4d \( \mathcal{N}=1 \) from 6d \( \mathcal{N}=\left(1,0\right) \) on a torus with fluxes

Journal of High Energy Physics, Jun 2017

Compactifying \( \mathcal{N}=\left(1,0\right) \) theories on a torus, with additional fluxes for global symmetries, we obtain \( \mathcal{N}=1 \) supersymmetric theories in four dimensions. It is shown that for many choices of flux these models are toric quiver gauge theories with singlet fields. In particular we compare the anomalies deduced from the description of the six-dimensional theory and the anomalies of the quiver gauge theories. We also give predictions for anomalies of four-dimensional theories corresponding to general compactifications of M5-branes probing \( {\mathrm{\mathbb{C}}}^2/{\mathrm{\mathbb{Z}}}_k \) singularities.

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4d \( \mathcal{N}=1 \) from 6d \( \mathcal{N}=\left(1,0\right) \) on a torus with fluxes

Received: March Published for SISSA by Springer Ibrahima Bah 1 2 4 6 8 9 10 11 Amihay Hanany 1 2 4 7 9 10 11 Kazunobu Maruyoshi 1 2 3 4 9 10 11 Shlomo S. Razamat 1 2 4 5 9 10 11 Open Access 1 2 4 9 10 11 c The Authors. 1 2 4 9 10 11 0 IPMU, University of Tokyo 1 Prince Concert Road, South Kensington , London, SW7 2AZ, U.K 2 3400 North Charles Street, Baltimore, MD 21218 , U.S.A 3 Faculty of Science and Technology, Seikei University 4 San Diego , La Jolla, CA 92093 U.S.A 5 Department of Physics , Technion 6 Department of Physics and Astronomy, Johns Hopkins University 7 Imperial College London , Blackett Laboratory 8 Department of Physics, University of California 9 Kashiwa , Chiba 277-8583 , Japan 10 Haifa , 32000 , Israel 11 3-3-1 Kichijoji-Kitamachi , Musashino-shi, Tokyo, 180-8633 , Japan Compactifying N = (1; 0) theories on a torus, with additional uxes for global symmetries, we obtain N = 1 supersymmetric theories in four dimensions. It is shown that for many choices of ux these models are toric quiver gauge theories with singlet elds. In particular we compare the anomalies deduced from the description of the sixdimensional theory and the anomalies of the quiver gauge theories. We also give predictions for anomalies of four-dimensional theories corresponding to general compacti cations of M5-branes probing C2=Zk singularities. Field Theories in Higher Dimensions; Supersymmetric Gauge Theory - = 1 from = (1; 0) on a torus with 1 Introduction 2 Anomalies from 6d 2.1 2.2 3.1 Anomaly polynomial of the 6d theory Mapping the charges in 6d, 5d and 4d 2.3 Anomaly polynomial from 6d 3 4d theories from tori Structure of the 4d quiver theories Anomalies of the 4d quiver theories 4 Case studies N = k = 2 4.2 N = 3; k = 2 5.2 4d analysis 4.3 N = 2; k = 3, the orbifold C3=Z2 5 Compacti cation with discrete twists A Anomalies of interacting trinions for general k; N B Fluxes for u(1)t symmetry for k; N = 2 Introduction Field theories in low dimensions can often be realized through compacti cations of higher dimensional models. This point of view clari es some of the well known properties of eld theories, and also predicts new properties and even new models; for example, the appearance of theories which do not have a known semiclassical limit. Such models are ubiquitous in compacti cations of six-dimensional supersymmetric theories to four dimensions [1]. In this paper we mainly study some of the simpler compacti cations. We consider theories living on the branes probing the singularity have in general some global symmetry, which in our case, for general values of N and k, is su(k) u(1). Upon compacti cation we might turn on uxes for abelian subalgebras of the global symmetry supported on the torus (see [2] for early work on the subject). Without the uxes the theories have an (see [4] for various ways to reduce on a torus without uxes). Turning the uxes on we turn on uxes only for sub-groups of su(k) su(k). In such a set-up the compacti cations give rise to theories with known Lagrangians. These turn out to be widely studied toric quiver theories, albeit with additional singlet elds. We thus obtain a novel parametrization of such theories labeling them with the number of M5-branes N , the order of the orbifold k, and the 2k 2 discrete numbers de ning the uxes through the torus. The theories in four dimensions are constructed by studying renormalization group (RG) ows of a quiver theory with su(N ) gauge nodes which together with the matter elds triangulate the torus and has k gauge groups winding around one of the cycles of the torus. The number of groups winding around the second cycle is related to the total ux through the torus. Turning on vacuum expectation values for baryonic operators in the setup one obtains theories which correspond to compacti cations on a torus with dictionary between the compacti cations and the four-dimensional models was suggested in [5]. For the dictionary to work one needs to introduce singlet elds coupled through superpotential terms to gauge invariant objects. These superpotential terms are in general irrelevant giving rise to free elds in the IR. Thus, although with non-trivial uxes all the gauge sectors are UV free, there are generally free chiral elds in the IR. The dictionary is checked in two main ways. First by showing that the anomalies of the compacti cation deduced by integrating the anomaly polynomial from six dimensions to four are consistent with the four-dimensional construction. Next, the global symmetry of the theory in four dimensions can be deduced from the compacti action details and we give examples of how this works. In addition to uxes for continuous symmetries we can turn on uxes for discrete symmetries of the six-dimensional model. The global structure of the avor symmetry is class of models in four dimensions. These uxes can materialize in di erent ways. One way is through fractional uxes whose quantization is consistent only for (SU(k) SU(k))=Zk. Another is by switching on almost commuting holonomies around the cycles of the torus, in the sense that the holonomies commute in (SU(k) SU(k))=Zk but do not in SU(k) SU(k). In four dimensions this procedure corresponds to constructing the torus by gluing a triangulated cylinder with a twist. We also discuss the eld theories one obtains with uxes for all possible u(1) subgroups in the special case of two M5-branes on Z2 singularity where the eld theoretic construction is known. Finally we give a prediction for anomalies of theories obtained from six dimensions for general choices of Riemann surfaces. We have no eld theoretic constructions in this case and this will serve as a prediction to be contrasted with future computations. The paper is organized as follows. In section 2, we discuss the computation of the anomalies from the six-dimensional vantage point. We consider the anomaly polynomial presence of general values of uxes. We then derive the anomaly polynomial for the fourdimensional models. The case of a torus is discussed in much detail. In section 3, we consider the construction in four dimensions which should result in theories corresponding to torus compacti cations. We compute the anomalies and see the agreement with the six-dimensional predictions. In section 4, we detail several examples deriving precise quiver diagrams and discussing symmetry properties which consistently enhance to match expectations from six dimensions. In section 5, we discuss compacti cations with Steifel-Whitney classes and the four-dimensional theories related to these. We have two appendices: in appendix A, we deduce some predictions from six dimensions for anomalies of four-dimensional SCFTs. The appendix B details eld theoretic constructions of strongly coupled models corresponding to compacti cations with general uxes. Anomalies from 6d We begin our discussion from the six-dimensional perspective. Consider taking N M5 six dimensions has su(k)b u(1)s symmetry for general value of k and N ; here Upon compacti cation we can choose an abelian subalgebra of this symmetry and turn uxes supported on the torus (see for example [5, 6]). As the rst Chern classes of uxes have to be properly quantized, the choice gives us models in four dimensions which are labeled by discrete parameters. We can compute the 't Hooft anomalies of the theories from the compacti cation setup by taking the anomaly eight-form polynomial and integrating this over the torus with the uxes turned on. This provides a prediction for the four-dimensional models which we will now deduce. Anomaly polynomial of the 6d theory eight-form polynomial I8, and can be computed using the methods developed in [7, 8], using the fact that on the tensor branch this theory becomes a linear quiver gauge theory with gauge group SU(k)N 1 . We use the normalization where the bifundamental hypermultiplets in the quiver have charge 1 under u(1)s. The resulting anomaly polynomial is I8 = c2(R)(Tr Fb2 + Tr Fc2) + c2(R)(4c2(R) + p1(T )) (Ivec(b) + Ivec(c)) Tr Fb2 + Tr Fc2 c1(s)2 + 2 N N 3 c2(R)c1(s)2 + the global symmetry. where p1(T ) and p2(T ) are the rst and second Pontryagin classes of the tangent bundle, c2(R) and c1(s) are the second and the rst Chern classes of the su(2)R and of the u(1)s bundles of the 6d theory, respectively, Tr Fbn and Tr Fcn are parametrized below by Chern Ivec(b) = 1)c2(R)p1(T ) Itensor = Mapping the charges in 6d, 5d and 4d We will match anomalies for various symmetries by performing computations in di erent dimensions, thus we will start by matching the symmetries between di erent dimensions. Let us map here the charges from 6d to lower dimensions. We have su(k)b u(1)s as the avor symmetry, in addition to the su(2)R transform as a bifundamental representation of the global symmetry. Let us say that in six dimensions the avor symmetry bundles split, and the Chern roots are given by b1; : : : ; bk; c1; : : : ; ck; s the line bundle with the Chern class ai bj + s. The Chern class c1(s) used in (2.1) is identi ed with this s. In our normalization, Tr Fb2 = c2(R) = Trfund Fb4 = TrfundFb3 = for the su(2)R bundle with Chern roots (x; x). 5d. Let us put the 6d theory on S1 with a nontrivial holonomy for the avor symmetry. Then in the infrared, the 5d theory is dual, in the sense described in [9] as continuation past in nite coupling, to the circular quiver su(N )k. Call Ii the Chern root for the instanton number symmetry of su(N )i and t + Hi the Chern root for the baryon number discussion in [10], we know that the one-instanton operator of su(N )i, that becomes the raising/lowering operators of the su(k)2 avor symmetry currents, couples to a line bundle with the Chern class This is to be identi ed with bi cj 1. Therefore, we see N Hi = bi Next, to relate t and s, it is useful to consider the Higgs branch of the theory, when we separate N M5-branes. In the following, we will write down some key invariants on the Higgs branch and specify some of the relations they satisfy. This will be su cient to derive the quantum numbers. In the 5d description, one can use the Kronheimer construction for su(N )i+1 as i and ~ i, let diag(za) = i ~ i, diag(xa) = k, diag(ya) = ~ k. Therefore, u and v have the u(1)t-charge we have z = Tr have u(1)s charge In six dimensions, again when N M5-branes are separated, the same Higgs branch can be found as explained in [11]. Namely, when we denote the su(k)2 bifundamental by , ~ , 1, thus x; y have charge k, thus u; v have charge can equate the u(1)s charge and u(1)t charge: s = t: The su(2)R symmetry in 5d and 6d can be naturally identi ed so the scalars in the bifundamental hypermultiplets in the su(N )k quiver are su(2)R doublets. Now we consider the situation in four dimensions. In the tube theory, most of the analysis above can be directly applied. The su(2)R symmetry is broken to the Cartan. We use the normalization where the supercharge has the charge 1 under the remaining u(1)R0 symmetry. Here R0 emphasizes that this is a natural R-symmetry coming from the six-dimensional construction; this generically will not be the superconformal R symmetry in the infrared, which needs to be determined by the a-maximization [12]. In any case, the bifundamentals in the su(N )k quiver, before the supersymmetry is broken by half, have the u(1)R0 -charge 1 and the u(1)t-charge 1. Then, the surviving chiral bifundamental in the su(N )k tube theory has the u(1)R0 -charge 1 and the u(1)tcharge 1. Together with (2.8), this data on the tube theory is enough to nd the charge assignment in the Lagrangian class Sk theory, as we will see in the next section. Anomaly polynomial from 6d We now compute the anomaly polynomial of the compacti ed theory from the 6d point of view. Let Nbi , Nci , and Ns be the numbers of uxes of the u(1)bi , u(1)ci , and u(1)s respectively. Let us also denote the rst Chern classes of line bundles in 4d as c1(R0), c1(t), c1( i) and c1( i). The Chern roots introduced above are related as follows2 x = c1(R0) bi = N c1( i) s = c1(t) + Ns 2g ci = N c1( i) where c2(R) = x2, c2(s) = c1(s)2 = Cg t = 2 proceeding equations hold also for this case. By substituting these into the anomaly eight-form and performing the integral over the Riemann surface RCg I8, we get I6 = 2)c1(R0)p1(T4) + Nsc1(t)p1(T4) 1)(k2(N 2 + N Nsc1(R0)2c1(t) 2)c1(R0)c1(t)2 1)c1(R0)2 + kN 2c1(t)2 X(Nbic1( i) + Ncic1( i)) kN X (Nbic1( i) + Ncic1( i)) p1(T4) Nbi)c1(t)c1( i)2 + (N Ns + Nci)c1(t)c1( i) Ns)c1( i)3 + (Nci + Ns)c1( i)3) (X c1( i)2)(X Ncj c1( j)) + (X c1( i)2)(X Nbj c1( j)) ; ed 4d theory. the following anomaly polynomial: N 2(N 1) (X c1( i)2)(X Nbj c1( j))+ N 2(N 1) (X c1( i)2)(X Ncj c1( j)) kN X (Nbic1( i)+Ncic1( i)) p1(T4) (X c1( i)2)(X Ncj c1( j))+(X c1( i)2)(X Nbj c1( j)) : I6 = kN (N 1)c1(R0)2 +kN 2c1(t)2 X(Nbic1( i)+Ncic1( i)) Nbic1(t)c1( i)2 +Ncic1(t)c1( i) where Pik=1 c1( i) = 0 and Pik=01 c1( i) = 0. The triangle anomaly in the more traditional form Tr xyz can be read o substituting c1( k) = 4d theories from tori Structure of the 4d quiver theories We consider now the eld theory construction corresponding to tori with uxes with no punctures [5]. The models are constructed by rst starting from a toric quiver built from some number of free trinion theories ( gure 1), and then by higgsing some of the symmetries. The free trinion theory corresponds to a sphere with three punctures: two are maximal and one is minimal, and the u(1)s maximal and the minimal punctures are associated with su(N )k and u(1) avor symmetries respectively. The former is known to be labeled by the color c 2 Zk and the sign have the same signs other charges are denoted in gure 1. Gluing of two maximal punctures corresponds to gauging of the su(N )k symmetry of both punctures. Depending on the signs of the punctures we have two gluings [5, 13, 14]: gluing: when the two punctures have the same sign, say = +1, we add an gauge factors cyclically, with superpotential coupling of the bifundamentals and the mesonic operators associated to the punctures; the quiver in gure 1 represents multiplet with the superpotential coupling of two mesonic operators coming from the These are associated to the theories on a tube without any ux. (See appendix for the tube theory with u(1)s ux.) As already noticed in [6] one can see that the charge assignment is consistent with the discussion in section 2.2 from 6d. If the two punctures have the same color this preserves all internal symmetries. However if the two punctures have di erent colors then all u(1) 's are broken. We here only focus on the gluings and construct a quiver theory associated to a torus with only minimal punctures from a collection of kl free trinions, as in gure 1. When l is an integer we can always glue two punctures that have the same color preserving all the internal symmetries, so the global symmetry of this model consists of u(1)k 1 kl u(1) j symmetries which are associated to minimal punctures. We obtain models with no punctures by giving vacuum expectation values to kl baryonic operators charged under u(1) j symmetries and introducing certain gauge-singlet chithese as bifundamentals of two copies of su(N )k. In the picture the circles are su(N ) groups and one has k groups winding around a cylinder. The trinion is associated to a compacti cation on a sphere with two maximal punctures (of di erent color) and a minimal puncture. On the right we glue trinions together to triangulate a torus. We have lk trinions combined with every introducing bi-fundamental elds which appear as vertical lines in the diagram. ral multiplets ipping some of the other baryons,3 as sketched in gure 2. There are choices to be made as to which baryons the vacuum expectation values are given and this choice maps to a choice of uxes in six dimensions [5]. We will write down the exact correspon. After higging all the u(1) j do not in general have known regular Lagrangians (see for example the discussion in the Let us make several general observations about these models. The quivers correspond to tiling of the torus with triangular and square faces. The exact details depend on the uxes and in four-dimensional language on the ways we close the minimal punctures. The theory with the minimal punctures we utilize as a starting point of the construction triangulates the torus. Importantly we supplement the quiver with singlet elds, some of which might be free and some coupled to gauge singlet combinations of elds through additional superpotential terms. Such theories were widely studied in various contexts about ten years ago [18{20]. It is convenient to think about the theories in terms of zigzag paths on the torus. Each symmetry factor u(1) (with exception of u(1)t) corresponds to a loop, zigzag path, winding around the cycles of the torus. Let us call the cycle around which we have, in the theory in 3By ipping an operator O we mean the procedure of adding a chiral eld MO to the model with superpotential W = MO O. In every free trinion we give a vacuum expectation value to one of the baryons. The choice of the baryons is related to the ux and in general di erent choices lead to di erent theories in the IR. The baryons which do not receive a vev but are charged with same charge under the minimal puncture symmetry as the baryon which does receive vev, are ipped. In the diagrams the elds with a cross are the baryons which are ipped. In the picture the baryons which receive vacuum expectation value are weighed as t 2 1N = N , t N2 2N N the UV, k gauge groups cycle A and the other one cycle B. The UV model then has loops winding once around cycle A corresponding to puncture symmetries and k loops winding l times around each one of the cycles A and B, and l loops winding l times around cycle A and l times around cycle B. The ow initiated by closing the punctures preserves the symmetries not associated to the punctures and breaks symmetries associated to punctures. The pattern of winding of the di erent lines can be translated to the uxes. For example, the torus with no ux is formally mapped to con gurations with all windings vanishing. Anomalies of the 4d quiver theories The anomalies for these models can be rather easily derived. For the sake of computation of the anomalies we do not need to gure out the quiver diagram in the IR of the triggered by vacuum expectation values turned on for baryonic operators when closing minimal punctures. We can compute these in the UV, making sure to use the symmetries surviving in the IR and decoupling the relevant Goldstone chiral multiplets. Let us give the algorithm for computing the anomalies. We will encode all anomalies in the trial a conformal central charge and in the trace of a trial R symmetry where we will keep dependence on possible mixing parameters with all the abelian symmetries. We denote the R charge as where qi , qi , qt and qu are the charges of u(1)k 1, u(1)k 1, u(1)t and u(1) j . We have the constraint Pjk=1 sj = Pk R-charge z and of a vector multiplet of group of dimension h are given by a (z) = av(h) = Then the anomaly of the free trinion is at(N; k; s ; s ; st; s ) = N 2 X a s i = 12 + 12 st si and ip the baryons. The speci cation of the fugacities re ects regarding u(1) i . We denote by Q = Pik=1 Qi , Q = Pk u=1 Qu. the fact that the operators receiving the vacuum expectation values have all their charges vanishing in the IR. The function F (i) is an arbitrary function mapping (1; ; k) to itself. We denote by Q i the number of minimal punctures closed with si , and Qi is the same We note that the color of the maximal punctures is not important in the computation of the anomaly of the free trinion as it only determines the sequence in which the di erent chiral elds are organized together. The anomalies of each gluing of maximal punctures of color c are easily seen to be ag(N; k; c; s ; s ; st) = kav(N 2 3 X((su where the indices are summed mod k. These elds are charged under both symmetries and thus the color of the puncture we glue is important for the anomaly. The anomaly of torus built from kl free trinions is then N 2k(st +1)l(S2 +S2 ) The conformal anomaly a for the torus with no punctures but with uxes determined by a choice of F is then ator;F (N; k; Qi ; Qj ; s ; s ; st) = ator i=1 u=1 X Qua (2 X Qua (2 i=1 u=1 where in the second line we have the contribution of the chiral elds which ip the baryons and in the rst line we specialize the parameters of the torus with punctures to be consistent the Goldstone bosons appearing in the ow as we break some symmetries. This evaluates to vector whose components are (si ; )n. i denotes the inner product of k-dimensional vectors and (s ; )n stands for the This expression has several nice features. If one shifts all Qi (or Qi ) by some integer the above does not change. This corresponds to completely closing minimal punctures exchanging all of Q Q . This is consistent with the expectation that the group here is enhanced to su(2k) as below these combinations are identi ed with the uxes. The anomalies here are in agreement with the anomalies computed from six dimensions. The map between the parameters is Nbi = Q Nci = Ns = 0 : For the free trinion we obtain that btrin: = kN 2(st gauging we obtain bg = N 2kst. This is independent of the puncture symmetries and the close the minimal punctures we have to ip the baryons and the anomaly is k2l. When we b(Qi ; Qi ) = k(Q + Q ) + k(Q + Q ) = N k(hQ ; s i + hQ ; s i) : We observe that with our identi cation of uxes with multiplicities of the various choices of closures of minimal punctures all anomalies agree between 4d and 6d. We can use the trial a-anomaly we have obtained to compute the conformal anomalies of the theories. Here we have to be careful as in general the elds which ip the baryons are coupled through irrelevant interactions and thus are free. One then needs to take this into account in the computation of the conformal anomalies. Case studies We construct examples of various quiver gauge theories of class Sk type associated to a torus for small values of k. We will study in greater detail some of their properties. Speci cally we calculate the superconformal index and test the global symmetry of the 4d xed point with that predicted from the 6d construction. The superconformal index also allows us to compute the dimension of the conformal manifold of the 4d theory. This can also be predicted based on the 6d construction as done an exactly marginal operator for each complex structure modulus, and at connections for the global symmetries. For the case of a torus, we always have a single complex structure modulus. In addition we can also have at connections for the global symmetries with nontrivial values around each of the two cycles of the torus. These must be abelian due to the homotopy group relation of the torus. Thus we see that we get 2(2k 1) real parameters 1 complex marginal deformations. So to conclude, we expect: dim(M) = 2k: We can use the 4d superconformal index to check this prediction. when considered as the AN 1 (2; 0) theory compacti ed on a torus. In that case, (4.1) gives N = k = 2 Consider taking two free trinions and connecting maximal punctures of the same color together. This results in a torus with two minimal punctures. Then by closing the minimal punctures, we can get a theory corresponding only to a torus, as discussed in the previous section. First we begin with the theory corresponding to a torus with two minimal punctures we get by connecting two free trinions. For the purpose of constructing these theories we will leave N general, setting it to the desired value at the end. The quiver diagram of the theory is shown in gure 3. It has a cubic superpotential for any triangle. This is the theory that lives on N D3-branes probing a C3=Z2 in various contexts. See e.g. [21] and references therein. Z2 singularity and has been studied elds under all the non R-symmetries: the internal u(1) ; u(1) ; u(1)t, and the minimal puncture ones u(1) ; u(1) . Additionally there is a cubic superpotential for every triangle. Alternatively it is given by the most general cubic superpotential that is gauge invariant and consistent with the symmetry allocation in the table. All elds have the free R-charge 23 . Let us denote the uxes of the theory associated with the surface by (Nb; Nc; Ns). As the free trinion has ux (0; 0; 12 ) [5, 6], this theory should correspond to ux (0; 0; 1). Next we give a vev to the baryon made from Q1. This corresponds to closing a minimal puncture. The resulting theory is associated to a toruAs with a 2minimal puncture Further we can give a vev to another baryon, associated with1the o1tht2er2 puncture. This will close the other puncture and leads us to a torus with no p2uncturte12s. We have three distinct choices for the baryon. These will di er by the ux on t3he tortu11s.22 One choice is to close with the baryon made from Q~2. This4 will leta22d to a torus with ux (0; 0; 0). This theory is somewhat singular and we shall refr5ain fr1omt2 1discussing it for now. We can also close the puncture with the baryon made from6 Q~ 1t.2 1T1h2is will lead to a ux (1; 0; 0). The quiver description of this theoryQi1s shtow1n1in2 gure 4. One can see that it resembles an a ne A1 quiver with additional Qsi2ngletst c11oupled through a superpotential. We shall refer to this theory as the a ne quiveQr.3 Next we can study some of its properties. Here we shall consider N cases to the next subsection. We begin with studying its anomalies. In this case the full superpotential, including the contribution from the ipping, is cubic, and performing a-maximization we indeed nd that under the correct u(1)R all chiral multiplets have the Next to the elds are their charges summarized through fugacities. We use mostly standard notation except for two points: lines from a group to itself represent N 2 hypermultiplets forming the adjoint plus singlet representations of the group; we write an X over a eld to represent the fact that the baryon of that eld is ipped. The theory has a cubic superpotential for every triangle which can also be derived by considering the most general cubic superpotential that is gauge invariant and consistent with the symmetry allocation. There is also the superpotential term which is not generally cubic coming from the ipping. All elds, save the ipping elds, have the free R-charge 23 . free eld R-charge 23 . Further we nd that: a = c = Note that these are the anomalies for the a ne A1 quiver in addition to two free hypermultiplets.4 This fact and the values for the R-charges suggest that this theory has a subspace on its conformal manifold where it is indeed the a ne A1 quiver gauge theory, where the supersymmetry enhances to N = 2. We can also evaluate the index of this theory. Particularly we consider the a ne quiver without the singlets as these are just free elds. The subgroup of so(7) which commutes with the ux is u(1) su(2)2 and so this is the expected global symmetry. Since without with global symmetry u(1) usp(4), the index in fact forms into characters of this group. We nd it is given by: IN=2;k=2 +pq 1 + [5]usp(4) [10]usp(4) + : : : ; where [4]usp(4) = t + t1 + t + t = [1; 0]usp(4). 4These are presumably the two chiral elds that accompany the adjoints plus the two chiral elds that are introduced for the ipping. 5.2.1 and in particular equation (5.19). Furthermore it was found to be the closure of the next to minimal orbit of usp(4) as in table 3 of [23] and tables 10 and 12 of [24], where another description sets it as the Z2 orbifold of the closure of the minimal nilpotent orbit of SL(4) (alternatively known as the reduced moduli space of 1 SU(4) instanton on C2). This emphasizes that the global symmetry on this part is indeed usp(4). The unre ned Hilbert Series takes the form HNA=2n;ek=Q2uiver( ) = and it admits the highest weight generating function [24, 25] HW GN=2;k=2 A ne Quiver( ; 1; 2) = with 1 and 2 the fugacities for the highest weights of usp(4). From this one deduces the re ned Hilbert series that admits a character expansion HNA=2n;ek=Q2uiver( ; t; ) = n1=0 n2=0 In addition there are the 4 singlets, which in this construction, two are given the charge 2 and two the charge 14 . Therefore their presence does not interfere with the global symmetry. Again it is reasonable to expect that at some sub-locus of the conformal manifold these are indeed free elds and so can be rotated separately leading to additional enhancement of symmetry. We also note that this structure is common in the so called \ugly" class S theories where the SCFT is accompanied by additional free hypers whose global symmetry is identi ed with part of the global symmetry of the SCFT. Next we can study the dimension of the conformal manifold for this theory. As the singlets cannot add additional directions [26], it can be directly read from the index without them (4.3). Particularly, we look at the pq order terms, which according to a result by [27], are just the marginal operators minus the conserved currents. For the case at hand we nd that there are 7 marginal operators where one is canceled against the u(1) conserved current in (4.3). Applying the logic of [26] we nd a dimension 3 conformal manifold, reproducing the result in section 3.2 of [28], along which the symmetry is broken to u(1) su(2)2. This similar to the story for k = 1. by the compacti cation of the A1 (2; 0) theory on a torus with two maximal punctures. Then the 6d analysis of [15] leads us to expect a three-dimensional conformal manifold, two directions of which preserve all the symmetries and correspond to the coupling constants su(2)2 global symmetry. This agrees with our observation. The class S and class S2 theories di er by the existence of the singlets. It is not di cult to see one can build marginal operators uncharged under the 6d apparent global elds are their charges summarized through fugacities. The theory has a bifundamentals Qi and R-charge 2 quartic superpotential involving the four bifundamentals as well as the superpotential coming from the ipping. There is also an R-symmetry where it is convenient to give R-charge 12 to the four symmetries. By the logic of [26] this should lead to exactly marginal operators. However, this fails as these operators in fact become free leading to the appearance of accidental symmetries invalidating the argument. Therefore we conclude that the 6d expectations regarding the conformal manifold are too naive, and like the anomaly analysis, can be modi ed due to the appearance of accidental symmetries. The \Klebanov-Witten" theory. We can also close the puncture with the baryon made from Q~4. This will lead to a torus with ux ( 12 ; 12 ; 0). The quiver description of this theory is shown in gure 5. One can see that it resembles the Klebanov-Witten model [29], but with additional singlets ; , coupled through superpotential terms. We shall refer to this theory as the KW case. Let us rst consider the theory without the singlets. In this case this model is known to go to an interacting xed point where the bifundamental elds have R-charge 12 [29]. Now consider adding the free elds and couple them through the superpotential. The behavior of the resulting theory depends on the value of N . For N > 2, this superpotental is irrelevant and the theory should ow to the same xed point, but with free singlets. ow to a new xed point. We shall now discuss the latter case in more detail. Note that for this special case of N = 2, Q1 is a 2 2 matrix and the notation Q21 stands for det Q1, and similarly for Q2. Each of these terms is invariant under a corresponding su(2) global symmetry and the global su(4) symmetry which is present in the absence of these terms [30] is broken to u(1) in the presence of these terms, where u(1) is the baryonic symmetry which acts as +1 on Q1 and Q2 and as 1 on Q3 and Q4. First we shall need to perform a-maximization to determine the superconformal Rsymmetry. It is straightforward to see that only the baryonic symmetry u(1) + u(1) can mix with the naive u(1)R of the KW model. Thus, we de ne: By performing a-maximization we nd 0:027, so the R-charges change only slightly compared to their naive value. One can check that all gauge invariant elds are above the unitary bound so this is consistent with the theory owing to an interacting We can next evaluate the anomalies for this theory. Particularly, for the conformal anomalies we nd: Next we evaluate the index for this theory. First we note that the subgroup of so(7) that commutes with the ux is u(1) usp(4) so this is the expected global symmetry, enhancing the global symmetry of su(2) u(1) found above. Indeed we nd that the index naturally forms into characters of this symmetry, where it is given by: INKW=2;k=2 = 1+p 21 q 12 u(1)0R = u(1)R + u(1) + u(1) a = c = +: : : (4.9) so the true R-charges of various operators should be shifted depending on their It is interesting that the singlets are necessary to get the enhanced global symmetry that is required from matching to 6d. They rst break su(4) to su(2) su(2) u(1) and then enhance to u(1) usp(4) which is evident from the index but not from the superpotential. being relevant only for N = 2. Another interesting computation is to check the moduli space for the conifold theory at su(4) and the Hilbert series was shown in [30] and particularly in equation 4.16 to admit a character expansion of the form n1=0 n2=0 or alternatively, using the fugacities for the highest weights 1; 2, a highest weight gener HW G ( ; 1; 2)CNo=n2ifold = These computations lead to a description of the moduli space as the set of all 4 by 4 complex symmetric matrices with rank at most 2. The natural guess after ipping and symmetry enhancement, with highest weight fugacities 1 and 2 for usp(4), is given by the highest weight generating function )KNW=2 = which leads to an unre ned Hilbert series H ( )KNW=2 = We can also calculate the dimension of the conformal manifold from the index. It is again given by the pq order terms under the true R-symmetry. For our case this translates to the pq order operators which are uncharged under u(1) . Thus we nd an 11 dimensional conformal manifold along which the usp(4) group is completely broken. This is greater than the 4 dimensional one we expect from 6d reasoning. This does not contradict the 6d reasoning since there could be 4d marginal operators with no clear interpretation in this with the 5 dimensional conformal manifold for the conifold theory which was found in section 2.2 of [28]. Another interesting observation regarding the index (4.9) is the appearance of the 4 dimensional representation of usp(4). This implies that the global symmetry is USp(4) and not SO(5). This group in 6d comes from breaking the SO(7) global symmetry of the 6d SCFT. Naively this suggests that this group must be Spin(7) and not SO(7). However, [6] found various 4d theories, matching 6d compacti cations with uxes that are consistent only with SO(7). These two observations suggest one of two scenarios. One, the 6d group is in fact Spin(7) which naturally explains the appearance of the 4 in the index (4.9). Then the 4d theories with non-standard quantization should be viewed as 4d theories with no valid 6d origin. An alternative explanation is that the 6d group is SO(7), which naturally t the observations of [6]. However in that case one must view the spinors in the 4d index (4.9) as 4d operators without a 6d origin, similarly to the excess marginal operators we seem to nd for this theory. This interpretation then implies that the 4d theory in fact undergoes an accidental discrete enhancement of symmetry SO(5) ! USp(4). N = 3; k = 2 special features that are not present in the general case. We shall now discuss the beindex calculations. ne quiver" theory. We start with the a ne A1 quiver case. The matter content and charges are as in gure 4. The models contain four singlet elds, two o which come from the ipping and are coupled through a superpotential. Without this superpotential, all elds have free R-charges and the theory is expected to sit on the involving the ipped elds is irrelevant and so the theory with these terms is expected to be additional symmetries rotating the free elds. symmetry for k = 2; N > 2 is su(2)t su(2) . This should be broken by the ux which is the symmetry we expect in the 4d theory. We indeed nd that the index forms characters of that symmetry. Ignoring the singlets, as these are just free elds, we nd the index to be: IN=3;k=2 A ne Quiver = 1 + p 23 q 23 2 4 + Additionally there are the 4 singlets which in this construction two are given the charge 2 and two the charge 16 . Therefore their presence does not interfere with the symmetry. We can also compare the dimension of the conformal manifold with the 6d expectations. Again we nd that the dimension of the conformal manifold is in fact greater than what is expected from 6d. The \Klebanov-Witten" theory. Next we consider the KW case. As we previously discussed for N > 2 the superpotential coupling the singlets is irrelevant and the theory should ow to the KW model with singlets. the 4d theory to preserve an su(2)t that the KW model shows an su(2)t global symmetry. In fact we shall see global symmetry which is broken have a considerable enhancement of symmetry in the IR. As the singlets decouple in the IR we concentrate only on the interacting part, for which we nd the index to be: INKW=3;k=2 = 1 + p 21 q 12 [2]su(2)t where [2]su(2)t = t + 1t and [2]su(2) In addition there are two free singlets with charges 16 and 16 . These are inconsistent with su(2) implying that it is broken to its Cartan by the superpotential only to return in the deep IR. This again resembles some situations in class S theories where the global symmetry of an interacting theory plus hypers is broken by mixing part of it with the symmetry rotating the hypers. The results of equation (4.15) agree with the computations in equation 3.83 of [31] but are still missing two essential operators that transform as [2]su(2)t [1]su(2) | the so called non factorizable baryons. It will be interesting to check if higher order computations produce these two essential contributions. Again we nd that the dimension of the conformal manifold is in fact greater than what is expected from 6d. We can in principle look at higher values of N and even the large N behavior. In fact both of the theories considered here, without the singlets, have well known large N gravity duals. As the number of singlets is order 1, it is reasonable that most of the properties of these theories will be well described by the gravity duals. In this regard it is interesting that both theories are reached by the compacti cation of the same 6d SCFT on the same surface di ering by order 1 uxes. The models we consider here correspond to having vanishing ux for u(1)t as in this case we have known Lagrangians. With the ux for u(1)t the models are expected to be strongly coupled, see appendix. The properties of these two types of models are qualitatively di erent. For example, the anomalies scale as N 2 for the gauge theories we consider here (when the singlet elds are appropriately taken into account), and are expected to scale as N 3 for the strongly coupled types as can be inferred from the six dimensional analysis. As from the six dimensional perspective we cannot infer existence of accidental symmetries this is just an expectation which can be invalidated in N = 2; k = 3, the orbifold C3=Z2 consider taking three free trinions, connecting them together and closing three minimal punctures. More speci cally we shall close two punctures with a vev to baryons charged under 1 with the same charge. Now we need to close the nal minimal puncture. We consider two di erent possibilities. First we consider closing the last puncture also with a vev to a baryon charged under 1 with the same charge as the last two. This is similar to how we got the a ne quiver in In this theory all gauge groups see 3N avors and so are conformal. Thus, without the ipping superpotential, all eld have the free R-charge 23 . The ipping superpotential is 23 under the superconformal R-symmetry. Again for N > 2 this entail an IR enhancement of symmetry due to the ipping elds becoming free. interacting part and ignore the singlets as these are free. The 6d global symmetry here is enhanced to su(6), but our choice of ux breaks it to su(3) Evaluating the index, we nd that it can be written as: INAl=l2;1k=3 = 1 + p 32 q 23 with only 1 ux, while on the right is a table summarizing the charges of the various elds. Note that several di erent elds have the same charges and so are represented with the same letter. This theory has a rather large cubic superpotential involving the 12 triangles in the diagram. Again these are most conveniently generated by taking all cubic terms consistent with the symmetries. Additionally there are the superpotential terms coming from the ipping, which in general are not cubic. It is again convenient to choose the R-symmetry so that all non- ipping elds have where [2]su(2) = and [3]su(3) = 12 + 22 + 121 22 . Additionally there are 6 singlet elds, 3 with charge 12 and 3 with charge 14 22. These can be written as 3 13 [2]su(2) and so are consistent with the 6d global symmetry. We can also consider the dimension of the conformal manifold. By the reasoning of [26], the terms appearing in (4.16) do not contribute any exactly marginal operators as one cannot form a 1 invariant from them. This leaves the marginal operators in the adjoint of the full global symmetry G which must be present to cancel the contribution of We can also consider closing the last puncture with a baryon charged under 2 ux leads to the symmetry breaking pattern su(6) ! su(4) we expect to be the 4d global symmetry. In the eld theory the vev leads to a quiver with an su(N ) group with N avors. This group con nes in the IR leading to the identi cation of the groups it's connected to and making the ipping elds massive. After the dust settles we end with the so called L222 [20] quiver theory in gure 7. This theory can also be derived from 4 NS branes on the circle, with two types of orientation. We next proceed to analyze this theory in detail. First we need to evaluate the conformal R-symmetry. By inspection one can see that there is only one u(1) that can mix with ux and one of 2 ux. Next to the elds are their charged summarized using fugacities. This theory has a combination of cubic and quartic superpotential terms. Again these are most conveniently generated by taking all terms consistent with the symmetries. Additionally there are superpotential terms coming from the ipping. The theory also has an R-symmetry, a convenient choice for which is to give all the bifundamentals R-charge 12 , and R-charge 1 for the adjoints and the R-symmetry, which in our notation is u(1) 1 . The remaining u(1)'s can be grouped into 4 baryonic u(1)'s, each rotating one of the four pairs of bifundamentals with opposite charges while the adjoints and their associated singlets being neutral. Three of these u(1)'s are combinations of u(1)t; u(1) 1 and u(1) 2 while the last is 2u(1) 2 Next we preform a-maximization. We take the R-symmetry to be: u(1)0R = u(1)R + u(1) 1 where we take u(1)R to rotate the bifundamentals with charge 12 and the adjoints and their associated singlets with charge 1. The ipping elds attached to the bifundamental then also have R charge 1 while those attached to the adjoits have R charge 0. Performing the a-maximization we nd that: = 3 6 5 . With this value we nd that a = 5 5 However with this R-charge the adjoint ipping elds are below the unitary bound. Therefore the natural conjecture is that these elds become free at some point along the ow leading to an accidental u(1) that mixes with the R-symmetry. We can now repeat the a-maximization, but taking these to be free elds where we nd: . We nd that all elds have dimensions above the unitary bound and that the superpotential coupling the ipping elds to the adjoints is irrelevant. All of these are consistent with our claim. We can also preform a-maximization considering all the ipping elds as free, where we indeed nd that, compared to that point, the superpotential coupling the ipping elds to the adjoints is irrelevant while the one coupling the ipping elds to the bifundamentals So to conclude we expect the theory in gure 7 to ow to an interacting xed point consisting of the quiver theory, without the adjoint ipping, plus two free chiral elds. We next want to evaluate the index of this xed point. Again for simplicity we shall rst ignore the two free chiral elds. From 6d we expect an su(4) u(1) global symmetry. We indeed nd that the index can be written in characters of this symmetry, where it reads: INM=ix2e;dk=3 = 1+p 21 q 21 4 2 +1+ 14 22 [5]su(2) + 14 22 [3]su(2) [6]su(4) + 14 22 [4]su(4) + [4]su(4) +2 [6]su(4) +2 [3]su(2) +: : : where [2]su(2) = 22 + 13 and [4]su(4) = pt 2 ( 12 + 22 + 121 22 ) + 2 we have used the naive R-symmetry, not the superconformal one, so the dimensions of operators are shifted based on their 1 charge. One can see that, as expected, the u(1) is given by u(1) 1 while 2u(1) 2 u(1) 1 forms the non-abelian part. This is apparent as the fugacity of the u(1) is 12 2 so states charged only under it are invariant under a 2u(1) 2 u(1) 1 transformation. Alternatively those charged only under the non-abelian part are invariant under u(1) 1 transformations. Finally we note that the dimension of the conformal manifold is again greater than what is expected from 6d. Compacti cation with discrete twists For now we were satis ed with the discussion of the symmetries of the models at the level of the algebra rather than the group. However, the global properties of the group in six dimensions have far reaching implications as far as the choice of uxes goes. As we will next discover the six dimensional constructions have a non trivial global structure which allows for discrete uxes to be switched on which in particular break some of the continuous global symmetry. This procedure has a four dimensional analogue. We turn our attention to this next. Global form of the avor symmetry. There is no doubt that the 6d N =(1; 0) theory we have been using in this paper has the avor symmetry su(k) u(1)t at the Lie algebra level. What is exactly the avor symmetry group? Let us for now concentrate our attention to the subgroup connected to the identity, neglecting the u(1)t part. On the generic point of the tensor branch, the theory becomes a linear quiver gauge theory. It contains a gauge-invariant operator which is a bifundamental of su(k) su(k), obtained by multiplying all the bifundamentals of the quiver. This suggests that the group is where the quotient is with respect to the diagonal combination of the two centers. If this is the case, on a compacti cation on T 2, we should be able to turn on the (generalized) Stiefel-Whitney class w2 2 Zk. In the next subsection, we will nd the corresponding operation in the 4d eld theory language, and will check that the anomaly computed in 4d agrees with the expectation from 6d. Before doing that, let us remind ourselves the basics of the (generalized) StiefelWhitney class w2 of bundles of non-simply-connected groups on T 2. (A detailed account readable for physicists can be found in [32, 33]. For complete generality, the reader should consult [34].) Let us say we construct a G bundle on T 2 by rst having a G-bundle on a rectangle by identifying two sets of parallel edges. This identi cation involves specifying the gauge transformation used in the gluing process along the boundary of the rectangle. This becomes a closed path within the group manifold, which is topologically classi ed by Zk. This is the (generalized) Stiefel-Whitney class w2. When the Stiefel-Whitney class of Flat bundles with nontrivial w2. First are the at bundles. In this case we have the holonomy gA along the A cycle and the holonomy gB along the B cycle. They should We can easily nd such a pair: take gA = diag(1; !; !2; : : : ; !k 1); ! = e2 i=k gB = BB for physicists these matrices were familiar from the work of 't Hooft [35]. There is no unbroken symmetry. !`gBgA such that the unbroken symmetry is su(m). This is done by rstly recalling that regarding the resulting gA;B as matrices in SU(k). As the SU(m) part is untouched, clearly the avor symmetry is su(m). Abelian bundles with nontrivial w2. Another are Abelian bundles. Let us denote by An Abelian SU(k) bundle is speci ed by a map U(1) ! T . In particular, specifying one on T 2 corresponds to specifying a point in a lattice of rank k 1 that can be naturally identi ed with the root lattice of SU(k). They can be thought of as k integers summing by a point in a lattice again of rank k 1, but now identi ed with the weight lattice of SU(k). If we use the Chern classes normalized to the subgroup of SU(k), they can now look rational, with denominator k. The Stiefel-Whitney class can be easily read o by consider (SU(k) Stiefel-Whitney class. For example, one can choose at bundles for both, Abelian bundles for both, or a at bundle for one and an Abelian bundle for the other. The computation of the 4d anomaly is straightforward: one just has to plug in the Chern classes of the Abelian parts in the formulas we have been using. Stiefel-Whitney class and the symmetry of the quiver graph. Now, to bridge our discussion here to the 4d analysis below, consider rst the compacti cation to 5d. Let us put the SU(k) holonomy gA (5.2) around S1. Then we have a circular SU(N )k quiver with the same gauge coupling for all groups. Now, the operation gB (5.3) naturally corresponds to the symmetry of the circular quiver shifting the node by one. Therefore, by compactifying the 5d theory on an additional S1 with a twist rotating the circular quiver, we can realize the compacti cation of the 6d theory with a nontrivial Stiefel-Whitney class. We have constructed tori theories by combining free trinions in multiples of k. This way we always glued punctures of the same color and preserve all the internal symmetries. It is however possible to glue punctures of di erent colors at the price of breaking some of the internal symmetry. For example, constructing a loop out of l free trinions the group is su(gcd(k; l)) . We can then close the punctures and try to identify the six dimensional compacti cation leading to such theories. Gluing two punctures of a single trinion. Let us consider taking a free trinion and gluing the two maximal punctures to each other. The symmetry preserved here is u(1)t and the puncture symmetry u(1) . The theory is the a ne quiver with k nodes and with a singlet eld associated to every node coupling to charged same manner as the adjoint chirals. The theory is superconformal with the coupling of the singlet elds being marginally irrelevant leading to them decoupling in the IR as free elds and the symmetry enhancing in the IR to su(k) rotating the free elds. The adjoint chiral elds are charged t 2 u(k) with the last factor 1, and the bifundmental elds between i 1 and i node have charges t 21 i 1 and ptq i. We can close the minimal puncture giving a vacuum expectation value to one of the baryons weighed t 2 j Closure of the minimal puncture also entails ipping the rest of the baryons charged under bifundamental chirals are charged i 1 j, ptq i. The singlets are k 1 nodes and with singlets. The 1 having charges t j as the chiral adjoint elds, and k 1 chiral elds which ip the baryons and have charges pq iN j N . The chiral elds couple through either irrelevant or marginally irrelevant terms the minimal one. and thus ow to free elds in the IR. The theory in the IR is then a collection of 2k free chiral elds and the a ne quiver with k 1 nodes (see gure 8). The anomalies of this model match the anomalies of six dimensional compacti cation k1 for the u(1)i with i not equal to j. Such of uxes to ows. This picture also predicts that the a ne quiver should have loci on conformal manifold with the avor group being su(k u(1)t. This is not trivial from the Lagrangian of the theory which for general number of branes exhibits u(1)k symmetry. The index will be consistent with this claim because the a ne quiver has dualities interchanging minimal punctures. This implies that the index will be invariant for permutations of i. This permutation symmetry implies that the index can be written in terms of characters of su(k 1)u(1) j parametrized by i . Moreover at order pq of the index (when one takes the free superconformal R charges to chiral elds) the index has the conformal manifold are such. Thus, we expect the index to be consistent with the enhanced symmetry. Gluing maximal punctures with a shift. Let us consider another operation we can legally do on the eld theory side. Gluing together two maximal punctures of a given theory we can twist them relatively to each other. The twist corresponds to matching i to i+u and i with u+i. This breaks the global symmetry to su(gcd(k; u)) These operations are considered for toric quivers in [36]. Performing both operation of gluing punctures with color di ering by l units and twisting with u we obtain the group, su(gcd(k; l; u)) We can also consider gluing punctures of same orientation which will exchange roles of For the anomaly polynomials that we have computed the e ect of the twist and the gluing is simple. We only have to remove the parameters which correspond to symmetries Thus we conclude that the anomaly polynomial is still given by the same expressions. This also means that the matching between the 6d and 4d analysis persists also for these cases. Matching with 6d. We are now in a position to merge the 4d and 6d observations in this section to a consistent picture. On the 4d side we observed that we can form a torus by gluing together l free trinions and closing the minimal punctures. When gluing to form the torus we may also twist the gluing by say u units so that in the last gluing we match i to i+u. Either of these breaks the global symmetry unless l; n are a multiplet of k. For simplicity let us consider each of these separately. First we have seen that if we take polynomial calculations in 4d and 6d agree. The matching, speci cally equation (3.9), implies that closing a minimal puncture shifts the ux in a U(1) embedded inside one of the SU(k)'s so that its commutant is SU(k normalization the ux for (SU(k) describes an SU(k) SU(k) consistent ux. However if the uxes of both SU(k)'s are fractional then we have a non-trivial Stiefel-Whitney class on the torus that is materialized through abelian uxes for both SU(k)'s. This in general breaks the global symmetry to it's Cartan subalgebra. case the global symmetry is broken down at least to su(gcd(k; u)) and may be further broken to the Cartan due to the uxes. In this case we identify n with a non-trivial Stiefel-Whitney class on the torus that is manifested using at connections for both SU(k)'s. This naturally accounts for the global symmetry breaking pattern. Also it is quite natural from the 5d viewpoint. It is also consistent with the matching of the anomaly polynomial between 6d and 4d as besides the breaking of symmetries this does not e ect either of the calculations. Note that again if the uxes of both SU(k)'s are fractional then we have an additional non-trivial Stiefel-Whitney class so that the total Stiefel-Whitney class, which may be trivial, is materialized partially by at connections and partially by abelian uxes. broken down at least to su(gcd(k; l)). We identify l with a non-trivial Stiefel-Whitney class on the torus that is manifested using at connections for one SU(k), the broken one, and uxes for the other. This correctly accounts for the symmetry breaking pattern, and also agrees with the matching of the anomaly polynomial since the two match where we have fractional uxes for the unbroken symmetry. Note that the at connections are necessary so that the total con guration be consistent with (SU(k) SU(k))=Zk. Finally we can consider an arbitrary con guration with any value of l and n. This should correspond to the 6d theory on torus with uxes determined by l through (3.9) and with an n Stiefel-Whitney class manifested using at connections. From the discussion so far it is apparent that the symmetry breaking pattern as well as the matching of the anomaly polynomial are consistent with this. Note that in the generic case the total StiefelWhitney class is manifested using both at connections and abelian uxes and may vanish specify the 6d con guration we must enumerate the Stiefel-Whitney class in addition to the uxes. Acknowledgments We are grateful to K. Intriligator, H. C. Kim, Z. Komargodski, and C. Vafa for useful conversations. The work of Y.T. is supported in part by JSPS Grant-in-Aid for Scienti c Research No. 25870159. The work of Y.T. and G.Z. is supported by World Premier International Research Center Initiative (WPI Initiative), 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. The Work of I.B. is supported by UC president's post-doctoral fellowship and in part by DOE grant DE-SC0009919. A. H. is supported by STFC Consolidated Grant ST/J0003533/1, and by EPSRC Programme Grant EP/K034456/1. Anomalies of interacting trinions for general k; N In section 2, we have obtain the anomaly coe cients of class Sk theories from the anomaly polynomial of the six-dimensional theory. By using these, we now predict the anomalies of the trinion models in this section. genus g Riemann surface First of all, let us reproduce here the anomalies of the class Sk theory associated to Tr R0 = Tr R03 = Tr R0t2 = 1)(k2(N 2 + N Tr t = Tr R02t = Tr t3 = We omitted the anomalies involving i and i which depend on the su(k) for simplicity. By subtracting the contribution of the 3g 3 tubes from these, one would get the anomalies of 2g 2 trinions from which one can deduce the single contribution. g 1. There is a duality frame where the four-dimensional theory consists of 2g = +1, combined gluing corresponding to the SU(N )k vector multiplets and the bifundamental chiral multiplets between them. Thus one tube contributes to the anomalies as Tr R0 = Tr R03 = k(N 2 1); Tr t = Tr t3 = Tr R02t = Tr R0t2 = 0: Tr R0 = Tr R03 = Tr R0t2 = Tr R0 = Tr R03 = Tr R0t2 = Tr R0 = Tr t = 8; Tr R03 = 2; Tr R02t = 4; Tr R0t2 = Tr t3 = Interacting trinions for this case were constructed and studied in [6]. Particularly there are three theories appearing there that have this u(1)s ux, with di erent su(k) su(k) uxes. These were dubbed TA, TB and the so(5) trinion that has a Lagrangian description. In all three cases the anomalies agree with the result above. With these anomalies one can obtain those of the trinion with arbitrarily ux Ns (k2 2)(N 1)+3k(N 2 1) (N 1)(k2(N 2 +N 1)+2) 3k(N 2 1) Tr R02t = Tr t3 = k2N 3n 3kN 2 (k2 2)(N 1)+3k(N 2 1) (N 1)(k2(N 2 +N 1)+2) 3k(N 2 1) Tr R02t = Tr t3 = By subtracting these tube contributions from (A.1) and dividing by (2g the anomalies of the trinion theory with Ns = 12 : Note that these anomalies are independent of the su(k) su(k) uxes. For example when N = k = 2, these are 1 can be easily obtained. The answer is simply (A.3) where the signs of the anomalies involving odd power of t are changed. We can now subject this construction to the following consistency conditions. Take 1) trinions, each with some value of the u(1)s ux ni, and glue them together. In this manner we get a genus g Riemann surface with ux Pi ni. Consistency now demands that 1) trinions and 3(g It is straightforward to show that this is indeed true. gluings we should recover (A.1) with Ns = Pi ni. We can further complicate by adding the conjugate trinion. Consider taking 2g 2 a surface with uxes ni and a conjugate trinions with uxes nj to build a genus g Riemann Pj nj. Punctures of opposite sign are glued together by S gluing. Now when constructing the Riemann surface we use some combination of of punctures with a positive sign, which contribute the anomalies in (A.2), punctures with a negative sign, which contribute the anomalies in (A.2) but with t ! and S gluing which only contributes to the R symmetry anomalies where it gives the same contribution as in (A.2). In fact we can construct the same theory in di erent ways using di erent combinations of the above. Particularly say we use b gluings of punctures with a negative sign then we must use 3a punctures with a positive sign. It is straightforward to show that due to the structure of the contributions b will drop out as required and further that summing all contribution we indeed recover (A.1). It is straightforward to generalize this to more complicated cases. First we can consider trinions with punctures of di erent signs. We can also consider anomalies involving the u(1) i and u(1) i symmetries. These will be sensitive also to the uxes under these symmetries, and to the colors of the punctures. These generalizations should be messy, but straightforward and we shall not carry them out here. of certain trinions under the assumption that these trinion have certain symmetries. It might happen that the puncture symmetries in certain situations are inconsistent with preserving some of the u(1)k 1 u(1)k 1 symmetries. Evidence for this was found in [6] where certain trinions were possible to construct only under assumptions that some of the symmetries are broken. This should be related to issues with discrete have studied in the previous section and we do not study this question in the current Let us here mention a caveat of the construction. We try to predict the anomalies Fluxes for u(1)t symmetry for k; N = 2 In the case of two M5 branes and Z2 singularity we can construct eld theories associated to non vanishing ux of u(1)t. These theories do not have a regular Lagrangian description, rather are described by Lagrangians with parameters ne tuned in strong coupling domain. To construct a general model we rst build a theory corresponding to sphere with two maximal punctures of same color and having a ux corresponding to u(1)t. The construction is based on singular Lagrangians one can obtain for models in this class derived in [6] following [37]. To obtain such a tube model we start with the TA trinion of [6] which has uxes ( 14 ; 14 ; 1) for (u(1) ; u(1) ; u(1)t) associated to the surface. Then we ip the sign of one of the punctures by ipping the mesons corresponding to it [5] (see [16, 17, 27, 38, 39] uxes. Then we close the ipped maximal puncture rst to minimal with vacuum expectation value for a meson shifting uxes with ( 14 ; 14 ; 12 ). Finally we close the minimal puncture with vacuum expectation value for a baryon shifting theory in the end has two maximal punctures of the same color and ux (0; 0; 2). We can then insert this theory in our construction of torus models together with free trinions and close the minimal punctures to obtain theories with ux for all three symmetries. We now construct the tube model in more detail. We refer the interested readers for details to [6] and here we just summarize the construction of the TA trinion. The TA trinion can be built by taking a sphere with two minimal and two maximal punctures and (0; 0; 1), which is constructed by combining together two free trinions, and tuning to the point on the conformal manifold of the model where the abelian symmetries coming from minimal punctures enhance to su(2) u(1)c and gauging the su(2) with special choice of matter. With that choice of matter the u(1)c symmetry enhances also to su(2) and one obtains additiona su(2) factor rotating the additional matter elds. We thus obtain model with three factors of su(2) su(2) symmetry associated to a triplet of maximal punctures. We can thus begin our procedure of ipping and closing maximal puncture by closing the puncture of the sphere with two minimal and two maximal punctures to obtain a sphere with two minimal and one maximal punctures and then perform the gauging needed to obtain the interacting trinion. First we ip the sign of one of the maximal punctures. This is done by ipping the mesons associated to that puncture [5]. We add singlet elds mi and couple them through a superpotential to the mesons, W = m M . Next we close the puncture by giving vacuum expectation values to a particular combinations of m. As the mesons in the sphere with two minimal and two maximal punctures are built from QQe combinations of chiral elds, the vacuum expectation values induce mass terms for some of the avors. The sphere with one maximal and two minimal punctures we obtain thus has Lagrangian in terms of two su(2) gauge groups each having ve avors and a bunch of singlets. Let us gure this out in complete detail. The discussion is most easily performed at the level of the index as it captures all the relevant information. The index of the sphere with two maximal and two minimal punctures is Iz;v;a;b = (p; p)2 (q; q)2 We ip the z puncture, Then we give vacuum expectation values for the meson weighed ptq (z1z2) 1. In the index . We also need to introduce new chiral elds coupling them through superpotential [5]. In the index this amounts to multiplying with 2) ee(( ppttqq 22 2)2)e(ep(tqptq 2 2) . Then we need to give a vacuum 2 2) expectation value to a speci c baryonic operator which amounts to setting We also need to ip the second baryon charged under u(1) by multiplying the index with e(pq 4). After all the above manipulations index of the sphere with two minimal and one = ( ptq ) 12 maximal puncture we obtain is, Iv;a;b = e Ifz1= ;z2= ptq g;v;a;b ; form mass term and decouple. Thus the index becomes, Ifz1= ;z2= ptq g;v;a;b = (p; p)2 (q; q)2 Which is the index of two copies of su(2) SQCD with ve avors coupled together through superpotential terms and bifundamental elds. Finally the index of the tube theory with ux for u(1)t and two maximal punctures of the same color is obtained by gauging the su(2) enhances group of u(1)a and u(1)b with appropriate chiral elds, Iv;c = e t (p; p) (q; q) z 1v1 1 Ic;pzv2;pv2=z : smoothly becomes to be one the above index Ifz1! ;z2! ptq g;c;pzv2;pv2=z j !1 : Smooth deformation of this sort breaking symmetry might correspond to marginal deformations. Indeed we do expect to have marginal deformations of this sort which break puncture symmetries down to the Cartan [6]. We thus have evidence that the tube theory corresponds to sphere with four minimal punctures and ux two for u(1)t when we tune the couplings to point where the abelian symmetries enhance. The anomalies of this theory are easily computed from Lagrangian implied by this index. Let us write the trial a anomaly for the model one obtains gluing together Q + Q Q =2 units of ux, and 2Qt units of t ux. The anomalies are given, a = 2Qtst 18s 2 + 18s 2 + 6st2 from which we determine the cubic anomalies which depend on ux of u(1)t, kttt = We can compute the dependence of T rR on Qt and get that it is equal to 8stQt which tells us that Tr u(1)t = 8Qt. All other anomalies do not depend on Qt. The anomalies here coincide with the ones deduced from (2.11). 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Ibrahima Bah, Amihay Hanany, Kazunobu Maruyoshi, Shlomo S. Razamat, Yuji Tachikawa, Gabi Zafrir. 4d \( \mathcal{N}=1 \) from 6d \( \mathcal{N}=\left(1,0\right) \) on a torus with fluxes, Journal of High Energy Physics, 2017, 1-35, DOI: 10.1007/JHEP06(2017)022