Lifting 4d dualities to 5d

Journal of High Energy Physics, Apr 2015

In this paper we set out to further explore the connection between isolated \( \mathcal{N}=2 \) SCFT’s in four dimensions and \( \mathcal{N}=1 \) SCFT’s in five dimensions. Using 5-brane webs we are able to provide IR Lagrangian descriptions in terms of 5d gauge theories for several classes of theories including the so-called T N theories. In many of these we find multiple dual gauge theory descriptions. The connection to 4d theories is then used to lift 4d \( \mathcal{N}=2 \) S-dualities that involve weakly-gauging isolated theories to 5d gauge theory dualities. The 5d description allows one to study the spectrum of BPS operators directly, using for example the superconformal index. This provides additional non-trivial checks of enhanced global symmetries and 4d dualities.

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Lifting 4d dualities to 5d

Received: January Lifting 4d dualities to 5d 0 Open Access , c The Authors 1 Haifa , 32000 , Israel 2 Department of Physics, Technion, Israel Institute of Technology In this paper we set out to further explore the connection between isolated N = 2 SCFT's in four dimensions and N = 1 SCFT's in five dimensions. Using 5-brane webs we are able to provide IR Lagrangian descriptions in terms of 5d gauge theories for several classes of theories including the so-called TN theories. In many of these we find multiple dual gauge theory descriptions. The connection to 4d theories is then used to lift 4d N = 2 S-dualities that involve weakly-gauging isolated theories to 5d gauge theory dualities. The 5d description allows one to study the spectrum of BPS operators directly, using for example the superconformal index. This provides additional non-trivial checks of enhanced global symmetries and 4d dualities. Brane Dynamics in Gauge Theories; Duality in Gauge Field Theories; Field 1 Introduction 2 5d gauge theories for 4d SCFTs 2.1 The TN theories Enhanced symmetry The R0,N theories Superconformal index SN (and E7) SU(N ) + 2N SU(N ) + 4 + 2 N = 2n N = 2n + 1 N + SU(N )k + N 3 5d duality for 4d duality SU(3) + 6 and Argyres-Seiberg duality Comparing superconformal indices More on the 5d SCFT A The 5d superconformal index B Flavors in webs Webs for antisymmetric matter perconformal field theories (SCFTs) that have no marginal coupling and no Lagrangian description. These theories are described only through their Seiberg-Witten curves, and can be characterized by their global symmetry, and by the dimensions of their Coulomb and Higgs branches. For example, there is a series of such theories with E6, E7 and E8 theories is in terms of M5-branes wrapping punctured Riemann surfaces [2]. In particular, 3-punctured spheres with various types of punctures give isolated SCFTs in 4d. Although they are a-priori isolated in the space of 4d SCFTs, some of these theories can be continuously connected to more ordinary superconformal gauge theories by gauging a subgroup of their global symmetry, whereby they provide an S-dual description of strong coupling limits of the gauge theories [3]. In the simplest example, the E6 theory with SU(3) and 6 flavors. The realization of 4d SCFTs in terms of punctured Riemann surfaces generalizes this idea, by realizing different weak-coupling limits of a given SCFT as different degenerations of the corresponding Riemann surface. This leads to many examples of Sduality between superconformal gauge theories and weakly gauged isolated SCFTs [4]. However the lack of an explicit Lagrangian description for these theories makes it difficult to test the dualities in detail. providing a Lagrangian description. For example, there are 5d rank one SCFTs with E6, n versions of these theories, corresponding to USp(2n) with the same number of flavor hypermultiplets plus an additional hypermultiplet in the antisymmetric representation of the gauge group. These reduce to the corresponding 4d SCFTs by compactifying on a circle in the limit of vanishing radius. The 5d Lagrangian description in principle allows one to determine the complete chiral ring of the theory, although some of the BPS states involve non-perturbative instanton particles. Indeed these states provide the necessary charges for the enhanced global symmetries, as can be seen, for example, from their contributions to the superconformal index [7]. A useful way to visualize 5d SCFTs in general is by (p, q) 5-brane webs in Type IIB string theory [8]. This construction makes manifest all the mass parameters and moduli of the theory, realized geometrically as the relative motions of the external and internal 5branes, respectively. In many cases, the 5-brane web can be mass-deformed to exhibit a 5d in correspondence with the classification of [6]. The mass in these cases corresponds to an inverse square gauge coupling of the gauge theory. In fact, there may exist different mass deformations leading to different IR gauge theories. This is somewhat analogous to Seiberg duality in 4d, except that in 5d there are two, or more, IR gauge theories that come from the same CFT in the UV, whereas in 4d there are two, or more, UV gauge theories that flow to the same CFT in the IR. The different 5d gauge theories are in a sense continuations past infinite gauge coupling of one another, since one has to go through a massless point in connecting them. From the point of view of the 5-brane web this usually entails an SL(2, Z) transformation exchanging D5-branes and NS5-branes [8, 9]. These types of dualities were further explored and generalized in [10, 11]. This connection was further studied in [13, 14]. In principle, this should allow one to identify the 5d IR gauge theory by suitably deforming the 5-brane web [14, 15]. Then, by looking at various limits on the Higgs branch of these theories, as described by the 5-brane webs, we will also find 5d gauge theories for several other 5d SCFTs that reduce to isolated 4d SCFTs, such as the ones considered in [4]. In some cases we will find dual gauge theories for the same fixed point theory. Our second goal, in section 3, is to relate the S-dualities associated with weakly-gauging these 4d SCFTs to dualities between 5d gauge theories associated with the same 5d SCFT in the UV. In particular, this allows us to use localization to compute the superconformal index using either gauge theory, and thereby obtain the explicit dictionary relating the BPS states of the two 4d theories. We will exhibit this in a number of examples, starting with the Argyres-Seiberg duality involving the E6 theory. Section 4 contains our conclusions. We have also included three appendices. In appendix A we give a brief review of the 5d superconformal index, and in particular of how various issues in the computation of instanton contributions are resolved. In appendix B we discuss the different representations of flavor degrees of freedom in 5-brane webs, and in appendix C we describe how to incorporate antisymmetric matter in 5-brane webs. A word on notation. We will denote global symmetries associated with matter in the fundamental representation (flavor) by an F subscript, those associated with matter in the bi-fundamantal representation by a BF subscript, and those associated with matter in the 2-index antisymmetric representation by an A subscript. In addition, we will use SU(N )F U(1)B flavor symmetry), and an I subscript for the topological (instanton) U(1) symmetries. Subscripts on gauge symmetries will denote either the CS level or the value where there is a product of several identical groups, will denote their order of appearance in the product. The TN theories 5d gauge theories for 4d SCFTs The 4d TN theory corresponds to M5-branes wrapping a 2-sphere with three maximal punctures, namely punctures labelled by the fully symmetrized N -box Young tableau [2]. This theory has no marginal couplings. The global symmetry is (at least) SU(N )3, and background vector multiplets associated with the global symmetry. The dimensions of the the Coulomb branch. theory. The rank 1 E7 and E8 theories can be realized as particular limits on the Higgs branch of the T4 and T6 theories, respectively. We will mention these below. The 5d version of the TN theory is described by a collapsed 5-brane web, or 5-brane junction, with N external D5-branes, N external NS5-branes, and N external (1, 1)5branes (figure 1a) [12]. This is the 5-brane configuration resulting from the reduction of the M5-brane configuration in M theory to Type IIB string theory. In describing the 5d theory, it is useful to have each external 5-brane end on an appropriate type of 7-brane. One can read off the basic properties of the theory from this configuration. The mass parameters (real in 5d) correspond to the relative positions of the 7-branes, so there are in figure 1b, and the Higgs moduli correspond to transverse deformations, where parts of the web separate along the 7-branes. The counting of the web deformations reproduce the dimensions of the Coulomb and Higgs branches. The gauge theory interpretation of the web in figure 1 is not completely obvious. However we can manipulate the web so that the gauge theory becomes apparent. Moving the 7-brane in the lower right corner upward across all the (1, 1)5-branes we obtain, via multiple Hanany-Witten (brane-creation) transitions, the web shown in figure 2a. In this web due to the s-rule [12, 16]. The 5d IR gauge theory becomes apparent when we mass-deform origin of the Coulomb branch, as shown in figure 2b. In the limit of large mass we get a gauge group is SU(4) SU(3) SU(2), and there is a single massless hypermultiplet in the bi-fundamental representation of each pair of adjacent groups, five in the fundamental repthe gauge theory (the shaded regions correspond to the gauge groups) (c) The S-dual web. resentation of SU(4), and two in the fundamental representation of SU(2). More generally, corresponding quiver diagram is shown in figure 3. S-duality gives the web shown in figure 2c. This actually describes the same quiver gauge theory. This is not immediately obvious, due to the avoided intersections involving the D5-branes. To see the flavor structure more clearly, one can go through a series of 7-brane motions, as described in appendix B, which basically brings us back to (an S-dual of) the web of figure 1b. To completely fix the gauge theory we also need to specify the CS levels of the SU(n) 3. Now deform the sub-web so as to give all the flavors a mass with the same sign. This the other hand, the renormalized CS level is easily read-off from the resulting pure SU(n) Enhanced symmetry U(1)IN2, where the first two factors are associated to the flavors at the two edges, the U(1)BF factors to the bi-fundamental fields, and the U(1)I factors to the instanton number currents. This should be enhanced at the UV fixed point to SU(N )3 by instantons. global symmetry is SO(10)F U(1)I . The full global symmetry in this case is E6, which is consistent with the fact that E6 is the unique rank six group which has SU(3)3 and SO(10) U(1) as subgroups. The enhancement to E6 was explicitly demonstrated in [7] by computing the superconformal index, including instanton contributions. The verification of the SU(N )3 symmetry in the more general case is technically harder, since it involves instantons with charges under two gauge groups. We did this explicitly for T4 and T5. The IR gauge theory corresponding to T4 is the linear quiver 4 + SU0(3) SU(2) + 2. A standard calculation of the perturbative superconformal index gives (see appendix A) Ip4e+rStU(3)SU(2)+2 = 1 + x2 4 + (15,1,1) + (1,3,1) + (1,1,3) where x, y are the superconformal fugacities, z is the fugacity associated to the bifundamental field, b is the baryonic fugacity associated with the U(1)B subgroup of the 3 there are also contributions from (1, 0), (0, 1) and (1, 1) instantons. The calculation of the instanton partition functions turns out to be simpler if we treat SU(2) as USp(2). For the SU(3) instanton we must use the U(N ) formalism and mod out the U(1) part. In general this procedure leaves some U(1) remnants that must be removed by hand (see appendix A for a discussion). In this case the remnant states correspond to a D1-brane between the parallel external NS5-branes in figure 4, and are removed by correcting the instanton partition function as Zc = P E (1 xy) 1 y where q1 is the SU(3) instanton fugacity. Note that the correction factor is not invariant under x 1/x, which is part of the conformal symmetry. This corrects a similar lack of invariance in the instanton partition function, due to a pole at zero in the integral over the dual gauge group (see appendix A). The resulting instanton contribution is given by (we present the result only to order x2, although we computed to order x3) I(41+,0S)U+((30),1)S+U((12,1)+)2 = x q1q2z q1q2z where q2 is the SU(2) instanton fugacity. Together with the perturbative contribution, the full index in terms of characters of SU(4)3 (now including the x3 terms): x2 terms exhibit an enhancement of SO(4)F U(1)4 SU(4)2, and we can express the IT4 = 1 + x2 (15,1,1) + (1,15,1) + (1,1,15) For T5, the gauge theory is 5 + SU0(4) SU0(3) SU(2) + 2, and the perturbative contribution to the superconformal index is IpTe5rt = 1 + x2 6 + (24,1,1) + (1,3,1) + (1,1,3) In this case the classical global symmetry is SU(5)F SO(4)F U(1)B U(1)2BF U(1)I3. The instanton part is again computed by treating SU(2) as USp(2). The U(1)-remnant states, which can be read-off from the 5 + SU(4) SU(3) + 2 sub-web, are removed by the z1b5 z2 + 1 Zc = P E where q1 and q2 are the instanton fugacities of SU(4) and SU(3), respectively, and z1 and z2 are the bi-fundamental fugacities for SU(4) SU(3) and SU(3) SU(2), respectively. As a consistency check, we verified that all the instanton partition functions we evaluated are x 1/x invariant. To order x3 there are contributions from the (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (0, 1, 1) and (1, 1, 1) instantons. These give IiTn5st = x2 1 + x y + q3q2z2 + q3q2z2 q1q2q3z1b5 z2 z2 q1q2q3z1b5 pz1b5z2 + z2 z2 q1q2q3z2 + q1q2q3z2 full index in terms of SU(5)3 characters: In this case we get an enhancement of SO(4)F U(1)6 SU(5)2, and we can express the IT5 = 1 + x2 1 + x y + The general picture that appears to emerge is that the SO(4)F flavor symmetry of the hard as N increases and instantons of increasingly higher orders are needed. Other theories Other isolated theories in 4d are described by 3-punctured spheres with non-maximal punctures, and correspond to various limits on the Higgs branch of the TN theories. The Young diagram associated with a puncture corresponds to the pattern of symmetry breaking of the corresponding SU(N ). The 5d lifts of these theories are obtained by deformations of the TN web in which some of the 5-branes break, and subsequently share a 7-brane boundary (figure 5) [12]. The broken 5-brane pieces can then be moved away along the 7-branes. Each column in a Young diagram of a puncture corresponds to a 7-brane in the N -junction picture, and the number of boxes in the column corresponds to the number of 5-branes that end on that 7-brane. We can use this procedure to obtain 5d Lagrangian descriptions for many other isolated Well concentrate on a series of isolated theories considered in [4]. The main properties of these theories are summarized in table 1. The R0,N theories puncture gives a series of theories known as the R0,N theories [4]. The R0,N theory has Clearly R0,3 is the same as T3. on them coincide, one of the 5-branes can break. global symmetry SU(2N ) SU(2) SU(N + 2) SU(2) U(1)2 SO(4n + 6) U(1) SU(N + 2) SU(3) U(1) 12 (N 1)(N 2) N 2 N 2 N 3 N 2) 2n2 + 5n + 4 SU(N )2 SU(k + 1) U(1) k(N 1) 21 k(k + 1) case, as well as in subsequent cases, we find it useful to move this 7-brane to the right. The resulting 5-brane web, after an appropriate mass-deformation, is shown in figure 6b. This describes the quiver gauge theory with 2 + SU(2) SU(2) SU(2) + 3. The general described by the S-dual 5-brane web in figure 6c. For R0,5 this web gives SU(4) + 9. More The bare CS level of second theory (for N > 3) can be computed as before. The factors. These can be computed in the same way as the CS level, namely by deforming the web so as to give mass to all the matter (fundamental and bi-fundamental) fields. Then for all of them. In either gauge theory description, seeing the full global symmetry of the R0,N theory requires instantons. We can however guess the form of the full symmetry by comparing the symmetry of the 5-brane junction, SU(N )2 SU(2) U(1), with that of second gauge of these as subgroups is SU(2N ) SU(2). Superconformal index Let us verify this in the simplest case of R0,4 by computing the superconformal index. We begin with the 3 + SU(2)(1) SU(2)(2) + 2 description, which has a classical global symmetry SO(6)F SU(2)BF SO(4)F U(1)I2. The calculation is similar to the one for the T4 theory, where we treat SU(2)(1) as USp(2) and SU(2)(2) as U(2)/U(1). therefore need to remove the decoupled states associated to the second SU(2) factor, which are described by a D1-brane between the corresponding pair of external NS5-branes in the web of figure 6b. The required correction of the instanton partition function is Zc = P E (1 xy) 1 y tively, and q2 is the instanton fugacity of SU(2)(2). Note that in addition to the lack of x 1/x invariance, the correction factor cannot be expressed in terms of characters of SU(2)BF and SU(2)F1 . This corrects another U(1) remnant: for U(2) the global symmetries associated with the bi-fundamental and fundamentals are U(1)BF and U(2)F , and the instanton partition function respects only these. The full SU(2)BF and SO(4)F are recovered only after the correction factor is included. To order x3 there are contributions from the (1, 0), (0, 1), (1, 1) and (2, 1) instantons, and the combined result for the index is (presented to order x2 for conciseness) This exhibits the enhancement of SO(6)F SU(2)BF SU(2)F1 U(1)I2 IR0,4 = 1+x2 (1,3) +(63,1) +x3 We can also demonstrate the enhancement in the dual SU(3) 1 + 7 description.1 The classical global symmetry in this case is SU(7)F U(1)B U(1)I . As before, the instanton partition function is not invariant under x 1/x, and does not respect the full classical global symmetry. The spectrum of U(1) remnant states that must be removed in this case is a bit more involved. We can get an idea for what they are from the 5-brane web (see figure 8). There are three types coming from (a) the D1-brane between the separated external NS-branes, (b) fundamental strings between the separated external D5-brane and the flavor D5-branes, and (c) a 3-string junction in the upper part of the web. The second and third types are novel. The state corresponding to the fundamental strings is clearly charged under SU(7)F . What is less obvious is that it also carries an instanton charge even though it is described by a fundamental string, which is why it contributes to the instanton partition function. This can be understood from the fact that its mass depends on the value of the gauge coupling, which is seen geometrically by the upward motion of the separated external D5-brane as the external NS5-branes move apart. The 3-string junction state carries both an instanton charge and an SU(3) gauge charge, so unlike the other two states, it is not decoupled from the gauge theory. This complicates the procedure for correcting the partition function, since this state is not expected to plethystically exponentiate. Unlike D1-branes or fundamental strings, one can merge multiple string junctions to make a new state, as suggested by the fact that these states interact. We can therefore only determine the correction to the 1Note that, like the quiver gauge theory, this gauge theory is also outside the regime of theories classified in [6], since the low energy effective theory has a singularity on the Coulomb branch. This is apparent in the 5-brane web in figure 6c, where the upper triangular part shrinks to zero size at a finite distance from the origin of the Coulomb branch. However, as in other cases, the singularity is resolved by taking into account the additional light instantonic states. More generally, we would conjecture that there exists a UV 1-instanton partition function in this case. Actually, the simplest setting in which this state appears is in the 5-brane web shown in figure 8b, which describes an empty SU(1)2 theory. The 1-instanton partition function computed for this theory using U(1)2 is precisely the contribution of this state, and is given by ZSU(1)2 = Z1c = Z1 + where q0 is the SU(3) instanton fugacity. The first, second and third terms in the numerator correspond respectively to the three remnant states above. This correction restores the x 1/x invariance to the 1-instanton partition function. One can also check that for gives the same 1-instanton partition function as when using USp(2). The full index including the 1-instanton contribution, expressed in terms of SU(7)F characters, is then given by IR0,4 = 1 + x2 b3 + ! The full correction to the 1-instanton partition function for the SU(3)+7 theory is therefore SU(2) is spanned by q0 b 47 , and the SU(8) by SU(8) = This exhibits the enhancement of SU(7)F U(1)B U(1)I SU(8) SU(2), where the expressed in terms of SU(8) SU(2) characters, the superconformal index is then given again by (2.11), in agreement with that of the dual theory. A more general class of theories that will be useful below is gotten by replacing one of call these theories k . For k = N 2 this is just TN . For k = 1 this is the R0,N theory. N symmetry that follows from the puncture structure is apparently SU(N )2SU(k+1)U(1). We can also read-off the structure of the Coulomb branch using the formulas in [4]. We find determine from the 5-brane junction describing the 5d lift of the theory, figure 9a. This gives dH = N 2 + k(k + 3)/2. for which the gauge theories are 5 + SU(5)0 [SU(5)0 + 1] SU(4)0 SU(3)0 SU(2) + 2 and 7 + SU(6)0 SU(5)0 SU(4)0 SU(3)0 + 3, respectively. The quiver diagrams for the two gauge theories in the general case are shown in figure 10. The computation of the superconformal index becomes technically challenging as N and k are increased, so we will not pursue it presently. More theories The mass-deformed web resulting from moving the D7-brane with the two D5-branes to the right, shown in figure 11b, describes the linear quiver theory 3 + SU(3) 12 SU(2) + 3. structure is made more manifest using the manipulations described in appendix B. The quiver diagrams for the general cases are shown in figure 12. particular the R1,6 SCFT admits the dual quiver gauge theory descriptions: 1 + SU(2) [SU(3) 1 + 1] SU(2) + 3 and 2 + SU(3) 12 SU(3)0 + 5. There is actually a third gauge 2 theory description that can be obtained by a rearrangement of the (0, 1) 7-branes at the bottom, figure 14. Going through some 7-brane gymnastics shows that this describes quiver diagrams for the general cases are shown in figure 15. 3 + SU(3)1 SU(3) 1 + 4. The S-dual web in this case yields the same gauge theory. The 2 One may wonder whether there exists a gauge theory for R1,N with a just single rank R1,5. Using SL(2, Z), the original 5-brane junction for R1,5 can be transformed to that shown in figure 16a. Through a series of HW transitions this is transformed to the web of figure 16b (see appendix C), which is in turn deformed into the web of figure 16c. The seven fundamental hypermultiplets, and one hypermultiplet in the rank 2 antisymmetric representation (see appendix C for details). More generally, we find that R1,N possesses theory, this description allows one to guess the full global symmetry of the R1,N theory, by comparing its classical global symmetry to that of the original 5-brane junction. The former is SU(N + 2)F U(1)B U(1)A U(1)I , and the latter is SU(N ) SU(2)2 U(1)3. The full global symmetry should therefore be at least SU(N + 2) SU(2) U(1)2, which + (N + 2) (figure 17). As in the R0,N is exactly what it is (except for N = 5). We can again use the gauge theory descriptions to compute the superconformal index, and verify the global symmetry. We will do this for R1,5, which should exhibit a further +7 description. The classical global symmetry of this gauge theory is SU(7)F SU(2)A U(1)B U(1)I (U(1)A is perturbatively enhanced to SU(2)A since the = 6 of SU(4) is real). The perturbative contribution to the index is given by To order x3 there is also a contribution from one-instanton states. As in other cases, the instanton partition function computed via U(4) does not respect the x 1/x symmetry or the full classical global symmetry (the SU(2)A part in this case), suggesting that there are U(1) remnant states that must be removed. The nature of these states is not obvious from the web in figure 16, but we can figure out the necessary correction factor by requiring x 1/x and SU(2)A invariance.2 The appropriate correction is given by Zc = P E (1 xy) 1 y and the resulting 1-instanton contribution is = x SU(3) U(1), where the SU(3) is spanned by SU(3) = q0 31 b0 6 S2U(2) + q0 3 b0 3 . 7 2 7 3 The x2 terms provide the additional currents to enhance SU(2)A U(1)B U(1)I As a further check, let us also compute the index in one of the other gauge theory descriptions, specifically in the 3+SU(3)SU(2)+3 theory (figure 12a). The classical global symmetry is SU(3)F SO(6)F U(1)B U(1)I2, and the perturbative contribution is now Ip3e+rStU(3)SU(2)+3 = 1 + x2 4 + (8,1) + (1,15) + x3 For the instanton calculation we again treat SU(2) as USp(2). An x 1/x invariant partition function is obtained by Zc = P E 2This can be derived from an alternative 5-brane web with O7-planes [17]. where q1, q2 are the SU(3) and SU(2) instanton fugacities, respectively. This apparently includes U(1) remnant states associated both with the SU(3) part, visible in the web of figure 11b in terms of a D1-brane between the external NS5-branes, as well as with the SU(2) part. The latter are not obvious in the given 5-brane web, but appear to be a necessary ingredient in computing the instanton partition function for USp(2) + 6 (see [18, 19]).3 With this, we find the instanton contributions (for conciseness we show to O(x2), but we computed to O x3 ) I(31+,0S)U+((30),1)S+U((12,1)+)+3(1,2) = x + q1q2b 23 z 21 . The instantons provide additional conserved currents enhancing SO(6)F U(1)B U(1)I2 Thus the full quantum symmetry in both descriptions is SU(7) SU(3) U(1). Furthermore the indices agree to O x3 , and both can be expressed using characters of the full symmetry as IR1,5 = 1 + x2 1 + (1,8) + (48,1) 0 0 2 + (1,8) + (48,1) + (73,1) + (7,1) + (21,3) +(211,3) 0 0 3 1 The superscript refers to the U(1) charge, given by 56 B0 31 I in the SU 1 (4) + 2 The R2,2n+1 theory is defined by replacing two of the maximal punctures by (n, n, 1). This theory has an n-dimensional Coulomb branch, a (2n2 + 5n + 4)-dimensional Higgs branch, and a global symmetry SO(4n+6)U(1). The corresponding 5-brane junction is shown for R2,5 in figure 18a. Note that this theory corresponds to a limit on the Higgs branch of the R1,5 (and more generally R1,2n+1) theory. By appropriately modifying the mass-deformed R1,5 webs we then get the R2,5 webs shown in figure 18b,c. In either case, the quiver gauge The single gauge group dual gauge theory can likewise be obtained from that of the R1,5 theory. In this case we notice that the Higgsing leading to R2,5 corresponds simply to a VEV for the SU(4) antisymmetric field. The gauge theory for R2,5 is therefore USp(4) + 7. More generally for R2,2n+1 this is USp(2n) + 2n + 3 (figure 19b). The global symmetry in this case is SO(4n + 6)F U(1)I , which is the full symmetry of the fixed point theory. In the quiver theory description the global symmetry should be enhanced by instantons. Let us demonstrate this explicitly with the superconformal index of the gauge theory. The 3This can also be derived from an alternative 5-brane web with O7-planes [17]. q1q2f perturbative contribution, expressed in terms of SU(2)BF SO(8)F characters and the U(1)F fugacity f , is Ip1e+rStU(2)SU(2)+4 = 1 + x2 3 + (3,1) + (1,28) The correction factor for the instanton partition function is quite similar to the one in the previous section for the 3 + SU(3) SU(2) + 3 theory, and is given by Zc = P E x2 q1zf + q22 + q1q22zf The resulting instanton contribution to the index is then (shown just to O(x2)) Ii1n+stSU(2)SU(2)+4 = x q1f The x2 terms provide the necessary conserved currents to enhance SU(2)BF SO(8) identify q0 as the U(1)I fugacity. q1q22f SN (and E7) The SN theory is defined by replacing two of the maximal punctures by (n, n) and further enhancement of symmetry to SU(10) and SU(4) SU(8) respectively. The 5-brane junction for S2n+1 (represented by S5) is shown in figure 20a. Moving the D7-brane with the three D5-branes attached to the right and mass-deforming leads to the web shown in figure 20b, which corresponds to 3 + SU(2) SU(2) + 3. The S-dual n + 1 + SU(n) 12 SU(n) 1 + n + 1, and the latter to 3 + SU(3)0n1 + 5. For n 3 we find (at 2 least) one more gauge theory description by rearranging the (0, 1) 7-branes at the bottom, case S2n+1 are shown in figure 21. The generalizations to S2n are shown in figure 23. corresponds to 2+SU(2) SU(3) 1 +5, and its S-dual in figure 22c to 4+SU(3)0 SU(2)+3. 2 As in previous cases, we also find a single-gauge-group theory for SN , figure 24. This E7 theory for N = 4. to (1, 0, 13). For all intents and purposes this gives the E7 theory, but the formula for the dimension of the Higgs branch is wrong. The E7 theory has a 17-dimensional Higgs branch. Since we already discussed the E6 and E7 theories as special cases of TN and SN , we close this section by discussing the 5d lift of the E8 theory [12]. The E8 theory is obtained from the T6 theory by replacing one of the maximal punctures with a (3, 3) puncture, and a second one by a (2, 2, 2) puncture, resulting in the 5-brane junction shown in figure 25a. An HW transition followed by an appropriate (and somewhat tricky) mass deformation (figure 25b), then reveals the known SU(2) + 7 gauge theory description. 5d duality for 4d duality several dual gauge theories associated to the same SCFT. In this section we will use this The 4d dualities usually involve weakly gauging an isolated SCFT of the type that we discussed in the previous section. The 5d gauge theoretic descriptions of these SCFTs allow us to describe also their weakly gauged versions in terms of weakly coupled quiver SU(3) + 6 and Argyres-Seiberg duality gauge coupling. It was conjectured in [3] that the strong coupling limit of this theory is dual to a weakly interacting SCFT, defined by weakly gauging an SU(2) subgroup of the E6 global symmetry of the E6 theory, and adding one flavor hypermultiplet, denoted in gauge coupling. The gauging breaks the global symmetry to SU(6), and the flavor provides an additional U(1), in agreement with the global symmetry of the SU(3) + 6 theory. We can relate this duality to a 5d duality between gauge theories as follows. Begin by lifting the 4d superconformal SU(3) + 6 gauge theory to 5d by constructing the appropriate 5-brane web. Here one has to make a choice of the CS level in the 5d theory. Not all choices are allowed. We are limited by the requirement for a UV fixed point to exist. Take the theory described by the 5-brane web in figure 26a. This is an SU(3) gauge matter fields massive and reading-off the renormalized CS level of the pure SU(3) theory. This gauge theory is the result of mass-deforming the 5d SCFT described by the 5brane junction in figure 26b, by moving apart the two 7-branes at the bottom. Reversing this motion corresponds to changing the sign of the mass, and therefore to a continuation past infinite coupling. This yields, after S-duality, the web in figure 26c. Note that this contains a sub-web corresponding to an interacting fixed point, which is in fact the E6 (or T3) theory. This web is the 5d realization of weakly gauging an SU(2) subgroup of the E6 theory. The two parallel external D5-branes have been fused into another sub-web corresponding to SU(2) with one flavor. The E6 module can be further deformed, as before, to an SU(2) + 5 gauge theory web, where now two of the fundamentals become bi-fundamentals of the total SU(2) SU(2) gauge symmetry (figure 26d). The dual 5d gauge theory is therefore the quiver theory with 3 + SU(2)(1) SU(2)(2) + 1. The 4d duality is then obtained from the 5d duality by dimensional reduction in a specific scaling limit. For the SU(3) theory, we compactify on a circle of radius R, and take R 0 and g5 0, holding the dimensionless combination E6 (d) 1 + SU(2)(1) SU(2)(2) + 3. For the SU(2)(1) fixed, but keep g5,1 fixed to the 5d UV cutoff. When we remove the 5d UV cutoff we end up with the dimensional reduction of the E6 theory with a weakly gauged SU(2) subgroup, with g4,2 as its marginal coupling, plus one flavor. Thus the 4d duality is equivalent to the 5d duality5 SU(3) + 6 1 + SU(2) E6 One can therefore strengthen the case for the 4d duality by providing evidence for the 5d duality. More significantly, one can use the 5d duality, where both sides have a Lagrangian description, to derive the precise mapping of the BPS states by comparing the superconformal indices of the two theories. To begin with, the two 5d gauge theories appear to have different global symmetries. The SU(3) + 6 theory has a global symmetry SU(6)F U(1)B U(1)I , whereas the dual theory has SO(6)F SU(2)BF U(1)F U(1)I1 U(1)I2 . Clearly there must be an instantonled enhancement in the latter (and potentially also in the former). We will show this explicitly by computing the superconformal index of the two theories. This will also allow us to determine the precise map between the global symmetry charges. But first let us give a more qualitative argument. Had SU(2)(2) not been gauged, we know that the classical SO(10)F U(1)I1 symmetry would be enhanced via the SU(2)(1) instanton to E6. The gauging breaks SO(10)F to SO(6)F SU(2)BF SU(2)(2), and E6 to SU(6)F , so we expect the (1, 0) instanton to enhance SO(6)F SU(2)BF U(1)I1 to The SU(2)(1) instanton is a spinor of SO(10)F , and therefore decomposes as (4, 2, 1) + (4, 1, 2) of SO(6)F SU(2)BF SU(2)(2). The gauge invariant piece, together with the anti-instanton piece, provide the extra conserved currents needed for SU(6)F , since the adjoint representation of SU(6)F decomposes as 35 = (15, 1)0 + (1, 3)0 + (1, 1)0 + (4, 2)1 + (4, 2)1 . We can also see how the U(1) factors map. The basic U(1)B-charged baryon operator of the SU(3) theory is in the 20 (rank 3 antisymmetric) of SU(6)F , and carries 3 units of charge 5We can regard the 5d duality as a generalization of the basic duality between SU(3) + 2 and SU(2) SU(2) [8] with additional flavors. In the SU(2)(1) SU(2)(2) theory, the (6, 2)0 state is formed by combining a flavor of SU(2)(1), the bi-fundamental, and the flavor of SU(2)(2) in a gauge invariant way. The (4, 1)1 state is formed by combining a (1, 0) instanton with the flavor of SU(2)(2), and the of the two theories are related as U(1)B U(1)F and U(1)I U(1)I2 . In fact, we see that the baryon charge on the SU(3) side is related to the U(1) flavor charge on the SU(2)2 side Comparing superconformal indices The perturbative part of the index of the SU(3) + 6 theory is given by IpSeUr(t3)+6 = 1 + x2 (2 + 35) + x3 For the 1 + SU(2)(1) SU(2)(2) + 3 theory, the perturbative contribution is given by Ip3e+rStU(2)2+1 = 1 + x2 3 + (1,3) + (15,1) + x3 where f denotes the U(1)F fugacity. The contribution of the (1, 0) instanton (computed by treating SU(2)(1) as USp(2)) is I(31+,0S)U(2)2+1 = x As anticipated, the (1, 0)-instanton provides the states needed to enhance SO(6)F SU(2)BF U(1)I1 to SU(6)F , where SU(6) = q2/3(1,2) + q11/3(4,1). Furthermore, the 6 1 This is the end of the story as far as the dimensionally-reduced theories (with the scaling limit mentioned above) are concerned. The main lesson here is the above relation between the U(1) charges. However, there is more to be learned about the underlying 5d fixed point theory. In particular, there is further enhancement of the global symmetry. More on the 5d SCFT From the point of view of the SU(2)(1) due to instantons carrying U(1)I2 charge. The contributions of these states is most easily SU(2)(2) theory, the additional enhancement is encounter U(1) remnants that must be removed from the instanton partition function, this time using Zc = P E z f (1 xy) 1 y To order x3 there are contributions from the (0, 1), (1, 1) and (2, 1) instantons: q2f q2q12f I(30+,1S)U+((21),21+)+1(2,1) = x Combining this with the previous contributions shows that the global symmetry at the fixed point is further enhanced to SU(7) U(1). The full index can be expressed concisely in terms of SU(7) U(1) characters: I3+SU(2)2+1 = 1 + x2(1 + 048) + x3 where the SU(7) is spanned by SU(7) = (q2q14/f )1/7(q21f + (1,2) + q1(4,1)), and the 7 SU(7) carries one unit of charge. From the point of view of the SU(3)1 + 6 theory this enhancement must be due to the SU(3) instanton. The situation here is similar to the one encountered in the SU(3) 1 + 7 theory in section 2.2, as one can see from the similarity of the 5-brane webs. In this case there are two remnant states: one corresponding to fundamental strings between the separated external D5-brane and the flavor D5-branes, and the other to a 3-string junction. The correction to the 1-instanton partition function is given by Z1c = Z1 + where q is the SU(3) instanton fugacity. This gives a 1-instanton contribution:6 ISU(3)+6 = x 6As a consistency check, we can also treat the theory as SU(3)1+5+ , since for SU(3) the antisymmetric section 2.4.1. The result agrees with what we find for SU(3) + 6. is identical to the anti-fundamental. The calculation is similar to the one we did for SU(4) 12 + 7 + fugacities as b3 = f and q = q2q12/3f 1/6 1 Together with the perturbative contribution, this exhibits an enhancement to SU(7) U(1), and (q/b5)3/7. The full index can again be expressed in terms of SU(7) U(1) characters as in (3.9). Comparing with the SU(2) SU(2) theory, we see that the duality relates the SU(N ) + 2N The first natural generalization of the SU(3) + 6 SCFT is to SU(N ) + 2N . In [2] Gaiotto proposed that at strong coupling this theory is related to an isolated 4d SCFT with a global symmetry SU(2) SU(2N ), later named R0,N [4], by weakly gauging the SU(2) factor and adding one flavor. This gives a theory with one marginal parameter and a global symmetry SU(2N ) U(1). We can lift this duality to five dimensions as before. Start with the 5-brane web in In the limit where the coupling of the last SU(2) is scaled with R, this corresponds to weakly gauging the SU(2) subgroup of the global symmetry of the R0,N theory, as described by the SU(2) factors follow from those of the R0,N theory. The 4d duality SU(N ) + 2N 1 + SU(2) R0,N is therefore equivalent to the 5d duality SU(N )1 + 2N 3 + SU(2) SU(2)0N3 As before, the duality requires an enhancement of the global symmetry, at least in U(1)IN1 suggests that this involves the topological symmetries of all but the last (single-flavored) SU(2) factor, namely U(1)Ik with k = 1, . . . , N 2. The additional conserved currents transform in all possible bi-fundamental representations of the non-abelian factors of the global symmetry, and carry charges under U(1)Ik . For of SU(4)F SU(2)BNF2 show that all the extra conserved currents arise from instantons with topological charges SU(4)F SU(2)BF1 SU(2)BF2 U(1)I1 U(1)I2 U(1)I3 U(1)F . In this case the SU(8) current decomposes as (we include also the trivial U(1)I3 charges, but not the trivial U(1)F + (4, 2, 1)(1,0,0) + (1, 2, 2)(0,1,0) + (4, 1, 2)(1,1,0) + c.c. . We can likewise relate the two remaining U(1) symmetries of the two gauge theories. The basic baryonic operator of the SU(N ) + 2N theory carries N units of charge under U(1)B, and transforms in the N -index antisymmetric representation of SU(2N )F . This operator contributes to the superconformal index at O(xN ). The dual operator in the perturbative part is simply the product of all the matter fields through the quiver. This SU(2)BNF2 by replacing some matter fields in the above chain (not including the flavor of the last 4-index antisymmetric representation of SU(8) is given by The (4, 1, 2)(1,0,0) state corresponds to the gauge invariant combination of the (1, 0, 0) instanton, the second bi-fundamental field and the flavor of SU(2)(3), and the (6, 1, 1)(0,1,0) state to the combination of the (0, 1, 0) instanton, the flavor of SU(2)(1) and the flavor of SU(2)(3). The (4, 2, 1)(1,1,0) and (1, 1, 1)(2,1,0) states are likewise gauge-invariant combinations of the SU(2)(3) flavor and the (1, 1) and (2, 1) instantons, respectively. Since all of these states carry one unit of U(1)F charge, we conclude that U(1)B U(1)I maps to Verifying all of this explicitly requires an index calculation, which quickly becomes technically difficult as we increase N . We shall not presently pursue it. We do however expect a further enhancement of the global symmetry at the 5d fixed point to SU(2N + 1) due to the instanton of the last SU(2) factor. The (0k, 1, 0Nk3) instanSO(8) SU(2)(k) SU(2)BFk SU(2)BFk+1 SU(2)(k+2). SU(N ) + 4 + 2 Another possible higher-rank generalization of SU(3) + 6 is to SU(N ) + 2 + 4. This too is an exact SCFT in four dimensions with one marginal parameter. In this case the Since R2,2n+1 has a global symmetry SO(4n + 6) U(1), the global symmetry in both cases In the SU(N ) theory, all the symmetries come from the matter fields. In particular one U(1) factor is the baryonic symmetry associated to the fundamentals, U(1)BF , and the U(1) is intrinsic to R2,2n+1, and the other is associated with the one flavor. We will now discuss the 5d lifts of the two cases separately. N = 2n relevant for us.8 Let us begin with the 5-brane junction for a 5d SU(2n) gauge theory with two antisym To lift the 4d SCFT we need to add four flavors to this. In terms of the 5-brane web there are several possibilities, resulting in theories with different bare CS levels. To motivate the correct choice, let us consider the S-dual web, figure 29a. This describes a a global symmetry SO(4n + 2), fully realized by the gauge theory. Gauging a USp(2n) 8One can also generalize to other CS levels, as shown in appendix C for the theory with a single the proposed 4d dual of SU(2n) + 2 + 4 would therefore seem to be the quiver theory should be added as shown in figure 29b. This web can be obtained by adding D7-branes in the appropriate places and following the procedures described in appendix B. S-dualizing back (rotating the web back by 90 degrees) we get SU(2n)1 + 2 namely the bare CS level is 1. Thus the 4d duality + 4 3 + USp(2n) R2,2n1 lifts to the 5d duality SU(2n)1 + 2 + 4 3 + USp(2n) USp(2n 2) + 1 . The global symmetries of the proposed 5d duals agree. On the SU(2n) side the symmetry is SU(4)F SU(2)A U(1)BF U(1)BA U(1)I , where BF is the baryon number associated to the flavors, and BA is the baryon number associated to the antisymmetric fields. On the U SpU Sp side it is SU(4)F SU(2)BF U(1)F U(1)I1 U(1)I2 . There is no enhancement in the 4d reduction (although there may be enhancement at the 5d fixed point). We can again derive the explicit map of the U(1) charges by finding dual descriptions of various charged operators. The simplest baryonic operator on the SU(2n) side is given by the gauge invariant product of two fundamentals and the conjugate of the antisymmetric The dual operator on the quiver theory side is given by the gauge invariant product of transforms in the (6, 2)(1,0,0) of SU(4)F SU(2)BF U(1)F U(1)I1 U(1)I2 . There are also three baryonic operators involving the Levi-Civita symbol, i1i2n Ai1i2 Ai2k1i2k qi2k+1 qi2n , respectively. In the dual quiver theory these operators involve the (0, 1) instanton. The , n 1 2 Pfaffian corresponds to the gauge-invariant component of the (0, 1) instanton, which trans( , n) 1 of USp(2n) SU(2)BF U(1)F , combined with a USp(2n) flavor, resulting in a 2 gauge-invariant operator in the (6, n)( 12 ,0,1) representation of the global symmetry. The (0, 1) instanton, 1 , with two USp(2n) flavors we get a gauge-invariant operator in the 15 of SU(4)F instead of a singlet. The correct operator combines the ( , n) 1 Comparing the U(1) charges of these operators in the two theories we conclude that the 5d charge map has the form: F = I2 = 2n 1 BF 2n the charge F corresponds to the U(1) in the embedding of USp(2n), and the charge I2 point of view. do this here. N = 2n + 1 stantonic operators in the SU(2n) theory and their duals in the quiver theory. We will not Most of the analysis here parallels that of the even N case, so we will be somewhat briefer. The 5-brane junction and mass-deformed web of the gauge theory SU(2n + 1) + 2 Following the same strategy as before, we consider the S-dual web, figure 31a. This describes a quiver gauge theory with USp(2n) USp(2n). Making use of the fact that the R2n+1 theory is deformable to a USp(2n) + 2n + 3 gauge theory, with an SO(4n + 6) global symmetry, and that gauging a USp(2n) subgroup of this leaves 3 + USp(2n) USp(2n), we conclude that we want to consider the quiver theory 3 + USp(2n) USp(2n) + 1. The web for this theory is shown in figure 31b. S-dualizing back we then get SU(2n + 1)1 + 2 SU(2)BF U(1)F this decomposes as 22n = (1, n + 1) 12 + ( , n) 1 + 2 SU(4)F it decomposes as 22n = (1, n, 4) + ( , n 1, 4) + , n 2, 4 + . 2 , n 1 Figure 31. 5-brane webs for (a) USp(4) USp(4), (b) 3 + USp(4) USp(4) + 1. In other words, the 4d duality lifts to the 5d duality SU(2n + 1) + 2 + 4 1 + USp(2n) R2,2n+1 SU(2n + 1)1 + 2 + 4 3 + USp(2n) USp(2n) + 1 . The global symmetries on both sides are the same as in the even N case. The mapping of the three U(1) charges follows in a very similar manner. The qAq operator is the same as before. The SU(2n + 1) theory has two additional baryonic operators: i1i2n+1 Ai1i2 Ai2k1i2k qi2k+1 qi2n+1 , mer is just the gauge-invariant component of the (1, 0) instanton, which transforms as (4, n + 1)(0,1,0). The dual of the latter is given by the gauge-invariant combination of the ( , n) component of the (1, 0) instanton and the flavor of the second USp(2n), which gives an operator in (4, n)(1,1,0). The 5d charge map then takes the form F = I1 = BF 2n + 1 The coefficients of I can again be determined by including SU(2n + 1) instantons, which intrinsic to R2,2n+1. N + SU(N )k + N As a further generalization of SU(N ) + 2N , let us consider the linear quiver gauge theory in 4d. The interesting limit to consider is when all the couplings become large. When only some of them become large the problem reduces to finding the dual of smaller quivers. it is a quiver theory with gauge group SU(2) SU(3), a fundamental of SU(2), and a bifundamental, where the SU(3) results from weakly gauging the corresponding part of the global symmetry of UN , SU(N )2 SU(3) U(1). This theory has two marginal couplings and a global symmetry SU(N )2 U(1)3, the same as the N + SU(N )2 + N quiver theory. There are two more special cases for which the S-dual theories have been identified. For [SU(N ) TN ] , (3.24) and that for k < N 1 it is given by 1 + SU(2) SU(3) SU(k) hSU(k + 1) N , k i kN is the class of isolated SCFTs that we introduced in section 2.3. The former 2N = UN , and to the example in [21] for k = N 2, since N2 = TN . N We will motivate these dualities by relating them to 5d dualities between gauge theories. But first let us do some 4d consistency checks. The global symmetry on both sides is the dual theories this comes about by summing the dimension of the Coulomb branch of the isolated SCFT and those of the gauge group factors in the product. For example, the Now let us consider the lift to five dimensions. We again have to make a choice regarding the CS levels in the SU(N )k theory. It turns out that the correct choice for the duality is N + SU(N )0k1 Figure 32. 5-brane web of 3 + SU(3)0 SU(3)0 SU(3)1 + 3 and its S-dual. Figure 34. 5-brane web of 4 + SU(4)0 SU(4)1 + 4 and its S-dual. shown in figure 32a. S-duality turns this into the web shown in figure 32b, which after some simple brane manipulations becomes the web shown in figure 32c. This describes the theory 1 + SU(2) [SU(3)0 + 1] SU(3) 12 SU(2) + 2. More generally the dual pair of 5d quiver gauge theories is shown in figure 33. The part on the r.h.s. of the second quiver diagram beginning with the last SU(N ) node corresponds precisely to the gauging of an SU(N ) subgroup of the global symmetry of the TN theory. This reduces to the 4d dual in (3.24). The web and its S-dual are shown in figure 34. We read-off the dual gauge theory as 1 + SU(2) SU(3) 12 [SU(3)0 + 1] SU(2) + 2. The dual pair for general N and k with theory. This reduces to the 4d dual claimed in (3.25). Obviously, the classical global symmetries of the two gauge theories (in both cases) are different, and we expect non-perturbative enhancement, at least in the theories described 1 2 ... k k + 1 ... k + 1 k + 1 k + 1 k ... 2 2 by the S-dual webs. The enhancement of symmetry, and more generally the operator map between the two theories, become increasingly harder to see for larger values of k due to the multitude of topological U(1) symmetries involved. have N + SU(N ) SU(N ) + N , which has a classical global symmetry SU(N )2 U(1)2B claim that the enhancement to SU(N )2 involves all the U(1) symmetries except those associated with the SU(2) SU(3) factor on the l.h.s. , namely U(1)F1 U(1)BF1 U(1)I1 U(1)I2 . For example, the instanton of the last SU(2) factor on the r.h.s. gives a partial usual enhancement of the global symmetry of the SU(2) + 5 theory to E6, with an SU(3) subgroup gauged. To get the next level of enhancement we include the next gauge group factor, SU(3). The (1, 0) and (1, 1) instantons of SU(3) SU(2) then lead to a further enhancement to SU(4)2, as we showed for the T4 theory in section 2.1.1. This suggests a pattern leading to SU(N )2 once all the gauge group factors except the SU(2) SU(3) on the l.h.s. have been included. This leaves the five U(1) symmetries, which in the dual theory are the four associated to the first SU(2) SU(3) factor plus one combination of the others. theories and 5d N In this paper we addressed two related aspects of the connection between 4d N = 2 SCFTs described by 5-brane webs in Type IIB string theory, which are in turn related by dimensional reduction to the 4d SCFTs. The theories considered in this paper are all of AN type, including the TN theories and theories obtained in limits on the Higgs branch of the TN theories. It would be interesting to generalize this to the DN type theories. In the second aspect, we showed that S-dualities in four dimensions, relating the strong another, lift to 5d dualities between different gauge theory deformations of the same 5d SCFT. It is important to stress that here it is the gauge theories themselves, not the 5d SCFT, that are dimensionally reduced to four dimensions. The examples studied in this paper include the Argyres-Seiberg duality involving SU(3) + 6, and the generalizations to SU(N ) + 2N , SU(N ) + 2 + 4 and N + SU(N )k + N . In the latter case the dual theories web corresponding to its 5d lift. An important question that our analysis raises is under what conditions does a duality between 5d supersymmetric gauge theories reduce to an S-duality between 4d superconformal gauge theories? We do not have a complete answer to this question. Clearly we need the reduction to produce a conformal theory, which will only happen for specific gauge groups and matter content, and in specific scaling limits. For example, the 4d superconformal gauge theory with SU(3) + 6 corresponds to the 5d gauge theory with same content, compactified on a circle in the limit R 0, with g 42 = g52/R held fixed. On the other hand the reduction of SU(3) + 5 with the same scaling limit gives SU(3) + 5 in 4d, which is not conformal. Also if we take the same content, SU(3) + 6, but a different scaling limit, say fixing mR, where m is the mass of one of the hypermultiplets, it is not clear what we get in four dimensions. A sufficient condition to get a 4d duality, other than obtaining a conformal theory in four dimensions, is that the 5d duality maps a YM coupling on one side to a YM coupling on the other. This is the coupling that is scaled in the reduction on both sides of the 5d duality, leading to an S-duality relative to the corresponding marginal coupling in 4d. Another important question that deserves further study is to determine how the duality maps the parameters of the dual theories in 5d. In particular, the relation between the scaled YM couplings should reduce to the S-duality map of the 4d couplings. The main quantitative tool used to study the 5d duality, the superconformal index, is insensitive to the values of the mass parameters, and in particular to the YM couplings. We must therefore look for a different approach. While this paper was being finalized, the paper [22] appeared, containing some overlap with section 3.4 of our paper, which discusses the dual of N + SU(N )k + N . 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 Scientific 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 Scientific Research and Development under grant no. 1156-124.7/2011, and the US-Israel Binational Science Foundation under grant no. 2012-041. The 5d superconformal index The superconformal index is a characteristic of superconformal field theories [23]. It counts BPS operators modulo the merging of pairs to form non-BPS operators. As such it is a rigid quantity, invariant under all continuous deformations of the theory that preserve the the highest weights of its SO(5) SU(2)R subgroup: j1, j2 and R. The superconformal index is defined as [7]: I = Tr (1)F x2 (j1+R) y2 j2 qQ , where x, y are the fugacities associated with the superconformal group, and q denotes collectively fugacities associated to other charges Q that commute with the chosen supercharge. These can include symmetries associated to matter fields, as well as topological (instantonic) U(1)I symmetries. If a Lagrangian description is available, the index can in principle be evaluated from the path integral via supersymmetric localization. This was done in a number of examples with SU(N ) and USp(2N ) gauge groups in [7]. The full index is the product of the perturbative index, corresponding to the one-loop determinant of the field theory, and a sum of instantonic contributions, integrated over the gauge group. For the perturbative part, each vector multiplet contributes hypermultiplet contributes wW i=1 where q is the instanton, i.e., U(1)I , fugacity. The k-instanton partition function Zk is the 5d version of the Nekrasov partition function for k instantons [24], expressed as an integral over the ADHM dual gauge group, where the integrand has contributions from both the gauge field degrees of freedom and the charged matter degrees of freedom. The gauge field contributions generically introduce poles, which must be dealt with by giving where eimi are the fugacities associated with the matter degrees of freedom, and the first sum is over the weights of the matter representations. These give the one-particle index. In order to evaluate the full perturbative contribution one needs to put this in a plethystic exponent, defined as P E[f ()] exp " X 1 where the represents all the variables in f . The instantonic contributions come from instantons localized at the north pole of the S4 and anti-instantons localized at the south pole, which also satisfy the supersymmetric localization conditions. These are computed by integrating over the full instanton partition function. The result can be expressed as a power series in the instanton number, inst = 1 + qZ1 + q2Z2 + , an appropriate prescription [7, 11, 18]. Matter fields in representations other than the fundamental introduce additional poles, and the correct prescription for dealing with them can be found in [18]. In some cases there are also poles at the origin or infinity. The prescription for these will be mentioned shortly. There are a number of subtle issues related to the instanton computation. Specifically for SU(N ), there are two issues related the fact that the computation really uses U(N ). Naively the diagonal U(1) part decouples, and one would assume that the result for SU(N ) are U(1) remnants that must be removed by hand. The first is just a sign given by are instantonic states that do not belong to the SU(N ) theory, but whose contribution nevertheless remains in the instanton partition function after removing the diagonal U(1). In some cases these states can be identified in the 5-brane web construction, where they correspond to D1-branes suspended between parallel external NS5-branes. They are removed by multiplying the partition function by the appropriate factor [11, 13, 14, 18], for example Zc = P E (1 xy) 1 y for a remnant instantonic state with F units of flavor charge. However a simple brane description is not always available. More generally, the presence of remnant states manifests itself as the lack of invariance under x 1/x, which is part of the superconformal group, and in some cases, the lack of invariance under the full classical global symmetry. The correction factor can then be determined by the requirement that the corrected partition function respect these symmetries. We have also checked that all the partition functions we used where invariant under these symmetries. The violation of x 1/x is intimately connected to the presence of poles at the origin or infinity. The prescription for such poles in the contour integral, namely the choice of whether or not to include them, is absorbed in the correction factor. In particular, in (A.6) all the poles enclosed by the contour are included. formulas for instanton partition functions of USp(2N ) are different, and in particular do not exhibit the U(1) remnants that the SU(N ) partition functions do. Explicit formulas for the instanton partition functions in many cases have appeared elsewhere, and we will not reproduce them here. We refer the reader to [7, 11]. The only expression we will need, which, as far as we know, has not appeared in the literature, is the contribution of a bifundamental hypermultiplet in SU(N1) USp(2N2). We evaluated this using the methods of [25]. The dual gauge group for a (k1, k2) instanton is U(k1) O(k2). components. Using z for the fugacity of the bi-fundamental symmetry U(1)BF , a, b for the instanton fugacities of SU(N1) and USp(2N2), respectively, ui, vj for the fugacities of the zai zai zai vj vj 1 1 zum + zum vj y vjy 1 1 m,j=1 zum + zum vj x vjx zum bn bn ZBUFUOSp = Y zai + zai zai vj vj 1 1 zum + zum vj y vjy 1 1 m,j=1 zum + zum vj x vjx zum bn bn for the O part and odd k2, and ZBUFUESp = Y zai zai z2u2m + z21u2m y2 y12 1 1 m=1 z2u2m + z2u2m x2 x2 ZBUF+USp = for the O+ part, N1,n21 zai vj vj zum bn bn 1 1 k1,n21 zum + zum vj y vjy 1 1 zum + zum vj x vjx for the O part and even k2. These contributions also add poles to the integral. The prescription for dealing with the integral assuming x, p, d 1, and only at the end returns to the original variables. Flavors in webs Matter in the fundamental representation of the gauge group is usually referred to as flavors. There are different, but equivalent, ways to represent flavor degrees of freedom in 5-brane webs. These are related by brane-creation, or Hanany-Witten (HW), transitions. Figure 36. Two representations of SU(2) + 1. In some representations, the counting of flavors is not obvious, so it is useful to be able to map to other representations in which the counting is obvious. We will illustrate this with four examples, which should make the general process clear. In the main body of the paper we will refer to this idea whenever we have a web in which the counting is unclear. In the first example, we add a single flavor to the pure SU(2) theory by adding a D7-brane, figure 36a. Moving the D7-brane to the right across the (1, 1) 5-brane we get an external D5-brane, figure 36b. These are two equivalent representation of a hypermultiplet in the fundamental representation of SU(2). The first corresponds to the D5-D7 strings, and the second to D5-D5 strings across the NS5-brane. The second example is a little more interesting. Now we begin with the 5-brane web shown in figure 37a. This also describes an SU(2) theory, but with how many flavors? Note that there is one avoided intersection due to the s-rule. There seem to be three independent external D5-branes, which leads us to conclude that there are three flavors. Let us verify this by going to a simpler representation. First we move the one D7-brane across both NS5-branes, leading to the configuration in figure 37b. Then we move the leftnow clearly visible. We can also move the two D7-branes on the left across the NS5-brane to get the representation in figure 37d. Our third example, figure 38, is an elaboration of the previous example. The steps are basically the same, showing that the original web describes an SU(2) gauge theory with Our final example involves a product group and a 7-brane with three attached 5-branes, figure 39. The steps are similar. Webs for antisymmetric matter There is no general prescription for including matter in 5-brane webs in representations other than fundamental or bi-fundamental. Nevertheless it is possible to incorporate some other representations in some cases. The most common cases are SU(N ) or USp(2N ) and matter in the rank 2 antisymmetric representation. The latter and some examples of the former were previously discussed in [11]. We do not have a precise (microscopic) understanding of how these fields arise (in terms of open strings), but it is possible to argue for their existence indirectly by going on the Higgs branch associated to them, and confirming the pattern of gauge symmetry breaking. Here we will consider SU(N ) with one antisymmetric plus fundamentals. We begin with the claim that the 5-brane junction shown in figure 40 corresponds to matter field. Roughly speaking, the matter multiplet is a degree of freedom associated with the two (1, 1) 7-branes. This can be confirmed by performing various deformations of the web. For simplicity we to a finite Yang-Mills coupling is shown in figure 41a. Then going on the Higgs branch pure USp(4) theory (see [11]), which is consistent with a VEV for a single matter field in the antisymmetric representation of SU(4). The CS level can be determined by turning on a mass for the antisymmetric field, described by the web in figure 41c. The remaining antisymmetric matter = 0 + N4 (the cubic Casimir of the antisymmetric representation 2 for the web in figure 40. We can add flavors, i.e., hypermultiplets in the fundamental representation, by attaching D5-branes (ending on D7-branes) on the r.h.s. of the web. Note that it makes a difference whether we attach a D5-brane to the top or the bottom part of the web. This determines the sign of the mass, and therefore affects the value of the bare CS level for the massless theory. Also, there is a limit to the number of flavors we can add. Beyond some number, external 5-branes will intersect, which in principle means that the corresponding fixed point theory does not exist. Some amount of intersection is however resolvable via HW transitions (see for example the webs for the TN theories). As a concrete example, let us find a 5-brane web for SU(4) 1 with an antisymmetric , we need to add one D5-brane at the top and six at the bottom (figure 42a). Then 0 = + 12 26 = 2 . Two HW transitions involving the lowest D7-brane lead to 1 the web in figure 42b, and then a couple involving the (0, 1) 7-brane at the bottom lead to the web in figure 42c. Repeating these steps for the next D7-brane leads to the web in 2 figure 42d. The latter is related by SL(2, Z) to the R1,5 web. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. dynamics, Phys. 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Oren Bergman, Gabi Zafrir. Lifting 4d dualities to 5d, Journal of High Energy Physics, 2015, 141, DOI: 10.1007/JHEP04(2015)141