Duality walls and defects in 5d \( \mathcal{N}=1 \) theories

Journal of High Energy Physics, Jan 2017

We propose an explicit description of “duality walls” which encode at low energy the global symmetry enhancement expected in the UV completion of certain five-dimensional gauge theories. The proposal is supported by explicit localization computations and implies that the instanton partition function of these theories satisfies novel and unexpected integral equations.

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Duality walls and defects in 5d \( \mathcal{N}=1 \) theories

Received: August Published for SISSA by Springer Open Access 0 1 c The Authors. 0 1 0 31 Caroline Street North , Waterloo, ON N2L 2Y5 Canada 1 Perimeter Institute for Theoretical Physics We propose an explicit description of duality walls" which encode at low energy the global symmetry enhancement expected in the UV completion of certain dimensional gauge theories. The proposal is supported by explicit localization computations and implies that the instanton partition function of these theories satis es novel and unexpected integral equations. Duality in Gauge Field Theories; Field Theories in Higher Dimensions; Su- - Duality walls and defects in 5d = 1 theories 3 Index calculations 4 Wilson loops SU(N )N theories Example: SU(N )N Nf =2 theories with Nf avors Hemispheres with Wilson loop insertions Example: SU(2) theories Example: SU(3) theories Duality and 't Hooft surfaces Higgsing IN;N Codimension 2 defects Higgsing in the absence of a duality wall Higgsing in the presence of a duality wall Enhanced symmetry of SU(N + 1) theory From Sp(N ) to exotic SU(N + 1) Duality walls between Sp(N ) and SU(N + 1) theories 1 Introduction Duality walls between SU(N ) gauge theories Pure N = 1 SU(N )N gauge theory Domain wall actions SU(N )N Nf =2 SQCD with Nf < 2N Duality walls for SU(N ) with Nf = 2N Exceptional symmetries in SU(2) theories A 5d Nekrasov's instanton partition function A.1 SU(N ) partition function A.2 Sp(N ) partition function A.3 Hypermultiplets B Partition functions of exotic SU(3) theory B.1 Superconformal indices Five-dimensional super-conformal eld theories are a particularly rich subject of investigation (see [1{5] for seminal work on the subject). The only constructions available for these theories involve brane constructions, in particular quarter-BPS webs of ve-branes in IIB string theory. Some of the ve-dimensional SCFTs admit mass deformations to ve-dimensional gauge theories, with the inverse gauge coupling playing the role of mass deformation parameter. Several protected quantities in the ve-dimensional SCFT are computable directly from the low-energy gauge-theory description [6]. More precisely, the space of mass deformations of the UV SCFT is usually decomposed into chambers, which ow in the IR to distinct-looking gauge theories, or to the same gauge theory but with di erent identi cations of the parameters. With some abuse of language, these distinct IR theories may be thought of as being related by an \UV duality", in the sense that protected calculations in these IR theories should match [7]. In such a situation, one may de ne the notion of \duality walls" between the di erent IR theories [8]. These are half-BPS interfaces which we expect to arise from RG starting from Janus-like con gurations, where the mass deformation parameters vary continuously in the UV, interpolating between two chambers. Duality walls between di erent chambers should compose appropriately. Furthermore, if we have some BPS defect in the UV SCFT, we have in principle a distinct IR image of the defect in each chamber, each giving the same answer when inserted in protected quantities. The duality walls should intertwine, in an appropriate sense, between these images. In this paper we propose candidate duality walls for a large class of quiver gauge theories of unitary groups.1 The UV completion of these gauge theories has a conjectural enhanced global symmetry whose Cartan generators are the instanton number symmetries of the low-energy gauge theory. The chambers in the space of real mass deformations dual to these global symmetries are Weyl chambers and the duality walls generate Weyl re ections relating di erent chambers. The duality walls admit a Lagrangian description in the low energy gauge theory. The fusion of interfaces reproduces the expected relations for the Weyl group generators thanks to a beautiful collection of Seiberg dualities. This is the rst non-trivial check of our proposal. The second set of checks involve the computation of protected quantities. The duality walls we propose give a direct physical interpretation to a somewhat unfamiliar object: elliptic Fourier transforms (see [10] and references within). These are invertible integral transformations whose kernel is built out of elliptic gamma functions. We interpret the integral kernel as the superconformal index of the four-dimensional degrees of freedom sitting at the duality interface and the integral transform as the action of the duality interface on more general boundary conditions for the ve-dimensional gauge theory. The integral identity which encodes the invertibility of the elliptic Fourier transform follows from the corresponding Seiberg duality relations. 1Duality walls of the same kind, for 5d gauge theories endowed with a six-dimensional UV completion, appeared rst in [9]. It follows directly from the localization formulae on the S4 S1 and the de nition of a duality wall that the corresponding elliptic Fourier transform acting on the instanton partition function of the gauge theory should give back the same partition function, up to the Weyl re ection of the instanton fugacity. This is a surprising, counterintuitive integral relation which should be satis ed by the instanton partition function. Amazingly, we nd that this relation is indeed satis ed to any order in the instanton expansion we cared to check. This is a very strong test of our proposal. Experimentally, we nd that this is the rst example of an in nite series of integral identities, which control the duality symmetries of Wilson line operators. These relations suggest how to assemble naive gauge theory Wilson line operators into objects which can be expected to have an ancestor in the UV SCFT which is invariant under the full global symmetry group. We also identify a few boundary conditions and interfaces in the gauge theory which transform covariantly under the action of the duality interface and could thus be good candidates for symmetric defects in the UV SCFT. We brie y look at duality properties of defects in codimension two and three as well. Finally, we attempt to give a physical explanation to another instance of elliptic Fourier transform which we found in the literature, which schematically appears to represent an interface between an Sp(N ) and an SU(N + 1) gauge theories. We nd that the AC elliptic Fourier transform maps the instanton partition function of an Sp(N ) gauge theory into the instanton partition function of an exotic version of SU(N + 1) gauge theory with the same After this work was completed, we received [11, 12] which have some overlap with the number of avors. last section of this paper. Duality walls between SU(N ) gauge theories Pure N = 1 SU(N )N gauge theory Our rst and key example of duality wall encodes the UV symmetries of a pure This gauge theory is expected to be a low-energy description of a 5d SCFT with SU(2) global symmetry, deformed by a real mass associated to the Cartan generator of SU(2). In turn, the SCFT can be engineered by a BPS ve-brane web involving four semi-in nite external legs: two parallel NS5 branes, a ( 1; N ) and a ( 1; N ) vebranes. The SU(2) global symmetry is associated to the two parallel NS5 branes. See gure 1. The mass deformation breaks SU(2) to a U(1) subgroup, which is identi ed with the instanton U(1)in global symmetry of the SU(N ) gauge theory, whose current is the instanton The Weyl symmetry acts as m ! m and the corresponding duality wall should relate two copies of the same gauge theory, glued at the interface in such to preserve the antidiagonal combination of the U(1)in instanton global symmetries on the two sides of the The gauge theory is supported on the bundle of N parallel D5 branes. After removing the centre of mass, the only non-normalizable deformation is the separation m between the NS5 branes. sides of an interface as open circles and the bi-fundamental matter as an arrow between them. The extra baryonic coupling is denoted as a black dot over the arrow. We propose the following setup: a domain wall de ned by Neumann b.c. for the SU(N )N gauge theory on the two sides of the wall, together with a set of bi-fundamental 4d chiral multiplets q living at the wall, coupled to an extra chiral multiplet b by a 4d W = b det q : See gure 2 for a schematic depiction of the duality wall. This system is rife with potential gauge, mixed and global anomalies at the interface, which originate from the 4d degrees of freedom, from the boundary conditions of the 5d gauge elds and and from anomaly in ow from the bulk Chern-Simons couplings. The cubic gauge anomaly cancels out beautifully: the bi-fundamental chiral multiplets behave as N fundamental chiral multiplets for the gauge group on the right of the wall, giving N units of cubic anomaly, which cancel against the anomaly in ow from the N units of ve-dimensional Chern-Simons coupling. Similarly, we get N units of cubic anomaly for the gauge group on the left of the wall, which also cancel against the anomaly in ow from the N units of ve-dimensional Chern-Simons coupling. The bi-fundamental chiral multiplets also contribute to a mixed anomaly between the bulk gauge elds and the baryonic U(1)B symmetry which rotates the bi-fundamental elds involving the left gauge elds has the same sign and magnitude as the anomaly involving gauge theory has N avors and at low energy it glues the two 5d gauge groups together. the right gauge elds. Both are the same as the anomaly which would be associated to a single fundamental boundary chiral of charge 1. We can make a non-anomalous U(1) global symmetry by combining U(1)B with U(1)in from both sides of the wall. Under U(1) a boundary baryon operator will have the same charge as an instanton particle on the left side of the wall, or an anti-instanton particle on the right side of the wall. In particular, the proposed duality wall glues together U(1)in on the two sides of the wall with opposite signs and thus has a chance to implement the Z2 We can also de ne a non-anomalous R-symmetry by combining the Cartan generator of the bulk SU(2)R symmetry and a boundary symmetry which gives charge 0 to the bifundamentals, and thus charge 2 to b. The cancellation of the mixed gauge anomaly proceeds as follows: the bulk gauge elds with Neumann b.c. contribute half as much as 4d SU(N ) gauge elds would contribute and thus the R-symmetry assignment is the same as for a 4d SQCD with Nf = N . A neat check of this proposal is that two concatenated duality walls will annihilate in the IR. Far in the IR, a pair of consecutive duality walls looks like a single interface supporting four-dimensional SU(N ) gauge elds which arise from the compacti cation of ve-dimensional SU(N )N gauge theory on the interval. Together with the quarks associated to each duality wall, that gives us a four-dimensional S(N ) gauge theory with N avors, deformed by a superpotential coupling which sets to zero the two baryon operators det q and det q~. This four-dimensional theory has a well-known low-energy behaviour: it can be debaryons B = det q, B~ = det q~, subject to a constraint W = b det q + ~b det q~ BB~ = 2N : Because of the bB + ~bB~ superpotential couplings, we can restrict ourselves to the locus freedom required to Higgs the left and right ve-dimensional theories back together, and thus ow in the far IR back to a trivial interface. This is the expected behaviour for Z2 Domain wall actions We should be able to use the domain walls to de ne a Z2 duality action on U(1)in-preserving half-BPS boundary conditions for the SU(N )N ve-dimensional gauge theory. As the vedimensional gauge theories are IR free, we can describe most boundary conditions in terms of their boundary degrees of freedom, which are in general some four-dimensional SCFTs equipped with an SU(N ) and an U(1)in global symmetries with speci c cubic anomalies. The exceptions are boundary conditions which (partially) break the gauge symmetry at units of cubic 't Hooft anomaly, a U(1)@ global symmetry with a mixed 't Hooft anomaly with the SU(N ) global symmetry equal to the contribution of a single fundamental chiral eld of charge 1 and an R-symmetry with a mixed 't Hooft anomaly with the SU(N ) global symmetry equal to the contribution of N quarks of R-charge 0. Such a theory can be used to de ne a boundary condition for the 5d SU(N )N gauge theory which preserves a U(1) symmetry, diagonal combination of U(1)in and U(1)@, and an R-symmetry. The action of the duality wall on this boundary condition gives a new theory B0 built from B by adding N anti-fundamental chiral multiplets q of SU(N ), gauging the overall the same type of mixed 't Hooft anomalies as we required for B (involving a new choice of U(1)@ global symmetry). In case of boundary conditions which break the gauge group to some subgroup H, we can apply a similar transformation, which only gauges the H subgroup of SU(N ). For example, the duality wall maps Dirichlet boundary conditions, which fully break the gauge group at the boundary, to Neumann boundary conditions enriched by the set of N chiral We can provide a more entertaining example: a self-dual boundary condition. We de ne the boundary condition by coupling the ve-dimensional gauge elds to N +1 quarks q0 and a single anti-quark q~0. For future convenience, we also add N + 1 extra chiral multiplets M coupled by the superpotential Thus the boundary condition has an extra SU(N + 1) U(1)e global symmetries de ned at the boundary. The SU(N + 1) simply rotates q0 as anti-fundamentals and M as fundamentals. The non-anomalous R-symmetry assignments are akin to the ones for a 4d SQCD with N + 1 avors. The bulk instanton symmetry can be extended to a non-anomalous symmetry under 1=N . The remaining non 1=N and on M with charge 1. After acting with the duality interface, we nd at the boundary four-dimensional SU(N ) gauge theory, with N + 1 avors given by the quarks q0 and anti-quarks q and q~0. The theory has a Seiberg dual description in the IR, involving the mesons and baryons 2It may be possible to consider a larger set of boundary conditions, involving singular boundary conditions for the matter and gauge elds, akin to Nahm pole boundary conditions for maximally supersymmetric gauge theories [13, 14]. tential coupling for the closed loop of three arrows. det q anti-baryon. The remaining qq0 mesons give N + 1 new fundamental chiral at the boundary, the dual version of q0. The remaining anti-baryons give one anti-fundamental chiral, the dual version of q~0. The baryons give the dual version of M . We should keep track of the Abelian global symmetries. The dual quarks have instanand U(1)e charge In order for the self-duality to be apparent, we should re-de ne our instanton symmetry 1=(2N ), on M charges, but leave U(1)e una ected. It is natural to conjecture that this boundary condition descends from an SU(2)in-invariant boundary condition for the UV SCFT, equipped with an extra SU(N + 1) U(1)e global symmetry. We can generalize that to a duality-covariant interface IN;N0 between SU(N )N and SU(N 0)N0 , coupled to three sets of four-dimensional chiral elds: N + N 0 fundamentals w of SU(N ), N + N 0 anti-fundamentals u of SU(N 0) and a set of bi-fundamentals v of SU(N 0) If we act with an SU(N )N duality interface, we obtain a four-dimensional SU(N ) gauge theory with N + N 0 avors, fundamentals w and anti-fundamentals v and q. Applying Seiberg duality, we arrive to an SU(N 0) gauge theory with N + N 0 avors. The original superpotential lifts the u elds and the vw mesons. The b det q superpotential maps to a similar b det q_ involving the Seiberg-dual quarks which transform under the ve-dimensional SU(N 0)N0 gauge elds. The nal result is identical as what one would obtain by acting with the SU(N 0)N0 duality interface. The duality-covariant interfaces IN;N0 have interesting properties under composition. Consider the composition of IN;N0 and IN0;N00 : it supports a four-dimensional SU(N 0) gauge theory coupled to N + N 0 + N 00 avors, which include the N + N 0 anti-fundamentals u, N 0 + N 00 fundamentals w0, bifundamentals v and v0. If we apply Seiberg duality, we nd a new description of a composite interface, which is actually a modi cation of IN;N00 ! Indeed, we nd an SU(N + N 00) gauge group which is coupled to the 5d degrees of freedom just as the avor group of IN;N00 , and is furthermore coupled to N + N 0 fundamentals and N 0 + N 00 Nf =2 SQCD. The gauge theory is supported on the bundle of N parallel D5 branes. After removing the centre of mass, the non-normalizable deformation are the separation m between the NS5 branes and the vertical separation mf between the semi-in nite D5 branes and the intersection of one of the NS5 branes and the ( 1; N ) vebrane. The latter parameter is the overall mass parameter for the hypermultiplets. We drew the resolved vebrane web for positive and negative values of the overall hypermultiplet mass. The former is closely related, but not identical to the gauge coupling or mass for U(1)in. It is possible to argue that the instanton mass mi actually equals m + N2f mf . The standard IR gauge theory description is valid for m > 0 and m + Nf mf > 0. When m becomes negative and we ip its sign to go to a dual parameterization, we exchange the roles of the NS 5 branes and thus the role of mf and the auxiliary parameter m0f = mf + mN . Alternatively, we can use mf +m0f as a parameter, 2 which remains invariant under duality. anti-fundamentals with a superpotential coupling to (N + N 0) (N 0 + N 00) mesons. This is consistent with the duality-covariance of the interface. The interface IN;N0 clearly has an SU(N + N 0) global symmetry. We can also de ne an U(1)e non-anomalous global symmetry, acting with charge 1 on v, N N+N0 on u. The second U(1)in global symmetry can be taken to act with charge 1 on w, 1 on u and charge N + N 0 on instantons on the two sides. The IN;N duality-covariant interface is particularly interesting. It supports a baryon operator det v charged under U(1)e only. If we give it a vev, by a diagonal vev of v, we Higgs together the gauge elds on the two sides of the interface and the superpotential coupling gives a mass to u and w. We arrive to a trivial interface. Later on in section 5 we will use IN;N to study the duality properties of of 't Hooft surface defects. SU(N )N Nf =2 SQCD with Nf < 2N theories with Nf avors, with Nf < 2N . The SCFT can be engineered by a BPS ve-brane web involving Nf + 4 semi-in nite external legs: two parallel NS5 branes, a ( 1; N ) and a N ) vebranes, Nf D5 branes pointing to the left. The SU(2) global symmetry is associated again to the two parallel NS5 branes, while the Nf D5 branes support an U(Nf ) global symmetry. The vebrane webs and mass parameters are depicted in gure 6. As the gauge elds are IR free, we expect to be able to describe a typical halfBPS boundary condition for such gauge theories in terms of an SU(N )-preserving boundary conditions for the ve-dimensional hypermultiplets, with a weak gauging of the dimensional SU(N ) symmetry. Of course, it is also possible to only preserve, and gauge, at the boundary some smaller subgroup H of the ve-dimensional gauge group. An extreme example would be to give Dirichlet boundary conditions to the gauge elds. Half-BPS boundary conditions for ve-dimensional free hypermultiplets may yet be strongly coupled. On general grounds [15], it is always possible, up to D-terms, to describe such boundary conditions as deformations of simple boundary conditions which set a Lagrangian half of the hypermultiplet scalars (which we can denote as \Y") to zero at the boundary. The remaining hypers (which we can denote as \X") can be coupled to a boundary theory B by a linear superpotential coupling W = XO involving some boundary operator O. This gives a boundary condition which we could Conversely, if we are given some boundary condition BX for free hypermultiplets, we can produce a four-dimensional theory B by putting the 5d hypers on a segment, with free chiral multiplets sides of the interface. With these considerations in mind, we can evaluate the 't Hooft anomaly polynomial must be exactly half of the 't Hooft anomaly polynomial for a four-dimensional free chiral with the same quantum numbers as X. Our proposal for the duality interface generalizes the interface for pure SU(N )N gauge theory: we set to zero at the boundary the fundamental half X of the hypermultiplets on the right of the wall and anti-fundamental Y 0 on the left of the wall, with a boundary W = b det q + Tr X0qY : The combination of gauge anomalies from q and the boundary condition for the hypermultiplet precisely matches the desired bulk Chern-Simons level N Nf =2. We denote as X the elds which transform as anti-fundamentals of U(Nf ). In particular, we give them 1 under the diagonal U(1)f global symmetry in U(Nf ). A consecutive pair of these conjectural duality walls can be analyzed just as in the pure gauge theory case, as the boundary conditions prevent the ve-dimensional hypers on the interval from contributing extra light four-dimensional elds. They can be integrated away to give a Tr X00q~qY coupling. As the meson qq~ is identi es with the identity operator in the group which goes through the interface as a strip. The dashed arrows indicate which half of the bulk hypermultiplets survives at the wall. We include a superpotential coupling for the closed loop of three arrows. 4d SU(N ) gauge theory has N avors and at low energy it glues the two 5d gauge groups together. The theory includes a quartic superpotential coupling which arises from integrating away the hypermultiplets in the segment. In the IR, it glues together the hypermultiplets on the two sides of IR, the interface ows to a trivial interface for both the gauge elds and the hypermultiplets, up to D-terms. Thus the interface is a reasonable candidate for a duality wall. Next, we can look carefully at the anomaly cancellation conditions. It is useful to express the anomaly cancellation in terms of fugacities. If we ignore for a moment the R-charge and say that q has fugacity 1=N , X has fugacity x and X0 has fugacity x0, the superpotential imposes x = instanton fugacities on the right to ir = x Nf =2 and i` = 1(x0) Nf =2. We can re-cast the relation as a statement about one combination of bulk fugacity being inverted by the interface, inverted irxNf =2 2N = i`(x0)Nf =2 2N . = irxNf =2 and 1 = i`(x0)Nf =2, and one being not Although these relations may look unfamiliar, they can be understood in a straight As far as R-symmetry is concerned, the bulk R-symmetry only acts on the scalar elds in the hypermultiplets, with charge 1. Thus we expect that assigning R-symmetry 0 to q and 2 to b will both satisfy anomaly cancellation and be compatible with the superpotential It is straightforward to extend to SQCD the duality-covariant boundary conditions and interfaces proposed for pure SU(N ) gauge theory. We refer to gure 9 for the quiver description of the IN;M interface and to gure 10 for the Seiberg-duality proof of dualitycovariance. The composition of IN;M and IM;S can again be converted to a modi cation tential coupling for the closed loops of three arrows. After removing the centre of mass, the non-normalizable deformation are the separation m between the NS5 branes and the separation m~ between the ( 1; N ) vebranes. The vertical separation mf between the semi-in nite D5 branes and the intersection of one of the NS5 branes and the ( 1; N ) N mf , m0 = mi + N mf . Duality walls for SU(N ) with Nf = 2N The SU(N ) theory with 2N avors is rather special: in the UV, two distinct Abelian global symmetries are expected to be promoted to an SU(2). Essentially, they are the sum and di erence of the instanton and baryonic U(1) isometries. Correspondingly, we will nd two commuting duality walls. In the vebrane construction, the extra symmetry is due to two sets of parallel vebranes. See gure 11. The rst duality wall is de ned precisely as before, i.e. set to zero at the boundary the fundamental half X of the hypermultiplets on the right of the wall and anti-fundamental the closed loop of three arrows. Y 0 on the left of the wall, with a boundary superpotential W = b det q + Tr X0qY : For the second wall, we replace q with a set of bi-fundamental elds q~ in the opposite direction, and set to zero at the boundary the anti-fundamental half Y of the hypermultiplets on the right of the wall and fundamental X0 on the left of the wall, with a boundary W = ~b det q~ + Tr Xq~Y 0 : Both walls implement Z2 symmetries: the composition of two walls of the same type ows to the identity, and they re ect one of the two fugacities = irxN or ~ = irx N while leaving the other one xed. We can consider the concatenation of the two walls. That gives us a 4d SU(N ) gauge theory coupled to q, q~ and the surviving half of the bulk hypermultiplet in the interval. If we pick one of the two possible orders of the composition, we nd W = b det q + Tr X0qY + ~b det q~ + Tr X0q~Y 00 with X0 being a set of 2N fundamental chiral multiplets and q, q~ anti-fundamentals. If we concatenate the walls in the opposite order, we nd W = b det q + Tr X00qY 0 + ~b det q~ + Tr Xq~Y 0 with Y 0 being a set of 2N anti-fundamentals and q and q~ fundamentals of the 4d gauge The two possibilities are precisely related by Seiberg duality! The mesons produced by the duality implement the switch in the boundary conditions for the hypermultiplets, and the baryons are re-mixed so that the b and ~b couplings match as well. Thus the two duality walls commute, as expected. The duality walls we considered can be de ned with minor changes in quiver gauge theories where one or more nodes satisfy a balancing condition = Nc Nf =2. In the language of vebranes, if the quiver is engineered by a sequence of D5 brane stacks stretched between NS5 branes, the balancing condition insures that either the top pair of semi-in nite pairs are parallel. See gure 14 for an example. theory with N avors at the left node and M at the right node. The ve U(1) global symmetries (two instanton symmetries and three hypermultiplet masses) are enhanced to U(1)2 SU(2) SU(3) because of the two sets of parallel vebranes. The six mass deformations in the picture satisfy a relation: m0 = m + M mf + N m0f . A sequence of k balanced nodes is expected to be associated in the UV to an SU(k + 1) global symmetry, enhancing a certain combination of the instanton and bi-fundamental hypermultiplet charges for these nodes. We want to understand the e ect of a duality wall for a node of the quiver on the other nodes of the quiver, and gure out how the duality walls for di erent nodes match We can de ne the duality wall at a balanced node as we did for a single gauge group, leaving the other gauge groups and other hypermultiplets continuous at the interface. As the X0 and Y elds for a given node are charged under the gauge groups at nearby nodes, but have di erent Abelian charges, in order for the corresponding symmetries to remain non-anomalous, we need to correct these Abelian charges by the instanton charge at the nearby nodes on either sides of the interface. In terms of instanton fugacities, that means that the instanton fugacities at the nearby nodes will have to jump by the sum of the fugacities of X0 and Y , i.e. the fugacity of q. That makes sense: the duality wall permutes two consecutive semi-in nite branes and the instanton symmetries at the other nodes are associated to the relative distance of nearby vebranes. If we permute two vebranes whose distance is associated to the fugacity , the distances from other vebranes jump by plus or minus that distance and the fugacities jump by factors of Let's denote the domain walls associated to nodes a with positive balancing condition as Da+, and the ones associated to nodes a with negative balancing condition as Da . If It is easy to show that all Da+ commute with all the Da . It is more interesting to show that each sequence of consecutive walls with the same sign satisfy the relations of a rise to a four-dimensional SU(Na) gauge theory with Na + Na+1 avors. For example, the left hand side gives Seiberg duality appears to neatly exchange the interfaces corresponding to the two sides of the permutation group relation, up to a small mismatch concerning the b0 det q0 coupling for the intermediate interface in the composition: b0 appears to couple on the two sides to two di erent operators with the same fugacities. The mismatch can likely be explained away by the possibility of operator mixing under Seiberg duality. Exceptional symmetries in SU(2) theories The UV completion of SU(2) gauge theories with Nf avors is expected to have an enhanced ENf +1 global symmetry. This can be understood as a combination of the general UV enhancement for SU(N ) gauge theories and the enhancement of U(Nf ) to SO(2Nf ) due to the fact that the fundamental representation of SU(2) is pseudo-real. Indeed, the SU(2) enhancement involves a linear combination of U(1)in and the diagonal U(1) subgroup of U(Nf ) and thus it combines non-trivially with the enhancement of U(Nf ) to SO(2Nf ). Correspondingly, we can nd continuously many versions of our basic duality wall, each labelled by a choice of U(Nf ) subgroup in SO(2Nf ) and a splitting of the hypermultiplet scalars into N \X" and N \Y" complex scalar elds. It is most useful to look at domain walls which preserve a common Cartan sub-algebra of the global symmetry group, implementing Weyl re ections in the UV. If we denote the bulk quarks as Qi, i = 1; ; 2Nf , we can consider duality walls for which the X elds consist of Nf k quarks from the i = 1; ; Nf range and k quarks from the i = Nf + 1; overall fugacity of the X ; 2Nf range. If we denote as xa the fugacities of the quarks, the elds will be de ned as xNf = Q a2X xa. The domain walls invert It is important to point out that not all splittings are simultaneously possible. There are two disconnected classes of choices of X and Y elds among the Qi, distinguished by comparing the sign of their \orientation" dX1dY1dX2dY2 . Intuitively, in order to interpolate between boundary conditions in di erent classes we need to add a single chiral doublet at the boundary, which contributes one unit to the discrete Z2 gauge anomaly of SU(2). Thus either boundary conditions with even k are simultaneously non-anomalous, or boundary conditions with odd k are simultaneously non-anomalous, but not both. Notice that SU(2) gauge theories have no continuous theta angle, but have a discrete Z2-valued theta angle. One unit of discrete Z2 gauge anomaly at the boundary can be compensated by a shift of the bulk discrete theta angle. Thus we expect the two classes (even k and odd k) of boundary conditions to be associated to the two di erent choices of bulk theta angle. Thus we have 2Nf 1 basic domain walls. In general, composing two such domain walls associated to splittings (X; Y ) and (X0; Y 0) will give an interface supporting an 4d SU(2) gauge theory, with as many chiral quarks as the number of bulk avors which belong to X and Y 0 (or equivalently X0 and Y ). The relations in the Weyl group of ENf +1 must correspond to Seiberg-like dualities in the corresponding domain wall theories. For reasons of space, we will only verify these for the simplest non-trivial example, involving Q3 and Q4. Both preserve the same SU(2) subgroup of the SO(4) global group, and mix the instanton symmetry with the other SU(2) subgroup to an SU(3). If we concatenate the two walls, the intermediate SU(2) 4d gauge group will be coupled to three avors, i.e. the six doublets q, q~, Q1, Q2. In the IR, they will ow to a set of 15 mesons. Two of them will be lifted by b and ~b and eight simply ip the boundary condition on the left and right hypermultiplets so that we are left with Q1 and Q2 at both boundaries. The remaining ones give a set of bi-fundamental elds between the left and right gauge groups and a neutral singlet. The Pfa an superpotential involving the 15 mesons couples the singlet to the determinant of the bifundamental eld and couples the bi-fundamental to the boundary values of the hypermultiplet. nal result is again a duality wall, combined with a permutation of the Q1, Q2 quarks with the Q3, Q4 quarks on one side of the wall. If we denote the two original duality walls as D1 and D2, and the trivial duality wall permuting the two sets of quarks as D3, we nd the relations D1D2 = D2D3 = D3D1 ; D2D1 = D3D2 = D1D3 which agree well with the properties of the three permutations in S3, the Weyl group Index calculations In this section, we consider the superconformal index (SCI) and the hemisphere index of a 5d SCFT at the UV xed point. The superconformal index is a trace over the BPS operators in the CFT on RD, or over the BPS states on a sphere SD 1 times R via the I(wa; q; p; q) = Tr( 1)F pj1+Rqj2+R Y waFa qk : j1; j2 and R are the Cartan generators of the SO(5) SU(2)R bosonic algebra and p; q are their fugacities. Fa are the Cartans of the global symmetries visible in the classical Lagrangian and wa are the corresponding fugacities. k is the instanton number and its fugacity is q. This index can also be considered as a twisted partition function on S1 which was computed in [6, 17] using supersymmetric localization. The hemisphere index is the supersymmetric partition function on an half of the sphere S4 times S1 with a speci c boundary condition of the D4. We can also interpret it as an index counting the BPS states on S1 R4 with Omega deformation, introduced in [18]. e 2. Roughly speaking, this index is an half of the superconformal index and thus the full sphere index (or SCI) can be reconstructed by gluing two hemisphere indices. We will now use these indices to test our duality proposal. SU(N )N theories Let us begin by pure SU(N )N gauge theories. The hemisphere index with Dirichlet b.c. is The \gauge fugacity" zi becomes here the fugacity of the boundary global symmetry. ZiNnst is the singular instanton contribution localized at the center of the hemisphere. The gauge theory on the full sphere can be recovered from two hemispheres with Dirichlet boundary conditions by gauging the diagonal SU(N ) boundary global symmetry. So the full sphere index can be written as IN ( ; p; q) = hIIN jIIN i IIN (zi; ; p; q)IIN (zi; ; p; q) : (3.3) QiN6=j (zi=zj) i6=j The integrand includes the contribution of the 4d gauge multiplet, with IV being the contribution of the Cartan elements. The integration measure is simply 2dziz . The overline indicates a certain operation of \complex conjugation", which inverts all gauge/ avor fugacities. Other boundary conditions or interfaces can be obtained from Dirichlet boundary conditions by adding boundary/interface degrees of freedom and gauging the appropriate diagonal boundary global symmetries. For example, if I4d N;M (zi; zi0; p; q) is the superconformal index of some interface degrees of freedom for an interface between SU(N ) and SU(M ) gauge theories, the sphere index in the presence of the interface becomes3 IIN (zi; ; p; q)IN4d;M (zi; zi0; p; q)IIM (zi0; ; p; q) : QiN6=j (zi=zj ) QiN6=j (zi0=zj0 ) Hemisphere indices, or sphere indices with an interface insertion, can be thought of as counting the number of boundary or interface local operators in protected representations of the superconformal group. Before going on, we should spend a few words on how to compute the correct instanton contribution ZiNnst to the localization formula. The partition function is computed by equivariant localization on the moduli space of instantons. The instanton moduli spaces have singularities, whose regularization can be thought of as a choice of UV completion for the theory. The standard regularization for unitary gauge group is the resolution/deformation produced by a noncommutative background, or by turning on FI parameters in the ADHM quantum mechanics [19, 20]. In principle, the standard regularization may not be the correct one to make contact with the partition function of a given UV SCFT. For SCFTs associated to (p; q) vebrane webs, the standard regularization is expected to be almost OK [21, 22]: the correct instanton partition function is conjectured to be same as the standard instanton partition function up to some overall correction factor, independent of gauge fugacities and precisely associated to the global symmetry enhancement of the UV SCFT: each pair of parallel ( 1; q) semi-in nite vebranes contributes a factor of4 Zextra( ; p; q) = PE to the correction factor, where is the fugacity for the global symmetry associated to the mass parameter corresponding to the separation between the parallel ( 1; q) semiin nite vebranes. This correction factor has been extensively tested against the expected global symmetry enhancement of the superconformal indices. It appears to account for the decoupling of the massive W-bosons living on the six-dimensional world-volume of the semi-in nite vebranes. 3One can bring the 4d index under the conjugation. The inversion of fugacities can be understood as the di erence in sign which appears when matching 5d and 4d fugacities for left or right boundary conditions. 4PE[f ] denotes the plethystic exponent of single-letter index f . The standard instanton partition function computed by using equivariant localization of [18, 23] result takes the following contour integral form ZQNM(zi; q; p; q) = PIk=1 I Zvec( I ; zi; p; q) ; is known that the integral should be performed by using the Je rey-Kirwan (JK) method, which is rst introduced in [24] and later derived in [25] for 2d elliptic genus calculations. See [26{28] for applications to 1d quantum mechanics and a detailed discussion of contour integrals. See also appendix A for details on instanton partition functions. The correction factor from the parallel semi-in nite NS5-branes is Zextra(q; p; q) = PE Let us leave a few comment on this correction factor. This factor can also be read o from the residues R 1 at in nity I = 1. R 1 are associated to the noncompact Coulomb branch parametrized by vevs I of the scalar elds in the vector multiplet. In fact, the above contour integral contains the contribution from the degrees of freedom along this Coulomb branch and it is somehow encoded in the R . The extra contribution is roughly an `half' . The residue at the in nity is in general given by a sum of several rational functions of p; q. The `half' here means that we take only an half of them such that it satis es two requirements: when we add it to the standard instanton partition function, 1) the full instanton partition function becomes invariant under inverting x 2) it starts with positive powers of x in x expansion. The second requirement follows from the fact that the BPS states captured by the instanton partition function have positive charges under the SU(2) associated to x. This half then gives the extra contribution from the Coulomb branch and it also coincides with the correction factor (3.7). We will see similar correction factors in the other examples below. Since the Coulomb branch of the ADHM quantum mechanics dose not belong to the instanton physics of the 5d QFT, we should remove its contribution to obtain a genuine 5d partition function. So the correct instanton partition function of the 5d SCFT is expected QiN;j=1 ( 1=N zi=zj0 ) ; with q = in this case. ZiNnst(zi; ; p; q) = ZQNM(zi; ; p; q)=Zextra( ; p; q) ; At this point, we are ready to study the duality interface. The easiest way to do so is to look at the boundary condition obtained by acting with the duality interface on a Dirichlet boundary, i.e. the dual of Dirichlet boundary conditions. This consists of the duality interface degrees of freedom coupled to a single SU(N )N gauge theory, with the second SU(N ) global symmetry left ungauged. More general con gurations can be obtained immediately by gauging that SU(N ) global symmetry. The 4d superconformal index of the duality interface degrees of freedom is simply where zi and zi0 are the fugacities for the gauge group on the left and right of the wall. The numerator factor comes from the bi-fundamental chiral multiplet q and the denominator is from the singlet chiral multiplet b. The anomaly-free U(1) symmetry, which is a linear combination of U(1)in instanton symmetry and U(1)B baryonic symmetry, rotates the of b is precisely the inverse of the contribution of a chiral multiplet with the same R-charge and fugacity as B. Thus the hemisphere index for dual Dirichlet boundary conditions is: 2 izi0 ( ) QiN6=j (zi0=zj0 ) If we have identi ed the correct duality interface, the hemisphere index for dual Dirichlet b.c. should actually coincide with the hemisphere index for Dirichlet b.c., up to a reection of U(1)in instanton charges, i.e. an inversion of the instanton fugacity This motivates us to propose the following relation: = 1+ which is checked up to x5 order. D^ IIN (zi; ; p; q) = IIN (zi; This is a highly nontrivial relation. The instanton partition function in the hemisphere index on the right side of the wall has a natural expansion by positive powers of the instanton fugacity . On the other hand, the instanton partition function on the left side of the wall is expanded by the negative powers of . This relation is a very stringent test of our conjectural duality wall. We can test this conjectural relation for small N and the rst few orders in the power series expansion in p; q. We nd that the relation holds with a particular choice of the integral contours. The contour should be chosen by the condition: jpj; jqj keeping the contour to be on a unit circle. One can then check the duality relation order by order in the series expansion of x For SU(2) case, one nds D^ IIN=2( 1) = IIN=2( ) = 1 + pp=q and r SU(N)(z) are the characters of dimension r representations of SU(N ) symmetry. We have actually checked this relation up to x7 order. Similarly, for SU(3)3 case, one nds D^ IIN=3( 1) = IIN=3( ) where j0i is the ground state tensored by the broken current supermultiplet. These states carry U(1) gauge charges (N 2)N; 0; +(N 2)N respectively. Among these three states, the rst and the third states carry appropriate SU(2)R charge for being a current multiplet. We also need to impose the U(1) gauge invariance. Therefore, the instanton operator provides a broken current supermultiplet when the classical CS-level satis es N = 0 : This supports the U(1)in ! SU(2) global symmetry enhancement of the SU(N ) N gauge theory at the UV We now consider SU(N ) gauge theory with fundamental hypermultiplets. The Nf fundamental hypermultiplets induce on the instanton moduli space Nf complex fermionic zero modes carrying the avor charges and U(1) gauge charge N 2. The quantization leads to Nf raising operators Ca; a = 1; ; Nf and they act on the states as which may signal the symmetry enhancement of the UV CFT. If we impose the standard bound j j Nf =2 as in [36], one nds that r should be 0 or Nf and the broken current multiplet exists only when r = 0 : Nf =2) ; r = Nf : = N Nf =2 : Nf . These states have U(1) gauge charge (N Nf =2) and avor charges. We can construct the instanton operators by tensoring these states with the above gaugino contribution and imposing U(1) gauge invariance. Then one can see that there exist candidate broken current supermultiplets having zero U(1) gauge charge when We nd that the broken current multiplets may exist if r n. The states n can survive when while the states with r n can survive when symmetry and carries the baryoninc U(1)B Nf =2 or Nf =2, respectively. U(1)in global symmetry will be enhanced as expected to at the UV xed point by the instantonic conserved currents, Thus the SU(Nf ) bound by n: symmetry is enhanced to SU(Nf ) If we relax the bound on , though, other possibilities occur. Suppose we violate the Nf =2 = 0 ; Nf =2 ; = N + r Nf =2 ; Nf =2 : These states provide candidate broken current multiplets in the rank r antisymmetric representation of the SU(Nf ) avor group. There is no symmetry group whose adjoint representation is decomposed into irreps involving any rank r > 2 antisymmetric representation of a subgroup. Thus we expect theories with n > 2 to be truly incompatible with an UV completion. The constraint (7.9) of the 5d CFTs in [11, 38, 39]. A similar analysis has been done in [12]. For r = 1 (or r = Nf = N + 1 Nf =2 (or 1 + Nf =2) the candidate broken currents transform in the (anti-)fundamental representation of the avor symmetry with the U(1)B charge Nf =2 + 1 (or Nf =2 an UV CFT may exist with enhanced global symmetry SU(Nf + 1) U(1). The current multiplet of the SU(Nf + 1) is in the adjoint representation which is decomposed by current multiplets in the adjoint and a fundamental and an anti-fundamental representation of the subgroup SU(Nf ). The fundamental and anti-fundamental current multiplets are generated by following the above procedure in the instanton background. In particular, when it gives a current multiplet which is a singlet under the SU(Nf ) avor symmetry. Thus in this case we have a bigger symmetry enhancement to SU(2N + 2) SU(2). Furthermore, = 12 = 0 and Nf = 2N + 2, both states r = 1 and r = Nf 1 survive and provide two broken current multiplets in the fundamental and anti-fundamental representations. Therefore the symmetry of the UV CFT may be enhanced to SU(2N + 4). Similarly, the instanton state with r = 2 (or r = Nf 2) generates the broken current multiplet in the antisymmetric representation of the SU(Nf ) when = N + 2 Nf =2 (or U(1) at the UV xed point. When = 1 (or 1) and Nf = 2N + 2, one is singlet under the SU(Nf ). So the enhanced symmetry of the UV xed point becomes SO(4N + 4) = 12 (or ) and Nf = 2N + 3, two states with r = 2 and r = Nf 1 (or r = 1 and r = Nf 2) can provide current multiplets in the antisymmetric and the fundamental representations of the SU(Nf ) with di erent U(1)B N + 12 and N + 12 respectively. So the enhanced global symmetry of the UV CFT is SO(4N + 8). Lastly, when = 0 and Nf = 2N + 4, two instanton states with 2 survive and they provide current multiplets in the rank 2 and r = 2 and r = Nf 2 antisymmetric representation of the avor symmetry. It has been conjectured is expected to be UV complete and has a 6d xed point. The corresponding 6d theory is the (DN+2; DN+2) minimal conformal matter theory [40, 41]. The discussion in this subsection strongly supports the duality proposed in this section. Following the fermion zero mode analysis above, the SU(N + 1) gauge theory with the CS= N +3 2N + 2 and SO(2Nf ) SU(2) when Nf = 2N + 3, which is the same as the expected UV global symmetry of the dual Sp(N ) gauge theory. SU(Nf + 1) SU(Nf + 1) SU(Nf + 2) SU(N ) (N+2 Nf =2) SO(2Nf + 2) From Sp(N ) to exotic SU(N + 1) We rst discuss the superconformal index and the instanton partition function of Sp(N ) gauge theory. The superconformal index of the Sp(N ) gauge theory with Nf fundamental avors takes the form ISNp;Nf (wa;qSp;p;q) = I YN i=1 2 izi QiN=1QaN=f1(ppqzi =wa;p;q)1 Zk=1 = The function ZSNp;;Ninfst is the instanton partition function of Sp(N ) gauge theory, which can be computed using localization of the path integral on the instanton moduli space given in [30, 42]. The 5d Sp(N ) instanton partition functions are studied in great detail in [6, 26]. The results are summarized in appendix A. The Sp(N ) gauge theory has O(k) dual gauge group in the ADHM quantum mechanics. At each instanton sector we will compute two partition functions Zk+ and Z and O(k) , respectively, Zk ( ; m; 1;2) = with k = 2n + = 0 for odd N + Nf and for even N + Nf while choosing the same mass signs for all matter elds for notational convenience. The k instanton partition function can be written as ZSkp(odd)( ; m; 1;2) = ZSkp(even)( ; m; 1;2) = given by sum of two partition functions Zk+=1 = p3=2q3=2 QaN=f1 wa 1=2( 1 + wa) q) QiN=1(1 for O(1)+ and O(1) , respectively. p3=2q3=2 QaN=f1 wa 1=2(1 + wa) q) QiN=1(1 + ppqzi ) For higher instantons, we need to evaluate the contour integral over O(k) Coulomb branch parameters using the JK-residue prescription. For example, the 2-instanton partition function has a contour integral over one variable 1 for O(2)+ sector, whereas has no integral for O(2) sector. The JK-prescription tells us that the poles we should pick up are i + + = 0 ; 2 1 + 1 = 0 ; 2 1 + 2 = 0 ; mod 2 ) : (7.16) ZSNp;;Ninfs=t8 = ZSNp;;NQfM=8=ZSNpf;e=x8tra ; where ZSp;QM is the standard instanton partition function before removing the extra factor. Next, we need to assemble the instanton partition function and 1-loop determinants into the hemisphere partition function for Dirichlet boundary conditions: IISNp;Nf (zi; wa; qSp; p; q) = QiN=1 QaN=f1(ppqzi =wa; p; q)1 The hemisphere index for the SU(N + 1) theory is similarly de ned and takes the form IISNU+1;Nf (zi; wa; qSU; p; q) = QiN6=+j 1(pqzi=zj ; p; q)1 QiN=+11 QaN=f1(ppqzi=wa; p; q)1 at in nity I = therefore de ned as The sum over the JK-residues plus the O(2) contribution gives the full 2-instanton parmechanics associated to a classical noncompact Coulomb branch. The partition function involves an extra contribution coming from this continuum which should be removed to obtain the correct QFT partition function. nd that the extra contribution takes ZSNpf;e=x8tra = PE The half-integral coe cient in the letter index obviously shows that this is coming from a continuum. This correction factor can also be obtained by taking a half of the residue 1 in the integral formula. The QFT instanton partition function is with an a-priory unknown instanton contribution ZSNU+;i1n;sNt f . The degrees of freedom on the duality wall have the 4d index contribution QiN=+11 QjN=1 ( where z0 and z are the fugacities for the bulk SU(N + 1) and Sp(N ) gauge groups. To couple this to the 5d index, we need to multiply the 4d Sp(N ) vector multiplet contribution and integrate the Sp(N ) gauge fugacities z. The result is given by D^ IISNp;Nf = where wa is the fugacity for U(Nf ) SO(2Nf ) avor symmetry and (C)(z; z0; ) = Our conjecture is that the duality action D^ on the hemisphere index of the Sp(N ) gauge theory converts it into the hemisphere index of the SU(N + 1) gauge theory in the other side of the wall. So the following relation is expected to hold D^ IISNp;Nf (zi; wa; qSp; p; q) = IISNU+1;Nf (zi0; wa0; qSU; p; q) : In this relation, the fugacities for the global symmetry in two sides of the wall should be identi ed as wa = 1=2wa0 ; qSp = (N+1)=2 Y(wa) 1=2 ; qSU = 1 Y(wa0) 1=2 : The rst relation comes from the the constraint of the 4d superpotential. mined the second and the third relations experimentally from the duality relations (7.24) and (7.29), but they agree with the relations expected from cancellation of the mixed 't Hooft anomalies for the duality wall. The simplest example would be the duality action between Sp(2) and SU(3) gauge theories with Nf avors. To evaluate the integral in (7.24) and see the duality relation, we should choose a particular contour. We take the contour to be along a unit circle while D^ IIS2p;0(zi; qSp) = 1 + SU(3) is the SU(3) character of the dimension r irrep with fugacities zi. We checked that the right hand side agrees with the perturbative part of the SU(3) hemisphere index and admits an expansion in non-negative powers of qSU, up to the order x5. For general Nf D^ IIS2p;Nf (zi;wa;qSp) = 1+ S3U(3) U(Nf )x+ U(Nf ) is the U(Nf ) character with fugacities (wa0) 1 of a irrep labeled by a Young Y tableau Y . We have identi ed the parameters as (7.25). The perturbative part on the right hand side agrees with that of the SU(3) theory and the other parts are expanded by non-negative powers of qSU. This relation has been checked at least up to x3 order. In appendix B, we shall suggest a UV prescription of the instanton moduli space of our exotic SU(3) theory with matter elds, whose partition function precisely reproduces the right hand side. In addition, we will explicitly compute the superconformal index of this SU(3) theory and show the desired global symmetry enhancement at the UV One can also consider the generalization to higher rank gauge theories. Acting with the duality wall on the hemisphere index of the Sp(3) theories, we obtain = 1+ S4U(4) U(Nf )x+ 3, by identifying the parameters as (7.25). We checked that the right hand side agrees with the perturbative part of the SU(4) hemisphere index and admits an expansion in non-negative powers of qSU, at least up to the order x4. Of course, the duality wall can also act in the opposite direction, from SU(N + 1) where the 4d index of the boundary degrees of freedom involving the 4d vector multiplet (A)(z0; z; ) = QjN=+11 ( QiN6=+j 1 (zi0=zj0 ) QiN>+j 1 ( zi =zj0 ) The contour is chosen along the unit circle with an assumption x parameters are matched as (7.25). Of course, this follows from the CA and AC inversion formula introduced in [10]: (C)(z; z0; )f (z) = f (x) ; d z0 (A)(z0; z; )f (z0) = f (x) : Note that the contours should be chosen along unit circles by assuming x for the A-type integral, but by assuming x < 1 for the C-type integral as speci ed In this subsection, we will study the properties of BPS Wilson loops under the conjectural duality in the previous sections. We will focus on the simplest cases: fundamental Wilson loops of the Sp(2) and SU(3) gauge theories meeting at the interface. The Wilson loops on two sides of the wall are connected at the boundary by the bi-fundamental chiral multiplet q. The chiral multiplet q has charge 12 under the non-anomalous U(1) global symmetry. To cancel the global charge when it couples to the Wilson loops, we combine the gauge Wilson loops with a avor Wilson loop for the U(1) , with from the similar argument in section 4. We will compute hemisphere indices and test this avor charge 12 , which follows duality property between two fundamental Wilson loops. We rst compute the hemisphere indices with fundamental Wilson loops inserted at the origin. We need to compute the instanton partition function in the presence of Wilson loops. As explained in section 4, Wilson loops are represented by equivariant Chern characters in the localization, and that for the fundamental Wilson loop is given in (A.13). Then the localized partition function can be written in terms of the equivariant Chern characters as For the Sp(N ) gauge theory, the equivariant Chern character for the fundamental Wilson loop can be written, at k-instantons, as Chf+und(e ; e ) = Chfund(e ; e ) = q)(pq) 1=2 X(e I + e I + ) ; q)(pq) 1=2 X(e I + e I + ei with k = 2n + = 0 or 1. Here the superscripts means those for O(k) sectors. Then the 1-instanton partition function can be written as Wk+=1 = Wk=1 = (pq)3=2 PiN=1(e i + e i ) q) QiN=1(1 q)(pq) 1=2 QaN=f1 2 sinh m2a (pq)3=2 PiN=1(e i + e i ) + (1 q)(pq) 1=2 QaN=f1 2 cos m2a q) QiN=1(1 + ppqe i ) There could be extra instanton corrections to the Wilson loop index as we have seen in section 4. For the cases in this section, however, we nd that there is no such corrections up to certain order in x expansion. Now we consider the duality wall action on the hemisphere index of the Sp(N ) theory with the fundamental Wilson loop. We propose that the fundamental Wilson loop partition function of the Sp(N ) theory is mapped to that of the SU(N + 1) theory after passing through the duality wall as follows: Sp(N);Nf (zi; wa; ) = 1=2Wfund with the parameter identi cation in (7.25). The duality action D^ is de ned in the same way as in (7.22), but the hemisphere indices IISNp;;NSUf in both sides are replaced by the Wilson avor Wilson loop. We compute the hemisphere indices of the Sp(2) gauge theories and test this duality. = 3 + S2U(2)(y) S3U(3)(z)x3+ S2U(2)(y) S3U(3)(z)2 D^ WfSupn(d2);0(zi; qSp) 1=2WfSuUnd(3);1(zi; qSU) for Nf = 0, and D^ WfSupn(d2);1(zi;w;qSp) 1=2WfSuUnd(3);1(zi;w0;qSU) = S3U(3)(z)+ (w10) 1 SU(3)(z)2+(w10) 1=2qSU x+ 3 + (w10) 2 SU(3)(z)+(w10) 2 SU(3)(z)+(w10) 1 SU(2)(y) S3U(3)(z)2 x2 10 8 2 expansion in non-negative powers of qSU and the perturbative part agrees with that of the SU(3) gauge theory, up to x4 order. It also turns out that the right hand sides agree up to x4 order with the hemisphere indices of the SU(3) theories with the fundamental Wilson loop whose instanton partition functions are computed using the UV prescription given in The research of DG and HK was supported by the Perimeter Institute for Theoretical Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. 5d Nekrasov's instanton partition function The moduli space of instantons has complicated singularities which are associated to one or more instantons shrinking to zero size. In the context of ve-dimensional supersymmetric gauge theories, these eld con gurations are outside the obvious regime of validity of the gauge theory description of the theory. Correspondingly, the de nition of the gauge theory instanton partition functions through equivariant localization on the instanton moduli spaces requires a prescription of how to deal with the singularities, which will depend on a choice of UV completion of the gauge theory. It is very challenging to work directly on the singular moduli spaces. Even in the absence of extra matter elds this was done only recently [43] using the technology of equivariant intersection cohomology. Extra matter elds, in the form of hypermultiplets transforming in some representation of the gauge group, provide additional fermion zero modes in the instanton background which are encoded into some appropriate characteristic class inserted in the equivariant integral. The correct description of these characteristic classes over the singular instanton moduli space is poorly understood. The standard alternative to working with the singular moduli spaces, available for classical groups only, is to employ the ADHM technology to provide a resolution of the singularities in the instanton moduli space. The ADHM construction has a clear motivation in terms of a string theory UV completion. It realizes the instantons as D0 branes in presence of other brane systems which engineer the gauge theory itself. It is important to realize that this is not obviously the same as the quantum theory UV completion we are after, which should involve some 5d SCFT or perhaps a 6d SCFT. Luckily, it appears that the answers computed by the ADHM construction can be easily corrected to sensible eld theory answers, as long as the matter content of the gauge theory does admit a reasonable string theory lift. When that is not the case, it is not obvious that a construction of the correct bundle of fermion zero modes will actually be available in the ADHM description of the moduli space. We will encounter some of these issues in the sections A.3 and B. When the ADHM construction for a gauge group G exists, it can be described as a one dimensional gauged linear sigma model of dual gauge group G^, called the ADHM quantum mechanic (ADHM QM). The Higgs branch of this theory coincides with the instanton moduli space. This theory has bosonic SU(2)1 SU(2)R symmetry and 4 real supercharges Q A_, where the SO(4) = SU(2)1 SU(2)2 corresponds to the spatial R4 rotation and the SU(2)R is the R-symmetry in 5d. The indices = 1; 2; _ = 1; 2; A = 1; 2 are the doublets of SU(2)1; SU(2)2; SU(2)R symmetries respectively. The ADHM QM consists of the (0,4) hypermultiplets ) in adjoint rep; ) in fundamental rep and the vector multiplet (At; ; A_). The bosonic elds in the hypermultiplets are called In order to apply the ADHM construction to a ve-dimensional gauge theory we need to nd within the ADHM quantum mechanics a construction of the bundle of fermionic zero modes associated to the hypermultiplets. Concretely, that means adding extra elds to the quantum mechanics which add the appropriate fermionic bundle on top of the Higgs branch of the theory. If a string theory description of the gauge theory is available, one can usually read o from it the required extra degrees of freedom. If the instanton moduli space was not singular, it would be possible to derive simple relationships between the characteristic classes in the equivariant integral associated to hypermultiplets in di erent representations. If a string theory construction is not available for some representation, one can try to guess an ADHM description for that representation by imposing the same relationship on the the corresponding characteristic classes/equivariant indices in the ADHM equivariant integral. Some equivariant indices for hypermultiplets in simple representations are given in [30]. We will present below the equivariant indices and partition functions for the hypermultiplets used in the main text and discuss the di culties associated to this naive choice of UV completion. The k instanton partition function takes the following integral expression Zk( ; m; 1;2) = 1 I PIk=1 I Zvec( ; ; 1;2) Y ZRa ( ; ; ma; 1;2) ; where ZRka is the contribution from a hypermultiplet in Ra representation and ma is the mass parameter. We will often use fugacities zi ema . The vector multiplet Zvec( ; ; 1;2) = QIk6=J 2 sinh I 2 J QIk;J 2 sinh I J2+2 + The instanton partition function takes the form of the instanton series expansion as with an instanton counting parameter q. The Zk is the k instanton partition function. It is the supersymmetric Witten index of the 1d ADHM QM. It also admits a path integral representation. The supersymmetric localization was employed to evaluate this path integral of the ADHM quantum mechanics in [18, 23]. See also [26{28] for 1d localization calculations. We will now summarize some results. SU(N ) partition function bulk 5d theory, one can also turn on a classical CS coupling 3. It induces a Chern-Simons coupling in the 1d quantum mechanics [44, 45]. Zinst = The hypermultiplet factor will be discussed later. We still have the contour integral to be evaluated. The contour integral of the instanton partition function should be performed using the Je rey-Kirwan method [26]. If the hypermultiplet factor has only fermionic contributions, as our naive expectation from the zero mode analysis in the 5d QFT, we need to take into account only the vector multiplet factor. The JK-prescription tells us that the residue sum of the following poles will give the nal result. i + + = 0 ; J + 1 = 0 ; J + 2 = 0 ; with I > J . However, we will see that the hypermultiplets can introduce extra bosonic degrees for the UV completion of their zero modes. Thus they can also provide nontrivial JK-poles above the poles from the vector multiplet. We will discuss some examples below. Sp(N ) partition function Since the O(k) group has two disconnected components O(k)+ and O(k) , we will get two partition functions Zk+ and Z k at each instanton sector. The k instanton partition function is then given by a sum of these two functions. In addition, the Sp(N ) gauge theory has a angle associated with 4 (Sp(N )) = Z2 [3]. Two possible parameters lead to the following two di erent combinations [7, 46]: ZSp = <8 21 (Zk+ + Zk ) ; = 0 When the theory couples to more than one fundamental hypermultiplet, the comes unphysical because it can be e ectively absorbed by ipping the sign of a single mass of one fundamental matter. The k instanton partition function takes the form Zk ( ; m; 1;2) = Yn d2 Ii Zvec( ; ; 1;2) Y ZRa ( ; ; ma; 1;2) ; with k = 2n + = 0 or 1. The Weyl factor is given by =0 = =1 = =0 = =1 = The vector multiplet for O(k)+ sector gives the contribution 2+ + QiN=12sinh 2+ + QiN=12sinh I 2 i+ + QIn=12sinh 2 I I J +2 + Zvec = Zvec = with k = 2n. with k = 2n + 1 and For O(k) sector, the vector multiplet contribution is 2+ + QiN=12cosh + +)QiN=12sinh( i+ +) I=1 I=1 2sinh 2+ + QiN=12sinh I 2 i+ + QIn=112sinh 2 I A hypermultiplet develops fermion zero modes in the instanton background. The presence of the fermion zero modes can be observed using an index theorem. Accordingly, it is expected that the bulk hypermultiplets induce fermionic degrees on the instanton moduli space. When we attempt to engineer an ADHM quantum mechanics description of these fermionic zero modes on the Higgs branch, however, extra bosonic degrees of freedom are in general required. Often these bosonic zero modes give rise to extra classical branches of vacua in the ADHM quantum mechanics, or extra continuum contributions to the spectrum, which may be spurious from the point of view of the 5d gauge theory. In string theory constructions, they may describe D0 branes moving away from the brane system which engineers the 5d gauge theory. These spurious branches of vacua must be carefully subtracted from the nal answer. We can give a few simple examples of this phenomenon. The instanton moduli space of a 5d gauge theory with an adjoint hypermultiplet has a string theory embedding. The instanton states can be interpreted as the D0/D4-brane bound states in this case. The 1d gauge theory living on the D0-branes is described by the ADHM quantum mechanics with additional matter elds corresponding to the bulk adjoint hypermultipet. This theory involves extra real 4 dimensional bosonic elds that parametrize the 4 transverse directions to the D4-branes in which the 5d gauge theory supports. The non-commutativity parameter (or FI parameter) in the 1d QM generally make these directions massive. However, when the commutativity is restored, these branches of vacua open up D0-branes (or instantons) can escape to in nity. Similarly, the UV completion of instanton dynamics in Sp(N ) gauge theory with an antisymmetric and fundamental hypermultiplets has extra bosonic degrees of freedom from the hypermultiplets. Its string theory embedding is given by D0-D4-D8-O8 brane system [47]. The extra bosonic modes again parametrize the transverse directions to the D4-branes. In particular, the ADHM for this theory does not have noncommutative deformation of the space. Hence the observables computed using this UV completion in general involves extra contributions to be subtracted o . One can nd examples in [26]. Next, we can describe our guess for the contribution of hypermultiplets in tensor powers of the fundamental representation, based on the prescription given in [30]. If we could ignore the singularities, the hypermultiplets introduce vector bundles on the instanton moduli space, and the vector bundles are constructed by tensor products of an universal bundle E . The tensor product structure of the vector bundle inherits that of the representation of the 5d hypermultiplet. We will now pretend that the same prescription can be applied to the ADHM-resolved moduli space of instantons. In [30], it was suggested that the equivariant index for the hypermultiplet can be computed by taking tensor product of the equivariant Chern character of the bundle E , which is given by [30, 48] ChE (e ; e ; p; q) = fund(e i ) q)(pq) 1=2 where fund(e i ) and fund(e I ) denote the character of the fundamental representations of the guage group G and the dual gauge group G^, respectively. For example, the equivariant indices for the hypermultiplets in the fundamental, symmetric, antisymmetric and adjoint representations are given by, respectively, indfund(e ; e ; p; q) = indsym(e ; e ; p; q) = indanti(e ; e ; p; q) = indadj(e ; e ; p; q) = Zfund = Y 2 sinh where the tensor product of the Chern character is de ned using the usual tensor product ChE E (e ; e ; p; q) = Ch^2E (e ; e ; p; q) = ChE (e ; e ; p; q)2 + ChE (e2 ; e2 ; p2; q2)i ; ChE E (e ; e ; p; q) = ChE (e ; e ; p; q) The equivariant indices in other representations can be obtained in the similar manner. The resulting index computed in this way contains terms independent of the fugacity e I for G^. These terms amount to the perturbative contribution, so we will ignore them when we compute the instanton partition function. The contribution to the instanton partition function of the hypermultiplets can be easily obtained using the relevant equivariant indices. There is a conversion rule for 5d indR = ZR = Y h2 sinh zi ini Thus the plethystic exponential of the equivariant index yields the instanton partition function contribution of the hypermultiplet. One can check that the contribution from an adjoint hypermultiplet computed using this prescription agrees with that from the localization of the ADHM quantum mechanics in [49]. Let us present explicit expressions for the hypermultiplets discussed in the main context. For SU(N ) gauge theory, the fundamental hypermultiplet contribution is with a mass parameter m. The antisymmetric hyper has the following contribution Zasym = QiN=1QIk=12sinh I + 2i m QIk>J 2sinh I + J 2 m I J +m I J +m For Sp(N ) gauge theory, the fundamental representation has the contribution for O(k)+, and with k = 2n + 1, and Zf+und = 2 sinh Zfund = 2 cosh Zfund = 2 sinh Next, we can assemble a modi cation of the bare ADHM quantum mechanics which would reproduce these modi cations to the equivariant integrand. The contribution for the fundamental hypermultiplet implies that a fundamental matter induces a (0; 4) fermi multiplet in fundamental representation of G^ in the ADHM QM. This agrees with our expectation that the hypermultiplet develops fermion zero modes in the instanton background. On the other hand, the contribution from the antisymmetric hyper has factors in denominator as well as the factors in numerator. The numerator factors correspond to a fermi multiplet in the bifundamental representation of G fermi multiplets in the antisymmetric representation of G^. While, the denominator factors corresponds to a pair of (0; 4) hypermultiplets in the symmetric representation of G^. This means that the UV completion of the zero modes acquires nontrivial bosonic degrees which G^ and a conjugate pair of are not present in the zero mode analysis of the 5d QFT. The computation of the 1d equivariant integral requires both an integrand and a choice of integration contour/prescription. The latter, in a sense, can be used to include or exclude the contribution of certain classical branches of vacua, by selecting which poles should be picked by the contour integral. The standard prescription in 1d localization computations is the JK-prescription. To read the relevant poles from the JK-prescription, we should know the exact representations of the extra bosonic degrees under G^ rotation. However, although the recipe given in [30] and in this section allows us to know the matter contents in the ADHM QM, it yet has an ambiguity in the exact representations of the multiplets. More precisely, it cannot distinguish a certain complex representation R and its conjugation, prescriptions for it case by case in the main context. '. Since we could not resolve this issue, we will give Further spurious contributions included by the standard JK-prescription have to be removed on a case-by-case basis. See [26] for few examples. Partition functions of exotic SU(3) theory In this appendix, we propose a prescription to compute the instanton partition functions of the exotic SU(3) theories with matters. With these results, we compute the hemisphere indices and then show that they agree with the hemisphere indices obtained in section 7.2 using the duality wall action on the Sp(2) hemisphere indices. We are interested in the SU(3) SQCD with = 5 Nf =2, which obviously violates bound, the localization integral of the instanton partition function from the usual ADHM are associated to the classical Coulomb branch of vacua in the ADHM quantum mechanics and not to the to the instanton moduli space which is described by the Higgs branch. Unfortunately, we do not know how to remove these spurious contributions when the 1. These poles degree of the pole is higher than 1. In what follows, we will explain how to avoid having higher degree poles at in nity by introducing `pseudo' hypermultiplets in the instanton background. We will add two (or more) `pseudo' hypermultiplets and integrate them out at the end. This will allow us to evaluate the instanton partition function without having the problem of the higher degree poles at in nity. Let us rst discuss the `pseudo' hypermultiplet and the ADHM quantum mechanics. The `pseudo' hypermultiplet is simply the hypermultiplet in the antisymmetric representation of the SU(3). It should be equivalent to the fundamental hypermultiplet for the SU(3) gauge theory. This is indeed the case for the perturbative analysis. However, the antisymmetric hypermultiplet a ects the ADHM quantum mechanics in a di erent way from that of the fundamental hypermultiplet. Strictly speaking, the ADHM quantum mechanics is designed for the U(N ) gauge theory since it involves singular U(1) instantons which is regularized by introducing extra UV degrees of freedom. Therefore, fermion zero modes from the antisymmetric hypermultiplet has a rather di erent UV completion than those from the fundamental hypermultiplet in the ADHM QM. The fermionic zero modes from the antisymmetric hypermultiplet provide many nontrivial multiplets, not just fermi multiplets but possibly also hypermultiplets including extra bosonic zero modes, in the ADHM QM as depicted in gure 16. The ADHM quantum SU(2)R symmetry. See appendix A for details. We then add a bi-fundamental chiral fermion (black dashed arrow) of U(k) and SU(3) groups, and a (0; 4) fermi multiplet (blue dashed arrow), which is a doublet under the SU(2)1 and in the antisymmetric representation of U(k), and a hypermultiplet (red solid arrow) in the symmetric representation of U(k). This is equivalent to add to the instanton moduli space a vector bundle given by the antisymmetric product of the universal bundle in the fundamental representation. We consider the SU(3) gauge theory with two `pseudo' hypermultiplets and Nf fundamental hypermultiplets. The k-instanton partition function from the ADHM QM can be where Zvkec; Zfkund; Zaksym are given in (A.5), (A.17), (A.18), respectively. We will set the = 4 1, which is now controllable. ZQM;k = at in nity I = We are essentially interested in the theory with = 5 Nf =2 and without the `pseudo' matters. This theory can be obtained by integrating out two `pseudo' hypermultiplets. We will send their mass parameters ti to in nity. Then it will e ectively shift the bare CS-level by +1 and the low energy theory will have the CS-level = 5 Nf =2 as desired for our exotic theory. To avoid the higher degree poles at in nity, we shall integrate out the `pseudo' matters after evaluating the contour integrals. It thus allows us to compute the instanton partition function of the exotic SU(3) theory without facing higher degree poles at the in nity. This procedure can be interpreted as a UV prescription of the SU(3) instanton moduli space at the exotic CS-level. Here the `pseudo' hypermultiplets are used as a UV regulator. We will restrict ourselves to the cases with Nf consider the dual SCFT with Sp(2) gauge group. 8,5 for which we can The contour integral will be evaluated using the JK-residue prescription. One then notices that the `pseudo' matter contributions provide additional nonzero JK-residues. For example, at one instanton, the JK-residues at the following poles are nonzero: + = 0 ; Summing over all JK-residues including both from the vector multiplet and from the `pseudo' hypermultiplets, we can compute the partition function with `pseudo' matters. This is not quite our nal answer. To obtain the QFT partition function, we need to strip o some overall factor associated to the extra bosonic at directions introduced by the `pseudo' hypermultiplets. We conjecture that the extra factor is given by ZeNxftra;pseudo=PE qSUf Nf (wa; a;p;q) ; f Nf = p 1 2QaN=f1pwa (1 p)(1 q)(1 pq 1= 2)(1 pq 2= 1) +(pq)3=2( 1+ 2) 1+( 1 2) 1 U(Nf )(1=w)+( 1 2) 2 U(Nf )(1=w) ; 4 8 5One may notice that the integral has higher degree poles at in nity when Nf > 8. We may be able to resolve this by introducing one more `pseudo' hypermultiplet, but we will not discuss these cases. U(Nf ) is the character of the rank L antisymmetric irrep of the L Note that this extra factor is independent of the SU(3) gauge fugacities and thus it indeed corresponds to the degrees of freedom decoupled from the 5d QFT. We have checked that, after subtracting o this factor, the instanton partition function has no poles for a and is a nite polynomial in 1 and 2, as expected, at 1-instanton for all Nf and up to 2-instantons There is the usual correction factor coming from the continuum along the noncompact 1. We obtain ZeNxftra;cont = PE ZeNxftr=a8;cont = PE qSU QaN=f1 pwa 1 Y pwa + pqp 1 2 Y pwa The `correct' partition function can then be written as ZiNnsft(zi; wa; a; qSU; p; q) = ZQNMf=ZeNxftra ; where ZQM is the partition function of the ADHM QM evaluated with the JK-prescription. 1,6 and take the leading contribution. By rescaling the instanton fugacity as qSU We now integrate out the `pseudo' hypers. We will send their masses to in nity ta ! qSU, we will end up with the instanton partition function of the SU(3) theory with Nf avors and the CS-level ZiNnsft(zi; wa; a; qSU= Taking into account the extra factors carefully, we compute 1-instanton partition functions for Nf 8 and obtain Zi3n;sNt;fk=1 = Combining the 1-loop determinant, we have checked that the hemisphere index of our exotic SU(3) theory yields exactly the right hand side of the duality relation (7.27) between the Sp(2) and SU(3) theories, in all examples at least up to x3 order. This result supports the UV prescription of the exotic SU(3) theory in this section. Similarly, we can compute the Wilson loop index of the exotic SU(3) theories using the above UV prescription. An Wilson loop in a representation R inserts the corresponding Nf =2. equivariant Chern character into contour integral of the instanton partition function. At kinstantons, the Wilson loop index before integrating out the pseudo hypers can be written as WQM;k = I=1 2 i PIk=1 I Zvkec( ; )YZfkund( ;ma)YZaksym( ;ti); where ChR( ; ) is the equivariant Chern character of the vector bundle in the representation R. We will focus only on the Wilson loop in the fundamental representation whose equivariant Chern character is given in (A.13). The contour integral is again evaluated using the JK-prescription. Since the Wilson loop insertion does not change the pole structure of the integrand, we can pick up the same poles as before. As we have seen above, the partition function involves the correction factors from the Coulomb branch and the extra bosonic degrees of the `pseudo' matters given in (B.4) and (B.3), which we should subtract o . Due to the same reason as without Wilson loops, we expect the correct Wilson loop index has no poles for the mass parameter a of the `pseudo' matters. However, even after subtracting the correction factors in (B.4) and (B.3), we notice that the Wilson loop index still has poles for a. We nd that the Wilson loop receives an additional correction when Nf > 0. For example, if we de ne new Wilson loop indices taking the form Wf3u;n0d(z; w; ; qSU) = WQM;fund 3;0 ZeNxftr=a0 ; Wf3u;n1d(z; w; ; qSU) = WQM;fund 3;1 Wf3u;n2d(z; w; ; qSU) = WQM;fund 3;2 ZeNxftr=a1 + qSU (1 ZeNxftr=a2 + qSU where II3;Nf is the bare hemisphere index without Wilson loops, these new indices have no poles for a. We have checked this till 2-instantons. Thus we suggest that the `correct' Wilson loop index with `pseudo' matters should be this new index. Let us integrate out the `pseudo' hypers by rescaling the instanton fugacity as qSU ! qSU and taking the limit ta ! 1. It leads to the Wilson loop index of the exotic SU(3) theory, given by We have also checked that this Wilson loop index yields the results in (7.35) and (7.36) obtained from the duality wall action on the dual Sp(2) hemisphere indices, up to x4 order. Superconformal indices Now, we compute the superconformal indices for the Sp(2) and SU(3) theories and check the duality conjecture. Let us rst discuss the Sp(2) theories. The superconformal index IS2p;Nf = 1+ 1+ SaOdj(2Nf ) x2+ SO(2Nf ) is the character of the r irrep of SO(2Nf ) symmetry with fugacities wa and S denotes the conjugate spinor representation and S is the complex conjugation of SO(2Nf )(wa2) denotes the fundamental character with fugacities wa2. For Nf = 8, we fund IS2p;Nf =8 = 1 + SO(16) is the character of the rank 2 symmetric representation of SO(16). This theory has an enhanced SU(2) SO(16) global symmetry at the UV xed point. There are additional BPS states at x2 order corresponding to the conserved currents with instanton fugacity qSp and all BPS states properly arrange themselves to form representations of the enhanced symmetry. Thus the result is consistent with the symmetry enhancement. We now turn to the SU(3) theories. The superconformal index of the general SU(N ) SQCD can be written as 2 izi QiN=1 QiN6=j (zi=zj ; p; q)1 QaN=f1(ppqzi=wa; p; q)1 only enters in the instanton partition function. = N + 2 Nf =2 which For our SU(3) theories, the instanton partition functions are given in the previous section, so the superconformal index computation is straightforward. We nd that the results perfectly agree with the indices of the dual Sp(2) theories computed in (B.11) and (B.12), once we identify the fugacities of two dual theories as (7.25). This has been checked at least up to x4 orders. 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Davide Gaiotto, Hee-Cheol Kim. Duality walls and defects in 5d \( \mathcal{N}=1 \) theories, Journal of High Energy Physics, 2017, 19, DOI: 10.1007/JHEP01(2017)019