Brane webs, 5d gauge theories and 6d \( \mathcal{N}=\left(1,\;0\right) \) SCFT’s

Journal of High Energy Physics, Dec 2015

We study 5d gauge theories that go in the UV to 6d \( \mathcal{N}=\left(1,\;0\right) \) SCFT. We focus on these theories that can be engineered in string theory by brane webs. Given a theory in this class, we propose a method to determine the 6d SCFT it goes to. We also discuss the implication of this to the compactification of the resulting 6d SCFT on a torus to 4d. We test and demonstrate this method with a variety of examples.

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Brane webs, 5d gauge theories and 6d \( \mathcal{N}=\left(1,\;0\right) \) SCFT’s

HJE Brane webs, 5d gauge theories and 6d Gabi Zafrir 0 1 0 Haifa , 32000 , Israel 1 Department of Physics, Technion, Israel Institute of Technology We study 5d gauge theories that go in the UV to 6d N = (1; 0) SCFT. We focus on these theories that can be engineered in string theory by brane webs. Given a theory in this class, we propose a method to determine the 6d SCFT it goes to. We also discuss the implication of this to the compacti cation of the resulting 6d SCFT on a torus to 4d. We test and demonstrate this method with a variety of examples. Field Theories in Higher Dimensions; Brane Dynamics in Gauge Theories - SCFT's 1 Introduction 2 Preliminaries 2.1 2.2 Properties of the 6d theories Class S technology 3.1 Generalizations 3.1.1 3.1.2 A simple example Another example: the 5d TN theory with extra avors 4 Additional 5d theories 4.1 Quivers of SU groups 4.2 SU quivers with antisymmetric hypers 4.2.1 4.2.2 SU quivers with an antisymmetric hyper at one end SU quivers with an antisymmetric hyper at both ends 4.3 Cases with completely broken groups 5 Conclusions A Instanton counting for SU(N ) + 2AS + Nf F 3 The 5d (N + 2)F + SU0(N )k + (N + 2)F quiver and related theories ries can also be realized in string theory using brane webs and geometric engineering [4{7]. The picture emerging from these methods is that 5d SCFT's exist and that they sometimes posses mass deformations leading to 5d gauge theories, with the mass identi ed as the inverse gauge coupling squared, g 2. These theories also posses some quite interesting non-perturbative behavior. One such phenomenon is the occurrence of enhancement of symmetry, in which the xed point has a larger global symmetry than that perturbatively { 1 { exhibited in the gauge theory. An important ingredient in this is the existence of a topological U(1) conserved current, jT = Tr(F ^ F ), associated with every non-abelian gauge group. The particles charged under this current are instantons. These instantonic particles sometimes provide additional conserved currents leading to an enhancement of the perturbative global symmetry. A simple example is SU(2) gauge theory with Nf hypermultiplets in the doublet of SU(2). For Nf 7, this theory is known to ow to a 5d xed point, where the global symmetry is enhanced from U(1) SO(2Nf ) to ENf +1 by instantonic particles [1]. This can be argued from string theory constructions, and is further supported by the superconformal index [8, 9]. In many cases, a single 5d SCFT may have many di erent gauge theory deformations. theory [5, 10].1 By now a great many examples of this are known, see [10{15]. String theory methods, such as brane constructions, also suggest the existence of interacting 6d N = (1; 0) SCFT's [16{18]. These theories include massless tensor multiplets, in addition to hyper and vector multiplets. The tensor multiplets contain a scalar leading to a moduli space of vacua. In some cases, the low energy theory around a generic point in this space is a 6d gauge theory, where g 2 is identi ed with the scalar vev [19]. By now, a large number of such SCFT's are known. In fact, there exists a classi cation of N = (1; 0) SCFT's using F-theory [20, 21]. See also [22] for a classi cation of N = (1; 0) gauge theories. There is an interesting relationship between 5d gauge theories and 6d N theories, where, in some cases, a 5d gauge theory has a 6d N = (1; 0) UV completion. The best known example is 5d maximally supersymmetric Yang-Mills theory, which is believed to lift to the 6d (2; 0) theory [23, 24]. Yet another notable example is the 5d gauge theory with a USp(2N ) gauge group, a hypermultiplet in the antisymmetric representation, and 8 hypermultiplets in the fundamental representation, which is believed to lift to the 6d rank N E-string theory [25]. Recently, another example was given in [26]. There the 6d theory in question is known as the (DN+4; DN+4) conformal matter [27], which has a 6d gauge theory description as USp(2N ) + (2N + 8)F . This theory is suspected to be the UV completion of the 5d gauge theory SU0(N + 2) + (2N + 8)F . The purpose of this paper is to extend these results to a large class of 5d gauge theories with an expected 6d N = (1; 0) SCFT UV completion. We consider theories which can be represented as ordinary 5-brane webs. The starting point is to generalize the discussion of [26] to the class of 5d gauge theories of the form (N + 2)F + SU0(N )k + (N + 2)F . These were recently conjectured to lift to 6d SCFT [28]. Furthermore, in [29] a conjecture for this 6d SCFT appeared. We start by generalizing the method of [26] to give evidence for this conjecture. 1In 5d one can add a Chern-Simons (CS) term to any SU(N ) gauge theory, for N > 2, and we use a subscript under the gauge group to denote the CS level. For USp(2N ) groups, a CS term is not possible, but there is a discrete Z2 parameter, called the angle, which can be either 0 or [2]. We again use a subscript under the gauge group to denote it. Also, when denoting gauge theories we use F for matter in the fundamental representation and AS for matter in the antisymmetric representation. When writing hyper associated with every . Using this result we then go on to propose a technique to determine the answer for other 5d gauge theories, by thinking of them as a limit on the Higgs branch of a 5d gauge theory (N + 2)F + SU0(N )k + (N + 2)F for some N and k. Then we can determine the 6d SCFT by mapping the appropriate limit of the 5d Higgs branch to the corresponding one of the 6d theory. We consider a variety of examples, exhibiting both the advantages and limitations of this technique. As an application of these results, we also consider the compacti cation on a torus of the 6d SCFT's appearing as the lift of 5d gauge theories. For example, consider the compacti cation of the rank 1 E string theory on a torus, where we take the limit of zero area, keeping the 6d global symmetry unbroken. First compactifying to 5d, we get the 5d theory SU(2) + 8F . We now want to compactify to 4d taking the limit of zero torus area, but without breaking the E8 global symmetry. It turns out that the way to do this is by rst integrating out a avor, owing to SU(2) + 7F .2 This leads to a 5d SCFT with E8 global symmetry [1]. Compactifying this to 4d then leads to the rank 1 Minahan-Nemashansky E8 theory [30, 31]. For additional examples of the compacti cation of 6d N = (1; 0) SCFT's on a torus, see [32{34]. We can now adopt a similar strategy to understand the result of compacti cation on a torus of the 6d SCFT's we encounter. That is we rst compactify to 5d leading to the 5d gauge theory. Taking the R6 ! 0 limit, while keeping the 6d global symmetry, is then implemented by integrating out a avor. This leads to a 5d SCFT with a brane web description of the form of [35]. It is now straightforward to take the R5 ! 0 limit, leading to a class S isolated SCFT, as shown in [35]. Thus, we conjecture that reducing the class of 6d theories we consider on a torus leads to an isolated 4d SCFT. The main idea is summarized graphically in gure 1. We next seek to provide evidence for this relation. To this end we use the results of [32], who found a way to calculate the central charges of a 4d theory resulting from compacti cation of a 6d theory on a torus in terms of the anomaly polynomial of the 6d theory. We can now compute the 4d central charges rst using class S technology (see [36]), and second from the anomaly polynomial (using [37]), and compare the two. We indeed nd that these match. This then provides evidence also for the original 5d 6d relation. The structure of this article is as follows. Section 2 presents some preliminary discussions about the computation of the anomaly polynomial for the 6d SCFT's considered in this article, as well as the class S technology we use. In section 3 we consider the 5d theory (N + 2)F + SU0(N )k + (N + 2)F . We rst generalize the methods of [26] to test the conjecture of [29], and then go on to consider related theories. Section 4 deals with other 5d theories expected to lift to 6d, that are not of the form presented in section 3. We end with some conclusions. The appendix discusses symmetry enhancement for a class of 5d theories that play an important role in section 4, and which, to our knowledge, were not previously studied. like: g12 0 R6 g2 ! 0 limit. 2Note that this is a R6 ! 0 limit. This follows as one must keep the e ective coupling, which behaves constant jmj, well de ned. Therefore, when taking the m ! 1 limit, one must also take the { 3 { HJEP12(05)7 deform the 5d gauge theory, corresponding to taking the R6 ! 0 limit while keeping the 6d global symmetry intact. This leads to a 5d SCFT. We then compactify this SCFT on a circle of radius R5, and take R5 ! 0. This leads to a 4d class S SCFT, which can in turn be thought of as a result of compactifying a 6d (2; 0) SCFT on a Riemann sphere with three punctures. We can use this description as a consistency check by calculating the properties of this 4d SCFT when thought of as a compacti cation of a 6d (2; 0) SCFT, known as class S technology, and comparing against what is expected from the compacti cation of the 6d (1; 0) SCFT. A word on notation. Brane webs comprise an important part of our analysis and so they appear abundantly in this article. In many cases only the external legs are needed and not how they connect to one another. In these cases, for ease of presentation, we have only depicted the external legs, using a large black oval for the internal part of the diagram. Many of the diagrams also contain repeated parts shown by a sequence of black dots. This should not be confused with 7-branes. In brane webs one can also add 7-branes on which the 5-branes can end. We have in general suppressed the 7-branes, with the exception of two cases. One, when several 5-branes end on the same 7-brane. In this case we depicted the 7-brane as a black oval, the type of which is understood by the type of 5-branes ending on it. We in general also write the number of 5-branes ending on this 7-brane. If no number is given then it is the number visible in the picture. Any other numbers that appear stand for the number of 5-branes. The second case where we explicitly include 7-branes is if no 5-branes end on them. In this case we denote a (1; 0) 7-brane by an X and a (0; 1) 7-brane by a square. Any other 7-brane is denoted by a circle with the type written next to it. We generically suppress the monodromy line of the 7-branes. In the special cases when we do draw it, we use a dashed line. { 4 { This section discusses the type of 6d theories we encounter, the computation of the anomaly polynomials for these theories, and the class S technology used in this article. 2.1 Properties of the 6d theories We start by presenting the 6d gauge theories that we consider in this article. We rst present them in their gauge theory description, namely at a generic point on the tensor branch of the underlying 6d SCFT. In this description the gauge theory is made from a quiver of SU(Ni) groups with one end being just fundamental hypers while the other end being either a USp gauge group or an SU group with a hyper in the antisymmetric. The freedom in the choice of the theory is given by the ranks of the groups Ni. The number of avors for each group is uniquely determined by anomaly cancellation for each group. The quiver diagrams for the theories we consider are shown in gure 2. Next we wish to evaluate the anomaly polynomial that we use later. We concentrate only on the terms in the anomaly polynomial that we need. By using the results of [37, 38], we nd that the anomaly polynomial contains: where Fi is the eld strength of the i'th group (we always denote the USp or SU with the antisymmetric as i = 1), and a summation over repeated indices is implied. Also C2(R) { 5 { stands for the second Chern class of the R-symmetry bundle, and p1(T ); p2(T ) are the rst and second Pontryagin classes of the tangent bundle, respectively. We use nh; nv and nt for the number of hyper, vector and tensor multiplets respectively, and hGi for the dual Coexter number of the i'th group. Finally: . . . 2 tensor multiplet (see [19] for the details). For the case at hand this adds the following to the anomaly polynomial: X 1 tr(Fi2) i i tr(Fi2+1) + C2(R) X hGj j=1 p1(T ) 4 32 5 2 collecting all the terms we nd: where C2(R)p1(T ) nv4d + dH 7p21(T ) 4p2(T ) 5760 (2.2) (2.3) (2.4) (2.5) where the sum i is over all the gauge groups. The labels we used were chosen with the compacti cation to 4d in mind. When compactifying to 4d on a torus we get some 4d SCFT in the IR. We can calculate the central charges, particularly the a and c conformal anomalies, of this SCFT using the results of [32].3 We nd that dH = 24(c a) and nv4d = 4(2a c). Thus, dH is the dimension of the Higgs branch, and n4d the e ective number of vector multiplets of the 4d SCFT resulting v from the compacti cation of the 6d SCFT on a torus. In that light the equation for dH has a rather nice interpretation as the classical dimension of the Higgs branch of the gauge theory, nh nv, plus the contribution of the tensor multiplets, each giving 29 dimensions, like the rank 1 E-string theory. Besides the a and c conformal anomalies, we also want to determine the central charges for avor symmetries, kFi , associated to the avors under the i'th gauge group. From the result of [32], this can be determined from the term 1k9F2i tr(Fg2lobali )p1(T ). Say we have a avor symmetry, the elds charged under it being avor of dimension under the group Gi. Then we nd that: we're considering are of this type. kFi = 12gi + 2d (2.6) 3For these results to hold, the 6d SCFT must be very-Higgsable, as described in [32]. All the 6d SCFT's { 6 { where gi = nG representation . i + 1, nG being the number of groups, and d is the dimension of the Before continuing we note that some of the theories we consider also include gauging the rank 1 E-string theory at one end of the quiver. This is a straightforward extension of the quiver theories with a USp(2N1) end to the N1 = 0 case. This follows from the fact that USp(2N1) + (2N1 + 8)F goes to a 6d SCFT known as the (DN1+4; DN1+4) conformal matter [27], so this class of theories can be regarded as gauging a part of the SO(4N1 + 16) global symmetry of (DN1+4; DN1+4) conformal matter. In this description we can also consider the case of N1 = 0 relying on the fact that (D4; D4) conformal matter is the rank 1 E-string theory. Going over the computation of the anomaly polynomial, we nd that (2.5) is still valid, where we include the rank 1 E-string theory in the sum and take hE string = 1. kF1 = 12Q(nG + 1). Generically when gauging a part of a rank Q E-string theory, some of the E8 global symmetry remains unbroken and serves as a global symmetry. For these cases we nd Finally, while we generally employ the gauge theory description of these (1; 0) SCFT's, it is worthwhile to also specify their description as an F-theory compacti cation. In this language the theory is described as a long 1 2 2 curves and a USp or SU type group on the For the details on the meaning of this notation we refer the reader to [21]. In a nutshell, specifying a 6d SCFT requires enumerating its hyper, vector and tensor content. The numbers represent the type of tensor multiplet, where a 2 curve represents a single free N = (2; 0), tensor and a 1 curve the rank 1 E-string theory. The sequence of numbers represents several tensor multiplets. For example, 2 An 1 theory where n is the number of 2 curves, and 2 1 2 : : : 2 2 gives the N = (2; 0) 2 : : : 2 gives the rank n + 1 E-string theory. One can add vector multiplets on these curves. When these are added, the theory on the tensor branch acquires a gauge theory description. For a 2 curve, adding an SU(N ) type group, leads to an SU(N ) + 2N F gauge theory on the tensor branch. For a 1 curve, adding a USp(2N ) type group leads to a USp(2N ) + (2N + 8)F gauge theory, while adding an SU(N ) type group, leads to an SU(N ) + 1AS + (N + 8)F gauge theory at a generic point on the tensor branch.4 It is now apparent that going to a generic point on the tensor branch indeed gives the gauge theories we consider. We can also consider the reverse process of removing vector multiplets from a curve. This describes a Higgs branch limit of the 6d SCFT in which some of the vector multiplets become massive and the theory ows to a di erent IR SCFT. Note in particular, that completely breaking a group, corresponding to removing all the vector multiplets from that curve, still leaves the associated tensor multiplet. The resulting IR SCFT generically has no complete Lagrangian description, but can still be described by a gauge theory gauging part of the avor symmetry of a non-Lagrangian part. We shall encounter several examples of this later. 4For SU( 6 ) there is an additional option giving an SU( 6 ) + 12 20 + 15F gauge theory at a generic point on the tensor branch. We brie y encounter this option later in this paper. { 7 { Sometimes gauge theory physics is insu cient to fully determine the properties of the SCFT. For example, in some cases the global symmetry naively exhibited by the gauge theory, is larger than the one of the SCFT. We encounter some cases where this occurs, and then it is useful to have an F-theory description. The results obtained from the 6d anomaly polynomial can be compared to the ones obtained using class S technology. Speci cally, the theories we consider are all isolated SCFT's, that can be represented as the compacti cation of an A type (2; 0) theory on a Riemann sphere with 3 punctures. We also have a 5d brane web representation using [35]. It is known how to calculate the central charges of such SCFT's from the form of the punctures. The explicit formula used to calculate dH ; nv4d and kF can be found in [36, 39]. In practice, it is usually simpler to calculate dH directly from the web. We also want to determine the global symmetry of the SCFT. In general this can be read of from the punctures, but in some cases the global symmetry can be larger than is visible from the punctures [40]. One way to determine this is using the 5d description either directly from the web, or using the gauge theory description. A more intricate method is to use the 4d superconformal index. Since conserved currents are BPS operators they contribute to the index, and so knowledge of the index allows us to determine the global symmetry of the theory. In practice we do not need the full superconformal index, just the rst few terms in a reduced form of the index called the Hall-Littlewood index [41]. An expression for the 4d superconformal index for class S theories was conjectured in [41{43], and one can use their results to determine the global symmetry. For more on this application see [44]. In cases where the global symmetry is bigger than what is visible from the punctures, we use the 4d superconformal index to show this. In cases where it is not di cult to argue this also from the 5d description, we also use this as a consistency check. 3 The 5d (N + 2)F + SU0(N )k + (N + 2)F quiver and related theories In this section we start analyzing the 6d lift of 5d theories. We start with the 5d quiver theory (N + 2)F + SU0(N )k + (N + 2)F . Since a conjecture for this theory was already given in [29], it is more convenient to start with the 6d theory. There are two slightly di erent cases to consider. First, we have the 6d SCFT whose quiver description is shown in gure 3. This theory can be realized in string theory by a system of D6-branes crossing an O8 plane and several NS5-branes, shown in gure 4. Note, that this is a generalization of the system in [26], by the addition of NS5-branes. We can now repeat the analysis of [26]. Since this is a simple generalization of their work we will be somewhat brief. We compactify a direction shared by all the branes and preform T-duality. The O8 plane becomes two O7 planes. Under strong coupling e ect, the O7 plane decomposes to a (1; 1) 7-brane and a (1; 1) 7-brane [45]. We then end up with the web of gure 5. This web describes the 5d gauge theory (N + 2)F + SU0(N )2l 1 + (N + 2)F , as shown in gure 6. Note that the number of groups in 5d must be odd, owing to the even number of NS5-branes. { 8 { HJEP12(05)7 part of the global symmetry of the shown 6d SCFT, in this case an SU( 8 ) subgroup of E8. gure 3. The horizontal lines represent D6-branes, and the number above the lines stand for the number of 6-branes. The black circles represent NS5-branes, and their number is given below. Finally, the vertical line stands for the O8 plane. The con guration also include 2N + 4l D8-branes, parallel to the O8 plane, on which the asymptotic D6-branes end. For clarity we have suppressed them in the gure. This suggests that to get an even number of 5d SU(N ) groups, we need to take an odd number of NS5-branes, which we do by adding a stuck NS5-brane on the O8 plane. The brane and quiver description of the resulting 6d theory is shown in gure 7. We can now repeat the analysis. After T-duality we get again two O7 planes with the stuck NS5-brane stretching between the two. Decomposing the O7 planes with the stuck NS5-brane, as shown in [14], we arrive at the web of gure 8. As shown in gure 9, this is the web of (N + 2)F + SU0(N )2l + (N + 2)F . This agrees with the conjecture of [29], that this 6d theory is the UV completion of the 5d gauge theory (N + 2)F + SU0(N )2l + (N + 2)F . Note that in the 6d theories covered so far we have assumed that N > 2l 1. Naively, this implies the same limitations on the 5d theories. However, it is not di cult to see that performing S-duality on the web for (N +2)F +SU0(N )k 1 +(N +2)F results in the one for (k + 2)F + SU0(k)N 1 + (k + 2)F . Thus, by doing an S-duality, one can map any 5d linear { 9 { and resolving the O7 planes. SU(N ) quiver to the required form. Also note that when k = N 1, both descriptions are of this form, and indeed the two 6d SCFT's are the same. We now wish to employ this relation to the compacti cation of the 6d SCFT on a torus, 2 preserving the global symmetry. Inspired by the E-string theory example, we are lead to consider an in nite mass deformation limit of the related 5d theory. The natural candidate is integrating out a fundamental avor. We have only one possibility, corresponding to the 5d theory (N + 2)F + SU0(N )k 2 SU 1 (N ) + (N + 1)F whose web is shown in gure 10. This theory does give a 5d xed point shown in gure 10. We note that this web is of the form of [35]. We can now employ class S technology to determine the global symmetry of this theory, nding that its global symmetry is U(1) SU(2) SU(2N + 2k) when N = k. SU(2N + 2k) when N 6= k and Note that this is exactly the same as the global symmetry of the 6d theory. The avors at the end give the SU(2N + 2k) part. The remaining U(1) is the anomaly-free combination of the various baryonic and bifundamental U(1)'s. The case of N = k indeed has an enhancement of symmetry to SU(2). For k = 2l + 1, this comes about because the antisymmetric representation of SU(4) is real while for k = 2l, this comes about as the gauging of SU( 8 ) E8 preserves an SU(2), since SU( 8 ) E7 E7 SU(2) E8. We now conjecture that compactifying this 5d theory to 4d should give the compactication of the starting 6d theory on a torus. We know from the work of [35] that for the theory of gure 10, this leads to a 4d isolated SCFT that can be described by a compacti cation of the 6d (2; 0) theory of type A2N+2k 3 on the punctured sphere of gure 11. on the left through the leftmost NS5-brane leading to the web in (b). We can now push the (1; 1) 7-brane and (1; 1) 7-brane through the neighboring NS5-brane. This changes the asymptotic NS5brane to a D5-branes, and is accompanied by a Hanany-Witten transition generating an additional 5-brane ending on the 7-brane. This gives the web in (c). Repeating this on the neighboring l 2 NS5-branes, and also doing the same on the right hand side, we end up with the web in (d). This is the web of gure 5 after pulling out the internal 7-branes. We next wish to test this conjecture by comparing the central charges of this 4d SCFT with the ones expected from the compacti ed 6d (1; 0) SCFT which can be determined through (2.5), (2.6). From the 5d theory, using class S technology, we nd: k, kSU(2) being relevant only for the N = k case. The results for N < k can be generated from (3.1) by taking N $ k. resolving the O7 planes. SU 12 (N ) + (N + 1)F . The upper left shows the web in its gauge theory description. Moving rst the two shown (0; 1) 7-branes down to the other side and then pulling out the two upper (1; 0) 7-branes, doing Hanany-Witten transitions when necessary, leads to the web on the upper right. Further pulling the remaining (1; 0) 7-brane to the right, doing all the Hanany-Witten transitions, leads to the web on the lower right. Finally, exchanging the upper (0; 1) 7-brane with N 1 NS5-branes ending on it with the lower one with 2N + k 3 NS5-branes ending on it, and also moving the left (1; 0) 7-brane to the right leads us to the web in the lower left of the gure. From the 6d theory we see that: nv = nh = nt = l 8k(k 8k(k 1)(k 1)(k 3 3 2) 2) for the case of k = 2l, and: k 2 2 + (N k)(2N k + 2k2 2N 6k + 1); + 2k(N 2 k2 + 4k 8); nv = 8(k 1)(k 2)(k 3) 3 k 1 2 + 2(k 1)(N k + 2)(N + k 4); them to 4d on a torus leads to the isolated SCFT represented in the lower part. nh = nt = l 8(k 1)(k 2)(k 3) 3 + 2kN 2 + 3N 2k3 + 16k2 43k + 30; (3.3) for the case of k = 2l + 1. Using these in (2.5) and (2.6) we indeed recover (3.1). An interesting thing happens for k = 2. In that case the 6d theory becomes USp(2N 4) + (2N + 4)F , which is also known as (DN+2; DN+2) conforml matter [27]. The reduction of this theory to 4d on a torus was recently studied in [32]. They found that it leads to an isolated SCFT corresponding to compactifying the 6d (2; 0) theory of type DN+2 on a Riemann sphere with three punctures shown in gure 12 (b). If we are correct in our description then these two SCFT's must be identical. Indeed, using the results of [39] we can calculate the dimension of Coulomb branch operators and compare between the two theories. We nd a perfect match. on a torus should give the isolated 4d SCFT that is described by compactifying the 6d (2; 0) theory of type A2N+1 on this punctured Riemann sphere. (b) A di erent analysis, done in [32], suggests that the same theory compacti ed on a torus should give the isolated 4d SCFT that is described by compactifying the 6d (2; 0) theory of type DN+2 on this punctured Riemann sphere. Our analysis does imply that these two theories are in fact identical. Before moving on to discuss other 5d theories, there is one more 6d SCFT, closely related to the ones considered, that we would like to discuss. The quiver theory description is given in gure 13. We can repeat the previous analysis, now the di erence manifesting in the 6d brane construction by adding a stuck 6-brane. Upon performing T-duality this becomes a stuck D5-brane on one of the O7 planes. We can decompose the O7 planes as done in [14], to get the nal web picture. The entire process is shown in gure 14. This describes the 5d gauge theory of gure 15. One can see that the Coulomb branch dimensions agree, and using the results of [29], also the global symmetries agree, in particularly, we get an a ne A(21n)+8l. As a further test we consider the compacti cation to 4d, where we expect to get the theory of gure 16. Using class S technology we can indeed show that the 4d isolated theory in gure 16 (b) has the same global symmetry as the 6d quiver of gure 13. We can also calculate the central charges nding: dH = 2(n + 4l)2 + n + 6l; Using (2.5), (2.6), this indeed matches what we expect from the theory of gure 13. 3.1 Generalizations The next step is to consider generalizations to other 5d gauge theories with an expected 6d lift. Consider the 5d gauge theory given by a linear SU0(Ni) quiver with fundamental T-dualize to the web system in the bottom of the gure. matter, where each non edge group sees an e ective number of 2Ni avors. If in addition the two edge groups see an e ective number of 2Ni + 2 avors, then it was argued in [29] that this 5d theory should have an enhanced a ne A(1) symmetry. This strongly suggests that these also lift to a 6d SCFT. Note that the previously considered theories are also of this form. Naturally, we would like to know to which 6d SCFT these theories lift. As there is an in nite number of possibilities, a case by case study seems ine ective. Thus, we wish to determine a procedure by which, given such a 5d quiver, the 6d SCFT can be determined. To do this we can utilize the fact that any such quiver can be reached starting with the linear the brane web, and doing several manipulations, we arrive at the web of gure 14. SU(N ) quiver considered before, for some N and k, and going on the Higgs branch. Also, for theories with 8 supercharges, the Higgs branch does not receive quantum corrections, and so the 5d and 6d Higgs branches must agree. Therefore, one possible strategy is to start from one of the previous cases, where we know the 6d SCFT, and determine the Higgs branch limit needed to get the required 5d quiver. Then, by mapping this to the 6d SCFT, we can determine the 6d lift of the 5d quiver. To understand the mapping, we can again rely on the brane description. Starting with the 5d case, the Higgs branch limits we are interested in are represented, in the brane web, by forcing a group of 5-branes to end on the same 7-brane. For example consider a group of N parallel 5-branes, crossing some NS5-branes, each ending on a di erent 7-brane, see gure 17 (a). This describes a quiver tail of the form N F + SU0(N ) SU0(N ) : : :. If we force two 5-brane to end on the same 7-brane then, because of the S-rule, one Coulomb modulus of the edge SU(N ) group is lost. Thus, this describes the Higgs branch breaking N F + SU0(N ) SU0(N ) to (N 2)F + SU0(N 1) SU0(N ) + 1F (see gure 17 (b)). is integrated out. Opening out the web we get to the presentation of (b). We could have also integrated out any other avor, and obtained the same theory. We can of course repeat this and force two other 5-branes to end on the same 7-brane. This leads to a similar breaking on the new quiver (see gure 17 (c)). However, we can also consider forcing an additional 5-brane to end on the same 7-brane, so as to have three 5-branes ending on it (see gure 17 (d)). Now the S-rule not only eliminates a Coulomb moduli of the edge SU(N ) group, but also one from the adjacent group. This describes the Higgs branch breaking associated with giving a vev to the gauge invariant made from a avor of the edge group, the bifundamental, and the avor from the adjacent group. The quiver left after this breaking is shown in (see gure 17 (d)). It is now straightforward to generalize to an arbitrary con guration. Before moving to the corresponding limits in the 6d theory, we note that this correspondence may not hold when completely breaking a gauge group. In general, the topological symmetry of the broken group survives the breaking and remains in the resulting theory, sometimes manifesting as extra avors. In these cases, perturbative reasoning alone may be inadequate to determine the answer. For our purposes, this can always be avoided. Also note, that this can be related to the classi cation of 4d quiver tails of [40] by using the results of [35]. This is an alternative way to argue this mapping. Next, we consider the implications of this on the 6d theory. Under T-duality, the D5-branes are mapped to D6-branes and the D7-branes to D8-branes, so the analogous breaking on the 6d side is represented in the brane con guration by forcing a group of D6-branes to end on the same D8-brane. If the breaking is not too extreme, this translates to a limit on the perturbative Higgs branch of the 6d SCFT. In fact, as the S-rule is the same as in the 5d case, we nd that this induces exactly the same e ect on the quiver tail. The only di erence is that now there is only one quiver tail. Each action performed on any of the two tails of the 5d quiver is mapped to the corresponding action done on the single 6d tail. Nevertheless, complications can arise in some instances, for example, when the 6d SCFT has a tensor multiplet without an associated gauge theory. For example, consider the 6d quiver of gure 3, for N = 2l. In that case the 6d SCFT has a non-Lagrangian part, the rank 1 E-string theory, possessing a 29 dimensional Higgs branch. Some of the breaking we consider may be mapped to the Higgs branch of the E-string theory, where we have no perturbative description. This can happen even in cases where the initial theory has a complete Lagrangian description, but on the Higgs branch limit the gauge group is completely broken leaving its associated tensor multiplet.5 Note that this method can still be used to determine the 6d SCFT, but is somewhat complicated as the Higgs branch limits may not be perturbatively realized. Thus, determining the resulting 6d SCFT will probably require string theoretic methods like the ones in [46]. 3.1.1 A simple example We next wish to illustrate this with a simple example. First, consider the 5d theories shown in gure 18. We can get these theories from the one in gure 6 (a) by going on the Higgs branch. On the web system this is manifested by breaking two pairs of 5-branes so that each of them end on the same 7-brane, the di erence between them being whether the pair are on the same side or opposite sides. In the 6d theory these are mapped to the same breaking, indicating that these two quivers are dual, in the sense of both lifting to the same 6d SCFT. Taking the corresponding limit in 6d, we get to the quiver of gure 19, which is the desired 6d SCFT. By construction, we are now assured that doing the T-duality on the brane system of this 6d quiver leads to the webs in gure 18. We can also consider compactifcation of the 6d theory to 4d. As the Higgs branch limit and dimensional reduction should commute, we again expect the resulting 4d theory to be given by the class S theory whose 5d analogue is given by integrating out a avor from the theories of 18. Naively, we have several di erent choices of which avor to integrate out, but we nd these all lead to the same class S theory, shown in gure 20 (c). 5This is manifested in the web when one is forced to coalesce 2 NS5-branes or an NS5-brane and the O8 plane due to the constraints of the S-rule. some of the 7 branes, we arrive at the spiraling con guration shown on the bottom left. (b) Starting with the same con guration, doing some 7-brane gymnastics, we get to the web on the right. One can note that this is a Higgs branch limit of a theory of the form of gure 6 (a). presented in section 3. Besides supporting the claim that this theory lifts to 6d, we can, by taking the required Higgs branch limit on the 6d lifts given in section 3, also argue that the quivers given in gure 33 are indeed the required 6d lifts. This is shown for the k > N case in gure 34, and for the k < N case in gures 35. We can again consider the reduction to 4d on a torus. We expect the 4d theory to be described by the case with one less avor shown in gure 36. The punctures suggests a global symmetry of SU(2N + 2k + 1) U(1)3 except in some special cases, for example, when k = N or k = N 1 where the symmetry enhances to SU(2N + 2k + 1) SU(3) SU(2) enhancement of U(1) when k = N or k = N U(1)2. From the 4d superconformal index we see that there is a further SU(2)2 ! SU(4), which becomes U(1) SU(2) SU(3) ! SU(5) 1. This enhancements, including the special cases with enhanced symmetry, exactly matches the ones expected from the 6d SCFT of gure 33. We can also this brane web. One can see that it is in the form of [35] so compacti cation to 4d will yield the isolated SCFT of (b). theory of gure 32. Higgs branch limit of the theories in gure 6 (a) given by the S-dual of the web on the right. ; ; (4.3) (4.4) Higgs branch limit of the theories in gure 8. calculate the central charges of this theory nding: dH = 2N 2 + 2k2 + 4kN + 17N + 9k; (2N 1)(6k2 + 4N 4N 2 + 9k + 18kN ) kSU(2N+2k+1) = 2(2N + 2k + 3); kSU(4) = 4k + 8N for k N where for k = N SU(4) ! SU(5), and dH = 2N 2 + 2k2 + 4kN + 17k + 9N + 4; k(12N 2 + 24N 13 18k + 36kN ) 3 8k2 3 kSU(2N+2k+1) = 2(2N + 2k + 3); kSU(4) = 4N + 8k + 2 for N > k. This indeed matches the results we get from (2.5) and (2.6). 4.2.2 SU quivers with an antisymmetric hyper at both ends We can next consider the case where both ends are SU groups with an antisymmetric so we have the quiver theory of gure 37. We conjecture the 6d lift to be the one shown in gure 38. We can repeat the same steps as before, rst deform the web to give a Higgs branch limit of a theory of gure 6 (a). This gives the web shown in gure 39. By series of HW transitions, we get to the brane web in the bottom left. One can see that it is in the form of [35] so compacti cation to 4d will yield the appropriate isolated SCFT. This is the form most suited to the k N case. For the N > k case, the one in (b), gotten from (a) by shu ing some of the 7-branes, is more adequate. implementing the required breaking on the 6d SCFT of gure 3 we indeed get the quiver of gure 38. As an additional test, we can again consider the reduction to 4d on a torus. We expect the 4d theory to be described by the case with one less avor shown in gure 40. Higgs branch limit of the web in gure 6 (a). The global symmetry visible from the punctures is SU(2k + 2) SU(4) which is further enhanced when k = 0 or N = 2. When k 6= 0, we can show from the superconformal index that there is an enhancement of SU(4) U(1) ! SO(12). This, including the enhancement when N = 2, exactly matches what is expected from the 6d global symmetry. However, the k = 0 case, the 5d SCFT of which corresponds to the 5d gauge theory 2 SU 1 (2N )+2AS+7F , has some puzzling features. First, let's start with the global symmetry for the SCFT of gure 40. As argued in the appendix, instanton counting methods suggests this theory has an E7 SU(2)3 global symmetry which is further enhanced to E7 SO(7) for N = 2. This is further supported by the 4d superconformal index. Note that the N = 2 case discussed here is identical to the N = 2 case for the theory in gure 27 (b), which provides a dual gauge theory description for the same xed point. Comparing with the 6d side, we naively encounter a contradiction. When k = 0 we have a long quiver of SU(2) groups leading to an enhancement of the U(1) bifundamental global symmetries to SU(2)'s. More importantly the mixed anomalies leading to the breaking of most of these U(1)'s now vanish so we naively expect to have an SU(2)N+1 global symmetry contradicting the global symmetry suggested by the 5d description. The issue appears to be the discrepancies between the global symmetry suggested from the gauge theory and the one that actually exists in the SCFT mentioned in section 2. To truly understand the 6d SCFT we should consider a string theory realization of it. Fortunately, the 6d SCFT at hand was considered in [21]. They considered a class of theories engineered in string theory by a group of M5-branes probing a C2=Z2k+2 orbifold and an M9-plane. One of the theories in this class is the theory with gauge theory description given in gure 38. This is no coincidence as the original 5d gauge theory, shown in gure 37, can be engineered by a group of D4-branes probing a C2=Z2k+2 orbifold and an O8 plane [47] so it is natural to expect the 6d lift to be of this form. According to the analysis of [21], the non-abelian global symmetry of this 6d SCFT is indeed SO(12) SU(2k + 2) SU(2). The case k = 0 is special: the non-abelian global symmetry is actually E7 SU(2)3. The extra SU(2) is there since the orbifold C2=Z2 preserves the full SO(4) symmetry, while C2=Z2k+2 breaks one of the SU(2)'s.7 So this appears to agree with what we see from the instanton counting analysis done in the appendix. The case k = 0; N = 2 is more special. Then the 6d theory is known as the (E7; SO(7)) conformal matter [27]. Again the gauge theory shows an SO( 8 ) global symmetry, while it is known the SCFT only has SO(7). This indeed agrees with the results from instanton counting done in the appendix. We can also to calculate the central charges of this theory nding: dH = 2k2 + 30N + 19k + 3; This indeed matches the results we get from (2.5) and (2.6), supporting the claim that compactifying the 6d SCFT of gure 38 on a torus leads to the isolated 4d SCFT of gure 40. Incidentally, the compacti cation of the (E7; SO(7)) conformal matter on a torus was already considered in [32]. They conjectured that the resulting theory is given in terms of a compacti cation of the E6 (2; 0) theory on a Riemann sphere with three punctures labeled: 0; 2A1; E6(a1) (see [44], for a discussion on the meaning of the notation and for properties of this SCFT). They further compared the central charges of this theory to the ones expected from the compacti cation, nding an exact match. Consistency of these two approaches then suggests that these two theories are in fact the same theory. Indeed, we calculated the central charges and spectrum of Coulomb branch operators of the theory in gure 41, nding exact matching to the previously mentioned SCFT from compactifying the E6 (2; 0) theory. We can also consider the even rank case shown in gure 42. While this can be gured out from the previous case by going on the Higgs branch, we will mention this case. We expect the 6d theory to be the one shown in gure 43. This can be argued by manipulating the brane web into a form, shown in gure 44, as a Higgs branch limit of the theory of gure 8. We can again consider the reduction to 4d on a torus. We expect the 4d theory to be described by the case with one less avor shown in gure 45. The discussion is quite similar to the odd rank case. The global symmetry visible from the punctures is SU(2) SU( 6 ) SU(2k + 2) U(1)2 which gets further enhanced when N = 1 or k = 1; 0. 7I am grateful for J. J. Heckman for making his work known to me and for discussing this point. this brane web. One can see that it is in the form of [35] so compacti cation to 4d will yield the appropriate isolated SCFT. 2 sentation of its associated 4d SCFT as a compacti cation of an A type (2; 0) theory on a three punctured sphere. theory of gure 42. limit of the web in gure 8. This is given by the S-dual of the rightmost web. From the 4d superconformal index we nd an enhancement of SU(2) SU( 6 ) U(1) ! SU( 8 ) which is further enhanced to SU(2k + 10) for N = 1. This agrees with what is seen from the gauge theory description of gure 43 except for the case of k = 0. In this case the 2 gauge theory is SU 1 (5) + 2AS + 7F and as discussed in the appendix, we expect to have an SO(16) SU(2)2 global symmetry. This is also con rmed from the 4d superconformal index. In the 6d theory we again encounter a series of SU(2) groups and we naively have a problem with matching the global symmetry. However, this theory was also considered in [21], as expected since the 5d theory is related to the previous one by adding D4-branes stuck on the orbifold and so should lift to a 6d SCFT of this type. The analysis of [21] suggests the non-abelian global symmetry of this theory is indeed SU( 8 ) SU(2k + 2). The k = 0 case is again special, and then the non-abelian global symmetry should indeed be SO(16) SU(2)2. We can also calculate the central charges of this theory nding: dH = 2k2 + 30N + 19k + 3; kSU(2k+2) = 4k + 16; kSU( 8 ) = 12N + 4k + 4 This indeed matches the results we get from (2.5) and (2.6). Cases with completely broken groups Finally, we wish to consider several additional cases. The common thread in all of them is that they involve completely breaking a 6d gauge group leaving a tensor multiplet. As our rst example we consider the case of USp(2N ) + AS + 8F . As mentioned in the introduction, this theory is known to lift to the rank N E-string theory. It also has a brane web description given in gure 46 (a) [14]. We can now recast this web as a Higgs branch limit of the theory in gure 6 (a). Carrying out this breaking on the 6d SCFT, one nds that this completely breaks the gauge symmetry leaving only the tensor multiplets. Indeed, as mentioned in section 2, the theory described by such a structure of tensor multiplets is the rank N E-string theory. Next we consider a case in which only part of the gauge theory is broken. Take the 5d gauge theory Nf F + USp(2N + 4) USp0(2N ) whose web is shown in gure 47 (a). First, this brane web. One can see that it is in the form of [35] so compacti cation to 4d will yield the appropriate isolated SCFT. limit of the theory in gure 6 (a). let us analyze the global symmetry of this theory. Instanton counting methods suggest that the (0; 1) instantons should lead to an enhancement of the USp0(2N ) topological U(1) to SU(2) [14]. In addition we expect an enhancement of U(1) SO(2Nf ) to ENf +1. This is most notable from the gauge symmetry on the 7-branes using the results of [48, 49]. Thus, we conclude that this theory has an ENf +1 SU(2)2 global symmetry. The case of N = 1 is exceptional as then there is an additional enhancement of SU(2)2 ! G2 [12] so in that case the global symmetry is ENf +1 G2. USp0(2). The generalization to USp(2N + 4) USp0(2N ) is apparent and we only show the shape of the external legs, shown on the right. The generalization to Nf F + USp(2N + 4) USp0(2N ) is also straightforward and is done by adding 7-branes. For example consider the web of (b) describing 8F + USp(2N + 4) USp0(2N ). By manipulating the 7-branes we can get to the web on the right which is in the form as a Higgs branch limit of the web in gure 6 (a). E8 gauge theory. In the case of Nf = 8 we get an E(1) global symmetry and the theory is expected to 8 lift to 6d. Indeed, as shown in gure 47 (b), the web for this theory can be cast into a form as a Higgs branch limit of the web in gure 6 (a). We can now implement this breaking on the 6d theory. Doing this one can see that we are left with the two free tensor multiplets of type 1 2. This gives the rank 2 E-string theory. The remaining quiver connects to this theory by gauging the SU(2) subgroup of the SU(2) E8 global symmetry of this 6d SCFT. This leaves an E8 global symmetry, as expected from the 5d theory. The explicit 6d theory we get is shown in gure 48. Like in previous cases, we expect most of the SU(2) global symmetries to be anomalous even though this is not visible in the gauge theory. The case of N = 1 is known as the (E8; G2) conformal matter [27] and there it is known that the global symmetry of the SCFT is actually E8 G2 and not the SO(7) visible from the gauge theory. This indeed matches what is expected from the number stands for an odd number of half-hypers, possible since the group is SU(2). We can also consider compacti cation to 4d on a torus. For simplicity, we only consider the N = 1 case. We expect the resulting 4d theory to be the one described by reducing the SU0(2), shown in gure 49, on a circle. This indeed preserves the 6d global symmetry. We can further test this by matching the central charges of the 4d SCFT with the one expected from the 6d theory. Using class S technology, we nd that this theory has Coulomb branch operators of dimensions: 6; 8; 12; 18. We further nd: dH = 92; nv4d = 84; kE8 = 36; kG2 = 16 (4.7) Using the methods of [37], we nd that this indeed matches the result we expect from (E8; G2) conformal matter. Like the previous case, the compacti cation of the (E8; G2) conformal matter on a torus was already considered in [33]. They conjectured that the resulting theory is given in terms of a compacti cation of a speci c E8 (2; 0) theory on a Riemann sphere with three punctures. Consistency of these two approaches then suggests that these two theories are in fact the same theory. Since the class S analysis for compacti cation of E8 (2; 0) theory is not yet available we cannot compare the two theories. It will be interesting to check this if the classi cation becomes available. The last case we wish to consider involves a 2 type tensor multiplet. Consider the 2 2 5d theories SU 3 (2N ) + 2AS + 7F and SU 3 (2N + 1) + 2AS + 7F . The instanton analysis calculation, done in the appendix, suggests these have an enhanced a ne global symmetry and so may lift to 6d. For simplicity, we concentrate on the N = 2 case, the generalization to other N being straightforward. Figure 50 shows the brane webs for these theories, and how they can be cast as a Higgs branch limit of the theories of gure 14. Implementing this breaking on the appropriate 6d SCFT yields the theories described in gure 51 which are the appropriate 6d lifts. One can see that indeed the theory of gure 51 (a) has the SO(19) symmetry expected from the 5d description. However, the one of gure 51 (b) shows an E7 SO(7), the E7 agreeing with the gauge theory expectations. We expect the SCFT to not posses the SO(7) global symmetry, but only have the G2 subgroup, like the (E8; G2) conformal matter case. It would be interesting to test this using the F-theory description. Figure 49. The brane web for 7F + USp( 6 ) SU0(2). From this one can arrive at the repre sentation of its associated 4d SCFT as a compacti cation of an A type (2; 0) theory on a three punctured sphere. 5 Conclusions In this article we studied 5d gauge theories that are expected to lift to 6d SCFT's. Given such a 5d gauge theory, we are interested in determining its 6d lift. We have proposed a method to do this for 5d gauge theories with an ordinary brane web description. We have provided several examples of these, showcasing its usefulness as well as its limitations. One such limitation is that to properly utilize it, one must be able to cast the web as a Higgs branch limit of a known theory. It is not immediately clear if this can be done for an arbitrary theory. However, we have checked a number of examples in which this appears to be true. This leads us to conjecture that all 5d gauge theories with an ordinary brane web description that lift to 6d, lift to the family of theories discussed in section 2. It will be interesting to further explore this. Another direction is to nd further evidence for the relations proposed in this article. One possible direction is to compute a quantity in the 5d theory and compare it against the expected result from the 6d SCFT. Such a thing was done, for example, in the case of the rank 1 E-string case in [28, 50], the quantity in question being the 5d superconformal index. It is interesting if this can also be carried out for some of the examples presented here. limit of the web in 14. 2 2 of the web in 14. (b) The brane web for SU 3 (5) + 2AS + 7F converted to a form as a Higgs branch It is also interesting to consider other 5d gauge theories. While it is not yet completely clear what gauge and matter content are allowed for the theory to posses 5d or 6d xed points, there are several cases that can be engineered in string theory and thus are known to exist. In particular one can generalize brane webs by adding O7 planes [14] or O5 planes [51] leading to additional possibilities. Some theories in these classes are known to have an enhancement to an a ne symmetry and so are expected to lift to 6d [29, 52]. It will be interesting to also determine the 6d SCFT's in these cases. Acknowledgments I would like to thank Oren Bergman, Soek Kim, Kimyeong Lee, Hee-Choel Kim, Kazuya Yonekura, Shlomo S. Razamat and Jonathan J. Heckman 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 { 41 { lift of the 5d gauge theory SU 3 (5) + 2AS + 7F . The rightmost circle in both quivers, corresponds to a single N = (2; 0) tensor multiplet where an SU(2) subgroup of the USp(4) (2; 0) R-symmetry 2 is gauged. A Instanton counting for SU(N ) + 2AS + Nf F In this appendix we consider symmetry enhancement in theories of the form SU(N ) + 2AS + Nf F . The method we employ borrows signi cantly from [53]. The essential idea is to identify the states, coming from 1 instanton con gurations, that are conserved currents. This sometimes allows one to determine what the enhanced symmetry is. The methods relies on the following observations of [53]: 1. The 1 instanton of SU(2), when properly quantizing the zero modes coming from the gaugino, forms a multiplet which is exactly the one associated to a broken current supermultiplet. 2. Any 1 instanton of some Lie group G can be embedded in an SU(2) subgroup of G. Therefore, to determine the spectrum of 1 instanton con gurations of arbitrary G it is su cient to decompose it to SU(2) representations. Particularly, for our case we consider gauge group SU(N ) with matter in the fundamental or antisymmetric. The case of SU(N ) with matter in the fundamental was studied already in [53] and later in [29], which also discussed antisymmetric matter. Yet, to our knowledge, a complete analysis of the case of SU(N ) + 2AS + Nf F was not done, even though the building blocks are in essence already known. Consider a 1 instanton of SU(N ) + 2AS + Nf F . It breaks the SU(N ) gauge symmetry to U(1) SU(N 2). We can decompose all fermionic matter under the reduced gauge symmetry and determine the zero modes provided by them. Particularly, there is only one state in the adjoint of SU(2) whose quantization provides the broken current supermultiplet. The remaining elds are all in the fundamental of SU(2) and so provide one raising operator B C A N 2 4) N N 2 2 2 2 1 1 Nf SU(N ) + 2AS + Nf F . The B operators come from the gaugino, the C from the fundamentals and A from the antisymmetrics. per fermion. By either doing the decomposition, or simply burrowing the results of [53], we nd the zero modes spectrum given in table 1. The full spectrum is now given by acting with these operators on the ground state, j0i, whose charges are: QUG(1) = (N 2)( Nf 2 4), QUB(1) = N2f and QUAS(1) = N 2, where is the CS level. Furthermore, recall that the ground state is a broken current supermultiplet. Thus, to get a conserved current we need to enforce two conditions: 1. The state must be gauge invariant under the unbroken UG(1) SU(N 2) gauge symmetry. 2. The state must remain a broken current supermultiplet, particularly, it must have as the lowest component, a triplet of scalar operators under SUR(2). The implications of these two conditions is that we must look at all operators made from the elds in table 1 that are SU(N 2) and SUR(2) singlets. The application of any combination of these on the ground state gives an SU(N 2) invariant broken current supermultiplet. Next, one must enforce UG(1) invariance. Going over table 1 we see that the only SU(N 2) and SUR(2) singlets are: B2(N 2), AN 2 , C and ( AlBN 2 l)2 for l = 1; 2 : : : ; N 1, where the SU(N 2) indices are contracted with the epsilon symbol. Before looking at all these operators, we should discuss under what conditions we expect a xed point. We answer this question by analyzing brane webs. We nd two cases with a spiral tau type diagram, or alternatively, a web description as a Higgs branch limit of a 6d lifting theory. These suggest that these theories lift to 6d. The cases are 2 SU0(N ) + 2AS + 8F (see gure 39 for the web in the N even case and gure 44 in the N odd case) and SU 3 (N ) + 2AS + 7F (see gure 50 for the web in the N = 4; 5 cases). Integrating out avors from these theories gives well de ned webs leading us to believe that this class of theories indeed go to a 5d xed point. Next, we want to determine what conserved currents are provided by the 1 instanton con guration in these cases. First, let's look at all gauge invariant states made by applying A and B on the ground state. These are: j0i ; B2(N 2) j0i ; AN 2 j0i ; ( AN 2)2 j0i ; AN 2 B2(N 2) j0i ; ( AN 2)2 B2(N 2) j0i ; ( AlBN 2 l)2 j0i (A.1) where in the last term l = 1; 2 : : : ; N 1. We can also act on each of these states with k C operators for k = 0; 1 : : : ; Nf . Next, we need to determine when each of these states j0i ; ( AN 2)2 B2(N 2) j0i can contribute if 2j j + Nf the only contribution can come from ( AlBN 2 l)2 j0i. is UG(1) invariant and thus give a conserved current. We only consider theories in the previously discussed class. We also assume N > 3 as the other choices reduce to known N = 4, AN 2 j0i and AN 2 B2(N 2) j0i can contribute if 2j j + Nf cases.8 We nd that B2(N 2) j0i and ( AN 2)2 j0i can only contribute if 2j j + Nf 8 and N = 4; 5 and 8 and 8. Thus, as long as 2j j + Nf < 8 The behavior of these changes depending on whether N is even or odd. If N is even then we can nd a conserved current from the l = N2 2 case, ( A N2 2 B N2 2 )2 j0i. This contribute conserved currents when = 0; Nf = 0. When avors are added then we can also conserved currents from the l = N 2 1 case. still get conserved currents by acting with C operators. If 2j j + Nf 2 If N is odd then we can nd a conserved current from the l = N 1 and l = N 3 cases. The rst contribute when = 2; Nf = 0 while the second when when avors are added then we can still get conserved currents by acting with C operators. We next need to go over all cases, and see what conserved currents we get. This tells us whether symmetry enhancement occurs in the theory, and if so, helps us determine the enhanced symmetry. Since we only see contributions from the 1 instanton, there can sometimes be further enhancements coming from higher instantons. In fact, the need to complete a Lie group sometimes necessitates the existence of conserved currents from higher order instantons. In the following, when writing the global symmetry of a theory, we write the minimal one consistent with the conserved currents we observe. We write our results for N > 5 odd in table 2, and for N > 4 even in table 3. As is clear already from the analysis of the currents the N = 4; 5 cases are special. In the N = 4 case this is manifested already at the perturbative level as the antisymmetric representation is real and the SUAS(2) UAS(1) symmetry is enhanced to USp(4). Then there are also further conserved currents completing the SUAS(2) UAS(1) representations to USp(4) 2 2 2; Nf = 0. Again, ones. We write our ndings for this case in table 4. In the N = 5 case, the di erence only arises when Nf + 2j j = 10. In this case we nd that there is a further enhancement of SU(2) SU(2) ! G2. This is related to the enhancement to G2 in the USp( 6 ) SU(2) theories mentioned in section 4.3 as, by manipulating brane webs, we nd that the theories SU 9 Nf (2n + 1) + 2AS + (Nf + 1)F and Nf F + USp (2n + 2) USp(2n 2) + 1F are dual (the angle for USp(2n + 2) is relevant only in the Nf = 0 case). One implication of this is that, besides the enhancement revealed from the 1 instanton analysis, there should be an additional enhancement of U(1) ! SU(2) coming from higher instantons. This is also apparent in the N = 2 case as this is necessary 2 8 then there can to complete the Lie group G2. For general N , this can be argued from the Nf F + USp (2n + 2) USp(2n 2) + 1F description. According to the results of [14], as the USp(2n 2) group e ectively sees 2n+3 8For N = 2 the antisymmetric completely decouples and we just get the rank 1 ENf +1 theories. For N = 3 the antisymmetric is just the anti-fundamental so the problem reduces to analyzing SU(3) with fundamentals where this analysis was done in [15, 26, 29] expect the case of Nf + 2j j = 12. However, the brane webs describing these theories are identical to the rank 2 E~Nf theory so these are just dual descriptions of known xed points. = 0 U(1)2 U(1)3 U(1)2 U(1)2 U(1) SU(2)4 U(1) SU(2)2 SU(3) U(1)2 SU(2) SU(5) U(1) SU(2)2 U(1) SU(2)3 SU(4) U(1) SU(2) SU( 8 )a SO(16)b SU(2) SU(3) U(1) SU(2)2 SU(2) SU(4) U(1) SU( 6 ) SO(10) SU(2)2 U(1) SU(2)2 SU(4) SU(2)2 SO(12)d SU(2)2 SU(2)3 SU(4) of SU(2)2 ! G2. Also note that for Nf + 2j j = 10 one of the SU(2) results from contributions of higher instantons and is inferred from a dual description of the xed point. (a) To get this global symmetry requires also two conserved currents that are avor singlet with instanton number (b) To get this global symmetry requires also two conserved currents with instanton number 2 that are in the 7 of SU(7). (c) To get this global symmetry requires also two conserved currents with instanton number 2 that are SU(4) singlets. (d) To get this global symmetry requires also two conserved currents with instanton number 2 that are in the 5 of SU(5). (e) To get this global symmetry requires also several conserved currents with instanton number 2 that are in the 1 and 15 of SU( 6 ), and another two with instanton number 3 that are in the 6 of SU(6). avors, the (0; 2) instanton should provide two conserved currents with charges 1 under SOF (2). These lead to an enhancement of at least U(1)2 ! SU(2)2. Furthermore, as argued in section 4, when Nf > 0 we expect a further enhancement of at least SO(2Nf ) ENf +1, where the U(1) containing the USp(2n + 2) topological symmetry. The minimal implication of these on the SU description is that a further enhancement of U(1) ! SU(2) should occur in this theory. Note that this argument does not hold for the pure case, SU5(2n + 1) + 2AS. Nevertheless, since this enhancement appears to be una ected by integrating out avors, as long as Nf + 2j j = 10, we conjecture that it should occur also U(1) ! for this case, and have included it in table 2. Finally, we want to discuss the cases where we expect a 6d xed point. First we have SU0(2n + 1) + 2AS + 8F , where we nd several conserved currents with the charges: (1; 28; 1; 2); (1; 28; 1; 2); (1; 1; N 2; 4) and (1; 1; (N 2); 4), under SU( 8 ) UAS (1) UB(1). All these currents cannot form a nite Lie group. The rst two seem to suggest that U(1)2 SU( 8 ) is enhanced to the a ne D8(1). The last two then imply that the remaining U(1) should also form an a ne group. SUAS (2) does not appear to be a nized at least at this level. For SU0(2n) + 2AS + 8F , the conserved currents are a bit di erent. First there is one current in the 70 of SU( 8 ). This cannot lead to any nite Lie group, but can form an a ne = 0 U(1) U(1)2 U(1)2 U(1)3 U(1) U(1) U(1)2 U(1)2 SU(2)4 U(1) SU(2) SU(2) SU(4) SO( 8 ) SU(5) SU(2) SO(10)e U(1)2 SU(2) Ea 6 SU(2) SO(10) SU(2)3 Eb 7 SU( 6 ) SO(12)f (a) To get this global symmetry requires also two conserved currents that are avor singlets with instanton number instanton number 2. (b) To get this global symmetry requires also two conserved currents with 2 that are in the 7 of SU(7). (c) To get this global symmetry requires also two conserved currents with instanton number 2 that are SUF (2) singlets. (d) To get this global symmetry requires also two conserved currents with instanton number 2 that are in the 3 of SU(3). (e) To get this global symmetry requires also two conserved currents with instanton number 2 that are in the 6 of SU(4). (f ) To get this global symmetry requires also two conserved currents with instanton number 2 that are in the 10 of SU(5). (g) To get this global symmetry requires also several conserved currents with instanton number 2 that are in the 1; 1 and 15 of SU(6), and another two with instanton number 3 that are in the 6 of SU(6). one E7(1). If n 6= 2 then we also have 4 additional currents, which are singlets of SU(2) SU(8), with charges (4; N 2); ( 4; (N 2)); (4; 2) and ( 4; 2) under UB(1) UAS (1). In light of the enhancement of SU( 8 ) to an a ne group, we also expect these currents to enhance U(1)2 to an a ne group. If n = 2 then we get two conserved currents in the UB(1). These indeed cannot t in a nite Lie group, but Next, we consider the case of SU 3 (2n) + 2AS + 7F . First, we nd a conserved current 2 The last case we consider is SU 3 (2n+1)+2AS +7F . We nd conserved currents in the (1; 35; 1; 12 ), (1; 7; (N 2); 52 ), and (1; 1; 1; 72 ) under SUAS (2) SU(7) UAS(1) UB(1). The rst two cannot t in a nite group, rather forming the a ne E7(1). Like in the other case, we expect the last current to a nize the remaining U(1). In the N = 5 case, there is (5; 1; 4) of USpAS(4) can form an a ne one, B3(1). SU( 8 ) 32 ), under SUAS (2) 2 additional currents in the (1; 7; 2n (5; 7; 52 ) of USpAS (4) can form an a ne one, B9(1). SU(7) to the a ne group D8(1). SUAS (2) does not appear to be a nized at least at this level. If n = 2 then these two currents merge with additional currents to form one current in the UB(1). These indeed cannot t in a nite Lie group, but 2 SU(7) 2; 52 ) and (1; 7; 2; 52 ). These suggest an enhancement UAS (1) UB(1). If n 6= 2 then we also have U(1) SU(3) USp(4) U(1) USp(4) SU(2) SO(7) U(1) USp(4) USp(4) SU(4) SO(13)d SO( 8 ) SU(5) SO(7) U(1) U(1) SO(7) SO(19)f USp(4) Ea 6 USp(4) USp(4) SU(4) USp(4) SO(10) SO(7) Eb 7 USp(4) SU(2) USp(4) U(3) SO(7) USp(4) U(1) SO(7) SU(2) SU( 6 ) SO(15)e = 0 SU(2) USp(4) U(1) = 5 U(1) USp(4) = 1 U(1) = 2 U(1) = 3 U(1) = 4 SO(7) U(1) U(1)2 U(1) U(1) U(1) U(1) USp(4) global symmetry requires also two conserved currents that are avor singlets with instanton number 2. (b) To get this global symmetry requires also two conserved currents with instanton number 2 that are in the 7 of SU(7). (c) To get this global symmetry requires also two conserved currents with instanton number 2 that are in the 3 of SU(3). (d) To get this global symmetry requires also two conserved currents with instanton number 2 that are in the 6 of SU(4). (e) To get this global symmetry requires also two conserved currents with instanton number 2 that are in the 10 of SU(5). 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Gabi Zafrir. Brane webs, 5d gauge theories and 6d \( \mathcal{N}=\left(1,\;0\right) \) SCFT’s, Journal of High Energy Physics, 2015, 1-51, DOI: 10.1007/JHEP12(2015)157