doublet and is in the adjoint representation of the gauge group. Thus its addition to the anomaly polynomials receives contributions from the tangent, gauge and R-symmetry bundles. Note that the chirality of the spinor is opposite to that of the fermion in the hyper and tensor multiplets, and so contribute to the anomaly polynomial with a minus sign. Using the formulas presented in this section, it is given by: p1(T )C2(G)Ad p1(T )C2(R) where we use C2(R) for the second Chern class of the su(2)R bundle in the doublet repre We consider a single tensor multiplet. It contains a single Weyl fermion which is an su(2)R doublet. In addition it contains a self dual tensor which is also chiral and thus contribute to the gravitational part of the anomaly, where the exact contribution was evaluated in [55]. Using this and the formulas presented in this section, one nds: p1(T )C2(R) Besides the contributions of the eld content one must also add the Green-Schwartz term. This term takes into account the e ect of modifying the Bianchi identity for the tensor multiplet. It is a complete square and is chosen so as to make all gauge anomalies vanish. The anomaly polynomial for the Z 2 orbifold of the A1 (2; 0) theory Using the above formulas it is now straightforward to calculate the anomaly polynomial 2 orbifold of the A1 (2; 0) theory using its gauge theory description. The matter content includes: a single tensor multiplet, a vector multiplet in the adjoint of su(2)G and 8 half-hyper multiplets in the doublet representation of su(2)G. These are rotated with an so(7) global symmetry where they transform as the 8 dimensional spinor representation Using the previous formulas we nd after a little algebra: I8 eld = C2(R)p1(T ) C2(so(7))8p1(T ) C2(R)C2(su(2)) C22(so(7))8 2C4(so(7))8 sentation r, to convert to a second Chern class that is independent of the representation. To this we need to add the Green-Schwartz term given by: C2(so(7))8 Summing both terms we nally get: I8 = I8 eld + I8GS 7C22(so(7))8 C2(R)p1(T ) C2(so(7))8p1(T ) C2(R)C2(so(7))8 Computation of anomalies and indices from In this appendix we give the details of the computations of the anomalies and indices from the eld theory side using the \strongly coupled" Lagrangians. Here we use the assignment of R-charges giving R charge 1 to the elds. The assignments giving other convenient charges, 2 or 23 , can be obtained from this by admixing a proper multiple of u(1)t charge. Anomalies. Let us de ne, a (R) = c (R) = av(G) = cv(G) = dimG ; For anomalies of chiral elds of R charge R and vector elds for group G. The superconformal R charge will be denoted, Rc(R; q ; q ; qt) = R + `1q + `2q + `3qt ; and is a function of three variables `i to be determined by a maximization for each theory. We also de ne, as the anomalies for contributions of elds, bifundamental chirals and vectors, introduced when we glue two punctures of the same color. The subscript denotes the color of puncture. We warm up by computing the anomalies of the orbifold theory. The chiral matter, i and Q0i , of table (4.16) determine the anomalies to be aOrb = av + 8 a 3`33 + 9`32 + 36`1`2`3 cOrb = cv + 8 c 9`33 + 27`32 + 108`1`2`3 From now on we will only consider the a anomaly as c can be obtained in a similar manner. We start from the rst non trivial theory, the TA trinion. It is constructed from the orbifold theory by gauging an su(2) group with chiral elds listed in table (5.5) and the elds 0 which are ipping the 0 elds and thus have opposite charges and their R charges sum up to two. The central charge is then given by, + a (Rc(1; 2; 0; 1)) + a (Rc(1; 0; 2; 1)) + a (Rc(2; 2; 2; 0)) : Extremizing this with respect to `i we obtain the result reported in the bulk of the paper (5.15). For the TB trinion we obtain again from the Lagrangian in section ve, +a (Rc(1; 2; 0; 1))+a (Rc(1; 2; 0; 1)) +a (Rc(2; 2; 2; 0)) : This upon maximization reproduces (5.15). We can compute the anomalies for theories with enhanced symmetry. The four puncfree trinions with subsequent closure of minimal punctures. It has the following confor+a (Rc(1; 1; 1; 1))+a (Rc(1; 1; 1; 1))+a (Rc(0; 2; 0; 0))+a (Rc(0; 0; 2; 0)) +a (Rc(2; 0; 4; 0))+a (Rc(2; 4; 0; 0)) 12`33 +18`32 16`3 +36(1+`3)(`12 +`22)+72`3`2`1 11 : Upon maximization it produces the anomalies (6.2). Moreover the conformal R symmetry one discovers that these are consistent with the gauge coupling being exactly marginal. For the so(7) theory of gure 10 we glue TA theory with plus sign to TA theory with minus sign. The contribution to the anomaly of the minus theory is the same as the plus one with `i ipping signs. The anomaly is then 18(2`12 +2`22 +`32) 17 : Let us now consider the anomaly for the su(3)u(1) example we discussed in section Tso(5). We have di erent types of punctures glues together and have to be careful about that. Combining the ingredients we obtain the following anomaly, +6av+(`1; `2; `3)+2av ( `1; `2; `3)+2av+( `1; `2; `3) 120`13 +24`33 +90`32 20(`3 +2`1)+36(`12 +`22)(5+2`3)+9`1(`32 +`22) 85 : Maximizing this we obtain the `i given in (6.24) and anomalies of (6.21). We can compute anomalies for avor symmetries and mixed R-symmetry avor symmetry anomalies. Let us give several examples. Since the theories are built iteratively from orbifold theory we start with it. The computation is completely standard and we give (T rRu(1)t2)orb = (T rRu(1)t2 )orb = (T rR2u(1)t)orb = (T rR2u(1)t )orb = (T ru(1)t3)orb = (T ru(1)t3 )orb = (T rRu(1)2 )orb = 4(1 + `3) ; 4(3 + 4`2 + `3) ; 1 + 4`1`2) ; 4(`32 + 2(1 + 2`2)(`3 + 2`1) The symmetry u(1)t is the diagonal combination of u(1)t and u(1) . We have written down the dependence on mixing parameters since we will have to plug in the di erent values for di erent theories obtained through the maximization procedure. One should use the expressions we derived for the a anomaly above for various theories and change the functions a for other anomalies for the di erent ingredients. One point to be cautions about is that when a minus type theory is taken and anomaly of odd number of currents for avor u(1)s is considered then in addition to switching the signature of `i the anomaly has to be taken with a minus sign. Let us quote the results for subset of the di erent models and we mentioned till now and a sub-set of anomalies. For the TB trinion we obtain, (T rRu(1)t2)TB = 4(`1 (T rRu(1)t2 )TB = 4(4(`1 (T rR2u(1)t)TB = 4(`12 + `32 + `22 + 2`2(`1 + `3) 8`1(`2 + 1) 4(2`2 + 1)`3 + 1 (T rR2u(1)t )TB = 2 8`12 (T ru(1)t3)TB = (T ru(1)t3 )TB = 16 ; (T rRu(1)2 )TB = 4(`1 For TA trinion we have, (T rRu(1)t2)TA = (T rRu(1)t2 )TA = (T rR2u(1)t)TA = (T rR2u(1)t )TA = (T ru(1)t3)TA = (T ru(1)t3 )TA = (T rRu(1)2 )TA = 4(`1 + `3 + `2 + 1) ; 4(8(`1 + `2) + 4`3 + 3) ; 4(`1 + `3 + 5`2 + 2) : 4(`12 + `32 + `22 + 2`3(`2 + `1 + 1) + 6`2`1 2(12`12 + 8`1(1 + 2`2 + `3) + 4`3(`3 + 4`2 + 1) The so(5) theory for sphere with four maximal punctures gives, (T rRu(1)t2)Tso(5) = (T rRu(1)t2 )Tso(5) = (T rR2u(1)t)Tso(5) = 8(1 + 2`3) ; 8(4(`1 + `2 + `3) + 3) ; 8(2((`1 + `2)2 + `32) + 2`3 (T rR2u(1)t )Tso(5) = 8(2((`1 + `2)2 + `32) + 2`3(2`2 + 2`1 + 1) + 4`1 (T ru(1)t3)so(5) = (T ru(1)t3 )Tso(5) = (T rRu(1)2 )so(5) = 16(`3 + 2`2 + 1) : We can construct anomalies for genus g surface by gluing g 1 four punctured spheres together. Summing the contributions of the spheres and the vectors we derive, (T rRu(1)t2)Tso(5);g = (g 1)(T rRu(1)t2)Tso(5) + (2g (T rR2u(1)t)Tso(5);g = (g 1)(T rR2u(1)t)Tso(5) + (2g (T ru(1)t3)Tso(5);g = (g 1)(T ru(1)t3)Tso(5) + (2g 2)( 8) = is a matter of convention. This agrees with (7.22) up to relative normalization ( 12 ) of u(1)t and u(1)a charges which Taking all the results and computing the anomalies for the su(3) example of section 6 (T rR2u(1)t )Tsu(3) = (T ru(1)t3 )Tsu(3) = (T rRu(1)t2 )Tsu(3) = (T rRu(1)2 )su(3) = This is in perfect agreement with (7.31) up to sign for u(1)t which is again a matter The computation of the indices is structurally identical to the computation of the anomalies. We have already computed the indices of the trinions TA and TB in appendices A and B. The indices of the other theories in the paper are obtained by combining these using the usual rules of index computations. We state here the basic rules of the computations. Gluing two theories, indices of which are given by Ia(u) and Ib(u), depends on the colors and the signs of the two punctures. If the signs are the same we use the -gluing and the index of the combined theory is (here we assign R charge one to the contrast to two in appendices A and B), Ia b = (q; q)2 (p; p)2 Ia (u1; u2) Ib(u2; u1) Here the number C takes value in 1 and denotes the color of the maximal punctures. Number S takes also value in 1 and denotes the sign of the punctures. If the signs of the two punctures are opposite then the index of the combined theory takes the form, Ia b = (q; q)2 (p; p)2 If we are given a theory of a particular sign index of which is I(ul; ; ; t) the index of the theory of opposite sign is I(uly; 1 ; 1 ; t 1). Employing these simple rules all the indices reported in the le can be easily derived. There are several ways to study the conformal manifold of the orbifold theory. We can analyze it for example at the vicinity of free point. At that point the symmetry of the theory is H = su(8) Q2 = fQ1 ; Q+; Q01 ; Q02+g are in the 8 of one of the two su(8)s. The su(2) group rotates 2 s. The u(1)t was de ned in section four, and under it the s have charge and all the other chirals have charge 12 . We have two additional abelian symmetries which are anomalous once the gauge interactions are switched on. The marginal operators are, u(1)t. The elds Q1 = fQ1+; Q ; Q01+; Q02 g and 2 Q2. In addition we have two gauge couplings. The coupling are singlets of u(1)t and transform as (8; 8; 2) under the non abelian factors. We nd the dimension of computing its dimension. This amounts to counting independent holomorphic invariants s and the two gauge g1 and g2. This can be easily done. The invariants under the two su(8)s are the \baryons" built from the s. These baryons form the 9 of su(2). There are six independent invariant built from these \baryons" and this is the dimension of the conformal manifold. We have 130 couplings we start with and the dimension of the group including anomalous symmetries is 132. We have eight symmetries, all abelian, preserved on a generic point of the manifold. This gives us again conformal manifold of dimension six as the broken symmetries must be consumed by marginal operators. One of the exactly marginal directions preserves the su(4) u(1) symmetry mentioned in the le. At generic point of this manifold we have marginal deformations in (15; 1; 0; 2) and (1; 15; 0; 2). This can be deduced for example by writing down the index of the orbifold theory which at order pq gives, The negative terms are the conserved currents, the rst terms is the exactly marginal deformation preserving the symmetries. the rest of the terms are the marginal deformations. We again can perform the quotient looking for holomorphic invariants. We can construct six singlets of the su(4)s charged under the u(1)s plus and minus four, six and eight. From the singlets we can build ve invariants. With the addition of the deformation we started with this again gives us six dimensional manifold. Let us also count the dimension of the conformal manifold preserving (su(2)2)3 of the so(7) trinion. The E7 surprise theory has 1463 marginal deformations forming a single representation of E7. This decomposes into so(12) su(2) = , The di erent representations of so(12) decompose to su(2) 66 = (3; 1; 1) + (1; 3; 1) + (2; 2; 8v) + (1; 1; 28) ; 77 = (1; 1; 1) + (3; 3; 1) + (2; 2; 8v) + (1; 1; 35v) (2; 2; 2; 2) + (3; 3; 1; 1) + (1; 1; 3; 3) + (1; 1; 1; 1) Thus the 1463 has only two singlets, which we denote as and 0, of su(2)6 decomposition independent holomorphic invariants of su(2)6 su(2) = (T r 2, T r 02, and T r give the three deformations preserving the su(2)6 symmetry. These are the exactly marginal deformations of our interest. Anomalies for more general case In this appendix we consider the calculation of the 4d anomaly polynomial from the 6d previously only by the di erent (1; 0) 6d SCFT, here being the Zk orbifold of the 6d AN 1 type (2; 0) theory. This SCFT has an N 1 dimensional Tensor branch along which the theory can be e ectively described by an N F + su(k)N 1 + N F quiver gauge theory. For general N and k the theory has an su(k) u(1) global symmetry. This is enhanced We will need the 6d anomaly polynomial for this theory to perform the calculation, where for simplicity we shall set the curvature of the avor symmetries to zero. The anomaly polynomial for this case was derived in [37] and reads: 1)(2 + k2(N 2 + N 2)p1(T )C2(R) I8 = (7k2 + 30N 4(k2 + 30N a = c = Next we integrate this on the Riemann surface. For this we need to include the twist and decompose C2(R) and p1(T ); p2(T ) exactly as done in section 7. The result is: I6 = 1)(2+k2(N 2 +N 1)(k2 2)(g 1)p1(T )C1(F ) From which we nd that T r(R3) = (N 1)(2 + k2(N 2 + N 1); T r(R) = 1). Combining these we nd: 1)(12+k2(9N 2 +9N 6))(g 1); 1)(8+k2(9N 2 +9N 4))(g 1) : There is one nal issue we need to address. For general N and k the global symmetry of the resulting 4d theory includes a u(1) coming from the 6d avor one. Thus one might worry it can mix with the R-symmetry and so invalidates (F.3). However, by inspection one can see that the only terms in the 6d anomaly polynomial that can contribute to the 4d one are proportional to C2(R). Furthermore the only term that can appear in the 6d anomaly polynomial containing both C2(R) and the curvature for the u(1) symmetry, C1(u(1)t), is C2(R)C12(u(1)t). Therefore the only non-trivial anomaly for u(1)t in 4d will be T r(Ru(1)t2) implying that a will be extremized at zero mixing. The supersymmetric indices of theories discussed in this paper can be neatly written in terms of eigenfunctions of certain di erence operators [9]. This makes the duality properties of the theories manifest through a TQFT structure of the index [40, 42, 56]. The di erent properties of the theories are encoded in the index in a duality invariant way and imply certain mathematical identities. Let us for completeness present this structure here. We can write the index of the free trinion as [9], I(u; v; ; t; ; ) = Here we use the choice of R charge giving value two for the elds . Each one of the three functions appearing in this sum is associated to a puncture, with two s associated to the two di erent colors of maximal puncture [9], and to the minimal one. The functions are orthonormal eigenfunctions of a set of di erence and integral operators with the index parametrizing the set. For example de ning, T (v1; v2; ; ; t) = are eigenfunctions of S((0;;1)) f (v1; v2) = For more details we refer to [9]. The functions can be understood as de ned by the relation (G.1). The di erent functions satisfy some relations. For example, since minimal puncture can be obtained by RG ows from maximal punctures, taking residues of [9]. As additional neat feature let us consider the splitting of maximal punctures into pairs of minimal punctures when Gmax is so(5)u(1). When we go on the conformal manifold of these models and break the u(1) u(1) symmetry, the minimal punctures, as we have seen, look as half a maximal puncture. Conversely, the deformations breaking su(2) minimal punctures. Turning deformations splitting both colors breaks u(1) and u(1) . we expect to obtain relations between functions and . Here we take take both specializations, = 1, and obtain rst, ( ; t; 1; 1) ( ; t; 1; 1) = C (t) ( ; = ; t; 1; 1) : This is the manifestation of two simple punctures combining into a maximal one when in some limits of the fugacities (p or q vanish). The proportionality factor C can be xed ows triggered by vacuum expectation values. The index of the Gmax equal so(5)u(1) sphere with three maximal punctures is then, I = (w; t; 1; 1) : a;b= 1 we obtain that, ^ (u; t; 1; 1) ^ (v; t; 1; 1) = (q; q)2(p; p)2 We can take advantage of this non linear integral equation in principle to solve for the eigenfunctions in similar manner to what was done in [57]. The index of a general theory with Gmax being so(5)u(1) is, Ig;s = (uj; t; 1; 1) : 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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Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir. 4d \( \mathcal{N}=1 \) from 6d (1, 0), Journal of High Energy Physics, 2017, 64, DOI: 10.1007/JHEP04(2017)064