5d fixed points from brane webs and O7-planes

Journal of High Energy Physics, Dec 2015

We explore the properties of five-dimensional supersymmetric gauge theories living on 5-brane webs in orientifold 7-plane backgrounds. These include USp(2N) and SO(N) gauge theories with fundamental matter, as well as SU(N) gauge theories with symmetric and antisymmetric matter. We find a number of new 5d fixed point theories that feature enhanced global symmetries. We also exhibit a number of new 5d dualities.

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5d fixed points from brane webs and O7-planes

HJE xed points from brane webs and O7-planes Oren Bergman 0 1 Gabi Zafrir 0 1 0 Haifa , 32000 , Israel 1 Department of Physics, Technion, Israel Institute of Technology We explore the properties of ve-dimensional supersymmetric gauge theories living on 5-brane webs in orientifold 7-plane backgrounds. These include USp(2N ) and SO(N ) gauge theories with fundamental matter, as well as SU(N ) gauge theories with symmetric and antisymmetric matter. We nd a number of new 5d xed point theories that feature enhanced global symmetries. We also exhibit a number of new 5d dualities. Brane Dynamics in Gauge Theories; Field Theories in Higher Dimensions - 1 Introduction 3 O7+ and SO gauge theories 4 SU(N ) with an antisymmetric 5 SU(N ) with a symmetric 6 Conclusions A Instanton partition functions and extraneous states A.1 USp(2N ) A.2 SO(M ) A.3 SU(N ) 1 Introduction Although 5d gauge theories are perturbatively non-renormalizable, in many N = 1 supersymmetric cases they are UV complete [1{3]. Namely, there exist interacting 5d N = 1 superconformal theories with relevant deformations corresponding to an inverse Yang-Mills coupling of a 5d N = 1 supersymmetric gauge theory. The superconformal theory is in some sense the in nite coupling limit of the gauge theory. The existence of such xed point theories is also suggested by the existence of a (unique) 5d superconformal algebra F (4) (which has a bosonic subgroup SO(5; 2) do not admit a Lagrangian description. SU(2)R). However these UV xed point theories { 1 { A useful way to describe such theories is using 5-brane webs in Type IIB string theory [4, 5]. This realizes a 5d SCFT as an intersection of 5-branes at a point. The moduli and mass parameters of the SCFT are described as motions of the internal and external 5-branes, respectively. In particular, the 5d SCFT may posses a deformation leading to a low-energy gauge theory described by a simple 5-brane con guration containing stacks of D5-branes. This allows us to study various aspects of 5d superconformal theories and the associated gauge theories. First, it provides a new way to classify 5d gauge theories that have UV since the gauge groups and matter content are constrained by the brane construction. Not all 5d supersymmetric gauge theories can be described by a 5-brane web. For example the theory with SU(2) and NF = 9 does not have a 5-brane web realization, and correspondingly does not possess a UV xed point. For the SU(2) theory the 5-brane webs reproduce the eld-theoretic classi cation of [3].1 However in several other cases the 5-brane web classi cation extends the eld-theoretic one, and admits 5d supersymmetric gauge theories that are ruled out by [3]. A simple example of this is the SU(2) SU(2) quiver theory. This theory is ruled out by the eld-theoretic argument, since the e ective coupling of one gauge group always develops a singularity on the Coulomb moduli space of the other gauge group. On the other hand there is a 5-brane web construction of this theory [4]. The 5-brane web exhibits this singularity in the collision of a pair of internal NS5-branes. But this also suggests that it is resolved by the appearance of a new massless state described by a D1-brane between the two NS5-branes, corresponding to an instanton-particle in the gauge group with the diverging coupling. This state is not accounted for in the eldtheoretic computation. We do not have a precise understanding of how the singularities on the Coulomb branch get resolved in all cases where the 5-brane web construction admits additional theories, such as for example SU(N ) with NF = 2N + 1, but it is plausible that in all such cases there are massless non-perturbative states. 5-brane webs also allow us to uncover dual gauge theories in ve dimensions. By \duality" in the context of 5d supersymmetric gauge theories we mean two (or more) IR gauge theories resulting from mass deformations of the UV xed point theory. In particular, since in 5d masses are real, the IR theories obtained by positive and negative mass deformations may be di erent. In particular these may be di erent gauge theories related by a continuation of the Yang-Mills coupling past in nity. This can be seen in the 5-brane web construction by reversing the deformation leading to the original gauge theory, and using the SL(2; Z) symmetry of Type IIB string theory to transform the web into a con guration with D5-brane stacks. Finally, 5-brane webs can also motivate and assist in demonstrating non-perturbatively enhanced global symmetries. 5d gauge theories possess U(1) global symmetries associated to the instanton number currents, jI = Tr(F ^ F ), in each non-abelian gauge group factor. In some cases the instanton operators associated to these currents provide additional conserved currents, and lead to a larger global symmetry than is apparent in the gauge 1For SU(2), a 5d xed point exists if NF 7 [1]. However a simple 5-brane web construction exists only for NF 4 [4]. The 5-brane web realizations of 5 NF 7 were found in [6]. These involve avoided intersections of external 5-branes, of the type we shall also encounter here. { 2 { theory Lagrangian. Such is the case, for example, for the SU(2) theory with NF 7, where the classical SO(2NF ) U(1)I symmetry is enhanced by instantons to ENF +1 [1]. Since conserved currents belong to BPS multiplets of operators, enhancement of global symmetries is exhibited by the 5d superconformal index, which can in turn be computed for the IR gauge theory using localization [7{10]. The crucial ingredient is of course the instanton contribution, which can be obtained from known expressions for instanton partition functions in four dimensions. However in some cases corrections are required [8, 10{12]. In the 5-brane web description these corrections are identi ed with extraneous states, corresponding to strings external to the web whose contributions have to be removed by hand from the partition function. Most 5-brane web constructions so far have been for theories with SU(N ) gauge groups and matter elds in fundamental or bi-fundamental representations.2 Our main purpose in this paper is to extend the study of 5d N = 1 gauge theories and xed point theories via 5-brane webs to theories with USp(2N ) and SO(N ) gauge groups, as well as to SU(N ) theories with rank-two antisymmetric and symmetric matter. We will do this by including orientifold 7-planes. We will exhibit 5d xed point theories, some of them new, with relevant deformations given by gauge theories of the above form. We will also nd new 5d dualities, and examples of enhanced global symmetry involving the above types of theories. The outline for the rest of the paper is as follows. Section 2 is devoted to USp(2N ) theories, constructed using 5-brane webs in an O7 plane background. In section 3 we construct SO(M ) theories using an O7+ plane. In sections 4 and 5 we construct SU(N ) theories with rank 2 antisymmetric and symmetric matter, by adding a fractional NS5brane to the O7 and O7+ plane, respectively. We conclude in section 6. We also include an appendix containing a discussion of corrections to instanton partition functions in the relevant cases of USp(2N ) and SO(M ). 2 O7 and USp gauge theories A classical 5-brane web con guration for a pure USp(2N ) theory (without an antisymmetric matter eld) is shown in gure 1a. This includes an orientifold 7-plane of type O7 , which is parallel to the various 7-branes on which the external 5-branes end.3 This is essentially a 5d generalization of the 4d constructions in [14{16]. In the gure we show the covering space, including two copies of the reduced web related by a re ection across the origin. We have also included two copies of the cut associated with the monodromy of the O7 plane, MO7 = T 4, where HJEP12(05)63 2One exception being [13] which contains a short discussion on webs for USp(2N ) and SO(N ) gauge groups using O5 planes. 3That this web does not include an antisymmetric hypermultiplet is seen from the inability to separate { 3 { (2.1) (N + k + 2, 1) O7 (a) (1,1) The physical space is the upper half plane, with the left and right halves of the cut identi ed. The discontinuity of (p; q)5-brane charges across the cut corresponds to a clockwise action of the monodromy. The bare 5d Yang-Mills coupling g0 2 is given by the separation along HJEP12(05)63 the cut. this below. We are still left with a choice of an integer k, corresponding to the D5-brane charge of one external 5-brane. Naively the di erent choices are all related by T 2 SL(2; Z), which shifts k ! k + 1, and therefore all describe the same USp(2N ) gauge theory. On the other hand, we know that the USp(2N ) theory admits a discrete theta parameter associated with 4(USp(2N )) = Z2. Apparently, the O7 plane is not invariant under T , but only under T 2. The theta parameter is then related to the parity of k. We will say more about At the quantum level, the O7 plane is resolved into a pair of mutually non-local (p; q) 7-branes, whose combined monodromy is given by that of the O7 plane [17].4 There is some ambiguity in the choice of the (p; q) charges of the two 7-branes. In particular, the MO7 = element T 2 SL(2; Z) transforms (p; q) ! (p + q; q), but clearly leaves the total monodromy T 4 invariant. One common choice for the (p; q) charges of the 7-branes is f(2; 1); (0; 1)g. Another, related by T , is f(1; 1); (1; 1)g. We will now argue that these two resolutions correspond to physically distinct O7 planes, and more generally that there are two variants of the O7 plane related by T , at least in the presence of the 5-brane web. Taking the 7-brane charges to be f(1; 1); (1; 1)g, we get the brane web shown in gure 1b (in the reduced space). Note the change in the 5-brane charges across the two cuts. We can get a simpler presentation by moving the 7-branes outside. Accounting for brane creation and for the e ect of moving the cuts, we end up with the 5-brane web shown in gure 1c. In this presentation the fact that the gauge group is USp(2N ), rather than say SU(2N ), is not immediately obvious. It follows from the constraint imposed on the N -junctions by the s-rule. We can now understand the connection between k and the theta parameter. Consider the N = 1 case, namely the USp(2), or SU(2), theory. We identify the k = 1 web with the = theory, and the k = 2 web with the = 0 theory. Furthermore, shifting k ! k + 2 leaves invariant. If we instead resolve the orientifold plane into 7-branes with charges f(2; 1); (0; 1)g 4The monodromy associated with a (p; q)7-brane is given by Mp;q = 1 pq q2 1 + pq p2 ! N + k + 2 gure 1c, but with k ! k + 1. This describes the USp(2N ) theory with the other value of . We conclude from this that, in the presence of the 5-brane web, there are two physically distinct variants of the O7 plane, one of which is resolved to a 7-brane pair with charges f(1; 1); (1; 1)g up to an action of T 2n, and the other to a pair with charges f(2; 1); (0; 1)g up to an action of T 2n. It would be interesting to see this directly from the point of view of the orientifold. This is presumably also related to the transition between the two values of in the Type I' brane construction given in [18]. 2.1 Flavors Matter in the fundamental representation ( avor) can be added by attaching external D5branes. Requiring that external 5-branes do not intersect (which would lead to additional massless degrees of freedom) leads to the condition that NF 2N + 4 ( gure 2a), in agreement with the classi cation of [3]. We claim however that also with NF = 2N + 5 one remains in the realm of consistent 5d theories. In particular, the N = 1 case is the rank one E8 theory.5 The 5-brane web for NF = 2N + 5 is shown in gure 2b. The dangerous intersection is avoided as a consequence of the s-rule. This is similar to the situation for SU(N ) with NF = 2N + 1 [19], which is also one more avor than allowed by [3]. As before, the O7 -plane is resolved quantum mechanically into a pair of 7-branes, and one can obtain an alternative 5-brane web realization of the theories by Hanany-Witten transitions. In particular for N = 1 we get the familiar 5-brane webs for SU(2) with NF = 1; 2; 3 and 4 avors, as well as those describing NF = 5; 6 and 7 avors (the E6; E7 and E8 theories) [6]. The 5-brane webs shown in gure 2 suggest that the global symmetry is enhanced at the xed points in these cases. For NF = 2N + 4 the classical global symmetry of the IR gauge theory is SO(4N + 8)F U(1)I . The pair of parallel external legs suggests that the U(1)I factor is enhanced to SU(2), like in the case of SU(N )0 with NF = 2N [8]. For 5The higher rank E8 theories correspond to USp(2N ) with NF = 7 and an antisymmetric hypermultiplet. For N = 1 the antisymmetric eld is a singlet and decouples. { 5 { HJEP12(05)63 NF = 2N + 5 on the other hand, the parallel external legs, D5-branes in this case, suggest that the classical global symmetry SO(4N + 10)F U(1)I is enhanced to SO(4N + 12). This can be seen explicitly in the 5d superconformal index, the key ingredient of which is the contribution of instanton states corresponding to the 5d lift of the multi-instanton partition function [7]. In the above two cases the instanton computation exhibits the same kind of pathologies associated with extraneous \decoupled" states that were encountered in other cases in [8, 19], such as for SU(N ) with NF = 2N and NF = 2N + 1. The relevant states in the present cases are shown in red in gure 2. For NF = 2N + 4 it is a D-string between the external parallel NS5-branes, and for NF = 2N + 5 there is a fundamental string between the avor D5-branes (or their images) and the extra external D5-brane, and a 3-pronged string attached to the color D5-branes (or their images). Note that all of these carry two units of instanton charge.6 This follows from the fact that the D-string worldvolume gauge symmetry is O(2), which is the ADHM dual group for two instantons of USp(2N ). The minimally instanton-charged object corresponds to a fractional D-string that intersects the O7-plane, and cannot move away from it. The string states in the NF = 2N + 5 web are also charged under other symmetries. The fundamental string state is in the vector representation of SO(4N + 10)F , and the 3-pronged string is in the fundamental representation of USp(2N ). The contribution of these states to the 2-instanton term then has the form Z2 = x 2 (1 xy) 1 x y [charge factor] ; (2.3) where x and y are the fugacities associated with the Cartans of SO(4) SO(5; 2), and the [charge factor] results from the quantization of the fermionic zero modes originating from the matter hypermultiplets. This must be subtracted from the 2-instanton partition function to obtain a consistent result (see appendix for details). In the case of NF = 2N +4 we expect the subtraction to plethystically exponentiate to a correction factor for the full multi-instanton partition function7 Zc = P E 4 2 (1 x2q2 xy) 1 3 x 5 Z : y (2.4) We verify this up to the 4-instanton level in the appendix. In the NF = 2N + 5 case, the presence of the gauge-charged state complicates the counting, and we can only perform the subtraction at the basic 2-instanton level. Combining the corrected instanton partition functions with the perturbative contributions from the gauge and matter supermultiplets we nd that the superconformal indices are given by: INF =2N+4 = 1 + x2 1 + Ad SO(4N+8) + q2 + + O(x3) ; (2.5) 1 q2 bare YM coupling, i.e., the horizontal separation. 6The instanton charge of the fundamental string state can be seen by the dependence of its mass on the 7In the case of SU(2) + 6F , such a correction was also noticed in [10], and for USp(4) + 8F in [21]. { 6 { and SO(4N + 12). 2.2 The x2 terms correspond to the contributions of conserved current multiplets. We see that the 2-instanton states provide additional charged currents. For NF = 2N + 4 these lead to an enhancement of U(1)I to SU(2), and for NF = 2N + 5, of SO(4N + 10)F U(1)I to Since we do not have a simple (perturbative) description for the S-dual of the O7-plane, we cannot identify the S-dual gauge theory directly in these cases. However for N = 1 we already know the answer in some cases. The USp(2) = SU(2) theory with NF 7 is a self-dual gauge theory description of the ENF +1 xed point theory [20]. We can use this to determine the duals in higher rank cases. Let us consider the speci c class of theories with gauge group USp(2N ) and NF = 2N + 2 fundamental hypermultiplets. This is an interesting class of examples, since it is related by dimensional reduction, as in [19], to a duality in four dimensions. The simplest interesting case is USp(4) with NF = 6. The orientifold 5-brane web for this theory is shown in gure 3a. The S-dual web, gure 3b, corresponds to the same UV xed point, but describes a di erent IR gauge theory. The dual theory appears to be a linear quiver with gauge group SU(2) SU(2), but the matter content is not obvious. We can identify it more precisely by \ungauging" the rst SU(2) factor, leading to the web in gure 4a. The S-dual of this web, gure 4b, corresponds to SU(2) with NF = 6. Since this theory is self-dual, the original web also describes SU(2) with NF = 6, albeit with only an SU(2) SO(12)F exhibited manifestly. The dual of USp(4) with NF = 6, gure 3b, therefore corresponds to a gauging of SU(2) SO(12)F in the SU(2); NF = 6 theory. The resulting gauge theory is the linear quiver theory SU(2) SU(2) + 4. The remaining matter global symmetry is SO(8)F SU(2)BF , where SU(2)BF is associated to the bi-fundamental hypermultiplet.8 We can therefore describe the 5d SCFT corresponding to the webs of gure 3 as gauging an SU(2) E7 in the rank one E7 theory and owing to the UV. The dual gauge theory descriptions of the resulting rank two SCFT correspond to di erent embeddings of SU(2) in E7, in one case leaving SO(12)F and in the other SO(8)F SU(2)BF . This is actually the 5d lift of the second 4d N = 2 duality of Argyres and Seiberg [22], which relates the strong coupling limit of the 4d USp(4) + 6 superconformal gauge theory to the weak gauging of an SU(2) E7 in the Minahan-Nemeschansky E7 theory [23, 24]. Generalizing to USp(2N ) with NF = 2N + 2 is straightforward (see gure 5). The S-dual web corresponds to the linear quiver theory SU(2) SU(2)0N 2 SU(2) + 4. The shared UV xed point is a 5d rank N SCFT with a (2N + 3)-dimensional space of mass parameters, and corresponds to gauging an SU(2) inside a particular 5d rank N the web along the lines described in [8, 19]. The latter has dual IR descriptions as the quiver theory 2 + SU(2)0N 2 SU(2) + 4, or as a USp(2N 2) theory with NF = 2N + 2. The reduction to 4d then gives a duality between the 4d N = 2 superconformal gauge theory with gauge group USp(2N ) and NF = 2N + 2, and an SU(2) gauging of a particular isolated 4d N = 2 SCFT given by the dimensional reduction of the above 5d rank N SCFT. We can identify the 4d SCFT according to the classi cation of [25] in terms of a 1 3-punctured sphere with a speci c set of punctures, by manipulating the 5-brane web to a standard triple 5-brane junction form, as shown in gure 6. This represents the theory as a limit of the T2N theory, in which two of the maximal punctures (corresponding to the fully symmetrrized 2N -box Young tableau) are replaced by non-maximal punctures; an (N; N ) puncture (corresponding to a Young diagram with two columns of N boxes), and an (N 1; N 1; 1; 1) puncture. This 4d duality rst appeared in [26]. { 8 { N + 1 N + 1 N + 1 (1,N 1,N 1,1) 2N HJEP12(05)63 N +1 N 1 N +1 plane with an O7+ plane, we obtain a 5-brane web for an SO(2N ) gauge theory. The monodromy matrix of the O7+ plane is given by MO7+ = T 4. The 5-brane web for the pure SO(2N ) theory is shown in gure 7a. This is the exact quantum con guration; unlike the O7 -plane, the O7+-plane is not resolved into 7-branes. As before, k ! k + 1 under T . Since there is no additional parameter in the SO(2N ) theory, the web must describe the same theory for all k.9 This implies that, unlike the O7 plane, the O7+ plane must be invariant under T . A slight generalization of this web including a fractional D5-brane gives the pure SO(2N + 1) theory ( gure 7b). 3.1 Flavors The condition in [3] for an SO(M ) gauge theory to come from a 5d xed point is NV M 4. In the 5-brane webs this is again seen as the condition of no intersections. But, as in other cases, the 5-brane web construction appears to imply that one additional avor is allowed. The 5-brane webs for SO(2N ) with NV = 2N N +k 2 As before, the 5-brane webs in these cases exhibit extraneous instanton-charged states (shown in red in gure 8) that are not part of the 5d gauge theories, and whose contribution to the instanton partition function must be removed by hand. Here they carry the minimal instanton charge, since there is no fractional BPS D-string. This can also be understood from the fact that the D-string worldvolume gauge symmetry is SU(2), which is the ADHM dual gauge group for one instanton of SO(M ). For SO(M ) with NV = M 4, the proposed correction factor is identical to the one in (2.4), except with q2 replaced by q. This is veri ed in the appendix to the 2-instanton level. In special cases one can also compare with other approaches. For SO(4) we compared with the instanton partition function of SU(2) SU(2) (without matter elds) [7], nding complete agreement. We have also compared the result for SO(5) with NV = 1 with that of USp(4) with NA = 1 [7], again nding complete agreement. Going beyond the 2-instanton level is quite di cult. Taking into account the correction factor, the superconformal index takes the form, INV =M 4 = 1 + x2 1 + Ad USp(2M 8) + q + + O(x3) (3.1) 1 q exhibiting the enhancement of U(1)I to SU(2). For NV = M 3, the subtraction can only be carried out at the one-instanton level, and has the form of (2.3), with the fundamental string state contributing charges in the fundamental representation of the USp(2M 6) avor symmetry, and the 3-pronged string contributing charges in the vector representation of the SO(M ) gauge symmetry. The analysis for higher instanton level is made complicated by this gauge charge. See the appendix for details. For SO(4) with NV = 1, we compared with the 1-instanton partition function for SU(2) SU(2) with a bi-fundamental [8], and nd complete agreement. Including the corrected 1-instanton contribution, the superconformal index is given by INV =M 3 = 1 + x2 1 + Ad USp(2M 6) + q + The x2 terms in the fundamental representation of USp(4N 6)F point to an enhancement of USp(4N 6)F U(1)I to USp(4N 4). But this also requires the existence of avorsinglet currents carrying two units of instanton charge, which we are currently unable to Both of these enhancements agree with the results of [27] using simpli ed instanton demonstrate. analysis. As before, we can ask whether the continuation past in nite coupling yields a dual gauge theory. In the case of the O7+ plane we cannot describe the S-dual webs in terms of gauge theories, since we do not have a simple case in which there is a known dual gauge theory description. However we may still be able to provide an alternative formulation of the theory in terms of gauging a subgroup of the global symmetry in a lower rank SCFT, which in turn has an IR gauge theory description. Let us take the SCFT corresponding to SO(6) with NV = 2 as our starting point. The global symmetry of the UV xed point is SU(2) USp(4)F , with the SU(2) realized nonperturbatively in the gauge theory. In the 5-brane web, gure 9a, the SU(2) is associated with the pair of parallel external NS5-branes. Gauging it is described by the 5-brane web of gure 9b. This corresponds to a SCFT with an IR description as SO(8) with NV = 2. So although we do not have a dual gauge theory for the latter, we can describe it alternatively as an SU(2) gauging of a rank 3 SCFT which has an IR description as SO(6) with NV = 2. This can be generalized in a straightforward way to SO(2N ) with NV = 2N 6, leading to an alternative description as the linear quiver with gauge group factors SU(2)N 3, where the last factor gauges the SU(2) part of the global symmetry of the aforementioned rank 3 SCFT. 4 SU(N ) with an antisymmetric Incorporating matter in representations other than the fundamental or bi-fundamental in 5-brane webs is notoriously hard. Even in the simplest example of USp(2N ) with an antisymmetric hypermultiplet (corresponding to the rank N E1 or E~1 theories), where the 5-brane web is known, the existence of the antisymmetric eld is only understood indirectly, by analyzing the symmetry on the Higgs branch [28] (In fact there is a remaining puzzle in this case which we will review and resolve below). Using the same strategy, we also proposed a 5-brane web for SU(2N ) with an antisymmetric hypermultiplet in [19].10 Here we will give a more direct construction using the O7 plane, which makes the antisymmetric hypermultiplet manifest. The construction involves a fractional NS5-brane that is stuck on the O7 plane. Analogous constructions exist in 4d [ 16, 29 ]. 4.1 The orientifold 5-brane web for SU(2N ) with NA = 1 is shown in gure 10a. Now di erent values of k correspond to distinct theories since the NS5-brane is not invariant under T 2 10Special cases of webs with two antisymmetric hypermultiplets appeared in [8]. HJEP12(05)63 (0,1) N(0,1) sponding to the antisymmetric hypermultiplet. SL(2; Z). The integer k is the CS level of the theory. Note that for a vanishing CS level, k = 0, the web is re ection symmetric, corresponding to charge-conjugation symmetry (which the CS term breaks). The hypermultiplet in the antisymmetric representation corresponds to an open string connecting the D5-branes on either side of the NS5-brane. The position of the external NS5-brane in the plane transverse to the O7 plane corresponds to the mass of the hypermultiplet, and its position along the O7 plane corresponds to its VEV, namely to the Higgs branch of the theory. On the Higgs branch the web reduces to that of USp(2N ), as it should. The orientifold web with the NS5-brane appears to exhibit 2N Coulomb moduli, since it has 2N faces. However they are not independent. There is one constraint coming from the fact that the NS5-brane cannot detach from the O7 plane. The number of independent parameters is then 2N 1, the dimension of the Coulomb branch of SU(2N ). To our knowledge, the resolution of the O7 plane with a fractional NS5-brane has not been previously studied, but one can make a conjecture based on the resolution of the bare O7 plane. The only possibility that makes sense is that the combination is resolved into a (2; 1) 7-brane and a (0; 1) 7-brane with the NS5-brane ending on it ( gure 10b). Using this, together with a couple of HW transitions and a T 1 transformation, we arrive at the web shown in gure 10c. This is precisely the web originally proposed in [19]. As an aside, we can now resolve two puzzles from [28]. The rst has to do with the 5-brane web resolution of the classical brane con gurations for the theories corresponding to the orbifolds without vector structure. For example for the Z2 orbifold the theory is SU(2N )0 with NA = 2. The classical Type IIB brane con guration consists of 2N D5branes on a circle, with two O7 planes at antipodal points, each supporting a fractional NS5-brane, gure 11a. We can now identify the resolved con guration, gure 11b, and using manipulations similar to the ones above, obtain a 5-brane web construction, gure 11c. O7 O7 O7 N (0,1) N (0,1) di erently, replacing (1; 1) + (1; 1) by (2; 1) + (0; 1) for one. Replacing both yields The second puzzle has to do with the 5-brane web construction of the USp(2N ) theory with NA = 1 ( gure 12). While this web exhibits the correct Coulomb and Higgs branches, it does not admit a deformation corresponding to giving a mass to the hypermultiplet. It exhibits only one mass parameter, corresponding to the YM coupling. We can now provide an alternative 5-brane web for the same theory, which does admit the hypermultiplet mass deformation. If we attach a fractional NS5-brane to one of the O7 planes the theory remains the same; it is still USp(2N ) with NA = 1. This is the \Z1" orbifold. However the 5-brane web obtained from resolving this con guration is di erent, and now admits an extra deformation corresponding to the mass of the antisymmetric hypermultiplet ( gure 13). 4.2 The orientifold web for SU(2N + 1) with NA = 1 is obtained by adding a fractional D5brane ending on the fractional NS5-brane, gure 14a. Note that this does not a ect the NS5-brane, since the net number of D5-branes ending on it vanishes. This con guration would not be possible in the absence of the fractional NS5-brane; the consistency conditions of [30] would require an even number of D5-branes. The integer k again determines the CS level of the theory, which must now be half-odd-integer due to the parity anomaly associated with the antisymmetric matter multiplet.11 11In general, there is a parity anomaly if the cubic Casimir of the matter representation is odd. For the antisymmetric representation of SU(M ) the cubic Casimir is M (1,0) O7 (a) It is important here that the fractional D5-brane is \broken" on the fractional NS5brane. In particular, the fractional NS5-brane can now detach from the O7 plane by combining with the fractional D5-brane, as shown in gure 14b, giving an extra modulus for a total of 2N , in agreement with the dimension of the Coulomb branch for SU(2N + 1). This also implies that there is no Higgs branch, since the NS5-brane cannot be separated from the fractional D5-brane. This is also consistent with the gauge theory. For SU(2N ) the Higgs branch is spanned by the baryonic operator 1 SU(2N + 1) there is no such gauge invariant operator.12 2N A 1 2 A 2N 1 2N . For There is also an obvious proposal for the resolution of the O7 plane in this case, gure 14c. 4.3 We can again add matter in the fundamental representation by attaching external D5branes. The condition of [3] for a xed point to exist for SU(M ) with NA = 1 is NF + 8 M . In this case we nd a very di erent bound. The orientifold 5-brane web M + 5. The CS level is either k or k 12 , depending on whether NF + M is even or odd, respectively. In particular we can have an arbitrarily large rank. The bound is saturated by the web with the avoided intersection shown (for M = 2N and k = 0) in gure 15c. The two other interesting cases are NF + 2jkj = M + 3 and NF + 2jkj = M + 4, shown (again for M = 2N and k = 0) in gures 15a and 15b, respectively. These have parallel external NS5-branes. In all three cases we exhibit some of the extraneous instanton-charged states that should be removed from the instanton partition function. We expect these three theories to exhibit enhanced global symmetries in the UV, and in particular we expect the non-abelian avor symmetry to be enhanced in the maximally avored theory. In fact, these theories were recently studied in [31], using simpli ed instanton analysis, whose nding is indeed consistent with the enhancement suggested by these webs. 4.4 Duality In formulating duality conjectures in this case we will follow the strategy we used for the USp(2N ) theories. We begin with a theory whose dual we know, and which we can engineer as a 5-brane web, and then gauge part of its global symmetry (and maybe add avors) to obtain a new theory, whose dual we infer from the S-dual web. 12There would be a Higgs branch if we added a hypermultiplet in the fundamental representation. The corresponding operator is 1 2N+1 A 1 2 A 2N 1 2N 2N+1 . N N +2 N N N +2 N +2 N N NF = 2N + 5, CS = 12 . describes the same theory (we have taken k = 0). In our rst example we begin with the self-dual SU(2)+6 theory. This is an IR gauge theory description of the rank one E7 theory. The antisymmetric matter multiplet is neutral in this case. The web for this theory and its S-dual in the present description are shown in gure 16. The dual web makes manifest an SU(3) subgroup of the global symmetry, acting on the three avors corresponding to the three external D5-branes. Now consider the 5brane web shown in gure 17a. This corresponds to gauging this SU(3) and adding one SU(3) avor. The resulting gauge theory is the linear quiver 2 + SU(3)0 SU(2) + 3. The extra SU(3) avor (other than the one we added) corresponds to the fractional D5-brane. The fact that the CS level is zero follows from the symmetry of the web. We can now read-o the dual gauge theory from the S-dual web, gure 17b, as SU(4)0 with NA = 1 and NF = 6. We can generalize this construction by adding two more SU(3) avors, gauging the global SU(3) symmetry, repeating N 2 times, and in the end adding a single avor to the last SU(3) factor, gure 18a. The result is a quiver theory with the structure 1+SU(3)0N 2 [SU(3)0 +1] SU(2)+3. The S-dual web, gure 18b, describes SU(2N )0 with NA = 1 and NF = 2N +2. As a consistency check, note that on the Higgs branch associated with the fractional 5-brane this reduces at low energy to the duality of section 2.2, between USp(2N ) with NF = 2N +2 and the quiver SU(2)N +4. On the quiver side this corresponds to a VEV of an operator given by a product of the matter elds through the quiver, starting with the fundamental of the rst SU(3) and ending with the avor of the last SU(3). Our second example is slightly more intricate. We start with the self-dual SU(2) + 5 theory (the IR gauge theory description of the rank one E6 theory), described here by the web and its S-dual shown in gure 19. First we gauge the SU(2) SO(10)F associated with the two avors given by the external D5-branes on the left of the S-dual web, and add one avor, gure 20a. This results in the quiver 2 + SU(2) SU(2) + 3, where the extra avor on the left comes from the fractional D5-brane as before. As the next step, we would like to gauge the SU(2) global symmetry associated with the two external (1; 1)5-branes. This gives the web shown in gure 20b, with the S-dual shown in gure 20c. The latter describes SU(4) 1 with NA = 1 and NF = 6. But what theory is this theory the dual of? This is not immediately obvious since the SU(2) that we gauged is not a subgroup of the perturbative avor symmetry of the quiver theory, which is [SO(6) SO(4) U(1)2] SU(2). In fact the quiver theory corresponds to a known 5d xed point, the so-called R0;4 theory, in which the factor SO(6) SO(4) U(1)2 is enhanced to SU(8) [19].13 The SU(2) that we are gauging is therefore contained in the SU(8) factor. The R0;4 theory has another IR gauge theory description as SU(3) 1 + 7 [19]. In this case the perturbative global symmetry is SU(7) U(1)2, with one U(1) getting enhanced to SU(2), and SU(7) U(1) 2 getting enhanced to SU(8). We can then use SU(8) transformations to rotate the gauged SU(2) into the non-Abelian avor symmetry of either of the gauge theories. This leads to three possible quiver theory duals of the SU(4) 1 theory with NA = 1 and NF = 6: SU(2) SU(2) SU(2) + 3, 2 + SU(2) SU(2) SU(2) + 1, and SU(2) SU(3) 1 + 5. 2 13The 5-brane web description in [19] is presumably related to the one here by resolving the orientifold plane. describes the same theory. SO(10) and add a avor. (b) Then gauge a non-perturbative SU(2). (c) S-duality gives SU(4) 1 with NA = 1 and NF = 6. 2 + SU(3) 1 2 1 + SU(2) UV We can generalize this to higher rank as follows. Start with SU(2N ) 1 with NA = 1 and NF = 2N + 3. The 5-brane web describing this theory is a straightforward generalization of gure 19a. Performing the same manipulations on the web as before leads to a web that describes SU(2N + 2) with NA = 1 and NF = 2N + 4, generalizing gure 20c. Let us now nd the dual theory. The original gauge theory corresponds to the UV theory R1;2N+1, a SCFT with global symmetry given by SU(2N + 3) SU(2) xed point U(1)2 [19]. Comparing with the gauge theory, we see that the topological U(1) symmetry is enhanced to SU(2) at the xed point. The rst step is to gauge the SU(2) and add one avor, but we cannot do this in the gauge theory since the SU(2) cannot be rotated into the avor symmetry. However we can use the dual gauge theory for R1;2N+1 given by the quiver SU(2) + 3, the web for which is the generalization of gure 19b. Gauging the SU(2) avor symmetry and adding an SU(2) avor then gives xed point R1;2N+2.14 A dual gauge theory for R1;2N+2 is given by the quiver 2 + SU(3)0N 1 + 5 [19]. The nal step is to gauge the SU(2) avor symmetry, leading 4.4.3 Next let us give two examples that involve SU(2N + 1). Start with SU(3)0 with NA = 1 and NF = 7, gure 21a. This gauge theory corresponds to a 5d SCFT known as S5, which has an enhanced SU(10) global symmetry [19]. Gauging a speci c SU(3) SU(10) and 14For N = 1 this reduces to the previous example, since R1;4 and R0;4 are the same theory. now use reasoning similar to before to derive a dual for this theory. The S5 theory has two dual IR gauge theory descriptions. The rst is SU(3)0 with NF = 8, which is equivalent to our starting point since the rank two antisymmetric and fundamental representations are equivalent for SU(3). The other is the quiver 3 + SU(2) SU(2) + 3. The global symmetries exhibited by the two gauge theories are SU(8) U(1)2 and SU(4)2 SU(2) U(1)2, respectively. Both are enhanced to SU(10) by instantons. Using the full SU(10) symmetry of the xed point we can rotate the SU(3) that we gauge into the (non-abelian) avor symmetry of the IR gauge theory, obtaining 1 + SU(3) SU(3) + 5 in the rst case, and 1 + SU(3) SU(2) SU(2) + 3 in the second. We therefore claim that both of these theories are dual to SU(5)0 with NA = 1; NF = 7. Generalizing to higher rank, we nd that SU(2N + 1)0 with NA = 1 and NF = 2N + 3 corresponds to gauging an SU(3) in the S2N+1 SCFT and adding an SU(3) avor. The latter is a UV xed point with several IR gauge theory manifestations, one of which is SU(2N 1)0 with NA = 1 and NF = 2N + 3 [19]. The full global symmetry at the xed point (for N > 2) is SU(2N + 3) SU(3) U(1), and the SU(3) factor is precisely what is gauged. As before, we can obtain a dual of our starting theory by embedding SU(3) into one of the other IR gauge theories. For example, if we take the realization of S2N+1 as the quiver theory 3 + SU(3)0N 1 + 5 [19], and gauge the SU(3) avor symmetry and add one avor, we get 1 + SU(3)0N + 5. 4.4.4 Our second SU(2N +1) example is again more intricate, and will involve a two-step gauging of a SCFT. The starting point is the R1;4 theory, which has a gauge theory realization as SU(3) 1 with NA = 1; NF = 6 [19]. The global symmetry of the gauge theory is SU(7) U(1)I , since the antisymmetric and fundamental representations of SU(3) are equivalent. The full global symmetry at the xed point is SU(8) SU(2). (Actually R1;4 is identical to R0;4, but we will use the rst description since the more general case below will involve R1;N .) The 5-brane web for the gauge theory is shown in gure 22a. Now we gauge an SU(2) SU(8), and add one avor, resulting in the web shown in gure 22b. On the other hand, using SU(8) we can rotate the gauged SU(2) into the avor SU(7) symmetry of the IR gauge theory, yielding 1 + SU(2) SU(3) 1 + 5 (which we can sort of understand from the S-dual of the web in gure 22b). We recognize this quiver theory as an IR gauge 2 theory description of the SCFT R1;5 [19]. gives SU(5) 1 with NA = 1; NF = 7. 2 Next, we would like to gauge the SU(2) corresponding to the parallel external (1; 1)5branes in gure 22b, leading to gure 22c. This corresponds to the gauge theory SU(5)1 with NA = 1; NF = 7. As in example 2, this SU(2) is not contained in the avor symmetry of the quiver gauge theory. In fact it is a subgroup of the SU(3) part of the full SU(7) 2 SU(3) U(1) global symmetry of R1;5. On the other hand R1;5 has a dual gauge theory description as 3+SU(3) 1 SU(2)+3, in which this SU(3) corresponds to a avor symmetry. Gauging an SU(2) SU(3) in this description then gives SU(2) [SU(3) 1 + 1] SU(2) + 3. Generalizing to arbitrary rank N , we begin with SU(2N 1) 1 , NA = 1; NF = 2N + 2, which is an IR gauge theory description of R1;2N . The 5-brane web is a simple generalization of gure 22a. The full global symmetry of the SCFT (for N > 2) is SU(2N + 2) SU(2) U(1)2, whereas that of the gauge theory is SU(2N + 2) U(1)3. In this case just the topological U(1) symmetry is enhanced to SU(2). The two-step gauging procedure leads to a web describing SU(2N + 1)1 with NA = 1; NF = 2N + 3. The dual theory can be determined by considering the dual description of R1;2N as the quiver 2 + SU(3) 1 SU(3)0N 2 + 5 [19]. Gauguing the SU(2) avor symmetry and adding a avor leads to SU(3)N 1 + 5, which corresponds to the SCFT R1;2N+1. The other gauge theory description of this theory is in turn provided by 2 + SU(3)N 2 avor SU(2) symmetry in this description we nally get 2 2 SU(2) A summary of all the general rank dualities discussed in this section is shown in gure 23. These dualities appear to be related to the 4d duals of SU(N ) with NA = 1 and NF = N + 2 [ 32 ]. It would be interesting to study this further using, for example, the superconformal index. 5 SU(N ) with a symmetric To engineer a symmetric hypermultiplet we replace the O7 plane with an O7+ plane. The resulting 5-brane webs for SU(2N )k and SU(2N + 1)k 1 with NS = 1 are shown in gure 24. In this case the maximal number of avors that one can add for SU(M ) is 2 NF = M 3, which corresponds, as before, to an avoided intersection of external 5-branes. Also as before, for NF = M 3; M 4 and M 5, there are extraneous instanton-charged states that have to be removed from the partition function. Like in the case with an antisymmetric, the bound we nd is very di erent from the one in [3] where these gauge theories were ruled out. However, in the webs we nd one SU(2N)0 2N + 2 SU(2N)1 2N + 2 A A SU(2N + 1)0 2N + 3 SU(2N + 1)1 2N + 3 (0,1) O7+ (a) 1 3 0 ... 2 1/2 3 0 1 3 0 N 2 3 0 0 3 1 N 2 ... N ... 3 1 ⇡ 2 3 1/2 0 below the nodes denote the CS level or . (N k 2, 1) N (1,0) (N,1) N (1,0) (N +k 2,1) can take the limit where all the branes intersect the O7+ plane, corresponding to the xed point. This strongly suggests that these theories do exist as 5d xed point theories. As in the case of the SO(M ) theories, the S-dual webs in the present case do not seem to correspond to gauge theories, but we can use them to relate the original gauge theory to a partial gauging of a lower rank SCFT which does have a gauge theory description. For example, the theory with SU(8)0 and NS = 1; NF = 2 is obtained by gauging an SU(3)0 subgroup of the global symmetry of the xed point corresponding to SU(6)0 with NS = 1; NF = 2, and adding one avor, gure 25. The generalization to SU(2N )0 with NS = 1; NF = 2 leads to a linear quiver with 1 + SU(3)0N 3, with the last SU(3)0 factor embedded in the SCFT corresponding to SU(6)0 with NS = 1; NF = 2. Likewise, for SU(2N + 1)0 with NS = 1 and NF = 1 we nd a linear quiver 1 + SU(3)0N 3, with the last SU(3)0 embedded in the SCFT corresponding to SU(5)0 with NS = 1; NF = 1. 6 Conclusions In this paper we explored 5-brane web constructions in the presence of an orientifold 7plane. We have shown that these can be used to argue the existence of new motivate symmetry enhancement and duality relations, and to assist in index calculations. We have concentrated on a simple class of examples, mainly webs with only one full NS5brane. Yet, this method can be used also to construct more complicated webs with an arbitrary number of NS5-branes. This then allows us to describe a large class of SU(N ) linear quiver theories with edge groups USp(2N ), SO(M ), or SU(N ) with a hypermultiplet in the symmetric or antisymmetric representation. The studies done here can be straightforwardly generalized to these cases as well. Besides adding an O7-plane we can also add an O5-plane, parallel to the D5 branes, without breaking supersymmetry. This also allows a realization of U Sp and SO gauge theories. Furthermore, this method should also allows realization of SO and U Sp alternating quiver, which to our knowledge have not been studied in 5d. Thus, it will be interesting to examine these constructions as well [33]. The study of the brane realization of 5d gauge theories and comparing with eld theory data have also lead us to conjecture new results in string theory. Particularly, we expect that there are two varieties of the O7 plane connected by an SL(2; Z) T transformation. It will be interesting to see if this can also be understood from the string theory perspective. Acknowledgments We thank Diego Rodriguez-Gomez for useful conversations. 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. O.B. is supported in part by the Israel Science Foundation under grant no. 352/13, the German-Israeli Foundation for Scienti c Research and Development under grant no. 1156-124.7/2011, and the US-Israel Binational Science Foundation under grant no. 2012-041. A Instanton partition functions and extraneous states The instanton partition function is expressed as a contour integral over the Cartan subalgebra of the dual gauge group, which depends on the gauge group and instanton number. The integrand contains contributions from the gauge group and the matter sector, and is a function of the various fugacities. We denote by si gauge fugacities for the true gauge group, by ua gauge fugacities for the dual gauge group, by fm fundamental avor fugacities and by z fugacities for other representations. We also use x; y for the superconformal fugacities. The integral can be evaluated using the residue theorem once supplemented with the appropriate pole prescription which determines which poles should be included. The poles can be classi ed depending on whether they originate from the contributions of the gauge multiplet, matter hypermultiplets, or are poles at zero or in nity. The prescription for the poles associated with the gauge multiplets can be found in [7, 8, 10], and the prescription for those associated with matter mutiplets (in representations other than the fundamental) can be found in [10]. one SU(N ) instanton), Poles at zero (or in nity) lead to the violation of the x ! x1 symmetry (required by conformal invariance) in the partition function. In general, for a single integral (e.g. for Z1[x] Z1 = Res[u = 0] Res[u ! 1] : For two integrals (e.g. two SU(N ) instantons) we have 1 x Z2[x] Z2 = Res[u1 x; u2 = 0] + Res[u1 = 0; u2 x] Res[u1 x; u2 ! 1] Res[u1 ! 1; u2 Res[u1 ! 1; u2 ! 1] : x] + Res[u1 = 0; u2 = 0] These poles are associated with extraneous states, whose contributions must be removed from the instanton partition function. The restoration of x ! x1 symmetry serves as a useful test of this procedure. In this case we deal with gauge groups SO(N ) and Sp(N ) and consider only noncomplex representations. As a result the residues at zero and in nity di er solely by a sign (this can be seen by the u ! u1 invariance of the expressions for these groups). Thus, we can simplify to: Z1[x] Z2[x] Z1 Z2 1 x 1 x = 2Res[u = 0] = 2(Res[u1 x; u2 = 0] + Res[u1 = 0; u2 x]) and completely ignore poles at in nity. A.1 USp(2N ) The k-instanton partition function for USp(2N ) has two components, Z+ and Z , corresponding, respectively, to summing over holonomies of determinant +1 and 1 in the dual gauge group O(k). The latter can be viewed as the sector with one gauge QM instanton corresponding to the non-trivial element of 0(O(k)) = Z2. For k = 2n + 1 there are n independent holonmies, and therefore n contour integrals, in both cases. For k = 2n there are n independent holonmies of determinant +1, but only n 1 of determinant 1. The total partition function is given by Z+ Z , where the relative sign depends on the value of the discrete parameter. (A.1) (A.2) (A.3) (A.4) The expressions for the contributions of the gauge multiplet and fundamental hypermultiplets to each component can be found in the appendix of [8]. For k = 1 there is no 1 integral, and both components are invariant under x ! x . For k = 2 the determinant +1 component Z+ involves a single contour integral, and there are poles at zero or in nity when NF 2N + 4. For NF = 2N + 4 we nd that 2 ZUSp(2N)+(2N+4)[x] 2 ZUSp(2N)+(2N+4) 1 = (1 x2) (1 xy) 1 x y : Therefore the combination 2 ZSp(N)+(2N+4)[x] + (A.5) (A.6) (A.7) is x ! x1 invariant. This agrees with the 2-instanton correction term that we subtracted in section 2, eq. (2.3). For NF = 2N + 5 we nd 2 ZSp(N)+(2N+5)[x] ZSp(N)+(2N+5) 1 (1 x2) (1 xy) 1 2 x SO(4N+10) 4N+10 ) 2UNSp(2N) ; and therefore the combination 2 ZUSp(2N)+(2N+5)[x] + x 2 (1 xy) 1 x y 4N+10 x 2UNSp(2N) is x ! x1 invariant. This agrees with the 2-instanton correction term, eq. (2.3), for this case. For k = 3 both components of the partition function involve a single contour integral, and again poles at zero appear for NF 2N + 4. Let's concentrate on the case with NF = 2N + 4, for which we gave a proposal for the full multi-instanton correction factor in eq. (2.4). For NF = 2N + 4 we nd 3 ZUSp(2N)+(2N+4) 1 x implying that the combination 1 ZUSp(2N)+(2N+4)[x] ; (A.8) x 2 (1 xy) 1 = (x + x y 1 x = x 2 (1 x2) (1 xy) 1 x y (1 xy) 1 x y ZUSp(2N)+(2N+4)[x] + 1 ZUSp(2N)+(2N+4)[x] (A.9) is x ! x1 invariant. For k = 4, Z+ has two contour integrals and Z has one. Both involve poles at zero for NF 2N + 4. We focus again on the case NF = 2N + 4. Evaluating the double contour integral for Z4;+ requires a little more work. We can separate the non-zero poles into mixed ones (where u1 / u2) and non-mixed ones. The non-mixed ones come in identical pairs of (u1 = 0; u2 6= 0) and (u1 6= 0; u2 = 0). It is not di cult to see that 1 4(1 xy) 1 y Res[u1 = 0; u2] = Z2;+[u2] ; (A.10) where Z2;+[u2] is the integrand of Z2;+. Combining this with Res[u2 = 0; u1] and with Z4; , and using (A.4) we nd 1 2 xy) 1 y 2 (1 x2)2 2(1 xy)2 1 x 2 : HJEP12(05)63 In deriving this we used: 2 ZUSp(2N)+(2N+4)[x] = Res[u (1 x2) 2(1 xy)(1 y ) x ; (A.11) where the residues are for the 2 instanton integrand, Z2;+ + Z2; . To this we need to add the contributions of the four mixed poles, which give u2y xy u2 + Res[u1 = Combining everything we nd 4 ZUSp(2N)+(2N+4)[x] (1 (1 x2) x2) xy)(1 xy)(1 4 ZUSp(2N)+(2N+4)[x]+ 1 is invariant under x ! x One can then show that the combination 4 ZUSp(2N)+(2N+4) 1 x = y y ) 2 ) 2 x ZUSp(2N)+(2N+4)[x] x ZUSp(2N)+(2N+4)[x] + (1 x2)2 2(1 xy)2(1 y y y ) x )2 y x 4 : 2(1 x 2 y In fact the expressions in (A.6), (A.9) and (A.14) reproduce the multi-instanton correction factor for NF = 2N + 4, eq. (2.4), to instanton number four. The dual gauge group for k instantons of SO(M ) is USp(2k). The expressions di er slightly between even and odd M . We denote M = 2N + where = 0 or 1. The contributions (A.12) (A.13) x )2 y (A.14) of the gauge and fundamental (vector) matter multiplets can be lifted from the 4d results of [ 34, 35 ]. The gauge multiplet contributes ZgSaOu(gMe) = 1 2kk! (1 (1 ua x2)k (ua x )k k Y Y a=0 ua(x + x1 u1a ) QiN=0(ua + u1a (ua + u1a ub u1b )2(uaub + ua1ub u1a )2(u2a + u12a xsi 2 x12 ) x1si )(ua + u1a x12 )( ua + uuab ub si 2 si ) x12 ) a<b (uaub + ua1ub x1y )( uuab + uuab x1y )(uaub + ua1ub y xy )( uuab + uuab y and fundamental hypermultiplets contribute ZfSuOn(dM: ) = k Nf Y Y a=1 m=1 fm + 1 ua : 1 (1 x2) (1 xy)(1 x ; ) y 1 fm M ua = Poles at zero and in nity appear when Nf 4, starting at k = 1. For NF = N 4 we nd and the x ! x1 invariant combination is then given by 1 (1 x 2 xy)(1 x : ) y For NF = M 3 we nd 1 ZSO(M)+(M 3)[x] ZSO(M)+(M 3) 1 = 1 (1 x2) x x y ) x and the x ! x1 invariant combination is then 1 ZSO(M)+(M 3)[x] + (1 x 2 xy)(1 x y ) As in the USp(2N ) theory, we expect the corrections to the higher instanton contributions for NF = M 4 to sum to a simple form involving a plethystic exponential. The analysis of the 2-instanton contribution is similar to that of the 4-instanton contribution to Z+ in the USp(2N ) theory. There are two contour integrals, and the contributions to the non-invariance can be split into a mixed and non-mixed parts. The non-mixed parts give 1 x 2 2(1 xy)(1 y ) 1 (1 x2)2 4(1 xy)2(1 x )2 y { 25 { (A.15) y ; ) x (A.16) (A.17) (A.18) M SO(M) ; (A.19) where we have again used 1 ZSO(M)+(M 4)[x] = Res[u y ) (A.21) u2 residues now referring to the 1 instanton integrand, Z1. The mixed parts give +Res[u1 = u2y 1 4(1 x2y2)(1 2(1 x2) xy)(1 xy)(1 y x ZSO(M)+(M 4)[x] x ZSO(M)+(M 4)[x] + (1 x2)2 xy)2(1 y ) y x 4 : 2(1 The x ! x1 invariant combination is given by 2 (1 2 xy)(1 x ZSO(M)+(M 4)[x] + x4(1 + x2) The one and two-instanton expressions are consistent with a multi-instanton correction (A.22) y factor of the form A.3 SU(N ) P E x2q xy)(1 x y ) SO(M)+(M 4)F : The contributions from the gauge multiplet as well as from avor in the fundamental were given in [8]. As we do not us them in this paper, we have not reproduce them. However, we shall write the contribution of matter in the symmetric and antisymmetric representations of SU(N ). For the symmetric we can lift from the 4d results of [35], nding: ZSk[ua] = u2az + Y a=1 1 Yk (uaubz + uaubz 1 a<b (uaubz + uaubz 1 u2z a y y x 1 y N Y i=1 1 y 1 x ) ) puasiz 1 puasiz (A.24) The contribution for the antisymmetric of SU(N ) was already given in [8]. Unfortunately, there was a mistake there which we take the opportunity to correct. The correct contribution is: zAk[ua] = Y a=1 k QiN=1 puasiz u2az + u12az pu1asiz 1 1 Yk (uaubz + uaubz 1 a<b (uaubz + uaubz x 1 ) ) (A.25) Both of these provide additional poles in the integrand. The prescription for dealing with them follows from the results of [10]. Speci cally, one de nes p = z1x and d = xz , calculates the integral assuming x; p; d 1, and only at the end return to the original variables. Note that as generically with matter contributions, the symmetric and antisymmetric provide negative powers of ua. Speci cally, they behaves as ZSk[ua] + sign is for the symmetric and the for the antisymmetric. Since a fundamental goes N 4 ua 2 , where the like ua 2 , we see that as far as the lowest u power and thus the appearance of zero poles is concerned, a symmetric contributes as N + 4 fundamentals and an antisymmetric as N 4 Finally a note regarding signs. 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Oren Bergman, Gabi Zafrir. 5d fixed points from brane webs and O7-planes, Journal of High Energy Physics, 2015, 163, DOI: 10.1007/JHEP12(2015)163