D-type conformal matter and SU/USp quivers

Journal of High Energy Physics, Jun 2018

Abstract We discuss the four dimensional models obtained by compactifying a single M5 brane probing D N singularity (minimal D-type (1, 0) conformal matter in six dimensions) on a torus with flux for abelian subgroups of the SO(4N) flavor symmetry. We derive the resulting quiver field theories in four dimensions by first compactifying on a circle and relating the flux to duality domain walls in five dimensions. This leads to novel \( \mathcal{N}=1 \) dualities in 4 dimensions which arise from distinct five dimensional realizations of the circle compactifications of the D-type conformal matter.

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D-type conformal matter and SU/USp quivers

Revised: May D-type conformal matter and SU/USp quivers Hee-Cheol Kim 0 1 2 3 6 Shlomo S. Razamat 0 1 3 4 Cumrun Vafa 0 1 3 6 Gabi Zafrir 0 1 3 5 0 Kashiwa , Chiba 277-8583 , Japan 1 Haifa , 32000 , Israel 2 Department of Physics , POSTECH 3 Cambridge , MA 02138 , U.S.A 4 Department of Physics , Technion 5 IPMU, University of Tokyo 6 Je erson Physical Laboratory, Harvard University We discuss the four dimensional models obtained by compactifying a single M5 brane probing DN singularity (minimal D-type (1; 0) conformal matter in six dimensions) on a torus with ux for abelian subgroups of the SO(4N ) avor symmetry. We derive the resulting quiver eld theories in four dimensions by rst compactifying on a circle and relating the ux to duality domain walls in ve dimensions. This leads to novel N = 1 dualities in 4 dimensions which arise from distinct ve dimensional realizations of the circle compacti cations of the D-type conformal matter. Supersymmetric Gauge Theory; Duality in Gauge Field Theories; Supersym- 1 Introduction 2 Six dimensions 3 Five dimensions 4 Four dimensions A S gluing B A ne quivers and duality taster C Branching rules from which all these phenomena arise naturally. One such a setup is to realize the 4d theory through the compacti cation of a 6d SCFT. As a simple example consider compacti cation of the (2; 0) theory on a torus leading to N = 4 super Yang-Mills in 4d. As is well known this 4d theory is conformal and has a 1 dimensional conformal manifold which in the compacti cation construction is realized as the complex structure of the torus. This space is spanned by a single complex variable de ned in the upper half-plane. However, it is known that values of 's di ering by a modular transformation in fact de ne the same torus and thus the same compacti cation. This imply that 4d theories whose value of the coupling constant, , di ers by an SL(2; Z) transformation should de ne the same SCFT. This is the well known S-duality of N = 4 super Yang-Mills, which is a dynamical nonperturbative phenomena of the theory. Yet, we see that it can be easily motivated from the This idea was later on extended to N = 2 theories by Gaiotto, by considering the compacti cation of the (2; 0) theory on more general Riemann surfaces [1]. The class of theories constructed in this way is known as class S theories, and this method has been used { 1 { to better understand the landscape and dynamics of N = 2 theories. For instance it can be used to motivate Argyres-Seiberg [2] type dualities, and to construct various N = 2 SCFTs without a manifestly N = 2 preserving Lagrangian (the so called non-Lagrangian theories). A natural next step then is to try and apply this to N = 1 theories. One way is to consider again compacti cations of the (2; 0) theory, but now only preserving N = 1 supersymmetry. This leads to the so called N = 1 class S theories that have been studied by various people, see for example [3, 4]. However, we can also consider starting from less supersymmetric theories, particularly, 6d (1; 0) SCFTs. These are far more numerous than their (2; 0) cousins, and many also posses interesting global symmetries that can be exploited in the compacti cation, and further should lead to 4d theories inheriting 5d, where they may have a low-energy description as a 5d gauge theory [10, 18{22]. These recent developments set the stage for the study of the compacti cation of (1; 0) SCFTs to 4d, and will be employed in this article for that purpose. At this point in time, the study of the compacti cation of (1; 0) SCFTs to 4d has already been undertaken for speci c choices of (1; 0) SCFTs and Riemann surfaces [9{16, 19, 21]. In this article we shall continue to expand the landscape of known compacti cations of (1; 0) SCFTs to four dimensions. The speci c case that we shall consider here is that of the minimal (DN+3; DN+3) conformal matter compacti ed on a torus, or a sphere with two punctures, with uxes in its global symmetry. The method employed to determine the eld theories is to rst reduce to 5d where the theory has an e ective description as a 5d gauge theory. In fact it has at least three di erent e ective descriptions [20], and in this paper we will study two of those three with the remaining one to be discussed in a companion paper [17]. The ux is then realized using a duality domain wall interpolating between two such descriptions. As the theory is compacti ed one direction is either a circle, in the case of the torus, or an interval, in the case of a two-punctered sphere, with boundary conditions at the two edges that play the role of the punctures. The resulting 4d theory can then be read o from the 5d bulk matter, compacti ed on intervals, and interacting through 4d elds living on the domain wall. This type of construction was successfully used to study [16] compacti cations of the rank 1 E-string, and here we generalize it to the case of the minimal (DN+3; DN+3) conformal matter. The E string is the rst case in this set of models, the (D4; D4) minimal conformal matter, with the SO(16) global symmetry enhancing to E8. Once a conjecture for the 4d theory was generated using the 5d picture, we can put that conjecture to the test by performing various consistency checks. Notably, the 6d reduction leads to various predictions that should be satis ed by the 4d theory. The most direct of which are the global symmetries and its 't Hooft anomalies, which can be computed by { 2 { integrating the 6d anomaly polynomial on the Riemann surface [3]. Moreover inequivalent realizations of the circle compacti cations to 5d as well as moving the uxes around lead to novel 4d duality predictions. We performed these checks on many of these theories, and in any case we considered the theories we nd are in agreement with these conditions. The structure of this article is as follows. We begin in section 2 by presenting the 6d SCFT known as the minimal (DN+3; DN+3) conformal matter. In particular we review some of its properties that are relevant for the later sections. We also use the 6d anomaly polynomial to predict the anomalies of the 4d theories resulting from the compacti cation of this 6d SCFT on a Riemann surface. Section 3 is devoted to the ve dimensional story. Here we consider rst reducing the 6d SCFT to 5d and then employ the 5d low-energy gauge theory description to formulate a conjecture for the 4d theories. Section 4 deals with the four dimensional story. Here we study the theories conjectured in the previous section, and compare the results against the 6d expectations. Some aspects of the constructions are postponed to the appendices. In particular, there is an alternative way to think of the minimal (DN+3; DN+3) conformal matter in such a way that is naturally generalizable to the non-minimal as well as more general cases. This is part of a much larger story that will be considered in a di erent publication [17], but when applied to the case of the minimal (DN+3; DN+3) conformal matter, it leads to other 4d eld theories that should lead to dual descriptions of the same 4d SCFT. We discuss this aspect in appendix B. 2 Six dimensions We start by describing several known facts about the 6d SCFT called the (DN+3; DN+3) minimal conformal matter. First we shall start with how this theory is constructed. There are various di erent ways to construct this SCFT in string theory. One way is as a Z2N 2 orbifold of the E-string theory [15]. Alternatively it can be realized as the theory living on a single M5-brane probing a C2=DN+3 singularity [6]. In F-theory, it can be constructed by a 1 curve decorated with a USp(2N 2) gauge group [5]. One interesting aspect of this theory is that it can be thought of as an orbifold of both the E-string theory and the theory living on a single M5-brane. Therefore, it is naturally related to both the orbifold generalizations of the (2; 0) theory and the E-string theory. In this article we shall mostly adopt the rst viewpoint and think of it as a generalization of the E-string. However, the second viewpoint also exists and can be studied in a similar treatment, which we perform in a di erent publication [17]. Nevertheless, we shall remark in appendix B on some interesting dualities upon comparing the two approaches. This SCFT has a one-dimensional tensor branch along a generic point of which the SCFT reduces to a USp(2N 2) gauge theory with (2N + 6) avors. This leads to another useful description of the SCFT as the 6d UV completion of this gauge theory. We shall mostly employ this description to uncover various properties of this SCFT though we should state that most of them can also be seen from the other perspectives as well. This is important to bear in mind as properties of the 6d SCFT and its related IR gauge theory may di er [ 11 ]. { 3 { HJEP06(218)5 The rst important property we shall need is the global symmetry. From the gauge theory we can see that there is an SO(4N + 12) global symmetry. Another useful property to keep in mind is that the theory has a moduli space, the Higgs branch, associated with giving vevs to the hypermultiplets. Generic points on this initiates a ow that leads us from one value of N to lower values. Speci cally one can break the USp(2N 2) group entirely in this manner. However the theory then is not empty as one still has the tensor multiplet. The string theory construction suggests that the resulting theory is the rank one E string theory. This is one way in which one can see that the (DN+3; DN+3) minimal conformal matter can be thought of as a generalization of the rank one E-string, which from the eld theory viewpoint is done by the addition of vectors and hypermultiplets. We also note that the naive expected symmetry SO(16) in the E-string case is enhanced to E8. We shall later discuss the mechanism leading to this enhancement from the gauge theory viewpoint. The 6d SCFT should then contain the conserved current of the SO(4N + 12) global symmetry. This is one important operator we observe already at the level of the gauge theory. There is one more interesting state, charged under the SO(4N + 12) global symmetry, that exists in the SCFT. The gauge theory contains non-perturbative excitations associated with instanton strings. These are massive at nite gauge coupling, but become massless at the origin of the tensor branch. This type of states is thought to play an important role in the UV completion. The gauge theory contains fermions, and these have zero modes in the instanton background which cause the instanton con guration to acquire avor charges. From instanton counting one discovers that these instantons should then be in a chiral spinor of the SO(4N + 12) global symmetry. These lead to an additional BPS operator in the 6d SCFT that turns out to be a Higgs branch generator. Several aspects of the Higgs branch of this 6d SCFT, including the existence of this Higgs branch generator were discussed in [25]. We next review how this can be seen by compactifying the 6d SCFT to lower dimensions on circles. Circle compacti cation to lower dimensions. We start with the compacti cation of this theory to ve dimensions on a circle. This has been analyzed by various people in di erent contexts [10, 18{21], and here we collect some observations that will prove important later on. First we consider the problem of reduction to 5d on a circle with nite radius. We shall also allow turning on arbitrary holonomies for the SO(4N + 12) global symmetry, which correspond to additional mass parameters in the IR 5d theory. This problem was studied in [18, 20], where it was found that this theory reduces to some 5d gauge theory. The 5d gauge theory is not unique and in fact there are at least three di erent possible 5d gauge theories one can obtain depending on the holonomies one turns on. The two descriptions which will prove most useful us in this paper have only a single 5d gauge group. One description is a 5d USp(2N ) gauge theory with 2N + 6 fundamental hypermultiplets while the other is a 5d SU(N + 1)0 gauge theory (subscript denotes the Chern-Simons level which is 0) with 2N +6 avors. These only have one gauge group so they only involve one large mass that is identi ed with the coupling constant. From the reduction viewpoint, this mass is identi ed with the radius, possibly tuned with an holonomy. There { 4 { is another description involving a quiver gauge theory which is a linear quiver of N SU(2) gauge groups with bifundamental hypermultiplets connecting the groups and the gauge groups at the ends of the quiver connected to additional quartets of hypermultiplets. This involves N gauge couplings implying that we need at least N 1 holonomies to reach it. This also explains why the global symmetry is quite broken in it. We will not discuss this description here however this is the description which is the easiest to generalize to ADE minimal conformal matter and we will address these general setups in a separate paper [17]. We next want to consider taking the zero radius limit with no holonomy. This limit was studied in [19], where it was found that the theory reduces to a 5d SCFT. It is convenient to allow mass deformations for this SCFT causing it to ow to an IR gauge theory. We can HJEP06(218)5 get at least three di erent IR gauge theories depending on the choice of mass deformations. These are related to the three di erent 5d descriptions of the 6d SCFT by integrating out a avor.1 For our considerations it will su ce to concentrate on one of them, USp(2N ) gauge theory with 2N + 5 avors. It is interesting to study the BPS spectrum, particularly the Higgs branch chiral ring, of this theory as this can teach us about the operator spectrum of the original 6d SCFT. In our case, a detailed analysis of the Higgs branch was performed in [26] primarily using the 3d mirror quiver one gets when compactifying the 5d SCFT to 3d. We next review some aspects of this analysis while referring the reader interested in more in depth study of the Higgs branch to the reference. We can study the BPS spectrum of the 5d SCFT e ectively using the 5d superconformal index [ 31 ]. One contribution we get comes from the hypermultiplets. Particularly the mesons are part of the SO(4N + 10) conserved current multiplet. Besides these there are no other independent invariants. Another important type of states are the instanton particles. These are non-perturbative excitations of the 5d gauge theory that become massless at the SCFT point. One important contributions of these operators is that they provide additional conserved currents that enhance the classically visible U(1)I SO(4N +10) global symmetry to SO(4N + 12) which is the symmetry of the 5d SCFT that is inherited from 6d. These are expected to come from two-instanton contribution [27]. So we see that we get the SO(4N + 12) conserved current multiplet from the mesons, gaugino bilinear2 and the two-instanton sector. This is identi ed with the reduction of the 6d conserved current multiplet, and naturally is in the adjoint of SO(4N + 12) and in the 3 of SU(2)R. There is one more important contribution coming from the 1-instanton sector. This provides a BPS operator in the spinor of SO(4N + 10). Together with the anti-instanton these form a chiral spinor of SO(4N + 12) that is in the N + 2 of SU(2)R. This state is naturally identi ed as coming from the contribution of the instanton strings of the 6d SCFT. We can further compactify to 4d where the 5d SCFT reduces to an A-type class S theory. From here it is also straightforward to reduce to 3d where we have a Lagrangian 1For example consider the case of the rank one E-string theory, where the three description degenerate. Compactifying on a nite circle, this 6d SCFT ows to an SU(2) gauge theory with eight avors assuming a suitable holonomy is turned on. It is known that when we take the zero radius limit the 6d SCFT ows to the 5d rank one E8 SCFT [23]. This 5d SCFT has a mass deformation where it ows to the 5d SU(2) gauge theory with seven avors, which is related to the former theory by integrating out a avor. 2This provides the scalar in the U(1)I conserved current multiplet. { 5 { The 3d mirror dual of the SCFT one gets by compacti cation of the 6d SCFT (DN+3; DN+3) conformal matter on T 3 without uxes. mirror dual [24]. The speci c dual for this case is shown in gure 1. We rst note that except the leftmost node, all other nodes are balanced. Thus from the results of [28], we expect the symmetry on the Coulomb branch to enhance to SO(4N + 12) by monopole operators. These again match the 6d SO(4N + 12) conserved current multiplet. Next we turn to the unbalanced node. It too does contribute a monopole operator, but as it is unbalanced it won't be a conserved current. It is known that an unbalanced node would contribute a monopole operator in the NF 2NC + 3 of SU(2)R,3 and in the representation of the global symmetry corresponding to the node in the Dynkin diagram it is connected to. For the case at hand this means we have a monopole operator in the N + 2 of SU(2)R and in the chiral spinor of SO(4N + 12). So we see from 6d, 5d and 3d perspectives that the basic operators charged under the SO(4N + 12) global symmetry are the conserved current, in the adjoint, and a Higgs branch generator in a chiral spinor. We do not appear to observe additional operators, particularly in 6d, but also in lower dimensions. Importantly we do not observe operators in the vector and the other chiral spinor. This means that there is no contradiction with the 6d SCFT global symmetry group being more precisely Spin(4N + 12)=Z2. Here we remind the reader that Spin(4N + 12) has a Z2 Z2 center where each element acts as 1 on the vector and one of the chiral spinors. The Z2 we mod by is the one not acting on the chiral spinor that appears in the SCFT. This will be important next when we discuss the reduction with uxes. 6d expectations from the reduction. In this section we shall discuss toroidal compacti cation of the (D; D) minimal conformal matter to 4d with uxes under its global symmetry. Particularly we shall consider the computation of the anomalies of the resulting 4d theory from those of the mother 6d theory. For that we require the anomaly polynomial of the (D; D) conformal matter. This was computed in [8]. Alternatively it can be computed directly from the USp(2N 2) gauge theory description on the tensor branch. HJEP06(218)5 Either way it is found to be: IDDMCM = N (10N + 3) C22(R) + (N + 2) (N 1) 24 24 6 3Here we refer to the R-symmetry that acts on the Coulomb branch. p1(T )C2(SO(4N + 12))V + C4(SO(4N + 12))V + (29 + (N 1)(2N + 13)) 7p1(T ) 4p2(T ) 5760 N (2N + 9) 48 N 2 p1(T )C2(R) C2(R)C2(SO(4N + 12))V 24 (2N + 1) C22(SO(4N + 12))V (2.1) { 6 { We use the notation C2(R) for the second Chern class in the fundamental representation of the SU(2)R. We also employ the notation Cn(G)R for the n-th Chern class of the global symmetry G, evaluated in the representation R (here V stands for vector), and p1(T ); p2(T ) for the rst and second Pontryagin classes respectively. Next we consider compactifying the theory on a torus with uxes under U(1) subgroups of SO(4N +12). For simplicity we shall only consider the case of ux to a single U(1) though the generalization to more U(1)'s is straightforward. A basis for such choices is given by 2N +6 distinct U(1)'s inside SO(4N +12). These are just given by the Cartan subalgebra of SO(4N +12). To each U(1) we can associate a node in the Dynkin diagram of SO(4N +12). Then for each node we get a di erent embedding of a U(1) inside SO(4N + 12) where the commutant of the U(1) in SO(4N + 12) is given by the Dynkin diagram one is left with after removing that node. By examining the Dynkin diagram one sees that the possible embeddings preserve U(1) SU(r) 2r) for r = 1; 2; : : : ; 2N + 4 with r = 2N + 6 being special. The special thing in the r = 2N + 6 case is that there are two U(1)'s associated with this case. This choice corresponds to the U(1)'s associated with the spinor nodes in the Dynkin diagram and so appear twice. What distinguishes the two choices is how the spinors decompose under U(1) SU(2N + 6). The group SO(4N + 12) has two inequivalent self-conjugate spinor representations, and under the embedding of U(1) SO(4N + 12), one decomposes to all the even rank antisymmetric tensor representations of SU(2N + 6) while the other decomposes to all the odd rank ones. The two embeddings di er by which spinor decomposes to each choice. As we can exchange the two spinors by an outer-automorphism transformation on the generators of SO(4N + 12), the two embedding truly di er if the theory is not invariant under this transformation. Here we note that in the previous section it was demonstrated that the (D; D) conformal matter 6d SCFT has a state in one of the chiral spinors of SO(4N + 12), but not the other. How this spinor decomposes under the global symmetry preserved by the ux di ers between the two embeddings which in principle is distinguishable in the 4d theories. Therefore, we expect there to be two distinct ux choices, leading to distinct 4d theories, both preserving U(1) SU(2N + 6). As we shall see later, this leads to some rather surprising expectations in 4d. We next need to decompose the SO(4N + 12) characteristic classes into those of the commutant and the rst Chern class of the U(1). In this case we nd that: 2 C2(SO(4N +12))V = rC12(U(1)) + C2(SO(4N + 12 2r))V + 2C2(SU(r))F ; (2.2) C4(SO(4N +12))V = r(r 1) C14(U(1))+C22(SU(r))F +2C2(SU(r))F C2(SO(4N +12 2r))V rC12(U(1))C2(SO(4N + 12 2r))V +2(3 r)C12(U(1))C2(SU(r))F 6C1(U(1))C3(SU(r))F +C4(SO(4N + 12 2r))V +2C4(SU(r))F : Here C1(U(1)) is the rst Chern class of the U(1), normalized as in appendix C. Using this we can next compute the anomalies of the resulting 4d theories. { 7 { HJEP06(218)5 r Before moving on to the anomaly calculation, we wish to introduce a ux basis. Fluxes can be associated with vectors on the root lattice so a basis of the root lattice can be used as a basis for uxes. For the case of SO(4N + 12) this is simply given by a 2N + 6 vector built from the roots of SO(4N + 12), which are given by: ( 1; 1; 0; 0; 0; : : :) + permutations. In this basis a convenient choice to represent the uxes we introduced is given by (z z; z};|: : : ; z{; 0; 0; : : : ; 0), where there are other choices related by Weyl transformations. The U(1)'s associated with the spinor nodes are both given by r = 2N +6, but di er in whether the number of minus signs is even or odd. We expect that a ux is consistent if and only if it can be written as a sum of the roots of SO(4N + 12) with integer coe cients.4 As a result, if r is odd then z must be even as one cannot build the z odd vector using the SO(4N + 12) root vectors. This is related to the di erence in the quantization condition between r even and odd noted in appendix C. It should be noted that the issue of ux quantization can be quite subtle due to the potential non-triviality of the global symmetry. Particularly, the `only if' part in the initial statement would be true if the group was Spin(4N + 12). However, if the group is Spin(4N + 12)=Z2 then some apparently disallowed uxes are possible as the states that made them inconsistent do not exist. For instance the ux associated with the spinor node whose associated spinor is in Spin(4N + 12)=Z2 can have half-integer z. So, if we chose to associate it with the case of even number of minus signs, then the ux ( 12 ; 12 ; 21 ; : : : ; 12 ) is consistent. Anomalies with ux. Next we can consider compactifying the 6d theory on a Riemann surface with ux under the U(1), that is R C1(U(1)) = z where z is an integer. For simplicity let us concentrate on the case where is a torus. As the torus is at we do not need to twist to preserve SUSY. However, SUSY is still broken down to N = 1 in 4d by the ux. The 4d theory inherits a natural U(1)R R-symmetry from the Cartan of the SU(2)R though this in general is not the superconformal R-symmetry. Under the embedding of U(1)R SU(2)R, the characteristic classes decompose as: C2(R) = Next we need to decompose SO(4N + 12) to the subgroup preserved by the ux, done in (2.2). Finally we set: C1(U(1)) = zt + C1(U(1)R) + C1(U(1)F ). The rst term is the ux on the Riemann surface, where we use t for a unit ux two form on , that is R t = 1. The second term takes into account possible mixing of the 4d global U(1) with the superconformal R-symmetry, where is a parameter to be determined via a-maximization. C12(U(1)R). Finally the third term is the 4d curvature of the U(1). Next we plug these decompositions into (2.1) and integrate over the Riemann surface. This yields the 4d anomaly polynomial six form. From it we can read o the anomalies and nd: T r(U(1)R) = T r(U(1)3R) = T r(U(1)RU(1)2F ) = 0; 4Here when we refer to consistent ux we mean one for which all states in the system satisfy Dirac quantization condition, without the need for central uxes. Regarding that last part, we note that, with our normalization of ux, in some cases the value of the ux will be fractional and in those cases one also has ux turned on in the center of the group, see appendix C of [16] for a discussion on this type of e ects. { 8 { T r(U(1)F U(1)2R) = 2N rz T r(U(1)F ) = T r(U(1)3F ) = 2r(N + 2)z; (3r + 2N 2)rz; T r(U(1)RSO(4N + 12)2) = T r(U(1)RSU(r)2) = 0; T r(SU(r)3) = T r(U(1)F SO(4N + 12)2) = T r(U(1)F SU(r)2) = 1); 2z(N rz 2 ; (2N + r 2 2)z : Here U(1)R refers to the 6d R-symmetry. From this we can evaluate a and determine . We nd that: = sign(z) p p Using this we can evaluate a and c: a = 3 6 2N + 3r p (5N + 1) 2 rjzj ; 2 c = (11N + 4)p5N + 1rjzj ; 12p2N + 3r 2 As previously mentioned, for r = 2N + 6 there are two distinct choices of the embedding di ering by how the spinor in the theory decomposes. However, the anomalies are indi erent to this distinction. Therefore, this suggests that there should be two distinct 4d theories with the same anomalies, but with slight di erences in their matter spectrum. 3 Five dimensions As discussed in the previous section, circle compacti cations of the 6d (D; D) conformal matter lead to at least three di erent 5d gauge theories. In this section we study these 5d gauge theories when various uxes are turned on under the compacti cations. The 6d uxes naturally reduce to domain wall con gurations in the 5d gauge theories [16]. We will call this type of domain walls as ` ux domain walls'. We shall construct Lagrangians of such domain walls coupled to the 5d systems that realize various uxes along the global symmetry of the 6d (D; D) conformal matter theory. We split the 5d spacetime into two chambers and consider half-BPS interfaces interpolating between the two. Each chamber hosts a 5d gauge theory which is one of the three gauge theories from the 6d (D; D) conformal matter possibly with di erent avor holonomies. Two gauge theories and their boundary conditions will be connected by extra 4d degrees of freedom and superpotentials at the interface. When we properly choose the 4d elds in the domain wall as well as the 5d boundary conditions of the bulk 5d gauge theories and couple them through certain superpotentials, this domain wall can implement the ux domain wall of the 6d conformal matter theory. { 9 { (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) SU(N +1) domain wall. An interesting domain wall in the 5d reduction of the 6d (DN+3; DN+3) conformal matter was studied in [32]. This domain wall glues together the USp(2N ) gauge theory on one side to the SU(N + 1) gauge theory on the other side. This domain wall is called `duality domain wall' as it interpolates between two dual gauge theories. This type of domain wall exists for any number of avors, but we will focus on the cases with Nf = 2N + 6 fundamental avors in both gauge theories. This domain wall is a higher rank N > 1 generalization of the ux domain wall in the 5d E-string theory. When N = 1 this domain wall implements a basic avor ux in the 6d (D4; D4) conformal matter theory (or 6d E-string theory) as noted in [16]. Similarly, we conjecture that the rank N duality domain wall with Nf = 2N + 6 is a ux domain wall of the (DN+3; DN+3) conformal matter theory. More precisely, this domain wall corresponds to 1=4 ux which breaks the SO(4N + 12) global symmetry down to U(1) SU(2N + 6) symmetry. Let us brie y review the construction of the duality domain wall between two gauge theories of USp(2N ) and SU(N + 1) gauge groups in [32]. We will consider a 1/2 BPS interface located at x4 = 0 along one of the spatial directions. We put the USp(2N ) gauge theory in the left chamber and the SU(N + 1) gauge theory in the right chamber. These gauge theories should satisfy 1/2 BPS boundary conditions at the interface. For the vector multiplets, we will choose Neumann boundary condition which sets the gauge eld as A4jx4=0 = 0 : (3.1) The boundary condition for the hypermultiplets is more involved. First, in the USp(2N ) gauge theory there are 2N + 6 fundamental hypermultiplets, (X; Y y)i (i = 1; 2N + 6) where Xi and Y i are the fundamental chiral elds in the i-th hypermultiplet. For each USp(2N ) fundamental hypermultiplet, we have two choices of boundary conditions as either or (3.2) We will denote this boundary condition by a sign vector ( ; ; ; ) where + or at the i-th entry stands for the rst or the second boundary condition for the i-th hypermultiplet. So we have 22N+6 di erent boundary condition choices for the USp(2N ) hypermultiplets. Each choice preserves a di erent U(1) SU(2N + 6) subgroup of the global symmetry. On the other hand, the SU(N + 1) gauge theory has only two di erent choices of 1/2 BPS boundary conditions. As we want to preserve U(1) SU(2N + 6) global symmetry, we should choose the same boundary condition for all the hypermultiplets. We thus have the boundary condition for the SU(N + 1) hypermultiplets as 1) or 2) @4Yijx4=0 = Xijx4=0 = 0 for all i : (3.3) We will collectively call the chiral elds surviving at the interfaces as M and M 0 for the USp(2N ) and SU(N + 1) matters respectively. The bulk 5d boundary conditions couple to the 4d boundary degrees of freedom at the interface such that the entire 5d/4d coupled system preserves 4 real supersymmetries or 4d N = 1 supersymmetry. Since we give Neumann boundary conditions for the vector multiplets, the boundary 4d system has USp(2N ) introduce a bi-fundamental chiral multiplet q of USp(2N ) and SU(N + 1) gauge groups and an anti-symmetric chiral multiplet A of the SU(N + 1) gauge group. We then couple these 4d elds to the 5d boundary conditions through the following 4d superpotentials: Wjx4=0 = M q M 0 + q q A ; (3.4) where dot denotes the contraction of the gauge and avor indices in an appropriate manner. This con guration provides a consistent 5d/4d coupled system. At the 4d interface, the SU(N + 1) gauge theory has in general cubic gauge anomaly of N + 3 unit arising from the 2N + 6 chiral multiplets with the Neumann boundary condition. This cubic anomaly is canceled by the 4d boundary chiral elds q and A which contribute in total 2N +(N 3) = N 3 to the cubic anomaly. As an anomaly free 5d=4d con guration, the above domain wall can naturally interpolate between the USp(2N ) gauge theory and the SU(N +1) gauge theory coming from the 6d conformal matter theory on a circle. There are three gauge anomaly free abelian symmetries. One is the 6d U(1)R SU(2)R R-symmetry under which the bulk 5d elds X and Y transforms with charge +1. The 4d chiral elds q and A carry the U(1)R charge 0 and +2 respectively. The gauge-U(1)R mixed anomaly coming from q and A is canceled by the contributions from the 5d vector multiplets with Neumann boundary conditions, while X; Y do not contribute to this anomaly. Another abelian symmetry is the avor U(1)x symmetry acting on the elds (M; M 0; q; A) with charges (1; N2+1 ; NN++31 ; 2(NN++13) ). This anomaly free U(1)x symmetry can in principle mix with the U(1)R R-symmetry. The true R-symmetry of the low energy theory in the presence of the domain wall will be determined by a-maximization. The last abelian symmetry is the 6d Kaluza-Klein (KK) symmetry. This symmetry remains unbroken even in the domain wall background by mixing with other abelian symmetries acting on the boundary elds. Under the 4d reduction which we will discuss below, this KK symmetry will become an extra 4d global symmetry when compacti ed on a nite size interval or will be broken when compacti ed on a torus. So far we discussed the duality domain wall in [32] connecting the USp(2N ) and the SU(N + 1) gauge theories. The construction of the domain wall involves as described above the 5d boundary conditions, the extra 4d degrees of freedom, and the superpotential couplings. We claim that this domain wall, which is drawn in gure 2a, is the basic ux domain wall between the USp(2N ) and SU(N + 1) gauge theories. It introduces at the location of the domain wall the 1=4 unit ux preserving U(1) SU(2N +6) in the 6d (D; D) conformal matter on a circle. In the orthogonal root basis, this ux corresponds to the ux along ( 14 ; 1 ; 14 ; : : : ; 41 ). When there is only a single ux wall, all di erent choices of the 4 boundary conditions for the hypermultiplets can be set to give the same ux by the Weyl symmetry of the SO(4N + 12) bulk global symmetry. Note that this domain wall when N = 1 reduces to the basic ux domain wall in the 5d E-string theory in [16]. We can carry out a simple but non-trivial check for this ux domain wall using a 4d reduction of this 5d/4d system. Basically, we will check that the 4d reduction of this system has the desired 't Hooft anomalies for being the 4d reduction of 6d (D; D) conformal matter USp q (a) M 0 SU A SU M˜ 0 SU the 5d bulk gauge groups. The chiral elds M and M 0 are from the hypermultiplets with Neumann boundary conditions on the two sides of the wall. Figure (b) is the SU(N + 1) SU(N + 1) type domain wall with ux 1/2 preserving U(1) SU(2) SU(4N + 8) symmetry. SU(N+1) SU(N+1) symmetry are gauged by the 5d bulk gauge groups. The chiral elds M; M~ and M 0; M~ 0 are from the 5d hypermultiplets with Neumann boundary condition. There is a gauge singlet chiral eld denoted by `X' which couples to the baryonic operator of the bi-fundamental chiral eld q. theory with uxes. We rst put this system in an interval L x 4 L. In low energy below L1 , this system e ectively reduces to a 4d theory. From our 5d Lagrangian description of the domain wall, we can deduce the 4d Lagrangian of the 6d (D; D) conformal matter theory put on a tube (or a two punctured sphere) with uxes. The boundary conditions at x4 = L and x4 = L de ne the punctures at both ends of the tube. We will choose the boundary conditions such that all chiral elds of 5d hypermultiplets with Neumann boundary condition at the domain wall (at x4 = 0) also satisfy Neumann boundary condition at both ends. The vector multiplets however are chosen to satisfy Dirichlet boundary condition at two ends of the tube. This truncates the 5d bulk gauge elds at the end of the each chamber, so the 5d gauge symmetries become non-dynamical global symmetries which we regard as global symmetries assigned to two punctures. The USp(2N ) SU(N +1) system with a basic ux domain wall then gives rise to a 4d Wess-Zumino type theory drawn in gure 2a with USp(2N ) SU(N +1) global symmetries for the two punctures. We claim the theory in gure 2a is the 4d reduction of 6d (D; D) conformal matter with ux 14 on a tube with USp(2N ) SU(N + 1) puncture symmetries. Let us check this proposal by comparing 't Hooft anomalies of this theory against direct computations from the 6d theory. When we know a 5d Lagrangian description of a 6d theory, the 4d 't Hooft anomalies when compacti ed on a tube can be computed directly from the 6d anomaly polynomial together with anomaly in ow contributions coming from the 5d boundary conditions. See [16] for detailed discussions. This computation does not rely on explicit Lagrangian descriptions of the 4d reduction. Integrating out the 6d anomaly polynomial (2.1) on a tube with 1=4 ux, we nd the following 't Hooft anomalies: In addition, the 5d boundary conditions at both ends of the tube induce anomaly in ow contributions given by T r(U(1)R) = T r(U(1)3R) = 0 ; T r(U(1)RU(1)2x) = 0 ; T r(U(1)xU(1)2R) = N (N + 3) ; T r(U(1)x) = (N + 2)(N + 3) ; T r(U(1)3x) = 4(N + 2)(N + 3) ; T r(U(1)x SU(2N +6)2) = N + 1 2 : T r(U(1)R) = T r(U(1)3R) = N (N +1) ; 3 2 T r(U(1)RU(1)2x) = T r(U(1)xU(1)2R) = 0 ; T r(U(1)x) = 2(N + 1)(N + 3) ; T r(U(1)3x) = 2(N + 3)(N 3 + 2N 2 + N + 4) (N + 1)2 ; T r(U(1)R USp(2N )2) = T r(U(1)R SU(N +1)2) = T r(U(1)x USp(2N )2) = T r(U(1)x SU(N +1)2) = T r(U(1)x SU(2N +6)2) = N + 1 N + 1 2 2 ; ; N + 3 2 N + 3 N + 1 N + 1 2 ; ; ; (3.5) which come from the 't Hooft anomalies of 5d fermion elds with Neumann boundary conditions for both the vector and hypermultiplets at the two ends. Combining (3.5) and (3.6), the result perfectly agree with 't Hooft anomalies of the 4d tube theory we proposed above. This provides a non-trivial evidence for our conjecture of the 4d tube theory and therefore for our ux domain wall conjecture given in gure 2a. Domain walls for other general uxes can be constructed by combining more than one basic domain wall with appropriate boundary conditions. We shall now explain how to connect two domain walls, which will be enough to construct general domain walls. Suppose we locate two domain walls at x4 = t1 and x4 = t2 with L < t1 < t2 < L. This splits the 5d spacetime into three chambers. Each chamber hosts either a SU(N + 1) gauge theory or a USp(2N ) gauge theory with 2N + 6 hypermultiplets. The 5d gauge theories in these three chambers are glued by two interfaces. The boundary condition at each domain wall is the same as that of the single domain wall case: Neumann boundary conditions for vector multiplets and 1/2 BPS boundary conditions (3.2) or (3.3) for hypermultiplets. The 5d boundary condition couples the 4d chiral multiplets q; A and q0; A0 in the rst and the second interfaces respectively with cubic superpotentials as we discussed. The theory in the second chamber lives in a nite interval with two boundaries. Thus, at low energy, the 5d theory in the second chamber reduces to a 4d gauge theory. The hypermultiplets in the second chamber can have di erent boundary conditions at the two ends t1 and t2. When a hypermultiplet satis es the same boundary conditions at both ends, it produces a 4d chiral multiplet coupled to the 4d gauge theory in the interval. On the other hand, a hypermultiplet obeying di erent boundary conditions at both ends becomes massive and will be integrated out at low energy. Integrating out this massive hypermultiplet at the end will induce a quartic superpotential between the chiral elds coupled to this hypermultiplet. This procedure de nes our gluing rule between two or more ux domain walls. The total ux of these two domain walls is determined by the boundary conditions of the hypermultiplets in the second chamber. A domain wall turns on + 14 ux along the U(1)'s rotating the chiral elds with Neumann boundary condition. Thus, if a 5d hypermultiplet satis es the same boundary condition at both ends, this combination of two domain walls introduces 14 + 14 = 12 ux along the U(1) acting on the hypermultiplet. However, if a hypermultiplet satis es opposite boundary conditions at two ends, then the ux along the U(1) direction is cancelled, 14 uxes for the 2N + 6 hypermultiplets. 14 = 0. The total ux is a sum of these U(1) Let us begin with a domain wall con guration with USp(2N ) gauge theories in the rst and the third chambers and SU(N + 1) gauge theory in the second chamber. In this case we have two di erent combinations with ux 1) ( 14 ; 14 ; ; 14 ) + ( 1 ; 1 ; 4 4 ; 14 ) ; 2) ( 14 ; 14 ; ; 14 ) + ( 14 ; 1 4 ; ; 14 ) : (3.7) For the rst con guration, we choose the boundary condition (+; +; ; +) for both USp(2N ) theories and Xi = 0 (all i) for the SU(N + 1) theory at two interfaces. This boundary condition couples to the 4d boundary chiral elds q; A and q0; A0. We will eventually obtain a ux domain wall con guration depicted in gure 3a. This theory has two cubic superpotentials for each loop in the quiver diagram and another two cubic superpotentials of the form q2A and q02A0 for the anti-symmetric 4d matters. The SU(N + 1) gauge theory in the second chamber reduces to a 4d theory at low energy E also the chiral halves of the 5d hypermultiplets satisfying Neumann boundary conditions at both boundaries become 4d chiral multiplets. In fact, we can regard this combination of two domain walls as a single domain wall coupled to the 4d SU(N + 1) gauge theory with chiral matter as shown in gure 3a. This domain wall implements the ux ( 1 ; 1 ; 2 2 ; 1 ) 2 preserving U(1) SU(2N + 6) global symmetry of the 6d theory. On the other hand, the second con guration in (3.7) can be constructed with di erent boundary conditions. We choose (+; +; ; +) for the rst USp(2N ) theory at x and ( ; ; ; ) for the second USp(2N ) theory at x4 = t2. The hypermultiplets in the middle chamber satisfy X = 0 at x4 = t1 and Y = 0 at x4 = t2. Since the boundary 4 = t1 (t1 t2) 1, and M 0 A SU A0 (a) q0 ˜ M USp USp A SU SU A0 USp M (b) M˜10 ; 12 ). Figure (b) is the domain wall connecting two 5d SU(N + 1) gauge theories with ux conditions at the two ends are opposite, these hypermultiplets will be truncated at low energy while leaving a quartic superpotential of the form M qq0 M~ . This gives rise to a trivial interface with zero ux. We now consider another type of two domain walls gluing SU(N + 1) gauge theories in the rst and the third chamber and USp(2N ) gauge theory in the second chamber. Without loss of generality we can set the boundary conditions of the hypermultiplets in the USp(2N ) theory at t1 as (+; +; ; +). So the rst domain wall introduces a ux ( 14 ; 14 ; ; 14 ). The boundary conditions at t2 can be generically chosen as (z+; }| ; +{; ; ; ). The USp(2N ) theory in the second chamber reduces to a 4d gauge theory with 2r fundamental chiral multiplets, say M , at low energy. These chiral multiplets come from the 5d hypermultiplets with Neumann boundary conditions at both ends. The number of chiral multiplets M should be even due to the Z2 anomaly of the USp(2N ) gauge group [34]. The remaining 2N+6 2r hypermultiplets with opposite boundary conditions at the two ends are truncated in the IR. After all, this con guration reduces to the quiver diagram in gure 3b. Here, two SU(N + 1) symmetries are gauged by the 5d bulk gauge couplings. The domain wall has cubic superpotentials as M10 q M + M~ 10 q0 M + q2A + q02A0 and a quartic superptential M20 qq0 M~ 20 . This domain wall corresponds to the 6d ux (z12 ; 12 ;}| { ; 12 ; 0; 0; : : : ; 0). One interesting case is when r = 1. In this case, we can simplify the quiver gauge theory on the domain wall using a known 4d duality. We can perform Intriligator-Pouliot duality [30] on the USp(2N ) gauge theory in the second chamber. This leads to a rather simple Wess-Zumino type domain wall theory given in gure 2b. This is a ux domain wall with ux ( 12 ; 12 ; 0; 0; connecting two 5d SU(N + 1) gauge theories. ; 0). We can treat this domain wall as another basic domain wall Generalization to more than two domain walls is straightforward. We just consider a concatenation of di erent 5d gauge theories in several chambers connected by our ux domain walls discussed in this section. When we glue the two 5d theories we apply the above gluing rules, which include 5d boundary conditions, 4d chiral elds, and 4d superpotentials, at each interface such that the whole 5d/4d coupled system is consistent. The corresponding ux is automatically determined by the rules above. The total ux of the full 5d=4d system is the sum over uxes of individual domain walls. We expect that each con guration realizes a 5d reduction of the 6d conformal matter theory with (generically di erent) ux. We note that the 6d ux does not depend on its location along x4 direction. This suggests that there will be a large number of dualities between di erent domain wall con gurations which give rise to the same total ux up to a Weyl transformation of the global symmetry. Under 4d reduction which we will study in the next section, the 5d domain wall dualities reduce to dualities between 4d N = 1 theories. By putting the 5d systems with domain walls on an interval, we can obtain 4d Lagrangian theories of the 6d (D; D) conformal matter theory compacti ed on a tube with general uxes. For this we will give boundary conditions of the 5d theories at two ends of the interval appropriately and take the low energy limit. We choose Dirichlet boundary conditions for the vector multiplets. For the hypermultiplets, we give the 1/2 BPS boundary conditions such that chiral halves of the hypermultiplets survive in the rst and the last chambers. This yields a 4d reduction of the 6d conformal matter with ux at low energy. The ux of the 4d theory is the same as the total ux of the domain walls. The 1/2 BPS boundary conditions de ne punctures at two ends of the interval. The puncture de ned by this boundary condition hosts either USp(2N ) or SU(N + 1) global symmetry. We can also construct 4d theories of the 6d theory on a torus. We can glue the two ends of a tube by using the same gluing rules which we used to connect two or more domain walls. This will give rise to a 5d theory on a circle with ux domain walls. At low energy this theory reduces to a 4d Lagrangian theory corresponding to the 6d (D; D) conformal matter on a torus with ux. In the next section, we will extensively test the 4d theories obtained from the 5d theories with ux domain walls compacti ed on an interval or a circle. 4 Four dimensions We begin by studying the basic tube theory, which we associate with a sphere with two punctures and ux 14 under the U(1) breaking SO(4N + 12) ! U(1) SU(2N + 6), where the spinor appearing in the 6d SCFT decomposes as the S spinor as de ned in appendix C. As argued in the previous section, the theory is just the collection of free elds with a superpotential shown in gure 4 (a). The tube has SU(N + 1), USp(2N ) and SU(2N + 6) global symmetries. It also has a U(1) that remains unbroken when tubes are connected that we identify with the ux U(1). The SU(2N +6) is identi ed with the internal symmetry while the SU(N +1) and USp(2N ) are associated with the two punctures. This implies that the two punctures are completely di erent, though have same rank, as the symmetries are not subgroups of one another. They arise as they are both 5d gauge groups that lift to the 6d (D; D) conformal matter SCFT. This seems to suggests that there are at least two distinct variants of maximal punctures in this class of theories. More generally for arbitrary 6d (1; 0) theory we expect there to be at least as many maximal punctures as inequivalent realizations of its 5d circle compacti cation. There are chiral elds in the tube, denoted by M and M 0, that are charged only under the internal symmetries and puncture symmetries associated with one puncture. These play a role when we glue punctures as well as close them. We next consider gluing them. (a) shows the theory associated with a sphere with two punctures and , while (b) shows the theory associated with a torus and . The arrow from the SU group to itself stands ux 14 for an antisymmetric chiral eld. There are cubic superpotentials for every triangle and for every antisymmetric chiral coupling it to two bifundamentals. The theory has an U(1)x global symmetry as well as a U(1)R symmetry. For the U(1)R it is convenient to use the 6d Rsymmetry under which the bifundamentals have charge 0, the antisymmetrics have charge 2, and all the others have charge 1. The charges under U(1)x are shown using fugacities. Inspired by the structure in 5d we postulate that gluing the punctures is done by identifying and gauging their associated symmetries and introducing fundamental 2N + 6 chiral elds, , coupled via the superpotential M + M 0 . Essentially the punctures are glued using half the matter content of the corresponding 5d gauge theory. Integrating out would lead to identifying M and M 0 which is natural from the 5d viewpoint discussed in the previous section. This happens when the boundary conditions on the two sides of the tube are the same. This is the version of gluing for this theory [13], where we glue punctures of the same sign together. We expect there to be also a version of S gluing [13], we will discuss it in appendix A. We can now consider taking two tubes and connecting them together to form a torus. The resulting theory is shown in gure 4 (b). We associate it with a compacti cation on corresponds to a torus.5 We next perform a variety of checks on this theory. a torus with ux 12 preserving SU(2N + 6). This is the smallest theory we can get that We rst start by noting some of its properties. Besides the SU(2N + 6), it also has a single non-anomalous U(1)x, and a U(1)R symmetry, which can be identi ed with the Cartan of the 6d SU(2)R. Under this U(1)R the bifundamentals have charge 0, the antisymmetrics have charge 2, and all other elds have charge 1. This is identi ed with the 6d R-symmetry as T r(R) = T r(R3) = 0. Therefore, the 4d global symmetry matches that expected from 6d. Next we can compare anomalies. We have already noted that the 6d R-symmetry anomalies match. We next wish to check the other anomalies. By comparing the anomalies with the expressions (2.3){(2.6) we nd that these agree if we take z = 12 and U(1)x = 5Naively, we cannot close the tube on itself since the two punctures are di erent. It should be noted that ux 14 can be accommodated if we allow central uxes in SU(2N + 6). It is interesting if the associated 4d theory can be found. (4.1) HJEP06(218)5 (N + 1)U(1)F . The rst identi cation in particular suggests that the ux associated with it is indeed 12 . As explained in appendix C, even though it is fractional, this ux is the minimal one possible without introducing central uxes. We next consider the dynamics of this theory. We begin by looking for the superconformal R-symmetry of this theory. It should be written as U(1)SRC = U(1)6Rd + U(1)x. Performing a-maximization we nd that: = p5N + 1 3(N + 1)p2(N + 2) ; as expected from (2.7) and the mapping of the symmetries. We can next analyze the ow to the IR and inquire what is the behavior of the theory at the deep IR. The rst issue that arises is that computing the functions one nds that the USp(2N ) group is asymptotically free, but the SU(N + 1) group is IR free. Therefore, naively we may expect the SU(N + 1) group to become free at the deep IR, but one has to be careful since the USp(2N ) group do ow to strong coupling where the behavior of the function of the SU(N + 1) group may change. To deal with this issue we consider rst the same system, but without the N = 1 vector multiplet of the SU(N + 1) group, that is the same system in gure 4 (b) but with the SU(N + 1) ungauged. The analysis we performed changes in two ways. First we do not have the contribution of the vector multiplet. This shifts a, but does not a ect the a-maximization. The second change is that we are now no longer constrained by anomaly cancellation involving this SU(N + 1) group. Thus, in principle we may have new U(1)'s which can then mix with the R-symmetry changing the behavior of the ow. However, in this case we have no U(1)'s consistent with the superpotential and USp(2N ) anomaly cancellation so we conclude that the mixing for this theory is still given by (4.1). Now we note the following observations. First one can see that all gauge invariant operators are above the unitary bound. Thus, it is plausible that this theory ows to an SCFT in the IR. We note that T r(U(1)SRC SU(N + 1)2) = SU(N + 1) does not break U(1)SRC implying that it is a conformal gauging. We expect that gauging the SU(N + 1) will give a new SCFT. We identify this SCFT with the IR theory (N + 1) so gauging the that the quiver in gure 4 (b) ows to. We next wish to explore the operator spectrum of this theory. Naturally it will be di cult to do so for any N . However we shall rst present our expectations based on 6d reasoning and then compare them against the eld theory. Finally we shall analyze a speci c example, N = 2, in detail using the superconformal index. Let us here comment of what is expected from a comparison of operators between six and four dimensions. The idea is that upon compacti cation from six to four dimensions local operators in four dimensions can be traced to local operators in six or to surface operators wrapping the compacti cation Riemann surface. For similar e ects one can consider compactifying four dimensional theories on a circle with monopole operators in three dimensions coming from line operators in four dimensions. There are some natural low dimension operators in six dimensions which we can try and identify in four. For general values of genus and uxes we expect these operators will be also the lowest operators in four dimensions with the surface operators giving rise to four dimensional operators with high charges. However, for low genus, and torus discussed here is of that kind, we do expect that surface operators will contribute to local operators with low charges. In particular we might not even see some of the operators related to six dimensional local operators in index computations as they, in principle, can combine with the ones coming from surface operators to form unprotected multiplets. We do however expect to nd a spectrum of low dimension operators which resembles the naive expectations from six dimensions by just studying local operators. Finding such a pattern is another check of our statements, and again we stress that deviations from the pattern are expected. Let us also mention that such deviations are observed, conjecturally for same reasons, also in compacti cations of class S theories. It will be very interesting to understand this issue in detail. We will compare the spectrum of four dimensional operators to expectations from six dimensional operators and nd the expected similarity along with the unsurprising deviations. So rst we ask what operators do we expect from 6d. As discussed in section 2, the 6d SCFT contains two basic operators charged under the 6d global symmetry, one in the adjoint and the other in the spinor. Naively we expect that these should contribute operators in the 4d theory. We can easily read o their charges. First, as U(1)6Rd is just the Cartan of SU(2)R, the operators coming from the adjoint of SO(4N + 12) should have the non conformal (six dimensional) R charge two in four dimensions, while those coming from the spinor should get R charge N + 1. Their charges under the global symmetry could also be read o by decomposing the SO(4N + 12) into their U(1) SU(2N + 6) representations as given in the appendix A. As previously mentioned when decomposing the 6d spinor state we have two di erent choices. These lead to a di erent operator spectrum in 4d. We shall now argue by studying the spectrum of the 4d theory that the spinor here decomposes like the S spinor decomposition as stated in appendix C.6 But rst let us consider the states in the adjoint of SO(4N + 12). These should give as operators in the adjoint of SU(2N + 6), as well as ones in the antisymmetric. The adjoint SU(2N + 6) operators just come from the triangle, that is the invariant made from a U Sp avor, an SU avor and a bifundamental. These are in the singlet and adjoint of SU(2N + 6) and are marginal operators in the 4d theory. We can also have an invariant made from two avors of the U Sp group. This is a marginal operator under the 6d R-symmetry, but not under the superconformal one as it is charged under U(1)x. We can get another operator, marginal under the 6d R-symmetry, but with opposite charges, from the invariant made from two SU(N + 1) avors and two bifundamentals. These two operators form the\o -diagonal" parts in the decomposition of the SU(2N + 6) adjoint. More invariants that can be built are baryons made just from SU(N + 1) avors and antisymmetrics. Particularly we can consider an invariant made from k antisymmetrics 6The two choices also di er by ux quantization, where only this choice allows half-integer ux without breaking the global symmetry. and N + 1 2k avors. One can see that all of these have 6d R-charge N + 1, U(1)x charge 2(N + 1)(k + 1), and in the rank N + 5 + 2k antisymmetric representation of SU(2N + 6). These precisely form the representations appearing in the decomposition of the spinor with positive U(1)x charge. We can also use the anti-fundamental of SU(N + 1), constructed from a U Sp avor and a bifundamental, and the anti-antisymmetric, constructed from two bifundamentals, and construct similar invariants. These give all the representations appearing in the decomposition of the spinor with negative U(1)x charge. The middle rank N + 3 antisymmetric representation appears to be absent. So far we have identi ed states that can be linked to a 6d operator. However that p11 p 9 8 10 1 identi cation has been crude. We next want to consider one example and study the superconformal index. We shall see that the observations made before appear also in the index. For our test case we take N = 2 which is the simplest case after the E-string. For the purpose of index calculations we shall use the R-symmetry U(1)6Rd 110 U(1)x. Since 0:03, this R-symmetry is close to the superconformal R-symmetry. We then nd: I = 1 + x6(pq) 170 [45] + 2x12(pq) 190 [10] + 1 x12 (pq) 1101 [10] + x6(pq) 170 (p + q) [45] +x6(pq) 56 [120] We can see the two operators x6(pq) 170 [45] and x16 (pq) 1103 [45]. These are exactly the o -diagonal terms that appear in the decomposition of the adjoint. One can see that these contribute at order pq under the 6d R-charge. We also note that their number is given as expected from [ 29 ] (see [16] Appendix E ). There are no terms at order pq which again agrees with the expectations of [ 29 ] (and [16] Appendix E). We also have the operators x12(pq) 190 [10] and x6(pq) 56 [120] which we can identify as coming from the 6d spinor state. These are precisely the lowest order operators that appear in the decomposition of the SO(20) spinor to SU(10). One can also see that under the 6d R-charge, they contribute at order (pq) 2 as expected. We do note that we do not see the 252, which is expected at order pq. Incidentally, the number of these operators, 3 including the 252, is given also by the formula of [ 29 ] (and [16] Appendix E). Generalized tubes. In this section we discuss tubes corresponding to ux under the U(1) breaking SO(4N + 12) ! U(1) SU(r) gure 5 (a). We associate with these tubes normalization conventions in appendix C. The tubes have an SU(r) r) non-abelian global symmetry, and two non-anomalous U(1)'s that remain after closing the tubes. We identify these with the U(1) SU(r) expected from 6d. The tubes also have two SU(N + 1) global symmetries that we identify with the punctures. These tubes, thus, have two punctures of the same types. The bifundamentals connecting these puncture symmetries to the internal symmetries are identi ed with the elds M and M 0 that play a role in the gluing. 2r). All groups are SU except the ones with 2N that are symplectic, U Sp. Here r is even so that the USp(2N ) gauge group is not anomalous. For bifundamentals between SU groups we adopted a double arrow notation indicating whether the eld is in the fundamental or antifundamenal of each SU group. Like before the arrow from the SU group to itself stands for an antisymmetric chiral eld. (a) The theory corresponding to a tube with There are cubic superpotentials for every triangle and for every antisymmetric chiral coupling it to two bifundamentals. Additionally there is a quartic superpotential for the lower `triangle'. (b) Connecting two tubes leads to this theory. We can consider taking two tubes and connecting them to form a torus with z = a U(1) z = U(1)m 1. The resulting theory is shown in gure 5 (b). We associate this theory to SU(r) 2r) preserving compacti cation on a torus with The visible symmetries in the Lagrangian are SU(r) SU(2N + 6 r) U(1)y. We also have an R-symmetry, identi ed with the Cartan of the 6d SU(2)R, with the elds charged as in the basic tube. The anomalies of this theory matches the ones computed from 6d with z = 1 if we identity U(1) = (N + 1)U(1)m and 1 2 . ux ux (4.3) as expected from (2.7). Like in the previous case we have the problem that the SU(N + 1) groups are IR free. However the USp(2N ) groups are asymptotically free in the range of r. It is therefore possible that the ow of the SU(N + 1) groups is changed along the ow of the USp(2N ) groups. We can test this in a similar way as in the previous case. Here we further have SO(4N + 12 y (N+1) [2N + 6 2r) ! U(1)y 1 r]SU(2N+6 r) + y (N+1) [2N + 6 r]SU(2N+6 r). We next analyze some dynamical aspects of this theory, starting with the superconformal R-symmetry. It is not di cult to see that only U(1)m can mix with the 6d R-symmetry. Thus we set U(1)SRC = U(1)6Rd + U(1)m, where we nd: r) such that [4N + 12 are related as stated in gure 5. the complication that there are quartic superpotentials that are irrelevant in the UV. We shall rst ignore this issue and then return to it later. We consider the theory where the SU(N + 1) groups are global symmetries and inquire where such a theory ows to. One can show that relaxing the anomaly cancellation condition of the two SU(N + 1) groups, we have an additional U(1) rotating the two SU(N + 1) r) bifundamentals with opposite charges. This U(1) does not mix with the R-symmetry so the theory still has mixing given by (4.3). Particularly compared to that R-symmetry the SU(N + 1) groups have zero function and so gauging them does not initiates a ow. This leaves the question of the nature of the theory we ow to. Examining the operators dimension we nd that for r N + 1 all gauge invariant operators appear to be above the unitary bound so an SCFT is plausible. However, when r < N + 1 the USp(2N ) mesons go below the unitary bound. This can be attributed to the fact that the conformal window for a USp(2N ) gauge theory with 2Nf fundamental chiral elds ends when Nf < 3(N+1) . 2 We can then perform Intriligator-Pouliot duality [30] to get to a description with IR free USp(r 2) groups shown in gure 6. Recall that the theory has a superpotential coupling the SU(N + 1) antisymmetric and SU(r) bifundamental to the USp(2N ) mesons. After the duality where the mesons get promoted to basic elds, these become mass terms, and many of these elds get integrated out. After the dust settles we nd that the SU(N + 1) groups sees 2N + r + 2 e ective avors. Thus, for r N + 1, the SU(N + 1) are IR free and the USp(r 2) group asymptotically free exactly like the dual description. However when r < N + 1 the groups reverse roles in this description: the USp(r 2) groups become IR free while the SU(N + 1) groups become asymptotically free. This means that gauging the SU(N + 1) groups is a relevant deformation in this range and will initiate a ow that may change the behavior of the U Sp groups. gure 5 (b) when r = 2. Here there is a cubic superpotential along the upper triangle and an order N + 2 along the lower triangle. Additionally there are two singlet elds that ip the baryons made from the bifundamentals, which is represented by the `X' drawn on the bifundamentals lines. We can proceed to analyze this theory in a somewhat similar manner. We rst start by ungauging the U Sp groups. In this case we do not get any new symmetries so the theory ows as before. As noted when r < N + 1 some operators hit the unitary bound. In this description these are the SU(r) antisymmetric chiral elds that ip some of the USp(r 2) mesons. It is thus likely that these decouple and become free elds in the IR. We can proceed and try to analyze the theory while treating these elds as free. The result of the a maximization will now be di erent and in general is quite ugly. To simplify we can check some special cases. For instance in the rst non-trivial case, r = N , we explicitly evaluated the R-symmetry, under the assumption that only the SU(r) antisymmetric chiral elds become free, and we have not found operators violating the unitary bound. Therefore, at least for one case it is plausible that the theory ows to an SCFT plus free elds. In general we expect that as r decreases the dimension of operators decreases and that other operators may hit the unitary bound. We have thus also checked the r = 4 case, where we again do not nd any unitary bound violating operators. The r = 2 case is somewhat special as in this case we do not have U Sp gauge groups in the dual description, and the dual theory simpli es to the theory in gure 7. Now there are no IR free groups, but there is an irrelevant superpotential of the form FLBN FR associated with the lower triangle, as well as the irrelevant ipping superpotential. The later appears to stay irrelevant along the ow as the two ipping elds go below the unitary bound suggesting that they actually decouple in the IR. The former, however, appears to be marginal as turning it o does not lead to additional symmetries that can mix with the R-symmetry. We do gain an extra SU(2) symmetry that should appear as an accidental symmetry at some point on the conformal manifold. Since there are two bifundamentals, and so more than one such kind of superpotential, we expect that some of these marginal operators should be exactly marginal. We can also consider analyzing the operator spectrum assuming only the ipping elds decouple. Again we do not nd any other operator going below the unitary bound. So to conclude the discussion on the dynamics of these theories, we seem to nd no contradiction with them owing to an SCFT when r > N + 1 and to an SFT plus r(r 1) free elds when r N + 1. { 23 { Finally we return to the issue of the quartic superpotentials. We can attempt to address both these and the IR free gauge couplings by turning o these superpotentials in addition to ungauging the SU(N + 1) gauge groups. In this case we do get an additional U(1) that mixes with the R-symmetry so that the SU(N + 1) r) bifundamentals have free R-charge, as these become decoupled free elds once both the SU(N + 1) gauge couplings and superpotentials are turned o . We can again perform a maximization, while regarding the SU(N +1) SU(2N +6 r) bifundamentals as free elds. The results are again quite messy so we shall not write them down. However, one can show that with respect to that R-symmetry either the superpotential is relevant and the gauging is irrelevant or vice versa, with the only exception being r = N + 1 where they are both marginal. It seems that when r > N + 1 the gauging is relevant while the superpotential is irrelevant, and vice versa for r < N + 1. Since turning any one of them breaks the additional U(1) that mixes with the R-symmetry, as long as one of them is relevant the theory should ow to the interacting SCFT with the mixing as in (4.3). The r < N + 1 is somewhat complicated by the fact that the some mesons of the USp(2N ) groups become free. In that case we have seen that the theory is easier to analyze from the dual frame where the SU(N + 1) groups are asymptotically free while the USp(2N ) groups are IR free. In that frame the quartic superpotential become cubic, and so we do not expect any modi cation from the results of the previous analysis. So to conclude, we see no indication that the quartic superpotentials decouple in a way that will modify the previous conclusion. Next we want to analyze some aspects of the spectrum. Particularly we will look at the various BPS states in the theory and try to nd states expected to match those in 6d. Naturally, this will be somewhat crude and we shall later try to study some cases more explicitly using the superconformal index. Nevertheless we shall see that there are operators one can naturally identify with those expected from 6d reasoning. We expect to nd 4d operators coming from both the 6d conserved current and the spinor state. Here the 6d global symmetry is U(1) SU(r) expect the index to form characters of SO(4N + 12 2r). Let's start with the conserved current multiplet. It should contribute operators with 6d R-charge 2 and with charges as worked out in appendix A. We can indeed identify these in the 4d theory. First we have the SU(r) r) gauge invariant bifundamentals we can build from the SU(r) USp(2N ), USp(2N ) SU(N + 1) and SU(N + 1) r) bifundamentals. This indeed has 6d R-charge 2, U(1)m charge N + 1, in the fundamental of SU(r) and in the yN+1 [2N + 6 1 r]SU(2N+6 r) and yN+1 [2N + 6 r]SU(2N+6 r). Looking at the mapping expected from anomalies, these match the operator expected from (FSU(r); VSO(4N+12 2r))1. We also have the conjugate state from the SU(r) SU(N +1) and SU(N +1) SU(2N +6 r) bifundamentals connected through two USp(2N ) SU(N + 1) bifundamentals. The states in the antisymmetric of SU(r), coming from the SO(4N + 12) conserved current, can be identi ed with the USp(2N ) mesons associated with the SU(r) bifundamental. The conjugate is given from the USp(2N ) `meson' generated from the SU(r) SU(N + 1) and SU(N + 1) USp(2N ) bifundamentals. This completes the states that we see from the 6d conserved current. with the case of N = 2, r = 8. From (4.3) we see that U(1)SRC = U(1)6Rd For the purpose of index calculation we shall employ the R-symmetry U(1)6Rd which is reasonably close to the superconformal one since superconformal index we nd: p22 9p13 1 9 9pp2123 U(1)m. 19 U(1)m, 0:03. Evaluating the We can also see some of the states associated with the spinor. For instance we can r 2 consider the baryons, from the right SU(N + 1), group made from a SU(r) bifundamentals, b SU(2N + 6 r) SU(N + 1) bifundamentals and c antisymmetrics, where a + b + 2c = N + 1. All of these have 6d R-charge N + 1, U(1)m charge (N + 1)( 2r a), U(1)y charge (N + 1)(N + 3 b) and are in the rank r a antisymmetric of SU(r) and rank 2N + 6 r b antisymmetric of SU(2N + 6 r). Here a and b run from 0 to N + 1 or r for a and 2N + 6 r for b depending on which one is smaller. Also due to the constraint a + b + 2c = N + 1 the even or oddness of a and b are correlated. These states exactly match some of the states expected from the spinor. We also have the baryons from the left SU(N + 1) group, and the baryons made from the conjugate antisymmetric, made from two SU(N + 1) bifundamentals, and antifundamentals we get from the USp(2N ) fundamentals and USp(2N ) SU(N + 1) bifundamentals. The latter is forced to give the conjugate representations due to the chiral ring relations enforced by the superpotentials. Finally we shall examine the superconformal index for some selected cases. We start where we use [SU(2); SU(2)y; SU(8)] and 2SU(2)y = y3 + y13 . As can be seen from (4.4) the index can indeed be written in characters of the global symmetry expected from 6d, SU(2) SU(2)y SU(8) U(1)m. Furthermore, all terms appearing in (4.4) have a natural 6d origin as expected from the reasonings of [ 29 ] (see also [16] Appendix E) and the branching rules in appendix C. Speci cally, the terms m6(pq) 32 [1; 1; 28] and m3 [2; 2; 8] are the ones expected from the adjoint state, while the others are the ones expected from the spinor. We also note that the conserved current contributions exactly cancels against that of the marginal operators again in accordance with the 6d expectations. We next consider the case of N = r = 2. This case is easiest to approach from the dual description in gure 7. From (4.3) we see that U(1)SRC = U(1)6Rd However, using this the two ipping elds have R-charges below the unitary bound so these must decouple. Performing the a maximization again, under the assumption that 9 2 p11 U(1)m. p these are free elds, we now nd U(1)SRC = U(1)6Rd 158 U(1)m. With this R-symmetry all gauge invariant operators have R-charges above the unitary bound. We next proceed to evaluate the superconformal index. For that, it is convenient to work with the R-symmetry U(1)6Rd 14 U(1)m, instead of the superconformal one. Note that since 158 1 4 R-symmetries are quite close. Also since the two singlets decouple in the IR, we shall ignore 0:03, both them in this calculation. Evaluating the superconformal index we nd: I = 1 + m3(pq) 85 [2; 16] + 3 m6 (pq) 4 + ( [3; 1] +m3(pq) 89 [1; 1280] + 1 m3 (pq) 181 [2; 16] + 10 m12 (pq) 23 : : : ; 4 2 5 1)pq + m3(pq) 8 (p + q) [2; 16] 3 5 m6 (pq) 4 (p + q) + m6(pq) 4 ( [3; 135] + [1; 120] [3; 1]) (4.5) where here we write the index in characters of the global symmetry expected from 6d, SU(2) SO(16) U(1)m. We have ordered the character has [SU(2); SO(16)] and we have: Some of the operators appearing above have a 6d interpretation. Particularly the rst term is the one expected from the 6d conserved current multiplet. This multiplet 1 is also expected to give two singlets contributing as m6(pq) 4 , which are exactly the two ipping elds that we ignored in this calculation. We also have the term m3(pq) 89 [1; 1280] which matches part of the contribution expected from the spinor. The U(1)m independent contribution from the spinor is not observed, similarly to as in the previous cases. The remaining terms are either product of lower operators or ones that have no immediate 6d origin. Here, unlike the previous cases, the conserved currents and marginal operators do not cancel exactly. Particularly we have a marginal operator in the 3 of the SU(2) that is not expected from 6d. As a result the structure of the conformal manifold deviates from the 6d expectations. This operator comes from the superpotential term involving two bifundamentals and one SU(3) SU(2) avor from each of the SU(3) gauge groups, where the SU(2) contraction is done symmetrically. As we have two bifundamentals there are three di erent choices for this superpotentials. Two are canceled against the conserved currents of the two SU(2) that would be there were the superpotentials turned o , leaving only the single contribution seen in the index. What we want to stress in this analysis is that this extra contribution is related to the existence of the two bifundamentals which is a property of the low ux and so is not generic and we expect the deviation with the 6d expectation to vanish once the ux increases. Finally we wish to analyze a case where r = N + 1. Since r must be even, the simplest new case is N = 3, r = 4. This case is easiest to analyze from the dual picture in gure 6, where the theory is manifestly free. We can evaluate the superconformal index for this theory. We note that here there are 12 elds, ipping the mesons of the global SU(4), that are decoupled. For simplicity we shall ignore them in the calculation. We nd: 2 2 I = 1 + (pq) 3 m8 [6; 1] + m4 [4; 16] + : : : : (4.6) Here we have written the index in characters of the global symmetry expected from 6d, SU(4) U(1)m. We have ordered the character has [SU(4); SO(16)] and we have: 16SO(16) = y48SU(8) + y14 8SU(8). We have performed the analysis up to order pq whose contribution is found to be vanishing. Of the two states appearing in the index, the last one can be identi ed as coming from the 6d conserved currents. The other state is eliminated from the chiral ring by the singlets, which provide the second contribution coming from the 6d conserved currents. We can consider gluing tubes with di erent ux together to obtain tubes with uxes in more than single U(1). This amounts to splitting the internal symmetries in the appropriate manner. When glued together we can compute anomalies of the models and compare them with six dimensional compacti cations on a torus. We do nd agreement though the details become rather cumbersome. We just write down the tube in gure 8 which one gets by gluing tubes with two di erent uxes. If one considers gluing two such tubes together the N + 1 2N N + 1 2N N + 1 2N +6+l ux. One is with r = l and the other one with l + . The lines will follow the obvious orientation pattern from gure 5. superconformal R symmetry is obtained by maximizing, 3 32 a(s; h) = 3ls3(3l + 2N 2) + 18l hs2 +6 h3(3 + 2N 2) 9 lsh2 + 4ls(5N + 1) 8 h(5N + 1) : (4.7) Acknowledgments We would like to thank Patrick Je erson for useful discussions. We also like to thank SCGP summer workshop 2017 for hospitality during part of this work. The research of HK and CV is supported in part by NSF grant PHY-1067976. GZ is supported in part by World Premier International Research Center Initiative (WPI), MEXT, Japan. The research of SSR was supported by Israel Science Foundation under grant no. 1696/15 and by I-CORE Program of the Planning and Budgeting Committee. HK is also supported in part by the National Research Foundation of Korea (NRF) Grant 2018R1D1A1B07042934. A S gluing Let us comment here on gluing punctures of opposite sign. The punctures break the SO(4N + 12) symmetry to SU(2N + 6) U(1). One can de ne the color of the puncture as the choice of the embedding of SU(2N + 6) in SO(4N + 12). The punctures come with operators M charged under puncture symmetry and SU(2N + 6) U(1). Punctures of opposite sign have operators in conjugate representations of these symmetries. We can glue punctures of di erent sign. The procedure is called S gluing, see for example [13] for de nition in related set ups. To do so we introduce gauge elds for the puncture symmetry and turn on superpotential M M 0 coupling the operators coming from the glued punctures. We can de ne tubes with punctures of opposite signs by conjugating all representations. It is natural to associate to these tubes opposite ux. One can then perform a consistency check by gluing two such tubes to a general model T. The theory then should be the same as the model T because the value of ux does not change and the surfaces are the same after and before the addition of the tubes. We can verify that indeed the above occurs. 2N M M0 M00 2N + 6 M 2N + 6 N + 1 M00 2N + 6 2N I N + 1 I 2N M0 M00 M00 N + 1 I I 2N M 2N + 6 M 2N + 6 2N + 6 N + 1 M00 2N + 6 2N M00 N + 1 2N N + 1 M0 M00 M0 M00 2N I I I I The reason is that when two tubes of opposite ux are combined the superpotentials and matter content are cNon+s1istent with triggering a vacuum expectation value which Higgses the gauge group. There are two cases we can consider. One is when we combine the tube models of opposite ux with SU(N + 1) gauging and other with USp(2N ) gauging. In both cases the mesonic operators (in U Sp case they are in (N + 1) (N + 1) representation of two SU(N +1) groups, and in case of SU they are in 2N 2N of two U Sp groups) obtain vacuum expectation values, Higgsing the group following which the theory ows back to the model T. In the latter case the vacuum expectation is triggered by quantum e ects following from analysis of the U Sp gauge theory [30], and in the former the superpotential with the elds in antisymmetric and the bifundamentals is needed to trigger the ow. For example in case of N = 2 the SU(3) gauging has ve fundamental avors, as the antisymmetric is anti-fundamental here. Without the superpotential there is no vacuum expectation value generated, but with it the claim is that it will be. One piece of evidence in favor of this is that such a vacuum expectation value, inthe presence of the superpotential, is not forbidden by symmetries. It would be interesting to understand whether it is actually turned on by instanton e ects or not, and we conjecture for consistency of our picture that it is. This is also consistent with ux domain walls in the 5d theory discussed in section 3. A puncture is de ned by boundary conditions for the 5d hypermultiplets. Di erent boundary conditions de ne di erent puncture types. S gluing corresponds to a special gluing: gluing two punctures with opposite boundary condition (or opposite sign). The gluing in this case is performed by gauging the diagonal avor symmetry of two puncture symmetries which are either SU(N + 1) or USp(2N ). The resulting gauge theory connecting two punctures can be considered as the 5d conformal matter theory with SU(N +1) or USp(2N ) gauge group on an interval with opposite boundary condition for hypermultiplets at two ends. Due to the opposite boundary conditions, the 5d hypermultiplets become massive which can e ectively be described by the superpotential M M 0 above for a hypermultiplet = (M; M 0). This gluing for example occurs when we connect two domain walls with opposite ux like the second combination in (3.7) for SU(N + 1) or like when r = 0 for USp(2N ) in the second chamber of the ux wall con gurations discussed in section 3. When we put the 5d theory with this domain wall con guration on a tube, while preserving maximal puncture symmetry, the 5d theory reduces to the 4d tube model introduced above with either SU(N + 1) gauging or USp(2N ) gauging. Since two domain walls have opposite ux, the combination should give a trivial domain wall with zero ux. This implies the triviality of the 4d tube models with opposite ux in the 4d reduction. Mathematically the equality of indices of T with the two tubes and T without them is due to the (A; C) and (C; A) inversion formulas of Spiridonov and Warnaar [33]. In the case of U Sp gauging the formula has a proof where in the case of SU gauging it was conjectured. Consistency of our picture is a physical motivation for such a conjecture to hold and it is related to the physical conjecture here of generating the vacuum expectation value by quantum e ects. B A ne quivers and duality taster As we mentioned in the discussion of the reduction of the (D; D) conformal matter to ve dimensions in addition to the SU and U Sp gauge theory description one can obtain a gauge theory with SU(2)N gauge theory. The quiver theory turns out to take the form of the a ne Dynkin diagram of DN+3.7 In fact this is the description which generalizes to (ADE; ADE) conformal matter and we will discuss this in detail in a forthcoming publication [17]. All these di erent descriptions can be used to construct theories corresponding to torus compacti cations with ux. In particular with same value of ux they should give equivalent conformal eld theories in four dimensions. Let us here discuss a particular example of such an equivalence with the details of the derivation and generalizations postponed to [17]. The claim is that torus compacti cations of (DN+3; DN+3) minimal conformal matter with ux breaking the SO(4N + 12) symmetry to SO(2N + 10) U(1) SU(N + 1) can be obtained by gluing together the Wess-Zumino model of gure 11. This theory corresponds to ux of 1=(N + 1) in the U(1) direction. The gluing is performed by gauging the SU(2)N symmetry and adding bifundamental elds forming the a ne Dynkin diagram of DN+3, introducing bifundamental elds i;i+1 for i = 1; : : : ; N 1 and two elds in fundamental of the last ( rst) SU(2) and in fundamental of the rst (second) SU(4). In plain words gluing is identifying the edges of two glued Wess-Zumino models. Let us consider constructing a torus with ux one. To do so we need to glue N + 1 copies of the Wess Zumino model, see gure 12. There are two di erent cases, when N 7Naively the quiver appears to be linear and not shaped like a D type Dynkin diagram. However, when the nodes in the center of the diagram are SU(2) then the edge nodes need to be \SU(1)". From the study of brane webs and partition functions, it appears that these \SU(1)" factors should be interpreted as two fundamental hypermultiplets for the SU(2) they are connected to. Thus, the 4 avors at the two ends play the role of the edge nodes. See section 2:2 in [18] and references there in. 4 chiral elds ipping baryonic operators as in gure 3. is SU(4) 10) computed to be, is even and when N is odd. For N even some of the rank of the symmetry is broken and we need to combine 2(N + 1) copies to preserve all symmetry. In the case of N odd we preserve all symmetry.8 Note that the symmetry one can see explicitly in the quiver SU(4) U(1)2N . We claim that this symmetry enhances to U(1) SO(2N + SU(N + 1). The conformal R symmetry is the free one and the central charges are a = (5N + 1)(N + 1) 6 c = (11N + 4)(N + 1) 12 : (B.1) This agrees with the choice of r = N + 1 for the U(1) ux and with the charges of the model of gure 5. Note that the theory in that gure and the one discussed here look rather di erent but we claim that they should be dual. That would be a non trivial consequence of di erent, but dual in certain sense,9 ve dimensional reductions of the (DN+3; DN+3) minimal conformal matter. All the anomalies match between them and also match the computation from integrating anomaly polynomial from six dimensions, and also one can verify that indices match in examples. Let us quote the computation of the N = 3 example. The supersymmetric operators form representations of SU(4) U(1). Some basic operators can be read o the quiver easily. For example, the ipping operators form two copies of the rank two antisymmetric of SU(N + 1) times a singlet of SO(2N + 10), 2( N(N+1) ; 1). The quadratic gauge singlets in a single R charge one bifundamental eld build (N + 1; 2N + 10) . The operators stretching between the ends of the quiver build two copies of the spinor 2N+4. 2 8This is related to the di erence in quantization between even and odd r = N + 1 discussed in section 2. Particularly for odd r, this ux is not consistent unless one also introduces a central ux in the Z2 center of SO(2N +10) that acts on the two spinors. This central ux exists in the tube for any r, but when connecting r such tubes, is canceled out for r even. There is also a central ux in the ZN+1 center of SU(N + 1) which causes the breaking of some of that symmetry for torus compacti cation when the number of tubes is not an integer multiple of N + 1. 9The theories in models. ve dimensions are completed in the UV by the six dimensional theory on circle. They di er by mass deformations of that completion. That is the sense in which one can think of them as dual N + 1 4 a1 a2 2 2 2 2 elds are charged under U(1)t symmetry. The ipped elds have charge one. The un- ipped elds get charge minus a half, and the ipping elds charge minus two. The two SU(4) SO(6) symmetries together with N 1 SO(2)si symmetries build SO(2N + 10). We have QlN=+11 al = 1 and this gives us SU(N + 1). The symmetry which has the ux is U(1)t. The six dimensional R symmetry of the ip elds is two, of the ipped elds is zero, and of all other elds is one. In the case of N = 3 the index is using the conformal R symmetry and standard de nitions of the index, 1 2 2 1 + 2(6; 1)t 2(pq) 3 + t 4(Sym2(2 6); 1)(qp) 3 + t 1(4; 16)(qp) 3 2 +t 1(4; 16)(q + p)(qp) 3 + t 4(2(6; 1)) (2(6; 1))(q + p)(qp) 3 2 1 4 +2(6; 1)t 2(q2 + p2)(qp) 3 + t 8(Sym4(2 6); 1)(qp) 3 t(4; 16)(qp) 3 4 +t 5(Sym2(2 6) 4 4 4 4; 16)(qp) 3 + 2(1; 128)t 2(qp) 3 + (Sym2(4; 16))t 2(qp) 3 4 + 3t4(1; 1)(pq) 3 4 t 22(4; 1)(qp) 3 + : : : The boxed terms are generators and the rest are products and derivatives. Let us de ne the characters, (4; 1) = (1; 2N + 10) = (sjRslR) 1 + (sjLslL) 1 j : (B.2) N+1 X aj ; 2 j=1 N 1 X s 2 j=1 Here sL(sR) parametrize the left (right) SU(4). We also want to mention a connection of the torus models to models obtained from M5 branes probing A type singularity which can be deduced by observing our results and 4 X j6=l 2 M5 branes on Zz(N+1) (DN+3 , DN+3) comparing them to [ 9 ]. The statement will be that the models we nd here, (DN+3 ; DN+3) conformal matter on a torus with ux z for a particular U(1) symmetry with r = N + 1 is, up to ip elds, the same as two M5 branes probing Z(1+N)z singularity on sphere with two maximal and N + 1 minimal punctures. See gure 13. It will be very interesting to understand whether there is an M/F theory explanation of such a relation. C Branching rules In this appendix we summarize some branching rules that are useful in the study of compati cations of conformal matter with ux. These will be for the group SO(4N + 12) which is the global symmetry of the SCFT. In what follows we shall write the decomposition of the vector, adjoint and two spinors. The adjoint and spinors are useful as they appear in the 6d SCFT. The vector decomposition is mostly convenient to derive the decomposition of the characteristic classes. Throughout this section we shall use the following notations for representations: V stands for the vector representation of an SO group, F; F for the fundamental and anti-fundamental representations of an SU group respectively, rank i AS for the antisymmetric representation of rank i, Ad for the adjoint representation, and S; C for the two spinors of a D2n group. SU(2N + 6). First we consider the SO(4N + 12) ! U(1) SU(2N + 6) breaking. Here we have: 1 VSO(4N+12) ! FS1U(2N+6) + F SU(2N+6); AdSO(4N+12) ! Ad0SU(2N+6) + (rank 2 AS)2SU(2N+6) + (rank 2N + 4 AS)SU2(2N+6) + 10; SSO(4N+12) ! X i= (N+3); i even (rank N + 3 + i AS)iSU(2N+6); (C.1) (C.2) (C.3) CSO(4N+12) ! i= (N+3); i odd (rank N + 3 + i AS)iSU(2N+6): (C.4) Note that the minimal charge here is normalized to 1 and it is in the vector and C spinor. In the adjoint and S spinor the minimal charge is 2. Since in 6d we have only an adjoint and one of the spinors, in one compacti cation, where the spinor we have in 6d decomposes as the S spinor, the ux can be half-integer without relying on fractional uxes. In the special case of N = 2, the complete decomposition is: 20 ! 101 + 10 1; 190 ! 990 + 452 + 45 2 + 10; 512 ! 10 4 + 120 2 + 2520 + 1202 + 104; 5120 ! 1 5 + 45 3 + 210 1 + 2101 + 453 + 15: SO(4N + 12) ! U(1) SU(r) SO(4N + 12 2r). For the generic case we have: VSO(4N+12) ! (FSU(r); 1)1 + (F SU(r); 1) 1 + (1; VSO(4N+12 2r))0; AdSO(4N+12) ! (1; AdSO(4N+12 2r))0 + (AdSU(r); 1)0 + (1; 1)0 SSO(4N+12) ! (1; SSO(4N+12 2r)) r2 + (FSU(r); CSO(4N+12 2r)) r2 +1 + (rank 2 AS; SSO(4N+12 2r)) r2 +2 + : : : ; CSO(4N+12) ! (1; CSO(4N+12 2r)) r2 + (FSU(r); SSO(4N+12 2r)) r2 +1 + (rank 2 AS; CSO(4N+12 2r)) r2 +2 + : : : ; + (rank 2 AS; 1)2 + (rank r 2 AS; 1) 2 + (FSU(r); VSO(4N+12 2r))1 + (F SU(r); VSO(4N+12 2r)) 1; (C.10) Note that when r is even the spinors of SO(4N + 12 2r) are real and we have two distinct self conjugate spinors, but for r odd then CSO(4N+12 2r) = SSO(4N+12 2r), which is re ected in the decomposition. It is interesting to note that here there are states with charge 1 also in the adjoint. When r is even this is indeed the minimal charge. However when r is odd the minimal charge is 12 which appears in the spinor. Therefore, if we adopt the uniform charge normalization given by equation (C.9), then the ux is quantized so as to be half-integer, integer or even integer for the cases of U(1) SU(2N + 6) with spinor decomposing like S, U(1) SU(2N + 6) with spinor decomposing like C or r even and r odd, respectively. Finally we shall write the decomposition for special cases that appear in this article. First for N = 2, we consider the closely related cases of r = 8 and r = 2. 20 ! (2; 2; 1)0 + (1; 1; 8)1 + (1; 1; 8) 1; 190 ! (1; 1; 1)0 + (3; 1; 1)0 + (1; 3; 1)0 + (1; 1; 63)0 + (1; 1; 28)2 + (1; 1; 28) 2 + (2; 2; 8)1 + (2; 2; 8) 1; (C.5) (C.6) (C.7) (C.8) (C.9) (C.11) (C.12) (C.13) + (1; 2; 70)0 + (2; 1; 56)1 + (1; 2; 28)2 + (2; 1; 8)3 + (1; 2; 1)4; 5120 ! (2; 1; 1) 4 + (1; 2; 8) 3 + (2; 1; 28) 2 + (1; 2; 56) 1 + (2; 1; 70)0 + (1; 2; 56)1 + (2; 1; 28)2 + (1; 2; 8)3 + (2; 1; 1)4: Here we have ordered the global symmetry as (SU(2); SU(2); SU(8))U(1). 20 ! (2; 1)1 + (2; 1) 1 + (1; 16)0; 190 ! (1; 1)0 + (1; 1)2 + (1; 1) 2 + (3; 1)0 + (1; 120)0 512 ! (1; 128)1 + (1; 128) 1 + (2; 1280)0; 5120 ! (1; 1280)1 + (1; 1280) 1 + (2; 128)0: Here we have ordered the global symmetry as (SU(2); SO(16))U(1). Finally we consider the case of N = 3, r = 4. SO(24) ! U(1) SU(4) SO(16). 24 ! 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Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir. D-type conformal matter and SU/USp quivers, Journal of High Energy Physics, 2018, 58, DOI: 10.1007/JHEP06(2018)058