Compactifications of 5d SCFTs with a twist

Journal of High Energy Physics, Jan 2017

We study the compactification of 5d SCFTs to 4d on a circle with a twist in a discrete global symmetry element of these SCFTs. We present evidence that this leads to various 4d \( \mathcal{N} \) = 2 isolated SCFTs. These include many known theories as well as seemingly new ones. The known theories include the recently discovered rank 1 SU(4) SCFT and its mass deformations. One application of the new SCFTs is in the dual descriptions of the 4d gauge theory SU(N ) + 1S + (N − 2)F . Also interesting is the appearance of a theory with rank 1 and F 4 global symmetry.

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Compactifications of 5d SCFTs with a twist

Received: September Compacti cations of 5d SCFTs with a twist Gabi Zafrir 0 1 Open Access 0 1 c The Authors. 0 1 0 32000 , Haifa , Israel 1 Department of Physics , Technion 2 Israel Institute of Technology We study the compacti cation of 5d SCFTs to 4d on a circle with a twist in a discrete global symmetry element of these SCFTs. We present evidence that this leads to various 4d N = 2 isolated SCFTs. These include many known theories as well as seemingly new ones. The known theories include the recently discovered rank 1 SU(4) SCFT and its mass deformations. One application of the new SCFTs is in the dual descriptions of the 4d gauge theory SU(N ) + 1S + (N 2)F . Also interesting is the appearance of a theory with rank 1 and F4 global symmetry. Brane Dynamics in Gauge Theories; Field Theories in Higher Dimensions 1 Introduction 2 Z2 twist on the SU(N )2 U(1) 5d SCFT The N = 2 case and related theories The general case The N = 2; k = 3 case and related theories The N = 2, general k case and related theories Dualities for SU with symmetric matter Twisted compacti cation of TN theories Superconformal index Z3 twisted TN and related theories Example 1: the T2 theory Example 2: the T3 theory Example 3: the T4 theory Example 4: an A4 case Example 5: an A5 case Example 1: the T2 theory Example 2: the T3 theory Example 3: rank 1 E7 theory Example 1: the N = 2; k = 3 case Example 2: the N = 2; k = 4 case Example 3: the N = 3; k = 3 case Z2 twisted TN and related theories U(1) SCFT and related theories A The Hall-Littlewood index B Indices for 5d T4 theory and its mass deformations B.1 T4 theory U(1)3 theory In recent years there has been an increased interest in the compacti cation of higher dimensional eld theories in order to better understand lower dimensional ones. The most notable case being compacti cation of the 6d (2; 0) theory on a Riemann surface initiated of these theories. A nice feature of this construction is that it naturally leads to ArgyresSeiberg type dualities [2] which are manifested as di erent pair of pants decomposition of the same Riemann surface. This motivate the studying of compacti cation of other higher dimensional eld theories. One possibility is to study the compacti cation of 6d (1; 0) SCFTs with its richer selection of possible theories. Indeed this has been recently studied for selected types of 6d (1; 0) SCFTs [3{7]. Instead in this article we wish to concentrate on a di erent route, the compacti cation of 5d SCFTs on a circle. The existence of 5d SCFTs with 8 supercharges has rst been noted in [8]. These provide UV completions to various 5d gauge theories which are non-renormalizable as the inverse gauge coupling squared has dimension of mass. One can interpret the gauge theory as the low-energy description of the SCFT perturbed by a mass deformation identi ed with the inverse gauge coupling squared. Interestingly, the gauge theory seems to contain considerable information about the UV SCFT such as its BPS spectrum, where the massive states are realized as instantons in the gauge theory which are particles in 5d. Therefore we shall sometimes drop the \low-energy" term and simply refer to these as gauge theory descriptions of the UV SCFT. In general the dynamics of instanton particles play an important role in the UV completion of the gauge theory. A nice example for this is given by the phenomenon of enhancement of symmetry where the xed point has a larger global symmetry than the low-energy gauge theory. From the SCFT point of view the extra symmetry is broken by the mass deformation. The mass deformation itself can be identi ed as a vev to a scalar in a background vector multiplet associated with a global symmetry whose Cartan remains as a symmetry of the gauge theory. This symmetry is the topological symmetry whose conserved currents is the instanton number, jT = Tr(F ^ F ). The broken symmetry is manifested in the gauge theory by the appearance of additional conserved currents whose origin is these instantonic particles. Five dimensional SCFTs can in turn be studied by embedding them in string theory. This can be conveniently achieved using brane webs [9, 10] where the SCFT is realized as the low-energy theory on a group of 5-branes in type IIB string theory. For example consider the web shown in gure 1 (a). The low-energy theory living on the two D5-branes is an SU0(2) gauge theory.1 The mass deformation associated with the SU(2) coupling 1The subscript denotes the value of the SU(2) angle [11]. We shall employ this to denote the for U Sp group or Chern-Simons level for SU groups. When denoting gauge theories we shall also use F for matter in the fundamental representation, AS for matter in the antisymmetric representation and S for matter in the symmetric representation. When SO groups are involved we use V for matter in the vector representation. When writing quiver theories, we use the notation G1 G2 : : : where it is understood ing in the gauge theory to the inverse gauge coupling squared. (b) Taking the g12 ! 0 limit leads to the web describing the 5d SCFT. (c) The web for g12 < 0. Note that performing S-duality leads us back to the original theory so this limit also has a low-energy description as an SU0(2) gauge theory. constant is visible as the distance between the two pairs of (1; 1) and (1; 1) 5-branes. Also visible are the BPS spectrum of the theory. For example F-strings represent the W-boson while D-strings represent the instanton particles. We can consider taking the g12 ! 0 limit in the web. This leads to the web in gure 1 (b) where all the 5-branes intersect at a point. Now there is no mass scale in the problem so the theory living on the 5-branes is an SCFT. All mass parameters in 5d are real and can be both positive and negative. Particularly this means that we can deform the SCFT in a di erent way corresponding to the negative of the deformation associated with g12 . This is shown in gure 1 (c). Note that the resulting low-energy theory is identical to the original as can be seen by performing an S-duality. So we see that deforming this 5d SCFT by a positive deformation leads to an SU0(2) gauge theory with coupling g12 , while doing a negative deformation leads to an SU0(2) gauge theory with coupling g12 . This phenomenon, where di erent mass deformations of the same 5d SCFT can lead to di erent low-energy gauge theory descriptions is called a 5d duality. For the SU0(2) case, this is a self-duality yet there are many other known examples where di erent gauge theories are related in this way [9, 12{17]. One can now study the compacti cation of 5d SCFTs on a circle to 4d. This has been previously explored for various SCFTs in [5, 18, 19]. It can be used to realize various isolated non-Lagrangian 4d theories of the type considered in [1]. The 5d SCFT lift of these theories generally have a low-energy gauge theory description and so can be studied by conventional means. Also in some cases, the 4d Argyres-Seiberg dualities lift to 5d dualities between two low-energy gauge theory descriptions of the same 5d SCFT [15]. This then allows studying these dualities via conventional techniques. When compactifying a theory on a circle one can impose various twists under symmetries of the theory. For example, holonomies under continuous global symmetries are generally incorporated where in supersymmetric theories they lead to various mass parameters. In the case of compacti cation of 5d SCFTs these complete the real mass parameters that there is a single bifundamental hyper associated with every . to study the compacti cation where we perform a twist by a discrete symmetry. That is we consider compacti cation of 5d SCFTs on a circle imposing that upon traversing the circle the theory is transformed by a discrete element of its global symmetry group. We shall concentrate on 5d SCFTs with a brane web description, particularly the ones whose non-twisted reductions were discussed in [5, 18]. The 5d SCFT is a strongly coupled non-perturbative beast so direct evaluation is usually not possible. Instead, as common in this eld, we shall start by examining various simple cases, and by studying their properties, conjecture the resulting 4d theories, which in the cases at hand turns out to be isolated 4d SCFTs. This is then subjected to a variety of consistency checks. Once the simpler cases are understood, we can use them to study more general cases where we do not have a candidate 4d theory. This then suggests that this compacti cation leads to a variety of unknown 4d theories. We further study some of their properties and perform various consistency checks on our conjectures. The structure of this article is as follows. In section 2 we discuss twisted compactication of the 5d SCFT represented in string theory by the intersection of N coincident propose identi cations for the resulting theories among known class S theories. We then test these identi cations by studying dualities and mass deformations of these theories. We then move on to the general case where we conjecture the resulting theories to be new isolated SCFTs. We employ various dualities to study their properties and to serve as consistency checks. One application for this is to study the duality frames for 4d SU(N ) gauge theory with symmetric matter and N In section 3 we move on to study the twisted compacti cation of the 5d SCFT represented in string theory by N coincident 5-brane junctions. We consider two di erent twists one under a Z3 discrete element and one under a Z2 one. In the Z3 case we rst examine various low N cases identifying these with various known 4d SCFTs. Interestingly one of these is the recently discovered rank 1 SU(4) SCFT found in [20]. Besides providing an additional string theory construction for this theory, by examining its mass deformation, we get string theory constructions also for other rank 1 SCFT generated by mass deforming the SU(4) SCFT, originally introduced in [21]. We suspect the general case to lead to unknown isolated 4d SCFTs and we comment on some of their properties. The Z2 twist is more mysterious with a variety of seemingly unknown 4d theories including one with rank 1 and F4 global symmetry. In section 4 we use the known Hall-Littlewood index for class S theories [22{24] and properties of the compacti cation to conjecture an expression for the Hall-Littlewood index for the theories we presented. This is then checked for the cases where the conjectured 4d theory is known. We end with some conclusion. Appendix A gives a short review of the Hall-Littlewood index. Appendix B discuses aspects of 5d index calculations for the 5d T4 theory and related theories that have interesting application to the rank 1 SU(4) SCFT and related theories. Z2 twist on the SU(N )2 U(1) 5d SCFT We start by considering the 5d SCFT engineered in string theory by the intersection of N D5-branes and k NS5-branes, shown in gure 2. This 5d SCFT has an SU(N )2 U(1) global symmetry (enhanced to SU(2N ) SU(2)2 when k = 2 or SU(2k) by performing S-duality on the web in gure 2. It has two convenient low-energy gauge theory descriptions. One is given by an N F + SU(N )k 1 + N F quiver and is generated via a mass deformation breaking the SU(k)2 global symmetry. Alternatively performing a mass deformation breaking the SU(N )2 global symmetry leads to the low-energy quiver gauge theory kF + SU(k)N 1 + kF [12]. This theory has a Z2 Z2 discrete symmetry (enhanced to the dihedral group D4 when interested in the element which simultaneously exchanges the two SU(N ) and SU(k) groups, given in the web by a rotation in the plane. In both gauge theory descriptions it is given by a combination of charge conjugation and quiver re ection. In this section we shall investigate the 4d theory resulting from circle compacti cation of this 5d SCFT with a twist in this discrete element. In other words we reduce the 5d SCFT on a circle where we enforce that upon traversing the circle the theory return to itself acted by this discrete element. This is an interesting twist to consider as it can be naturally implemented in a brane construction of the SCFT. For this let's consider what a rotation of the web entails in string theory. First this includes a rotation of the spacetime plane where the web lives. This can also be interpreted as a re ection of the two coordinates spanning the plane. We shall call this operation I45. In addition the rotation of the web changes the charges of the 5-branes. Particularly, a (p; q) 5-branes is mapped under this operation to a ( p; q) 5-branes. Thus, in addition to the spacetime re ection I45, we must also perform the SL(2; Z) transformation I = This in turn is equal to ( 1)FL where is worldsheet parity and FL the left moving spacetime fermion number. Therefore a rotation of the web can be implemented in string deformation of the SCFT illustrating the 2F + SU(2) SU(2) + 2F gauge theory description. (b) The S-dual web now illustrating the SU0(3) + 6F gauge theory description. theory by the operation I45 ( 1)FL . The twisted compacti cation we consider then can be implemented by compactifying a direction common to all branes and enforce that upon traversing the circle we return to the system acted by I45 ( 1)FL . In fact this type of compacti cation is just a generalization of the Dabholkar-Park background [25]. This speci c twisted compacti cation was actually studied in [26] which considered the T-dual con guration (see also [27, 28]). They found that performing Tduality on the circle leads to type IIB on the dual circle in the presence of an O6+ and O6 -planes. In particular this also shows that this compacti cation preserves the same supersymmetry as an O7-plane, and so applying this twisted compacti cation on a brane web should lead to a 4d system with N = 2 supersymmetry. The N = 2 case and related theories We wish to begin by presenting some simple examples before discussing the general case. The N = 2; k = 3 case and related theories gure 2. It has an SU(2)2 SU(6) global symmetry and two convenient gauge theory descriptions, one being 2F + SU(2) SU(2) + 2F and the other SU0(3) + 6F . Like the other cases, it also has a Z2 discrete symmetry given in the web by a plane. This is identi ed with quiver re ection in the 2F + SU(2) SU(2) + 2F theory and charge conjugation in the SU0(3) + 6F theory. Now we want to consider compactifying it on a circle with a twist involving this Z2 discrete symmetry. We inquire as to what theory we get in 4d. This theory should have a 1-dimensional Coulomb branch as only one of the two Coulomb branch dimensions of the 5d SCFT is symmetric under this Z2 discrete symmetry. The two SU(2)2 are mapped to one another so only the symmetric combination survives. The SU(6) global symmetry should be broken to the Z2 invariant part which is USp(6). This can be seen as follows: we consider the action of the Z2 as mapping the two SU(3) subgroups of SU(6) with charge conjugation, as suggested by the web. Under the U(1) (1; 8)0 + (3; 3)2 + (3; 3) 2 + (1; 1)0. Under the Z2 action the adjoints of both SU(3) groups are mapped to one another so we get only one SU(3). The (3; 3) is projected to conserved currents 80 +62 +6 2 +10 which builds the adjoint of USp(6). Thus the resulting 4d SCFT should have an SU(2) USp(6) global symmetry. Further we can use the web to calculate the Higgs branch dimension of the resulting SCFT. The Higgs branch dimension is given by the number of possible motions of the 5-branes along the 7-branes. The 5d SCFT has a 12 dimensional Higgs branch. This is manifested in the web by the directions given by: breaking the 2 D5-branes on a D7-brane both on the left and the right of the web (this gives 1 direction for each side), breaking the 3 NS5-branes on the (0; 1) 7-branes both on the top and the bottom of the web (this gives 3 directions for each side), and nally separating the remaining 2 D5-branes and 3 NS5-branes along the 7-branes (this gives 4 directions, one for each brane modulo a global translation). To nd the Higgs branch dimension for the 4d theory resulting from the twisted compacti cation we must limit the counting to those motions invariant under the Z2 discrete symmetry. As the two sides are mapped to one another, the motion on both sides are identi ed. This leaves determining whether there are constraints when separating the 5branes along the 7-branes which is the direction xed by the orbifolding. In other words, we need to determine if it is consistent to have a brane mapped to itself. For this we performing T-duality which maps this con guration to a group of NS5-branes, D4-branes and D6-branes in the presence of an O6+ and O6 -planes. There is no impediment to separating the NS5-branes along the O6-planes. Also we can tune parameters so as to have the D4-branes sit on top of the O6+-plane where they can be separated along it. Thus, we conclude that we can separate 5-branes along this orbifold. We can now count all the possible breakings consistent with the Z2 discrete symmetry, where we nd: 1 direction from breaking the 2 D5-branes on a D7-brane simultaneously on both sides of the web, 3 directions from breaking the 3 NS5-branes on the (0; 1) 7-branes simultaneously on the top and bottom of the web and 4 directions from separating the remaining 2 D5-branes and 3 NS5-branes along the 7-branes. This gives an 8 dimensional Higgs branch. There is an isolated rank 1 4d SCFT with an SU(2) USp(6) global symmetry and an 8 dimensional Higgs branch [29]. Thus, the natural conjecture is that the preceding compacti cation leads to this theory. Next we wish to test this conjecture. As one piece of evidence for this conjecture, we shall show that we can recover 4d dualities involving the SU(2) USp(6) SCFT from the 5d construction. Consider gauging both SU(2) global symmetries of the 5d SCFT in gure 3 with the same coupling g5d so as to preserve the Z2 discrete symmetry. This leads to the SCFT shown in gure 4 (a). Now let's reduce this SCFT with the Z2 twist while taking the limit R ! 0, g5d ! 0 keeping gR52d xed. R1 the 5d theory is e ectively described by weakly gauging the SU(2)2 global symmetry of the 5d SCFT in gure 3 by two SU(2) gauge groups with identical couplings Let's consider rst doing the reduction in the R g52d > 0 limit. At energy scales of SCFT should reduce to the proposed 4d SU(2) USp(6) SCFT. So we see that in this limit E we get a 4d theory. Under the twist the two SU(2) gauge gauging both SU(2) global symmetries. (b) A mass deformation of the SCFT corresponding to the limit gSU2(2) ! 1 where gSU(2) is the coupling constant of both edge SU(2) gauge groups. (c) The mass deformation corresponding to the limit gSU2(2) ! 1 where we have also performed S-duality on the web. That this deformation is a continuation of the previous one is apparent as it preserves the U(6) global symmetry. In this limit the SU(2) quiver description is inadequate, but there is a di erent description as an SU0(5) + 6F gauge theory where this limit corresponds to gSU2(5) ! 1 for gSU(5) being the coupling constant of SU(5). double gauging an SU(3) + 1F into the SU(6) global symmetry. (b) A mass deformation of the SCFT corresponding to the limit gSU2(3) ! 1 where gSU(3) is the coupling constant of both edge SU(3) gauge groups. (b) The mass deformation corresponding to the limit gSU2(3) ! have also performed S-duality on the web. That this deformation is a continuation of the previous one is apparent as it preserves the U(2)2 U(1)2 global symmetries associated with the semi-in nite 5-branes. In this limit the SU(3) quiver description is inadequate, but there is a di erent description as an 2F + SU0(4) SU0(4) + 2F gauge theory where this limit corresponds to gSU2(4) ! 1 for gSU(4) the coupling constant of both SU(4) gauge groups. the resulting 4d theory is an SU(2) gauging of the SU(2) global symmetry of the proposed Consider approaching this limit from a di erent direction given by g52d < 0, shown in gure 4 (c). Now the SU(2) description is inadequate and we should switch to a di erent description of the SCFT given by performing S-duality on the brane web. In this description we have an SU0(5) + 6F gauge theory with coupling g52d > 0. We now ask what happens to this theory under the twisted reduction. The twist should project the SU(5) to SO(5) and the 6F to the 3V . This follows from the global symmetry as well as the Higgs branch analysis, both agreeing with the 4d gauge theory SO(5) + 3V . Thus, we arrive at the 4d gauge theory SO(5) + 3V with g42d pled. In fact this is a conformal theory with a marginal parameter g42d. So we see that we can compactify the same 5d SCFT in the same limit getting di erent weakly coupled descriptions in di erent ranges of the marginal parameter gR52d . This implies that these two theories are dual. That the 4d SU(2) USp(6) SCFT obeys such a duality is indeed known [29]. the limit gSU2(2) ! 1. (b) A di erent mass deformation, shown in the S-dual frame, corresponding 2 to the limit gSU(3) ! 1. (c) The 5d SCFT we get after a mass deformation corresponding to avor on both sides. There is another way to generate a 4d conformal theory by gauging part of the global symmetry of the SU(2) USp(6) SCFT which we can directly implement in the web. This is done by gauging the USp(6) global symmetry with an SU(3) + 1F gauge theory. On the 5d SCFT this should lift to a double SU(3) + 1F gauging of the SU(6) global symmetry. The resulting 5d SCFT is shown in gure 5 (a) where we have also shown the SCFT in the two limits of g5d ! 0 for g52d > 0 in gure 5 (b) and g52d < 0 in gure 5 (c). When g52d > 0 the SU(3) group is weakly coupled and we expect the 4d theory to be a weakly coupled SU(3) + 1F gauging the SU(2) 2 USp(6) SCFT. However when g5d < 0 the 5d SCFT is more appropriately described by an 2F + SU0(4) SU0(4) + 2F gauge theory with equal g52d. The two groups are identi ed under the Z2 symmetry. The identi cation and charge conjugation implies that the bifundamental matter should decompose to a symmetric and an antisymmetric of SU(4) of which only the symmetric is Z2 invariant. Therefore, when reduced to 4d, we expect to get a weakly coupled SU(4) + 1S + 2F gauge theory. Again these are two description of the same theory in di erent limits of a marginal operator and so describe a duality. This duality indeed appears in [29]. Mass deformations. As a nal piece of evidence we can consider mass deformations of this SCFT. For example, consider the deformation of the 5d SCFT shown in (a). This mass deformation breaks the SU(6) part of the global symmetry to SU(2)3, but leaves the SU(2)2 part unbroken. It is most convenient to describe this using the SU(2) + 2F gauge theory description where it corresponds to taking gSU2(2) ! 1. Reducing with the Z2 we see that the resulting 4d theory should be SU(2) + 1S + 2F which is an IR free gauge theory. Therefore the 4d SU(2) USp(6) SCFT, resulting from the Z2 twisted compacti cation of the 5d SCFT in gure 3, should have a mass deformation leading to the 4d gauge theory SU(2) + 1S + 2F . Recently the mass deformations of the USp(6) SCFT were analyzed using the Seiberg-Witten curve in [30] who found that it indeed possess such a mass deformation. We can also consider another mass deformation, shown in gure 6 (b), now breaking the SU(2)2 global symmetry while preserving the SU(6). This one is most conveniently addressed from the SU0(3) + 6F description where it corresponds to the limit gSU2(3) ! 1. Again the previous discussion leads use to conclude that the resulting 4d theory is SO(3) + 3V which is again IR free. Such a mass deformation of the SU(2) was also found in [30]. double gauging both SU(2) subgroups of its SU(4) global symmetry. Note that we have performed S-duality compared to the web shown in 6 (c). (b) A mass deformation of the SCFT corresponding to the limit gSU2(2) ! 1 where gSU(2) is the coupling constant of both edge SU(2) gauge groups. (c) The mass deformation corresponding to the limit gSU2(2) ! 1 where we have also performed S-duality on the web. That this deformation is a continuation of the previous one is apparent as it preserves the U(1)4 global symmetry associated with the semi-in nite 5-branes while breaking the SU(2)2 associated with the semi-in nite (1; 1) 5-branes. In this limit the previous description is inadequate, but there is a di erent description as an 1F +SU1(3) SU 1(3)+1F gauge theory where this limit corresponds to gSU2(3) ! 1 for gSU(3) the coupling constant of both SU(3) gauge groups. by a mass deformation. It has two gauge theory descriptions as an SU (2) SU (2) quiver gauge theory and an SU0(3) + 2F one. (b) A di erent brane web of another 5d SCFT generated from the one in gure 6 by the opposite mass deformation. It has a gauge description has an SU0(2) SU0(2) quiver gauge theory. There is an additional mass deformation we can consider, given in the web by integrating a avor on both sides. This leads to a new 5d SCFT, shown in gure 6 (c), with gauge theory descriptions of 1F + SU(2) SU(2) + 1F and SU0(3) + 4F . It has an SU(4) global symmetry so we expect the 4d theory to have USp(4) U(1) global symmetry. There is indeed a mass deformation of the SU(2) USp(6) SCFT with that pattern of symmetry breaking leading to an isolated 4d SCFT with USp(4) U(1) global symmetry [30]. It is natural to identify the resulting 4d SCFT with this theory. We can test this in the same spirit as the previous tests. First we can compute the dimension of the USp(4) U(1) SCFT. Second we can gauge various global symmetries and study the resulting dualities. In this case we can gauge an SU(2) USp(4) which indeed gives a 4d conformal theory. The two interesting limits of this gauging are shown in gures 7 (b)+(c). The limit of gure 7 (b) describes a double weak gauging of the USp(4) deformation of the SCFT illustrating the 3F + SU(2) SU(2) + 3F gauge theory description. (b) The S-dual web now illustrating the SU0(3) + 8F gauge theory description. SCFT by an SU(2) gauge group, while gure 7 (c) describes an SU(3) + 1S + 1F gauge theory. This suggests that these are dual as was discovered in [29, 31]. We can consider taking an additional mass deformation given by integrating an additional avor. We can get to two di erent 5d SCFTs depending on the sign of the mass deformation. The rst shown in gure 8 (a) has an SU(2) U(1)2 global symmetry and gauge theory descriptions of SU (2) SU (2) and SU0(3)+2F . The second, shown in gure 8 (b), has an SU(4) global symmetry and gauge theory description of SU0(2) SU0(2). When reduced to 4d with a twist these should lead to 4d theories with global symmetries of SU(2) and USp(4) respectively. Examining mass deformations of the USp(4) U(1) SCFT we nd two natural candidates for these theories: SU(2) + 1S + 1F for the SU(2) and SU(2) + 2S for the USp(4) theory. These are IR free gauge theories. We can further test this by comparing the dimension of the Higgs branch nding complete agreement. It is also interesting to consider the 5d SCFT we can get by adding avors to the 5d gauge theories in When reduced with a twist this should lead to a 4d theory with a mass deformation leading to the SU(2) USp(6) SCFT. Particularly consider the 5d SCFT shown in gure 9. It has an SU(10) global symmetry (see [17, 32, 33]) and two convenient gauge theory descriptions, one being 3F + SU(2) SU(2) + 3F and the other SU0(3) + 8F . It also has the Z2 discrete symmetry so we can compactify it on a circle with a twist under it. We expect this to lead to a rank 1 4d theory with USp(10) global symmetry. There is indeed a rank 1 isolated SCFT with USp(10) global symmetry, rst found in [29]. Furthermore this SCFT indeed has a mass deformation leading to the USp(6) SCFT [30]. We can also compute the Higgs branch dimension The N = 2, general k case and related theories In this subsection we generalize the previous discussion by the addition of NS5-branes. Like in the previous case, we can propose a known 4d SCFT as the result of the twisted compacti cation and test this using dualities. deformation of the SCFT illustrating the 2F + SU(2)k 1 + 2F gauge theory description. (b) The S-dual web now illustrating the SU0(k) + 2kF gauge theory description. gauging both SU(2) global symmetries. (b) A mass deformation of the SCFT corresponding to the gSU2(2) ! 1 where gSU(2) is the coupling constant of both edge SU(2) gauge groups. (b) The mass deformation corresponding to gSU2(2) ! 1 where we have also performed S-duality on the web. That this deformation is a continuation of the previous one is apparent as it preserves the U(2k) global symmetry. In this limit the SU(2) quiver description is inadequate, but there is a di erent description as SU0(k + 2) + 2kF where this limit corresponds to gSU2(k+2) ! 1 for gSU(k+2) the coupling constant of SU(k + 2). Consider the 5d SCFT shown in gure 10. It has an SU(2)2 SU(2k) global symmetry and two convenient gauge theory descriptions given by 2F + SU(2)k 1 + 2F and SU0(k) + 2kF . Reducing this 5d SCFT to 4d with a twist, we expect a 4d SCFT with an SU(2) USp(2k) global symmetry. Further, we can gauge the SU(2) global symmetry and consider xed limit, in two di erent regimes of gR2 . We nd one describes a weak SU(2) gauging of the aforementioned SCFT (see gure 11 (b)) while the other describing a weak SO(k + 2) + kV gauge theory (see gure 11 (c)). There is indeed a known duality of this form [34], leading us to identify the SU(2) appearing in these dualities with the one resulting from the twisted compacti cation. In the k even case, this theory can be constructed by the compacti cation of a D type 6d (2; 0) theory with twist [34] which allows determining its properties. We can perform some consistency checks on this identi cation. First we can calculate the Higgs branch dimension of these SCFTs from the web. These can be compared against the class S result for even k and against what is expected from the duality for general k nding complete agreement. Another check we can do is to consider gauging a part of the USp(2k) global symmetry and so consider a di erent duality. One option is to gauge it with an SU(k) + (k gauge group which is a conformal gauging. We shall consider this duality in the next plane here shown after it has been resolved to a (1; 1) and (1; 1) 7-branes. We can proceed by pulling out the 7-branes arriving at the con guration on the right, where a number next to a 7-brane stands for the number of 5-branes ending on that 7-brane. 1. Reducing with a twist leads to the theory shown below where we use grayed Young diagrams to represent twisted punctures, and black arrows to represent gauging the appropriate symmetry of the shown class S theory. Further we use two arrows as both USp(2k) groups, associated to the two punctures, are gauged. (b) This limit corresponds to gUS2p(2k) ! reducing it with a twist leads to the theory shown below. section when we discuss the general case. When k is even we can also consider a double gauging of the USp(2k) global symmetry by a USp(k) group which leads to an interesting duality.2 In the web this can be performed by adding an O7 plane and then resolving it as shown in gure 12. This leads to the duality shown in gure 13. We can now use the known properties of USp(2k) SCFT to check this duality by comparing the conformal anomalies, dimensions of Coulomb branch operators and global symmetries and their associated central charges nding complete agreement. Furthermore we can argue this duality from a class S construction, where we reduce the 6d Dk+1 (2; 0) theory on a torus with a single puncture whose associated Young diagram is shown in gure 13, being the ungrayed one on the 2There is a generalization of this that works for every k given by gauging with an SU(k) + 1AS gauge group. We will consider this in the next section. avor on both sides. to the limit gSU2(2) ! 1. (b) A di erent mass deformation, shown in the S-dual frame, corresponding 1. (c) A 5d SCFT we get after a mass deformation corresponding to theory we conjecture results from reducing the 5d SCFT in (a) with a Z2 twist when k is even. bottom left theory. In addition we add a Z2 twist in the outer automorphism of Dk+1 on one of the cycles of the torus. We then get both theories in the bottom of gure 13 as di erent pair of pants decompositions of this Riemann surface (see also [31] for an example of this type of dualities for the twisted A (2; 0) theory). Finally we can also use this to study mass deformations of these SCFT. For example, gure 14 suggests that they should have a mass deformation, breaking the SU(2) global symmetry, that leads to the IR free SO(k)+ kV gauge theory. There should also be another mass deformation, now breaking the USp(2k) global symmetry, that leads to an IR free SU(2) + 2F gauging the SU(2) global symmetry of the SU(2) 4) SCFT. It will be interesting to see if this can be veri ed by alternative means. We can also consider a mass deformation interpreted in the web by integrating a avor. This leads to the 5d SCFT shown in gure 14 (c) having a U(1)2 global symmetry. We can consider the result of reducing this SCFT to 4d with the twist where based on the previous example we expect a 4d SCFT with a U(1) global symmetry. When N is odd we can identify this theory with the class S theories introduced in [31]. One evidence for this is that the dimension of the Higgs branch agree. We shall give an additional piece of evidence in section 2:2:1. the theory shown below. gure 2. (a) This limit corresponds to gUS2p(2l) ! 1. Reducing with a twist leads to the theory 1 and reducing it with a twist leads to Finally we can consider the generalization of the USp(10) theory by the addition of NS5-branes. The 5d SCFT, shown in gure 15 (a), has an SU(2k) global symmetry so reducing it with a twist leads to a 4d SCFT with USp(2k) global symmetry. When k is even we can naturally identify it with the class S theory shown in 15 (b). This is supported as the global symmetry, dimension of the Coulomb branch and dimension of the Higgs branch all agree. Further we can again consider double gauging the USp(2k) global group with a USp(k) gauge group leading to the duality similar to this of gure 13 after forcing the two 5-branes to end on the same 7-brane. In the class S description this corresponds to changing the puncture with SO(3) symmetry to the minimal puncture. The rest works out exactly as in the duality of gure 13 so we won't elaborate on it. The general case In this section we turn to analyzing the general case. Particularly, we consider the compacti cation of the 5d SCFT whose brane description is given in gure 2. We wish to study its compacti cation to 4d with the Z2 twist. We shall argue that this leads to an isolated 4d SCFT. The basic tool we use to study this is the dualities of the type considered in the global symmetry by USp(2l). The two interesting limits of this gauging are shown in gures 16 (a)+(b). These suggest the duality shown in the lower part of gure 16. An important feature here is that the right side of the duality is given in terms of known theories allowing us to deduce the properties of the unknown theory. For this we rely on the properties of the class S theory appearing in the duality. Particularly we require the spectrum and dimensions of Coulomb branch operators, global symmetry and central charges. the middle summarizes the spectrum and dimensions of Coulomb branch operators. These are somewhat di erent depending on whether 2N k + 2 and whether k is even or odd. In the table i stands for the dimension of the operator and di for the number of such operators present in the SCFT. Also written are the global symmetry with the central charges, Higgs branch dimension and e ective number of vector multiplets. The global symmetry written is for the N; k > 2 case, and is further enhanced to SU(2) SO(4N + 4) for k = 2 and SU(2k + 4) for U(1) SCFT resulting from the twisted compacti cation of the 5d SCFT in gure 2. These can be determined from the dualities in gures 16 and 19. The table summarizes the spectrum and dimensions of Coulomb branch operators where we have assumed that k N , the other case given by exchanging N and k. The last entry in the middle table refers to the existence of one more Coulomb branch operator, in addition to the other ones appearing in the table. Also written are the global symmetry with the central charges, Higgs branch dimension and e ective number of vector multiplets. In the global symmetry we have These can be evaluated using the methods of [35], and as these play a prominent rule in the proceeding discussion we have summarized them in gure 17. For the central charges we use the Higgs branch dimension and e ective number of vector multiplets. These can be a); nv = 4(2a From these we see that the theory on the right hand side is an SCFT with a single marginal parameter. The duality suggests the theory on the left side should also be of this type. Therefore the U Sp gauging should be conformal and the SU(k) theory should be an isolated SCFT. From the duality we can determine its properties, at least when N is even, which are summarized in gure 18. This and the resulting duality are shown in gure 19. That this describes an SU(2l + 1) + SCFT in gure 2. (a) This limit corresponds to gSU2(2N+1) ! 1. Reducing with a twist leads to 1 and reducing it with a twist leads to the theory shown below. NS5-branes, we arrive to the right con guration. 1AS gauging can be reasoned by resolving an O7 -plane with a stuck NS5-brane (see gure 20). It is instructive to argue this also in an alternative way. We can interpret the system in gure 19 as an SU(2l + 1) gauging of, on one side the SU(2l + 1) of the U(1) SCFT, and on the other the 5d SCFT shown in gure 21 (a). Perfuming a series of 7-brane motions we can map it to the one in gure 21 (b) which is of the form considered in [18]. Thus, there is a class S theory associated with this SCFT which describes an antisymmetric hyper and two fundamentals under the SU(2l symmetry manifested in the punctures. Also note that we performed a transition of the type considered in [19] so there is an additional hyper in the theory of gure 21 (a). Therefore the theory in gure 21 (a) is a collection of l(2l+1) free hypers that transform 1) + (1; 1) under the SU(2) 1) subgroup of SU(2l + 1). This can only be consistent with this theory describing a single hyper in the antisymmetric of SU(2l + 1). 7-brane motions we arrive at the con guration in (b) which is of the form of [18]. theory shown below. 2)F gauging of an SU(N ) global symmetry of the SCFT in gure 2. (a) This limit corresponds to gSU2(N) ! 1. Reducing with a twist leads to the theory 1 and reducing it with a twist leads to the We can now use the duality in gure 19 to study the SU(N ) when both N and k are odd. This is again summarized in gure 18. We can perform several consistency checks on the properties we nd. First we nd that these are indeed invariant under the interchange of N and k as suggested by the web. This is also necessary as we could have performed the same dualities by gauging the SU(k) group instead and this structure guaranties that this as well is consistent. Another consistency check we can perform is to compare the Higgs branch dimension evaluated from the web against the one expected from the duality using dH = 24(a c) where we again nd agreement. We can also compare the dimension of the Coulomb branch required from the duality against the one expected from the web where again we nd agreement. In order to perform additional consistency checks we consider other dualities. For example we can gauge the SU(N ) group with an SU(N ) + (N 2)F gauge theory which leads to a 4d conformal theory. This gives to the duality shown in gure 22. We can now test this duality by matching central charges using the properties of the SU(N ) SCFT we determined from the previous dualities, and consistency now necessitates that these agree. We indeed nd that they agree. gauge theory description of the SCFT. (c) Dualizing the SU0(3)+4F part to 1F +SU(2) SU(2)+1F we get this theory where the dashed line standing for an half-hyper in the (2; 2; 2). gure 39. (b) A gauge theory description of the SCFT. symmetry. In [21] it was found that there is indeed a mass deformation leading to a 4d theory with an SU(2) U(1) global symmetry which is expected to be yet another rank 1 SCFT. It is natural to identify the resulting 4d theory with this SCFT. There are two additional pieces of evidence for this identi cation. First it has a mass deformation leading to an SU(2) + 12 4 + 1F IR free gauge theory as expected from the U(1) SCFT [21]. Second the web suggests it has a 3 dimensional Higgs branch, which agrees with the results of [21] if one uses dH = 24(c a). We can also consider the Hall-Littlewood chiral ring of this SCFT from the 5d index similarly to the previous case. We carry this out in appendix B. Finally we can consider another mass deformation now leading to the gauge theory in gure 40 (b).5 Again this gauge theory should originate from a 5d SCFT which we identify with the web in gure 40 (a). This should have a U(1)3 global symmetry leading us to expect the 4d SCFT to have a U(1) global symmetry. The resulting theory should also have a mass deformation leading to an SU(2) + 12 4 IR free gauge theory. Indeed in [21] it was found that the SU(2) U(1) SCFT has such a mass deformation leading to an supersymmetry [38]. As a supporting evidence we note that the Higgs branch dimension is 1 which agrees with the eld theory analysis of [38]. 5Naively there should be two di erent gauge theories depending on whether the SU(2) or . Yet we seem to nd only one Z3 symmetric brane web leading us to suspect that these are identical. In fact, due to the presence of the trifundamental, exchanging two SU(2) gauge groups is e ectively seen by the third as reversing the mass of one avor and so changes its Higher N and related theories. We can now continue to theories with higher N . For Coulomb branch. It should also have a 14 dimensional Higgs branch, along which it can be reduced to the rank 1 SU(4) SCFT. We also expect the Hall-Littlewood chiral to contain the operators: 2 [24]; 4 ( [17500]+ [17500]). To our awareness, no such 4d theory is known. The preceding thus suggests that there is host of possibly unknown 4d SCFTs given by the twisted compacti cation of the 5d TN theories with a Z3 twist. We can further generate additional theories by Higgs branch ows and mass deformations. Higgs branch ows are readily visible from the brane webs. First we can pullout a group of 5-brane junctions. This initiate a ow one TN to another one with lower N . Alternatively we can break some of the 5-branes on the 7-branes. As previously discussed this can be naturally implemented by associating to the SCFT a Young diagram with N boxes. The theory given by the compacti cation of the TN theory is then represented by the one row Young diagram, while other choices giving di erent theories. We can identify some of these theories with known 4d theories. As an example of a theory in this class that we can identify, consider the rank k E6 theory whose web is shown in gure 41. It is natural to conjecture that reducing it with a twist leads to the 4d rank k SO(8) theory which is just the gauge theory USp(2k) + 1AS + 4F . Indeed this theory has 1 dimensional Higgs branch agreeing with the web. In particular the one associated with the antisymmetric breaks the theory to k copies of SU(2) + 4F agreeing with what is expected from the web. We shall see another example of a theory in this class in the next We can also consider mass deformations. As seen in the T4 example, this may lead to new SCFT as well as non-conformal theories. We suspect that this will be true also for cases with higher N so besides the theories introduced so far there should be many additional theories that are mass deformations of these. As we shall now argue some of them are related to the ones introduced and to themselves via dualities. We can consider dualities of the class of theories we introduced in the same spirit as performed in section 2. As a simple example consider the Z3 symmetric gauging of the of the T4 theory. (b) The 5d SCFT in the limit of gSU2(4) ! 1. Bellow is the 4d theory resulting from a twisted compacti cation in this limit. (c) The 5d SCFT in the limit of gSU2(4) ! 1. In this limit there is a better description as an SU(3) gauging of the 5d SCFT in (d). Bellow is the 4d theory resulting from a twisted compacti cation in this limit. three SU(4) global symmetry groups of the T4 theory by an SU(4) + 1F gauge theory. The brane web for the resulting 5d SCFT is shown in gure 42 (a). That it is Z3 symmetric is most readily visible by noting it is invariant under T S. We can consider reducing this theory to 4d with the Z3 twist and taking the scaling limit gS2U(4) ! 0; R ! 0 keeping the We can examine the reduction in two di erent limits. First we can consider the limit gure 42 (b). In this limit the SU(4) + 1F gauge theory is weakly coupled and should reduce to an SU(4) + 1F gauging of the rank 1 SU(4) SCFT. This is a conformal gauging so the result is a 4d SCFT with a single marginal operator. gure 42 (c). Now the description as an SU(4) + 1F gauging is inadequate, but there is an alternative description given by a weakly coupled SU(3) gauging of the SCFT in gure 42 (d). This SCFT in turn is given by a Z3 symmetric mass deformation of the T5 theory and should have an SU(3)3 U(1)3 global symmetry and a 6 dimensional Higgs branch. Thus, when compacti ed with a Z3 twist should give a 4d theory with an SU(3) U(1) global symmetry. The result of the twisted compacti cation in the limit of gure 42 (c) should therefore be an SU(3) gauging of this theory. Since we know from the opposite limit that this theory is an SCFT with a single marginal operator, it is quite reasonable that the SU(3) gauging is in fact conformal and that the SU(3) U(1) theory is an SCFT. Since we do not know much about the SU(3) U(1) SCFT we cannot put to much tests on this duality. Yet it is apparent that the global symmetry agrees, both having a U(1) global symmetry. Also the Higgs branch dimension calculated from the duality using dH = 24(c a) agrees with that evaluated from the web. The dimension of the Coulomb branch also agrees as the 5d construction suggests the SU(3) U(1) SCFT having a 2 dimensional Coulomb branch. a Z3 twist we expect it to lead to a 4d theory with SU(N ) The web after a Z3 symmetric gauging by an SU(N ) + (N U(1) global symmetry. (b) 3)F gauge theory of the SCFT in (a), in the limit of gSU2(N) ! 1. Bellow is the 4d theory resulting from a twisted compacti cation in this limit. (c) The same web but now in the gSU2(4) ! 1 limit. In this limit there is a better description as an SU(k + 3) + kF gauging of the 5d SCFT in (a). Bellow is the 4d theory resulting from a twisted compacti cation in this limit. We can generalize this to other cases. Consider the 5d SCFT shown in (a). It has an SU(N )3 U(1)3 global symmetry as well as the Z3 discrete deformations of the 5d TN+2k SCFT or alternatively from N mass deformations of the 5d Tk+2N SCFT. When compactifyied with a Z3 twist we expect it to lead to a 4d theory with U(1) global symmetry. We can now consider weakly gauging the global SU(N ) symmetry by an SU(N ) + 3)F gauge theory. In the 5d description this can by done by performing the SU(N ) + 3)F gauging in a Z3 symmetric manner and consider the gSU2(N) ! 1 limit. This is shown in gure 43 (b). We can now consider taking the opposite limit gSU2(N) ! shown in gure 43 (c). When reduced to 4d this leads to an SU(k + 3) + kF gauging of the From the previous cases, we expect the two theories to be weakly coupled descriptions of one conformal theory on di erent points on its conformal manifold. Therefore we conjecture that the SU(N ) SU(k) U(1) theory appearing is an SCFT and the gauging is conformal. We can perform a few consistency checks. First we can compare the global symmetries and their central charges. One can see that the global symmetries matches. To compare the central charge under the avor symmetry we use the assumption that the SU(N )+(N 5d SCFT under the interchange of N and k, this e ectively determine the central charge also for SU(k). With this central charges we must have that the SU(k + 3) + kF gauging is conformal as well as matching of the central charges of global symmetries. This is indeed obeyed. We can also compare the conformal anomaly combination c a, where we use the a), to determine this combination for the SU(N ) U(1) theory. We indeed The construction done here is quite reminiscent of the one done in section 2 and can be considered as a generalization of it to a Z3 case. Similarly to that case we can consider more general dualities like the dualities in gure 23 also in this case. As this is a straightforward application of the things discussed here we won't carry it. Unfortunately, unlike the previous case, we do not nd a duality frame with purely known theories so we cannot use this to determine their properties. In this subsection we discuss compacti cation of the 5d TN theory and related theories on a circle with a Z2 S3 twist. The Z2 discrete symmetry we twist by is given by exchanging two of the SU(N ) global symmetry groups. Together with the previously discussed Z3 element, these generate the group S3. reduces to eight free half-hypermultiplets in the (2; 2; 2) of the SU(2)3 global symmetry. It is not di cult to carry out the reduction where we nd the twisted 4d theory to be that of six free half-hypermultiplets in the (3; 2) of the SU(2)2 global symmetry. This is again visible from the dimension of the Higgs branch consistent with the Z2 symmetry being 3. Again this assumes that the direction given by separating junctions along the 7-branes is not projected out. Alternatively this can be used to argue this is true which can then be applied to the higher N cases. N = 3 case. has an E6 global symmetry and so possesses moment map operators in the adjoint of E6 which decomposes to (8; 1; 1) + (1; 8; 1) + (1; 1; 8) + (3; 3; 3) + (3; 3; 3) under the SU(3)3 subgroup of E6. Implementing the Z2 projection on these states lead us to suspect the resulting 4d theory possesses moment map operators in the (8; 1) + (1; 8) + (6; 3) + (6; 3) under the visible SU(3)2 global symmetry. This in fact span the adjoint of F4 so we conclude that we get a rank 1 theory with F4 global symmetry. This can be also inferred as the operation exchanging two SU(3) subgroups in E6 is identical to its Z2 outer automorphism. It is well known that the invariant part in E6 under this outer automorphism is F4. It is also interesting to examine the Higgs branch of this theory. From the brane web we can determine that it has an 8 dimensional Higgs branch. Interestingly this is also the dimension of the 1 instanton moduli space of localized F4 instantons. Furthermore as E6 it is naturally embedded in the localized E6 1 instanton moduli space which is the Higgs branch of the T3 theory. So it is natural to conjecture that the resulting 4d theory has this space as its Higgs branch. To our knowledge, there is no known theory possessing these properties, and so we view this as a hint for the potential existence of such a theory. It is also natural to expect it to be an SCFT as an F4 global symmetry suggests strong interactions are involved and the low rank severely limits it from having additional scale dependent coupled parts. It will be interesting to further study this and see if additional evidence for the existence of such a theory can be uncovered. Higher N and related theories. We can also consider other theories by compactifying other TN theories or theories related to them by mass deformations or Higgs branch ows. unknown. It is interesting to look for known theories among them as this can hint as to whether or not these theories exist, and if so whether they are conformal or not. As an example consider the 5d rank 1 E7 theory whose web is shown in gure 44. This has a Z2 symmetry exchanging the two maximal punctures, given in the web by the 4 external (1; 1) and NS 5-branes. From the eld theory view point this corresponds to exchanging the two SU(4) parts in the SU(2) SU(4)2 classically visible global symmetry. We can compactify this theory with a Z2 twist and inquire as to properties of the resulting 4d theory. First we ask what is the global symmetry of the theory which we try to answer by studying the moment map operators that survive the compacti cation. The 133 of E7 decomposes under its SU(2) SU(4)2 subgroup as: (3; 1; 1) + (1; 15; 1) + (1; 1; 15) + (1; 6; 6)+(2; 4; 4)+(2; 4; 4). Enforcing the Z2 projection we get: (3; 1)+(1; 15)+(1; 200)+ (2; 10) + (2; 10), under the SU(2) SU(4) global symmetry.6 These span the adjoint of E6. In addition one can see that its Higgs branch is 11 dimensional, like the rank 1 E6 theory. All of these lead us to conjecture that the resulting theory is the rank 1 E6 theory. Superconformal index In the previous section, we have argued that we can infer some of the operators in the Hall-Littlewood chiral ring of a 4d theory, resulting from the twisted compacti cation of a 5d SCFT, from information on the spectrum of operators in the 5d SCFT. In this section we shall try to make this more accurate by conjecturing an exact expression for the full Hall-Littlewood index for some of the 4d theories we considered in this article (we refer the reader to appendix A for the de nition of the Hall-Littlewood index). The Hall-Littlewood index is particularly useful for this owing to the following observations: 6In projecting the (1; 6; 6) we have taken the traceless part. Like in the previous example involving T4 this amounts to a constraint on the operator with no analogue in the full space. 1. For the theories we consider, the Hall-Littlewood index should be identical to the Hilbert series of the Higgs branch. Furthermore, the Higgs branch is invariant under quantum corrections and so also under direct dimensional reduction. Therefore, it is conceivable that the Hall-Littlewood index of the twisted theory can be generated by twisting the 4d index of the direct dimensional reduction. 2. The Hall-Littlewood index is relatively easy to compute with known expressions abundant in the literature. 3. Due to points 1 and 2 there are ample expressions in the literature for theories we consider giving us direct expressions to compare with. We consider the 4d theories resulting from the Z2 or Z3 twisted compacti cation of the 5d TN and related theories, and the Z2 twisted compacti cation of the 5d SCFTs of gure 2 and related SCFTs. The strategy takes from point 1 above, that is we use the known expression for the Hall-Littlewood index of the 4d theory resulting from direct dimensional reduction as a basis for our conjecture. The direct dimensional reductions of the theories we consider are known to be comprised of A type class S isolated SCFTs. Therefore it is useful to rst review the expressions for the Hall-Littlewood index for these types of SCFTs. The Hall-Littlewood index for A type class S isolated SCFTs was determined in [22{ 24]. These are described by a compacti cation of the AN 1 6d (2; 0) theory on a Riemann sphere with three punctures. It is given as follows: IcHlaLss S = NN Qi3=1 K( 0i(ai)) NN is an overall normalization factor given by: NN = (1 The sum is over all the partitions of N = ( 1; : : : ; N 1; 0) corresponding to irreducible representations of SU(N ). The product is over the three punctures. K( 0i(ai)) are fugacity dependent factors associated with each puncture. The exact expression for them can be found in [23]. (xi) are the Hall-Littlewood polynomials given by: (xi) = N ( ) where N ( ) is a normalization factor given by: and m(i) is the number of rows in the Young diagram 2( ) = Y i=0 j=1 i(ai) is a list of N elements whose exact form depends on the type of puncture. The procedure for determining it in the general case can be found in [23]. Z3 twisted TN and related theories We shall rst start with the Z3 twisted TN type theories considered in section 3:1. These are generated by compactifying the 5d TN type SCFT with a Z3 twist. The untwisted dimensional reduction of these theories leads to a class S isolated SCFT corresponding to the compacti cation of the AN 1 6d (2; 0) theory on a Riemann sphere with three identical punctures. Its Hall-Littlewood index, which is identical to the Higgs branch Hilbert series of the 5d SCFT, is given by equation (4.1). The twist project the operators down to their Z3 invariants so as a minimalistic assumption it should identify the three K factors and project the three Hall-Littlewood , down to one for the completely symmetric product. Thus we conjecture that the index for the twisted 4d theory should have the form: IZH3Ltwisted = NN dependent normalization factors and we use 3 to mean the partition given by (3 1; 3 2; : : : ; 3 N 1; 0). In fact we further conjecture that: N N0 = (1 0 = N 2 = Y i=0 j=1 We next support our conjecture by testing this against the cases where we can identify the resulting 4d theory with a known theory. Example 1: the T2 theory The simplest example to start with is the T2 theory. From equation (4.5) we nd: ITH2Ltwisted = i=1 (1 + 2) (i;0)( ; 1 ) where we use a for the SU(2) fugacity. This can be evaluated explicitly. Using Mathematica ITH2Ltwisted = = P E This is indeed the Hall-Littlewood index of 4 free half-hypers in the 4 of SU(2). Let's now consider the T3 theory. From equation (4.5) we nd: ITH3Ltwisted = where we span the SU(3) global symmetry as 3 = s + 1r + rs . ITH3Ltwisted = 1 + 2 ( SU(3)[8] + SU(3)[10] + SU(3)[10]) + O( 4) The 2 terms give the contribution of the conserved current supermultiplets and so should be in the adjoint of the global symmetry. Indeed these form the adjoint of SO(8) where only an SU(3) subgroup is visible. Expanding up to order 6, we nd the index naturally forms SO(8) characters where it is given by: We can compare this against the known Hilbert series of the 1-instanton moduli space of localized SO(8) instantons evaluated in [37] nding perfect agreement. Recalling that the gauge theory SU(2) + 4F as this space as its Higgs branch and the identity between the Hall-Littlewood index and the Hilbert series, we see that this agrees with our expectations. We can further calculate the complete unre ned index, that is setting the SU(3) fugacities to 1. Using Mathematica we nd: ITH3Ltwunisrteedned = 1 + 18 2 + 65 4 + 65 6 + 18 8 + 10 This indeed agrees with the unre ned Hilbert series of the 1-instanton moduli space of localized SO(8) instantons [37]. Example 3: the T4 theory Let's now consider the T4 theory. From equation (4.5) we nd: ITH3Ltwisted = c ) Expanding this in a power series in ITH3Ltwisted = 1+ 2 SU(4)[15]+ 3 ( SU(4)[84]+ SU(4)[50] ( SU(4)[140] + SU(4)[140] + SU(4)[120] + SU(4)[120] + SU(4)[2000] + SU(4)[2000]) + O( 6) = P E[ 2 SU(4)[15] + 3 SCFT with an SO(8) U(1)2 global symmetry. This agrees with the Hall-Littlewood index for the rank 1 SU(4) SCFT computed Example 4: an A4 case Consider the 5d SCFT represented by the web in gure 45 (a). This theory describes the T4 SCFT with a single free hyper [6]. We can compactify this theory to 4d with a twist where we expect to get the rank 1 SU(4) SCFT with a free hyper. We can now use this as a further test on our index conjecture, now for a case with a non-maximal puncture. Applying equation (4.5) and expanding in a power series in IAH4Ltwisted = P E which is indeed the Hall-Littlewood index of the rank 1 SU(4) SCFT with an additional Example 5: an A5 case Consider the 5d SCFT represented by the web in gure 45 (b). The punctures show an U(1)3 global symmetry though the true global symmetry is SO(8) SU(2)3 as can be inferred from the superconformal index of the associated 4d class S theory. The Z3 discrete symmetry acts by permutating the three SU(2)'s, and act on the SO(8) as the Z3 element of its outer automorphism group. We can consider the 4d theory resulting from the Z3 twisted compacti cation. Applying equation (4.5) we nd: IAH5Ltwisted = 1+ 2 1 + [3; 1]SU(2)2 + [1; 3]SU(2)2 + q6 + where we use q for the fugacity of the U(1) global symmetry. The 2 terms show the conserved currents for the SU(2)2 U(1) global symmetry visible from the web and the puncture, but in addition there are additional conserved currents spanning the adjoint of SU(2). Expanding up to 4 we indeed nd that it forms characters of G2 + [1; 770] + [1; 27] + [1; 14] + 2 [3; 7] + 4) + O( 5) where we have written it in characters of the G2 SU(2) global symmetry ordered as We can compactify it to 4d with a twist, where we expect to get a 4d SCFT with a 12 dimensional Higgs branch and a G2 SU(2) global symmetry. We can in fact nd an appropriate candidate for this theory in a known theory being theory number 19 in [20]. This theory indeed has a G2 SU(2) global symmetry and 12 dimensional Higgs branch agreeing with the expectation from the index and the web. As a consistency check we can see from the web that it should have a Higgs branch direction leading to the rank 1 SU(4) SCFT and a di erent one leading to the gauge theory USp(4)+1AS+4F . These indeed exist also for the theory we identi ed. Furthermore using the expressions in [20] we can compute the Hall-Littlewood index for it, nding it matches (4.18) at least to the order we evaluate it. Z2 twisted TN and related theories We next move on to the case of the Z2 twisted TN type theories considered in section 3:2. These are generated by compactifying the 5d TN type SCFT with a Z2 twist. The untwisted dimensional reduction of these theories leads to a class S isolated SCFT corresponding to the compacti cation of the AN 1 6d (2; 0) theory on a Riemann sphere with two identical Its Hall-Littlewood index, which is identical to the Higgs branch Hilbert series of the 5d SCFT, is given by equation (4.1). The conjectured expression now takes the form: IZH2Ltwisted = N N00 X K( 01(a1))K( 02(a2)) 2 (a1) where N N00 and N 00 are dependent normalization factors and we use 2 to mean the partition given by (2 1; 2 2; : : : ; 2 N 1; 0). The values of N N00 and N 00 are further given by: N N00 = (1 N 00 = N We next support this conjecture by testing this against the cases where we can identify the resulting 4d theory with a known theory. Example 1: the T2 theory As the simplest example let's consider the T2 theory. Using equation (4.19) we nd that the index for the Z2 twisted theory is: ITH2Ltwisted = Using Mathematica we can perform the representation sum and nd that: ITH2Ltwisted = = P E This is indeed the Hall-Littlewood index for 6 half-hypers in the (3; 2) of SU(2) SU(2). Example 2: the T3 theory Let's next consider the T3 theory. We have argued that this should lead to a 4d theory with an F4 global symmetry and an 8 dimensional Higgs branch. This can be naturally accommodated if the Higgs branch is the moduli space of localized F4 1 instantons. We now wish to apply equation (4.19) to this case. Expanding the index in a power series in , we nd: + [15; 3]+ [15; 3]+ [6; 15]+ [6; 15]+ [6; 6]+ [6; 6]+ [6; 3]+ [6; 3] + [150; 6]+ [150; 6]+ [8; 8]+ [27; 1]+ [1; 27]+ [8; 1]+ [1; 8]+1)+O( 6) where we write the index in characters of the SU(3) SU(3) global symmetry. As previously mentioned, the 2 terms, which contains the contribution of the moment map operators, form the adjoint representation of F4. Furthermore looking at the 4 terms we see that they form the 10530 dimensional representation of F4. This is in fact the rst few terms in the Hilbert series of the localized 1 instanton moduli space of F4. Furthermore, for the ubre ned index we can perform the representation summation with Mathematica ITH3Ltwisted = 1 + 36 2 + 341 4 + 1208 6 + 1820 8 + 1208 10 + 341 12 + 36 14 + 16 This is indeed the unre ned Hilbert series of the localized 1 instanton moduli space of Example 3: rank 1 E7 theory Let's consider the rank 1 E7 theory. As previously discussed compactifying this theory to 4d with a Z2 twist, we expect to get the rank 1 E6 theory. We can use this to test equation (4.19) also for a case with a non-maximal puncture. Expanding the index in a power series in , we indeed nd it forms characters of E6, and is further given by: series of the localized 1 instanton moduli space of E6 [37]. k 1 is a class S theory whose de nition is given in [15]. The theory is somewhat di erent depending N is identical to the N > k case with N and k replaced. U(1) SCFT and related theories Finally we wish to attempt to extend the conjecture for the Hall-Littlewood index also for the Z2 twisted theories we originally considered in section 2. We shall adopt a similar strategy. We rst consider the index for the 4d theory resulting when compactifying the theory without the twist. We then use this to conjecture the form for the index with the twist. Finally we test this by comparing against cases where we can identify the 4d theory with a known one. Let's rst consider the 4d theory resulting from compactifying the 5d theory of gure 2 with no twist. This was considered in [5] and they found we get the IR free theory shown in gure 46. One can see that it has two AN 1 maximal punctures and two Ak 1 ones. These correspond to the groups of parallel 5-branes and closing them corresponds to forcing some of the 5-branes to end on the same 7-brane. This essentially gives the generalization of this to other theories generated by Higgs branch ows. The 4d theory shows the Z2 Z2 discrete symmetry of the 5d SCFT, given by exchanging the two pairs of punctures.7 Particularly the Z2 that we twist by corresponds to exchanging both punctures simultaneously. We can write the Hall-Littlewood index8 of these theories as: IHL = 2 SU(k)[k2 for the N > k case and IHL = 2 SU(k)[k2 and IHL and IHkL1 are the Hall-Littlewood indices of the Tk and Tk element does not commute with the Z2 Z2 discrete symmetry and together they form the dihedral group D4. 8Since the theory is IR free by index we mean the index of the two SCFTs with the gauge invariance constraint. Alternatively we can de ne it as the Hilbert series of the Higgs branch. generalization to cases with non-maximal punctures can be done by replacing the Tk and k 1 theories by their appropriate versions. We can now conjecture a form for the Hall-Littlewood index of the twisted theory. As mentioned the twist act by simultaneously exchanging both pairs of maximal punctures. As suggested in the previous subsection, implementing the twist on each on the two class S theories convert it to a 4d theory whose Hall-Littlewood index is given by equation (4.19). replaced with their twisted cousins: IHL = 2 SU(k)[k2 for the N > k case and IHL=Z 2 SU(k)[k2 this section we test this relation by considering various examples. Example 1: the N = 2; k = 3 case As a starting example let's consider the original theory we discussed in section 2 which is theory. We wish to use this case to test equation (4.29). In the case at hand, the Z2 twisted T2 theory is just 3 free hypermultiplets while the Z2 twisted 13 theory is just the F4 theory whose conjectured Hall-Littlewood index was given in equation (4.24). Thus, we conjecture the Hall-Littlewood index for the rank 1 SU(2) USp(6) theory to be: ISHUL(2) USp(6) = 2 SUG(2)[3]]IFH4L = 1 + 2( [3; 1] + [1; 21]) + 3 [3; 140] + 4( [5; 1] + [3; 21] + [1; 1260] + [1; 90] + 1) + 5( [5; 140] + [3; 216] + [3; 140]) + O( 6) We now want to compare (4.31) with the Hall-Littlewood index evaluated from a class S construction. This theory can also be realized, though accompanied by free hypermultiplets, in class S constructions by a twisted compacti cation of an A or D type (2; 0) theory [34, 39]. We can use this to calculate the Hall-Littlewood index of this theory, though the presence of the free hypers makes a high order calculation quite consuming. Using this we indeed nd (4.31), at least to the order we evaluated it. Example 2: the N = 2; k = 4 case to the rank 2 SU(2) USp(8) theory. This theory can be constructed by a twisted compacti cation of a type A or D (2; 0) theory [34, 39]. Using the A type construction we nd: where again we write the index in characters of the SU(2) USp(8) global symmetry ordered as [SU(2); USp(8)]. We can now compare this against the conjectured expression in (4.29). Again on one side we have the Z2 twisted T2 theory which is just 3 free hypermultiplets. The other side is the Z2 twisted 14 theory which we have not previously discussed. The Z2 twisted 14 theory as rank 2 and SU(2) USp(8) global symmetry which we may be tempted to identify with the SU(2) USp(8) SCFT we are considering. Furthermore the dimension of the Higgs branch also agrees. However this theory as a Higgs branch limit leading to the F4 theory in contrary to the known SU(2) USp(8) SCFT so it must be a di erent theory. Also using (4.19) we nd the following Hall-Littlewood index: IH14Ltwisted = 1 + 2( [3; 1] + [1; 36]) + 3 [2; 42] + 4( [5; 1] + [3; 36] + [1; 330] + [1; 308] + 1) + 5( [4; 42] + [2; 42] + [2; 1155]) + O( 6) which di ers from (4.32). the conjecture (4.29) where we nd: Returning to the index computation for the SU(2) USp(8) SCFT, we can now use ISHUL(2) USp(8) = 2 SUG(2)[3]]ISHUL(2) USp(8) = 1 + 2( [3; 1] + [1; 36]) + 4( [5; 1] + [3; 36] + [3; 42] + [1; 330] + [1; 308] + 1) + O( 6) This matches the explicit expression (4.33) to the order it was evaluated. Example 3: the N = 3; k = 3 case the conjecture (4.29) we nd: ISHUL(3) SU(3) U(1)= 2 SUG(3)[8] IFH4LIFH4L = 1+ 2(1+ [8; 1]+ [1; 8])+ 3 m [6; 1]+m [1; 6]+ [1; 24]+m [15; 1]+m [1; 15] [1; 15]+m [6; 8]+m [8; 6]+ [6; 6]+2m [6; 1]+2m [1; 6]+ where we write the index in characters of the SU(3) SU(3) global symmetry. To our knowledge this theory has not been realized before so we have nothing to compare this expression to. Nevertheless, the 5d construction suggests that this theory calculating the Hall-Littlewood index on both sides and comparing. Calculating to order 5 we indeed nd they match. In this article we have explored the dimensional reduction of 5d SCFTs to 4d with a twist in an element of their discrete global symmetry. We have concentrated on 5d SCFTs with a brane web representation particularly the SCFTs given by the intersection of NS and D5branes and the 5d TN theory. We have argued that this leads to various known 4d isolated SCFTs as well as a wealth of potentially new ones. We then used the 5d description to infer various properties of these SCFTs such as their Higgs branch, mass deformations We have also used this construction to conjecture an expression for the Hall-Littlewood index for these theories. It is interesting to see if we can nd additional evidence for the existence of the 4d theories we introduced. These may also teach us more about their properties. One interesting question is whether or not these can be incorporated into the known class S construction. Alternatively it is interesting if they can be constructed by alternative means such as compacti cation of (1; 0) 6d SCFTs or geometric engineering. For instance recently [40, 41] on Calabi-Yau 3-fold singularities. It is interesting to see if the theories found in this paper can also be constructed using this method. This is especially true for the theories introduced in section 3:2 particularly the rank 1 and it is interesting if this can support or disfavor its existence. Furthermore the existence of a rank 1 F4 SCFT was suspected from the superconformal bootstrap analysis in [43, 44], which also calculated some of its central charges. It will be interesting if we can calculate these also for our proposed rank 1 F4 theory and compare with their results. An additional angle is to try to generalize these constructions to more theories. One possibility is to study other 5d SCFTs. Another possibility is to twist by other discrete symmetries. We have seen that the Z3 twist discussed in section 3:1 can be thought of as a generalization of the Z2 twist discussed in section 2. In both of these the twisted discrete element can be represented by the action of an SL(2; Z) element on the brane web, being T S for the Z3 case, and I for the Z2 case. So a possible generalization is to consider other nite subgroups of SL(2; Z), for example the Z4 and Z6 subgroups appearing in the construction of S-folds [45, 46]. In fact the connection with S-folds itself appear to warrant Yet another interesting direction is to study the compacti cation of these theories to 3d. Besides the natural interest, the resulting theories are then 5d SCFT compacti ed on a torus with a twist on one of its cycles. Alternatively we can get the same theory by taking the untwisted 4d reduction and compactifying it to 3d with a twist. These should be related by a modular transformation on the torus which could potentially lead to interesting 3d Acknowledgments I would like to thank Oren Bergman, Shlomo S. Razamat and Leonardo Rastelli for useful comments and discussions. G.Z. is supported in part by the Israel Science Foundation under grant no. 352/13, and by the German-Israeli Foundation for Scienti c Research and Development under grant no. 1156-124.7/2011. The Hall-Littlewood index 4d superconformal index is the counting of all BPS operators in the theory, annihilated by a chosen supercharge, modulo the possible merging of BPS operators to form a non BPS multiplet. It can be further re ned so as to keep track of the representations of the operators under the superconformal and avor symmetries. Ur(1). The representations are then labeled by the highest weights of Ur(1) subgroup. We label the two weights of SO(4) as j1; j2, that of SUR(2) as R and that of Ur(1) as r. I = T r( 1)F pj1+j2 rqj2 j1 r 2R+2r Y afi where p; q and are fugacities associated with the superconformal algebra, ai are fugacities associated with the various avor symmetries whose Cartan charges are given by fi. the theory. Alternatively it is given directly by the following trace formula: where T rHL denotes trace over all operators obeying: j1 = 0, E r = 0 [22]. Indices for 5d T4 theory and its mass deformations In this appendix we discuss the 5d index for the T4 theory and its Z3 symmetric mass The 5d index for the T4 theory can be evaluated from its gauge theory description as SU(2) + 2F . This was done in [15] where it was indeed shown that there are additional instantonic conserved currents that enhance the classical SU(2)2 symmetry to SU(4)2. The index also contains order 3 operators in the (4; 4; 4) and (4; 4; 4), as expected from the 4d Hall-Littlewood chiral ring. Furthermore we expect an order 4 operator in the (6; 6; 6). Breaking the enhanced SU(4)2 group into their SU(2)2 representations, we nd: I = 1 + x2 5 + [3]SU(2) + q2 + (6; 6; 6) = where we used the notation of [15]. One can see that it is made of two perturbative contributions corresponding to the operators J ijZi Zj Q Q , qiZi Q Q and their conjugates, where we use Z for the bifundamental eld and Q and q for the avors of the SU(3) and SU(2) gauge groups respectively. The rest are instanton charged states, and we have also veri ed using instanton counting methods that these exist. U(1)3 theory In this section we deal with the 5d SU(2)3 U(1)3 SCFT. In particular, we calculate the index from the 2F + SU(3) SU(2) + 1F gauge theory description. We use the fugacity allocation shown in gure 47. We calculate the index to order x3. To that order, besides the perturbative contribution, we also get contributions from the (1; 0), (0; 1) and (1; 1) instantons. We nd: One can see from the x2 terms the conserved currents of the classically visible SU(2) characters of the SU(2)3 U(1)3 global symmetry: U(1)5 global symmetry as well as additional conserved currents, coming from the (0; 1) instanton, that lead to an enhancement of U(1)2 ! SU(2)2. The index can be written in I = 1 + x2(3 + [3; 1; 1]0;0;0 + [1; 3; 1]0;0;0 + [1; 1; 3]0;0;0) [2]y 4 + [3; 1; 1]0;0;0 + [1; 3; 1]0;0;0 + [1; 1; 3]0;0;0 + [2; 2; 2]1;1;1 + [2; 2; 2] 1; 1; 1 + [2; 1; 1]1;0;0 + [2; 1; 1] 1;0;0 + [1; 2; 1]0;1;0 + [1; 2; 1]0; 1;0 + [1; 1; 2]0;0;1 + [1; 1; 2]0;0; 1 where we use the notation [d1; d2; d3]q1;q2;q3 for an operator with in di dimensional representation under SU(2)i and charge qi under U(1)i. In term of the fugacities these are [3; 1; 1]1;0;0 = [1; 3; 1]0;1;0 = [1; 1; 3]0;0;1 = We can use this to try and guess the Hall-Littlewood chiral ring of the 4d SCFT where I4HdL = 2 [3]0 + 3( [4]3 + [4] 3 + [2]1 + [2] 1) + O( 4) It will be interesting to see how the enhancement as well as the full index arise from the SU(2)3 gauge theory description, both for this theory and the T4 itself. Unfortunately, there are technical issues in performing instanton counting due to the half trifundamrntal that impedes instanton counting in these theories. Since the U(1)3 SCFT, that we get by performing another Z3 symmetric mass deformation, as only this description, understanding this will allow us to repeat this analysis also for this theory. We reserve this for future work. 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Gabi Zafrir. Compactifications of 5d SCFTs with a twist, Journal of High Energy Physics, 2017, 97, DOI: 10.1007/JHEP01(2017)097