Brane webs in the presence of an O5−-plane and 4d class S theories of type D

Journal of High Energy Physics, Jul 2016

Abstract In this article we conjecture a relationship between 5d SCFT’s, that can be engineered by 5-brane webs in the presence of an O5−-plane, and 4d class S theories of type D. The specific relation is that compactification on a circle of the former leads to the latter. We present evidence for this conjecture. One piece of evidence, which is also an interesting application of this, is that it suggests identifications between different class S theories. This can in turn be tested by comparing their central charges.

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Brane webs in the presence of an O5−-plane and 4d class S theories of type D

Accepted: June -plane and 4d class S theories of type 0 Haifa 32000 , Israel 1 Department of Physics, Technion, Israel Institute of Technology In this article we conjecture a relationship between 5d SCFT's, that can be engineered by 5-brane webs in the presence of an O5 -plane, and 4d class S theories of type D. The speci c relation is that compacti cation on a circle of the former leads to the latter. We present evidence for this conjecture. One piece of evidence, which is also an interesting application of this, is that it suggests identi cations between di erent class S theories. This can in turn be tested by comparing their central charges. Brane Dynamics in Gauge Theories; Field Theories in Higher Dimensions - Brane O5 1 Introduction 2 Preliminaries 2.1 2.2 3.2 SCFT's with marginal deformations 4 Conclusions A Brane motions and free hypermultiplets 1 2 then study the 5d SCFT which can lead to interesting consequences also for the 4d SCFT, see [11] for an example. There are a large number of other 5d SCFT's, some of which can also be engineered by generalizations of brane webs through the addition of orientifold planes [12{15]. In this article we conjecture that compactifying a class of 5d SCFT's that can be engineered by brane webs in the presence of an O5 -plane leads to 4d type D class S theories. This can be motivated by studying the dimension of the Higgs branch, and the global symmetries manifest in the web description, and matching them against the properties of class S theories. The major piece of evidence we provide, which is also an interesting application of this relation, is by testing equivalences between di erent class S theories. Many of the 5d SCFT's we consider can be mass deformed to a 5d gauge theory. In some cases this gauge theory can also be realized by a di erent brane realization, either with the O5 -plane, or without it which leads to a web of the type considered in [4]. In both descriptions the SCFT is realized when all mass parameters vanish. Thus, from 5d reasoning we conclude that these are di erent string theory realizations of the same 5d SCFT. On one hand compactifying this 5d SCFT should lead to one 4d SCFT. On the other hand, as the string theory realizations are di erent, we get seemingly di erent class S theories. Consistency now necessitates that these class S theories are in fact identical. This can be checked by comparing the central charges of these class S theories. We indeed nd that they are equal in all cases checked. The structure of this article is as follows. Section 2 consists of a summary of properties of D type class S theories and brane webs in the presence of an O5 -plane that play a role in this article. In section 3 we present our conjecture and provide supporting evidence for it. We end with some conclusions. The appendix discusses the suspected creation of free hypermultiplets accompanying certain 7-brane manipulations. 2 Preliminaries We begin by reviewing several aspects of class S technology and brane webs in the presence of O5-planes that play an important role in the proceeding discussion. 2.1 Class S technology In this article we will be particularly interested in 4d class S SCFT's given by compactifying the type D (2; 0) theory on a Riemann sphere with punctures. There are known methods to compute various quantities of interest for this class of theories, and in this subsection we shall summarize the ones used in this article. All of these, and much more, can be found in [16, 17]. Like their A type cousins, punctures of D type class S theories are labeled by Young diagrams. For a DN theory, this is given by a Young diagram with 2N boxes, subject to the constraint that columns made of an even number of boxes must repeat with even multiplicity. Furthermore if all the columns are made of an even number of boxes then there are actually two di erent punctures associated with this Young diagram. These are { 2 { HJEP07(216)35 usually called very-even partitions, and the two punctures are usually distinguished by the color of the diagram, red and blue being the common choice. Each puncture has a global symmetry associated to it, where each group of nh columns with the some height h contribute a USp(nh) factor if h is even, or an SO(nh) factor if h is odd. Note that here the constraint that columns made of an even number of boxes must repeat with even multiplicity is important. The contributions from all the punctures then give part of the global symmetry of the SCFT, which may be further enhanced to a larger global symmetry. In some cases we will also want to determine the full global symmetry. In those cases we use the 4d superconformal index. Since conserved currents are BPS operators they contribute to the index, and so knowlcase in [21]. Occasionally a class S theory may contain free hypermultiplets in addition to, or even without, a strongly interacting part. The 4d index then provides an excellent tool for discovering if such a thing occurs, and to nd the number of such free hypers. A good review for these applications of the index of class S theories can be found in [22]. We will also need to evaluate various properties of the SCFT, notably, the spectrum and dimension of Coulomb branch operators, dimension of the Higgs branch, and central charges of the various global symmetries. Their evaluation from the Riemann surface is known and given in [17]. One quantity that will play an important role is the dimension of the Higgs branch for a SCFT arising from the compacti cation on a three punctured sphere. The Higgs branch dimension is then given by: where the compacti ed theory is DN , and the fi's are the contributions of the three punctures. These can be evaluated directly from the Young diagram, and are given by: dH = N + f1 + f2 + f3 f = 4 1 X r j 2 j 2 1 X rj j odd (2.1) (2.2) where rj is the length of the j'th row, and the rst sum is over all rows while the second is only over the odd numbered rows (j = 1; 3; 5 : : :). 2.2 Brane webs in the presence of O5-planes In this subsection we shall summarize some important results regarding brane webs in the presence of O5-planes. Such systems were rst introduced in [12], and further studied in [14, 15]. The particular system we are interested in consists of a group of D5-branes, an O5 -plane parallel to the D5-branes and several NS5-branes stuck on the orientifold 5-plane. Such a stuck NS5-brane leads to a change in the orientifold type: an O5 -plane changes to an O5+-plane and vice versa. We also take the number of such branes to be even so that the asymptotic orientifold 5-plane on both sides is an O5 -plane. We can further add 7-branes on which the 5-branes can end. { 3 { The gauge symmetry living on N D5-branes parallel to an O5 -plane (O5+-plane) is SO(2N ) (USp(2N )). Thus, the resulting 5d SCFT has mass deformations leading to a 5d quiver with alternating SO and USp gauge groups. The matter content includes half-bifundamentals between each USp=SO pair, fundamental hypermultiplets for the USp groups and vector matter for the SO groups. In some cases we can also incorporate spinor matter for some SO groups. We refer the reader to [14] for the details. Besides the USp=SO quiver, these SCFT's have an additional gauge theory description. In the brane web this can be seen by performing S-duality on the con guration where the O5 -plane becomes a perturbative orbifold. Speci cally, an O5 -plane with a full D5brane on top of it is S-dual to an object called an ON 0 plane, which is a C2=I4( 1 )FL orbifold [26, 27]. Studying the gauge theory living on the 5-branes in this background reveals that the 5d SCFT also possess mass deformations leading now to a 5d quiver of SU groups in the shape of a D type Dynkin diagram. We will also want to understand the global symmetry of the SCFT. This can be determined by considering the gauge symmetry living on the 7-branes, which is a global symmetry from the point of view of the 5-branes. For the system we will be considering, there are three contributions given by the D7-branes on each side of the orientifold as well as the (0; 1) 7-branes. At the xed point the D7-branes, on each side of the orientifold, merge on the O5 -plane while the (0; 1) 7-branes merge outside the O5 -plane. We want to consider the classical gauge symmetry on the D7-branes, which may also have an arbitrary number of D5-branes ending on them. To do this we envision moving them through several NS5-branes. Each such transition removes a D5-brane via the Hanany-Witten e ect, so as to arrive at a group of D7-branes with no D5-branes ending on them. Once this con guration is reached, the D7-branes sit on top of an O5+-plane (O5 -plane) if, in the initial con guration, the number of D5-branes ending on the D7-branes is odd (even). The gauge symmetry living on a group of k stuck D7-branes intersecting an O5 -plane (O5+-plane) is USp(k) (SO(k)) where k must be even for the O5 case. For the (0; 1) 7-branes each group of n such 7-branes with the same number of NS5branes ending on them contributes a U(n) global symmetry. Taking the direct product of all three contributions then gives the classical global symmetry. This may be further enhanced to a larger symmetry group. We also wish to determine the dimension of the Higgs branch of the SCFT. This can be done from the web by examining the number of motions of the 5-branes parallel to the 7-branes. For the system we consider, these can be broken into four parts. First we can separate the D7-branes across the O5 -plane, and pairwise separate D5-branes, stretched between neighboring D7-branes, from the orientifold (see the discussion in [14]). Doing this on either side of the orientifold comprises parts one and two, and doing so on the central 5-branes comprises part three. Finally we can break the NS5-branes on the (0; 1) 7-branes and pairwise separate them from the O5 -plane. This gives the fourth part. See gure 1 for an example. { 4 { D7-branes on one side of the orientifold. In this case there are two such directions for each side of the O5-plane, where there is an arrow representing the pair. There are also two directions given by pairwise breaking and separating the four D5-branes stretching between the two groups of D7branes on either side of the O5-plane. Finally we can pairwise break and separate the NS5-branes along the (0; 1) 7-branes. In this case there are two directions, one given by breaking the longer part of the right NS5-branes while the other by separating the remaining part of the NS5-branes from the O5-plane. 3 Statement of the conjecture In this section we present our conjecture that compactifying a class of 5d SCFT's on a circle leads to 4d class S SCFT's given by compactifying the type D (2; 0) theory on a Riemann sphere with punctures. We start with the cases related to compacti cation on a three punctured sphere, and then move on to the general case. 3.1 Isolated 4d SCFT's We start by presenting the class of 5d SCFT's we consider. These can be engineered by a web with an O5-plane, and a system of 5-branes ending on (1; 0) or (0; 1) 7-branes only. More speci cally, asymptotically the orientifold plane is O5 in both directions, and there are a total of 2N D5-branes (in the covering space) ending on the (1; 0) 7-branes whose number may be less than 2N . The number of (0; 1) 7-branes is at most two. If there is only one all asymptotic NS5-branes end on it, while if there are two then one NS5-brane ends on one of the (0; 1) 7-branes, and the rest end on the other. We claim that compactifying the 5d SCFT on a circle of radius R5 and taking the R5 ! 0 limit, without mass deformations, leads to a 4d class S theory given by compactifying the type DN (2; 0) theory on a Riemann sphere with three punctures. The three punctures are associated with the 7-branes, and the 5-branes ending on them, where one puncture is associated with the (0; 1) 7-branes and two with the (1; 0) 7-branes on either side of the orientifold. The punctures associated with the (1; 0) 7-branes can be determined as follows: for each such 7-brane with n D5-branes ending on it we associate a column with n boxes. Starting from highest n to lowest we can combine them to form a Young diagram which is the one associated with the punctures (see gure 2). It is not di cult to see that if { 5 { ciated puncture. In all cases the 6d theory is the (2; 0) theory of type DN where N is determined as explained in the text. there are no stuck 7-branes this generates a Young diagram of DN type on both sides. This also straightforwardly works when there are an even number of stuck 7-branes on both sides of the O5-plane. For an odd number of stuck 7-branes, besides not giving a D type Young diagram, in this case the number of D5-branes on both sides is di erent. Nevertheless, we can still associate a 4d class S theory with this case by moving one of the stuck 7-branes from the side with more D5-branes to the other, doing HananyWitten transitions when necessary. Note however that this may change the theory by the generation of free hypermultiplets. We refer the reader to the appendix for details. This leaves the question of the very-even partitions which should give two di erent punctures. We nd that we can only get one type of puncture. This appears to be related to the question of changing the angle for USp groups engineered by webs in the presence of O5-planes (see [14]). Presumably, if this problem can be resolved then one can also realize the other choice of puncture in the 5d web. { 6 { associated puncture. In all cases the 6d theory is the (2; 0) theory of type DN where N is determined as explained in the text. The punctures associated with the (0; 1) 7-branes are shown in gure 3. Basically if there is a single (0; 1) 7-brane with 2n NS5-branes ending on it then the corresponding puncture is given by a Young diagram with two columns of length 2N 2n 1 and 2n + 1. Alternatively, if there are two NS5-branes, one with a single NS5-brane ending on it and one with 2n 1 NS5-branes ending on it, then the Young diagram associated with this puncture has four columns, one of length 2N 2n 1, another of length 2n 1 and two of length 1. Next we want to provide evidence for this conjecture. First, note that the global symmetry generated by the (1; 0) 7-branes is in accordance with those given by the punctures. As discussed in the previous section, a group of k D7-branes with an even (odd) number of 5-branes ending on it contributes a USp(k) (SO(k)) global symmetry. This indeed agrees with the global symmetry associated with the punctures. This usually also works for the (0; 1) 7-branes. First note that one U( 1 ) global symmetry in the web, identi ed with translations along the O5-plane, decouples from the SCFT. So a single (0; 1) 7-brane has no associated global symmetry while two (0; 1) 7-branes have an associated U( 1 ) global symmetry which is enhanced to SU(2) when the number of 5-branes ending on them is equal. This indeed agrees with the global symmetry expected from the punctures with the exception of the case where the two longest columns in the Young diagram are equal. Then there is an extra U( 1 ) in the puncture, not appearing in the web. { 7 { We can also compare the Higgs branch dimension. As discussed in the previous section, we can break the Higgs branch of the web to 4 contributions. Likewise we can break the Higgs branch of the 4d SCFT to the 4 contributions appearing in equation (2.1). First there are the two contributions coming from breaking the D5-branes on the D7-branes on the two sides of the orientifold. The number of possibilities depends on the number of D5-branes ending on each 7-brane and so on the choice of puncture. We identify these with f1 and f2 in (2.1), where there are two for the two sides of the O5-plane. Next we need to show that these two quantities indeed agree, at least for the maximal puncture. Using (2.2) we nd that: HJEP07(216)35 Our mapping instructs us to map this to 2N (1; 0) 7-branes with one D5-brane ending on each one. The number of possible separations is: fMax = N (N 1) 2 N 1 ! X i = N (N 1) i=1 (3.1) (3.2) agreeing with the class S result. A second contribution comes from pairwise separating the 2N D5-branes stretching between the two stuck D7-branes closest to the NS5-branes. The number of possible separations is always N , and is identi ed with the N in equation (2.1). The nal contribution comes from breaking the NS5-branes across the (0; 1) 7-branes, and pairwise separating them from the O5-plane. The number of possibilities depends on the number of NS5branes ending on each 7-brane and so on the choice of the third puncture. It is thus natural to identify this with f3. Using equation (2.2) we can evaluate f3 for the punctures appearing in gure 3, and we nd that these indeed agree to what is expected from the web. For example consider the minimal puncture, which is the left uppermost puncture in gure 3. Using (2.2) we nd that f3min = 1. This indeed agrees with what is expected from the web, corresponding to the 1 dimensional Higgs branch given by pairwise separating the two NS5-branes from the O5-plane, along the (0; 1) 7-brane. It is straightforward to carry this over also to the other cases. So far we gave some rather general arguments, but next we want to put more stringent tests on this conjecture. Consider a 5d gauge theory which can be engineered by a web of the previously given form. In some cases this gauge theory may also be constructed by a brane web system without the orientifold. In that case the results of [4] suggest that compactifying it to 4d should lead to a class S theory of type A. However, our conjecture implies that compactifying the same theory, in the same limit, should yield a class S theory of type D. Consistency now requires that these in fact be the same SCFT which we can test using class S technology. Particularly, we can compute and compare their spectrum and dimensions of Coulomb branch operators, Higgs branch dimension, global symmetry, and central charges of the non-abelian global symmetries. If our conjecture is correct then these must match. { 8 { There are several types of theories where this can be done. One type are gauge theories of the form USp(2N ) + Nf F . When Nf = 2N + 4; 2N + 5 the webs using an O5-plane are indeed of the previously considered form, and so are conjectured to lead to a 4d D type class S theory (see gure 4). These theories can also be engineered using an O7 plane, which when decomposed leads to an ordinary brane web description [13]. Precisely when Nf = 2N + 4; 2N + 5 these webs are of the form of [4], and so lead to an A type class S theory (see gure 5). Note that the identi cation of the theories in gures 4(b) and 5(b) was already done in [23] while the identi cation of the theories in gures 4(a) and 5(a) is expected from the results of [24, 25]. As mentioned in the previous section, the S-dual of the webs considered here generally leads to D shaped quivers of SU groups. In the case of D3 the D shape degenerates to a linear quiver which can also be engineered using ordinary brane webs. This provides another case where we can compare two di erent constructions of the same gauge theory, where one is expected to give an A type class S theory, while the other a D type. Figures 6, 7 and 8 show two examples of theories in this class. Figure 6 shows the quiver diagram for the 5d gauge theories we consider. Figure 7 then shows the realization of these gauge theories using brane webs without orientifolds, which is possible since these are linear quivers of SU groups. Finally gure 8 shows the realization of these gauge theories using brane webs in the presence of an ON 0-plan, which is possible as these are D3 shaped quivers. This can be mapped to a con guration with an O5 -plane using S-duality. We can test these identi cations by comparing the spectrum and dimensions of Coulomb branch operators, Higgs branch dimension, global symmetry, and central charges of the non-abelian global symmetries of the two SCFT's.1 For the A type class S theory in gure 7(a) we nd using class S technology that it has a Higgs branch dimension of dH = 4N 2 + 10N + 15 and global symmetry SO(4N + 6)8N+4 SU(4)12 where the subscript denote the central charge. We also nd the spectrum of Coulomb branch operators to be: d2 = d3 = 0; d4 = 1 di = di = 1 0 ( 2 i even ) ( 1 i even ) i odd i odd for 4 < i 2N + 1 for 2N + 1 < i 4N + 2 where di stands for the number of such operators with dimension i, and we assumed N > 1 (for N = 1 the matching reduces to the N = 1 case of gures 4(b) and 5(b)). We can now compare these values with the ones for the D type class S theory in gure 8(a). Using class S technology we carried out this calculation nding complete agreement. For the A type class S theory in gure 7(b) we nd using class S technology that it has a Higgs branch dimension of dH = 4N 2 + 4N + 7 and global symmetry SO(4N + 4)8N 1We can also compare the a and c conformal anomalies of the SCFT's. However, for class S theories these are completely determined in terms of the spectrum and dimensions of Coulomb branch operators and the dimension of the Higgs branch [31], and so do not provide an independent check. { 9 { (b) using an O5-plane. The upper part shows the webs describing the 5d SCFT's. These webs are of the form we consider and so we conjecture that their compacti cation on a circle results in the D type class S theories shown below them. (b) using a resolved O7 -plane. The left part shows the webs one gets after resolving the O7 plane. By performing a series of 7-brane motions these can be recast into the the form of [4]. The resulting webs are shown in the middle part of the gure. Upon compacti cation to 4d, these are expected to give the type A class S theories shown on the right. and 8. All groups are of type SU with the CS level written above the group. gure 6(b). The left part shows the webs in a form where the gauge theory description is manifest. Note that we use black X's for (1; 0) 7-branes. By performing a series of 7-brane motions, these can be recast into the the form of [4]. The resulting webs are shown in the middle part of the gure. Upon compacti cation to 4d, these are expected to give the type A class S theories shown on the right. SU(2)28 U( 1 ). We also nd the spectrum of Coulomb branch operators to be: d2 = 0 di = di = ( ( 1 0 2 i even 1 i even i odd i odd ) ) for 2 < i 2N for 2N < i 4N We can now compare these values with the ones for the D type class S theory in gure 8(b). Using class S technology we carried out this calculation nding complete agreement. gure 6(a), now using an ON 0plane. (b) The brane web for the 5d gauge theory shown in gure 6(b), now using an ON 0-plane. The left part shows the webs using the ON 0-plane, which makes the D3 shaped quiver nature of the gauge theories manifest. Again we use black X's for (1; 0) 7-branes. Performing S-duality leads a web using an O5-plane where for ease of presentation we have suppressed the bending due to the change in O5-plane type. By performing a series of 7-brane motions these can be recast into the form we consider. Therefore we conjecture that their compacti cation on a circle results in the D type class S theories shown on the right. We can also consider SO(N ) gauge theories with spinor matter for N 6. In that range the theory can also be engineered using brane webs without the orientifold. Selected examples are shown in gure 9, for cases with a D4 class S theory, and gure 10 for cases with a D5 class S theory. All the class S theories of type D4 were analyzed in [17], and their results indeed agree with our expectation from the 5d SCFT. It is straightforward to also carry this analysis for the theories in gure 10. For the D type class S theory in gure 10(a) we nd it has an SU(4)10 SU(8)12 global symmetry, a 33 dimensional Higgs branch and Coulomb branch operators of dimensions 4; 5; 6. This indeed matches the properties of the A type class S theory in gure 10(a) evaluated using class S technology. We can do the same for the theory in gure 10(b) now nding an SU(2)9 (E7)16 global symmetry, a 35 dimensional Higgs branch and Coulomb branch operators of dimensions 4; 8. This indeed matches the properties of the rank 2 E7 theory as can be computed using class S technology from the construction of this theory with a free hyper given in [19]. We do note that sometimes the class S theory contains free hypers in excess of those expected from 7-brane motions (as explained in the appendix). This seems common for SO(3) and SO(4), and an example for this is given by the rst case in gure 9. Nevertheless, the Higgs branch of the class S theory with the free hypers agrees with that of the web. Thus, the natural interpretation is that the SCFT described by the web in these cases also contains free hypers. As mentioned in [14], the spinors are thought to come from instantons associated class S theories being a compacti cation of the D4 theory where we have used that SO(3) = SU(2), SO(4) = SU(2)2 and SO(5) = USp(4). Up to the free hypers in the rst case, these match what is expected from the gauge theory description, see [17]. of an SU(2) gauge theory broken by a motion on the Higgs branch. Thus, it is tempting to identify these free hypers as directions on that Higgs branch that somehow remain in the low-energy theory. A further consistency check is to match the global symmetry expected from the 5d description to the one of the 4d SCFT calculated using class S technology. As an example, consider an SO(N ) gauge theory with spinor matter. In [28] an analysis of the 1 instanton operators was done for these theories, which in turn suggests that enhancement of symmetries should occur in some cases. The class S theories which we conjecture result from compactifying these theories do not show that symmetry, and in many cases not even the classically visible symmetry. Consistency now requires that the 4d global symmetry is actually enhanced to a larger global symmetry than is visible from the punctures, which we associated D5 theory and the construction using ordinary brane webs leading to an A5 theory. (b) Shows another example now using the rank 2 E7 theory. can test in turn using class S technology. Several examples are shown in gures 11 and 12, where in all cases complete agreement with the results of [28] is found, and further that the Coulomb branch dimension of the class S theory agrees with that expected from the web. Also class S technology calculations are consistent with the two theories in gure 11 being the same theory up to a di ering number of free hypers. The webs are shown on the left, and upon moving the rightmost stuck D7-brane to the leftmost side of the web, they can be cast in the form we consider. Note that this 7-brane motion results in the creation of free hypers. We conjecture that compacti cation on a circle to 4d results in the D type class S theories shown on the right. Above each class S theory we have written its global symmetry evaluated using the superconformal index. These agree with the expected global symmetry of the 5d SCFT determined from the 1 instanton analysis of [28]. 3.2 SCFT's with marginal deformations Next we want to discuss theories related to the compacti cation of the D type (2; 0) theory on a sphere with more than three punctures. Unlike the previous case the reduction now involves several scaling limits where in addition to taking the R5 ! 0 limit one also takes the limit mi ! 1, for several mass parameters mi, while keeping miR5 xed. These then become the marginal deformations that exist in compacti cation of a (2; 0) theory on a sphere with more than three punctures. These mass parameters can be chosen as mi = g152d corresponding to the couplings of gauge groups. These then become marginal gauge group couplings in 4d which are indeed related to their 5d counterparts by g142d gR525d . The actual correspondence is a straightforward generalization of the previous case. The theories in this class are of the same form as the previous ones except that the number of (0; 1) 7-branes is unconstrained save for the demand that they can be partitioned into groups each one in the form of gure 3. We can then do the reduction in the limit where (upper part) and SO(8) + 4V + 2S + 1C (lower part). The webs are shown on the left, where the lower one is already in the form we consider. The upper web can also be cast in this form by moving the rightmost stuck D7-brane to the leftmost side of the web, though this leads to the creation of free hypers. We conjecture that compacti cation on a circle to 4d results in the D type class S theories shown on the right. Above each class S theory we have written its global symmetry evaluated using the superconformal index. These agree with the expected global symmetry of the 5d SCFT determined from the 1 instanton analysis of [28]. each group of (0; 1) 7-branes is in nitely separated from the other. This corresponds to a weakly coupled limit for the gauge theory living on the D5-branes stretched between neighboring groups of (0; 1) 7-branes. It is thus straightforward to conjecture that the resulting theory is given by a collection of isolated SCFT's connected by the gauge groups whose weak coupling limit was taken. This in turn is equal to the compacti cation of a D type (2; 0) theory on a Riemann sphere with punctures where each group of (0; 1) 7-branes and the (1; 0) 7-branes on the two sides of the O5-plane gives a puncture of the type explained in the previous subsection. As an example we consider SO gauge theories with spinors and vector matter, for example the SO(8) and SO(7) theories shown in gure 13. With the chosen matter content, and SO(8) + 4V + 2S (in (c)). These are in a form that we conjecture that when reduced to 4d, in the in nite weakly coupled limit, lead to a compact cation of the D4 theory on the shown four punctured sphere. Indeed the analogue 4d gauge theories are conformal, and as was found in [17] have the shown representation as a compacti cation of the D4 (2; 0) theory. Also note that the 4d theory in (a) also contains a free hyper that from the 5d point of view is generated when moving the right stuck 7-brane to the left side. the webs are of the form expected to lead to a four punctured sphere when reduced to 4d. Furthermore it is clear from the web that the reduction is done in the limit where the SO group becomes weakly coupled, and we expect it to descend to the corresponding SO group with marginal coupling. In fact the matter content is exactly the one required for the analogous 4d theory to be conformal. So we expect the resulting theory to have a completely perturbative description given by an SO gauge group with vector and spinor matter. In fact such 4d theories were constructed, using compacti cation of D type (2; 0) theories, in [17], and we can compare the class S construction they give with the one generated from the web by our prescription, nding complete agreement (see gure 13). This naturally leads to one frame of the many possible frames obtained by taking di erent pair of pants decompositions of the Riemann surface. One may ask whether the other are also manifest in the 5d description. Some of them, the ones corresponding to exchanging the punctures associated with a group of (0; 1) 7-branes, are manifest as we can exchange them also in the web. This leads to 5d lifts of Gaiotto dualities, similar to the cases studied in [9] for the A type theories. In the 4d theory we can exchange any of the punctures while in the 5d theories there does not appear to be a way to exchange the punctures associated with the (0; 1) 7-branes with the ones associated with the (1; 0) 7-branes. This is already apparent from the mapping, since we can essentially get any puncture from the (1; 0) 7-branes, but only a subset from the (0; 1) 7-branes. It would be interesting to nd a generalization of these mappings allowing a description of all types of punctures also with the (0; 1) 7-branes. Finally we wish to discuss the issue of very-even partitions. To do this we draw your attention to gure 13(c), in which there is an example of a class S theory with two veryeven partitions. In these cases there are two possible theories depending on whether the two very-even partitions are of the same or di ering colors. In both cases the theory is an SO(8) gauge theory with four hypermultiplets in the vector representation and two in the spinor. The spinors have the same chirality if the colors are the same, but di erent otherwise. So one can see that the choice of very-even partition is related to the chirality of the spinors. It is currently unclear how one can change the chirality of the spinors in the web, and so how the other choice of very-even partition can be realized. See [14] for a discussion of this point. 4 Conclusions In this article we have proposed a connection between a class of 5d SCFT's engineered by brane webs in the presence of O5-planes and 4d class S theories of type D. It will be interesting to further explore this, and see if further evidence for this proposal can be found. This has several interesting implications for the class S theories involved. First, the 5d theory may have one or more gauge theory descriptions. This implies that the corresponding class S theory has mass deformations leading to the analogous 4d gauge theory. We have also seen that this proposal motivates identi cations of apparently distinct class S theories. Using the work of [29], this then also implies an identi cation of the dual mirror theories, which may lead to interesting 3d dualities. Also there are several other intriguing questions that warrant further exploration. We have seen that there are class S theories of type D for which we do not know the 5d lift, notably the DN theory. It is interesting if there is some generalization of these constructions that may allow the realization of the 5d lifts of these theories as well. On the other hand, there are other brane webs system with an O5-plane that are not of the desired form. It will be interesting to also study their 4d limit. Acknowledgments I would like to thank Oren Bergman and Shlomo S. Razamat 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. after moving the two D7-branes past each other. (c) The brane con guration one gets after performing two T-dualities in the directions parallel to both the D7-branes and the D5-branes, followed by an S-duality, on the system of (a). The vertical wide black lines are NS5-branes and the thin horiozontal lines are D3-branes. (d) The resulting brane con guration after the same transformations are done on the system in (b). A Brane motions and free hypermultiplets In this appendix we wish to discuss the creation of free hypermultiplets as a consequence of brane motions. Consider the two systems shown in gures 14(a) and (b), related by a 7-brane motion. The system in (a) has a one dimensional Higgs branch, corresponding to breaking the extended D5-brane on the 7-brane. However, in (b) there is now a two dimensional Higgs branch so it appears that this motion has a ected the theory. Furthermore, as the Coulomb branch moduli hasn't changed, the most logical explanation is that a single free hyper was created in this motion. Next, we wish to show that this leads to known relations in other dimensions. Consider taking the systems in gure 14(a)+(b), performing two T-dualities and an S-duality, so that the 7-branes become NS5-branes and the D5-branes become D3-branes. This leads to the brane con guration of gures 14(c) and (d) which are again related by brane motion, now of the NS5-branes. These motions and the dualities they imply for the theories on the D3-branes, where considered in [30] who found that indeed the theory in (d) is the one in (c) with the addition of a free twisted hypermultiplet. As an additional example consider the theory shown in gure 15(a). This describes the 5d T4 theory. Figure 15(b) shows the system after several brane motions which also involves one motion of the type considered before. Thus, we conclude that the theory shown in gure 15(b) is the 5d T4 theory with a single free hyper. This theory is also in the form of [4] so it leads to 4d class S theory. This theory is indeed known to be T4 with a single free hyper [31]. at the con guration in (b). Note that these include a transition of the type shown in gure 14. (c) and (d) are the mirror duals of the 3d analogue class S theory for the theories in (a) and (b) respectively. Another interesting test is given by employing the relation of [29] between the 4d class S theory and the 3d mirror dual. These are shown in gure 15(c) and (d) for the two theories involved. The preceding imply that these two are identical up to a free twisted hyper, and indeed this is just an application of the previous duality of [30] to the central node. directions parallel to both the D7-branes and the D5-branes, and an S-duality. We can ask what happens in the more general case shown in gure 16(a). Counting the Higgs branch moduli we nd that it changes by jn kj so the natural expectation is that jn kj free hypers are generated. We can again consider the related 3d system shown in gure 16(b). If we are correct then these must be dual up to jn kj free twisted hypers. This has indeed been argued in [32] from 3d reasoning. We can also consider the analogue application for class S theories. Figure 17 shows similar motions as the ones done for the T4 theory now done for TN . This leads to identifying the two resulting class S theories up to N 3 free hypermultiplets. For N > 4 these theories are bad in the sense of [19] so index analysis is unknown for them. Also application of class S technology leads to a negative number of Coulomb branch operators and so is not applicable in this case. Yet, the preceding analysis suggests that these theories exist and are given by the TN theory with free hypermultiplets. One supporting evidence is to consider the 3d mirror duals shown in gure 17, which are indeed related by the duality of [32] on the central node. Finally we wish to discuss the implications of these on the systems considered in this article. This phenomenon still occurs with little changes due to the presence of the orientifold plane. For example consider the systems shown in gure 18(a) and (b). Counting the Higgs branch dimensions we see that in the case of (a) we get one free hyper while in the case of (b) we get two. It is straightforward to generalize this to other cases. We next ask whether we can test this also in this case against known phenomena in 4d or 3d. The discussion in section 3 implies that this should lead to similar relationships between di erent class S theories of type D, but it appears at least one of them is bad (again in the sense of [19]) making explicit comparison di cult. One interesting implication of this is a generalization of the dualities in [32] also to SO or USp gauge theories. To our knowledge these have not been previously discussed from purely 3d reasoning. It will be interesting to further explore this. leads to the web in (b). 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Gabi Zafrir. Brane webs in the presence of an O5−-plane and 4d class S theories of type D, Journal of High Energy Physics, 2016, 35, DOI: 10.1007/JHEP07(2016)035