Mirror Symmetry And Loop Operators

Wilson loops in gauge theories pose a fundamental challenge for dualities. Wilson loops are labeled by a representation of the gauge group and should map under duality to loop operators labeled by the same data, yet generically, dual theories have completely different gauge groups. In this paper we resolve this conundrum for three dimensional mirror symmetry. We show that Wilson loops are exchanged under mirror symmetry with Vortex loop operators, whose microscopic definition in terms of a supersymmetric quantum mechanics coupled to the theory encode in a non-trivial way a representation of the original gauge group, despite that the gauge groups of mirror theories can be radically different. Our predictions for the mirror map, which we derive guided by branes in string theory, are confirmed by the computation of the exact expectation value of Wilson and Vortex loop operators on the three-sphere.


Introduction
Wilson loop operators [1] play a central role in our understanding of gauge theories. A Wilson loop is specified by a curve γ in spacetime and by the choice of a representation R of the gauge group G W R (γ) = Tr R P exp γ iA µ dx µ .
In a certain sense, they are the most fundamental observables in gauge theories.
Wilson loops raise an immediate challenge to any conjectured duality, be it a field theory duality or a gauge/gravity duality: what is the dual description of Wilson loops? Even the more qualitative question of how the choice of a representation R of the gauge group G labeling a Wilson is encoded in the dual theory is challenging, and in general the answer is unknown. Indeed, the gauge groups of theories participating in a field theory duality can be drastically different, and in gauge/gravity dualities there is not even a Lie group G in sight in the dual bulk theory.
One may even argue that the existence of Wilson loops actually introduces a puzzle for dualities. While global symmetries and 't Hooft anomalies between dual theories must match, gauge symmetries between dual theories need not. In a sharp sense, gauge symmetries are not symmetries, but rather redundancies in our local description of particles of helicity one. Nevertheless, the gauge group G is not void of important physical information about the theory: Wilson loop operators are labeled by a representation R of G. And therefore, it is in this vain, that the gauge group is "physical" and its elusive representations must be found in the dual.
In this paper we identify the dual description of half-supersymmetric Wilson loop operators in gauge theories related by three dimensional mirror symmetry [2], an infrared (IR) duality whereby two different ultraviolet (UV) 3d N = 4 gauge theories flow in the IR to the same superconformal field theory (SCFT):

UV
Theory A Theory B

IR SCFT
We find that there is a rather intricate "mirror map" relating Wilson loop operators in one theory to Vortex loop operators in the mirror: The mirror map in non-abelian gauge theories is rather subtle. In abelian gauge theories it follows from the mapping of the abelian global symmetries of the mirror dual theories [3][4][5]. 1 We determine the mapping of loop operators under mirror symmetry for arbitrary 3d N = 4 gauge theories encoded by a quiver diagram of linear or circular topology (see figure 1).  Despite that the mirror dual 3d N = 4 gauge theories -denoted by Theory A and Theory Btypically have completely different gauge groups G A and G B , we are able to encode the choice of representation R of a Wilson loop in one theory in the precise choice of the 1d quiver gauge theory in figure 2 describing the mirror dual Vortex loop operator. The 1d quivers in this class are characterized by the ranks of their gauge nodes, the number of fundamental and anti-fundamental chirals in the last node and the presence or not of an adjoint chiral in each node. In particular, to a quiver gauge theory with all Fayet-Iliopoulos (FI) parameters η i = 0, for i = 1, . . . , p, we associate a Wilson loop representation as follows: 3 • R = S n 1 ⊗ S n 2 −n 1 ⊗ . . . ⊗ S np−n p−1 : for a quiver with one adjoint chiral in each gauge node , 2 Arrows encode bifundamental chiral multiplets of two gauge nodes (circles) or of a gauge and a flavor node (square). For the details of superpotential couplings see section 2.2.
3 See sections 3.1 and 5.3.2 for the general dictionary.
• R = A n 1 ⊗ A n 2 −n 1 ⊗ . . . ⊗ A np−n p−1 : for a quiver with one adjoint chiral for U (n i ) for i = 1, . . . , p − 1, but no adjoint chiral in the U (n p ) node, where A k and S k denote the k-th antisymmetric and symmetric representations of U (N ) L , if η p < 0, or U (N ) R , if η p > 0. A generic product of symmetric and antisymmetric representations like R = S n 1 ⊗ A n 2 −n 1 ⊗ A n 3 −n 2 ⊗ . . . ⊗ S np−n p−1 is obtained by removing an adjoint chiral multiplet in some nodes. We propose, but have less quantitative evidence for, that an arbitrary irreducible representation R characterized by a Young diagram can be obtained from the family of quivers in figure 2 by setting all but one of the FI parameters to zero -that is η p = 0 -, i.e. η 1 = η 2 = . . . η p−1 = 0. 4 A Vortex loop associated to a given gauge group factor with N fundamental hypermultiplets in the 3d N = 4 gauge theory is labeled by a representation R of the gauge group as well as by a splitting of N into two factors: N = M + (N − M ). A Vortex loop labeled by a representation R and by the integer M is constructed by gauging flavour symmetries of the 1d N = 4 quiver gauge theory with the bulk 3d N = 4 theory. The explicit 3d/1d theory obtained in this way can be encoded by a mixed 3d/1d quiver diagram, see e.g figure 3. The mirror map between loop operators that we uncover is rather intricate and rich. In particular, the map depends strongly on the choice of integer M labeling a Vortex loop. As  The same quiver but for 2d N = (2, 2) gauge theories was shown to be labeled by an irreducible representation R in [8], where they describe M2-brane surface defects, which are indeed labeled by a representation R. 5 As we shall see in where W U (M ) Rs is a Wilson loop in a representation R s of U (M ), W fl N,qs is an abelian flavour Wilson loop of charge q s and ∆ M is the set of representations that appear in the decomposition of R = ⊕ s∈∆ M (q s , R s ) under the embedding U (1) × U (M ) ⊂ U (N − 1). 6 With the algorithm we put forward in this paper, the mirror map between loop operators in arbitrary linear and circular quivers can be constructed, and we provide explicit representative examples for both types of quivers.
The key insight that allows us to construct the explicit mirror map between loop operators in linear and circular quivers is the identification of the brane realization of Wilson and Vortex loop operators in the Type IIB Hanany-Witten construction of 3d N = 4 gauge theories [9]. 7 In the string theory construction, mirror symmetry is realized as S-duality in Type IIB string theory [9]. By understanding the detailed physics of branes in string theory, the action of S-duality on our brane realization of loop operators allows us to find an explicit map between brane configurations, which in turn yields the mirror map between Wilson and Vortex loop operators.
We provide quantitative evidence for our mirror maps by computing the exact expectation value of circular Wilson and Vortex loop operators in N = 4 gauge theories on S 3 . 8 The computation of the expectation value of Vortex loops combines in a rather interesting way the computations of the S 3 partition function in [17] with the supersymmetric quantum mechanics index of [18,19]. The detailed matrix integral capturing the expectation value is obtained by understanding how to couple the 1d N = 4 gauge theory defining the Vortex loop on an S 1 ⊂ S 3 to the bulk gauge theory. All our computations confirm our brane-based predictions.
Our understanding of the action of duality on Wilson loops in the context of three dimensional mirror symmetry give us ideas and renewed confidence that this problem can also be tackled in other interesting dualities, like four dimensional Seiberg duality [20], where the gauge groups of the two dual theories are also different, and subject to the puzzles raised at the beginning. 9 The plan of the rest of the paper is as follows. In section 2 we study the classes of loop operators that can be defined in a 3d N = 4 SCFT as well as a UV gauge theory definition of the SCFT. This analysis leads to consider Wilson and Vortex loop operators. In section 3, after reviewing the brane construction of 3d N = 4 gauge theories we put forward a brane realization of Wilson and Vortex loop operators. We give at least two different UV descriptions for each Vortex loop operator, distinguished in particular by the gauging of the global symmetries of the 1d quiver gauge theory with bulk 3d fields. The explicit 3d/1d theories obtained this way can be encoded by mixed 3d/1d quiver diagrams as in figure 4. We then develop a brane-algorithm that allows us to use the action of S-duality on brane configurations to construct the mirror map for loop operators between dual gauge theories. This algorithm can be applied to an arbitrary 3d N = 4 gauge theory labeled by a linear or circular quiver. Section 4 presents the detailed mirror map for representative classes of gauge theories, including T [SU (N )], circular quivers with equal ranks on all nodes and supersymmetric QCD (SQCD). In section 5 we consider loop operators on S 3 and write down a matrix model representation of the exact expectation value of a Vortex loop operator in terms of the supersymmetric index of the SQM that defines it and the matrix model for the 3d theory. We perform explicit computations of the expectation value of Wilson and Vortex loop operators and confirm our brane-based predictions for the mirror map. 10 We also show that the distinct UV definitions of a given Vortex loop operator (see figure 4) give rise to the same operator in the IR, by showing that the expectation value of the two UV definitions coincide, and are related by hopping duality [21]. Some technical details are relegated to the appendices.

Loop Operators in 3d N = 4 Theories
An N = 4 SCFT is invariant under the 3d N = 4 superconformal symmetry OSp(4|4). The bosonic generators comprise those in the SO (3,2) Sp(4) conformal algebra and those that generate the SU (2) C ×SU (2) H R-symmetry of the SCFT. The supercharges in OSp(4|4) transform in the (4, 2, 2) representation of SO(3, 2)×SU (2) C ×SU (2) H . A UV Lagrangian definition of the IR SCFT preserves 3d N = 4 Poincaré supersymmetry, the subgroup of OSp(4|4) that closes into the isometries of flat space. 11 Mirror symmetry is the statement that a pair of UV gauge theories flow in the IR to the same SCFT with the roles of SU (2) C and SU (2) H exchanged:

UV
Theory A Theory B
A line defect in a UV Lagrangian description of a SCFT can be made invariant under four of the supercharges in the 3d N = 4 Poincaré supersymmetry algebra of the UV theory. The 3d N = 4 Poincaré supercharges transform in the (2, 2) representation of the SU (2) C × SU (2) H R-symmetry of the UV theory and obey We take the SO(2, 1) γ-matrices (γ 0 , γ 1 , γ 2 ) = (iτ 3 , τ 1 , τ 2 ), where τ a are the Pauli matrices. The charge conjugation matrix is C = τ 2 . In Lorentzian signature and in this basis the reality condition on the supercharges is Q † αAA = (τ 1 ) β α AB A B Q βBB .
There are two inequivalent 1d N = 4 supersymmetric quantum mechanics (SQM) subalgebras of the 3d N = 4 Poincaré superalgebra of a UV theory that can be preserved by a line defect. We denote them by SQM W and SQM V . SQM W preserves U (1) C × SU (2) H and the supercharges Q 11A and Q 22B , which anticommute to the generator of translations H along the defect The embedding of the SQM W and SQM V algebras in the 3d N = 4 Poincaré superalgebra has a U (1) commutant: R C − J 12 for SQM W and R H − J 12 for SQM V , where J 12 is the U (1) ⊥ rotation generator transverse to the defect and R C and R H are the Cartan generators of SU (2) C and SU (2) H respectively. 16 Therefore, up to shifts by the "flavour" symmetry R C − J 12 for SQM W and by R H − J 12 for SQM V the Cartan R-symmetry generators can be taken to be In summary, in a 3d N = 4 gauge theory there are two classes of line defects that can defined, one preserving SQM W and the other SQM V . These line defects flow in the IR to superconformal line defects preserving U (1, 1|2) W and U (1, 1|2) V respectively. Our analysis can be succinctly summarized by the following diagram: and and Both classes of UV supersymmetric line defects in a UV 3d N = 4 theory can be realized by coupling different 1d N = 4 SQM theories with four supercharges to the bulk 3d N = 4 theory. A canonical way of coupling a 1d N = 4 SQM theory to a 3d N = 4 theory is by gauging flavour symmetries of the SQM theory with 3d N = 4 vector multiplets. The gauging of the defect flavour symmetries with bulk vector multiplets is made supersymmetric by embedding the defect vector multiplet of the supersymmetry algebra preserved by the defect into the bulk 3d N = 4 vector multiplet at the position of the line defect. The embedding is found by identifying which combination of fields in the higher dimensional vector multiplet transform as the fields of the defect vector multiplet under the supersymmetry preserved by the defect. Replacing the defect vector multiplet fields in the gauged 1d N = 4 SQM theory with the proper combination of bulk vector multiplet fields ensures that the coupling of 1d fields to 3d fields is supersymmetric under the supersymmetry of the defect 1d N = 4 SQM theory. Superpotential couplings between defect and bulk matter multiplets may also be added when defect matter multiplets can be embedded in bulk hypermultiplets. Such couplings gauge defect flavour symmetries with bulk flavour or gauge symmetries, depending on which symmetries of the bulk matter multiplet are global and which are gauged.
• SQM V : 2d N = (2, 2) → 1d N = 4 SQM Superpotential couplings between defect and bulk fields also play an important role in the construction of defects.

Wilson Loop Operators
A line defect in a UV 3d N = 4 gauge theory invariant under SQM W is the Wilson line operator, which is labeled by a representation R of the gauge group. It is given by where σ ≡ σ 3 is the scalar field in the N = 2 vector multiplet inside the N = 4 vector multiplet. This operator manifestly breaks the SU (2) C symmetry acting on the three scalars σ = (σ 1 , σ 2 , σ 3 ) in the N = 4 vector multiplet down to U (1) C , 17 while preserving SU (2) H . If the operator is supported on a straight line 18 , it preserves the 1d N = 4 SQM subalgebra SQM W of the 3d N = 4 theory and its U (1) C × SU (2) H R-symmetry. In the IR a Wilson line operator flows to a conformal line operator in the SCFT preserving U (1, 1|2) W .
The supersymmetric Wilson line operator (2.5) can be realized by coupling a 1d N = 4 SQM theory living on the line with the bulk 3d N = 4 gauge theory. This coupling preserves the SQM W supersymmetry algebra. The defect 1d N = 4 SQM that represents a Wilson loop operator is the theory of a 1d N = 4 fermi multiplet, obtained by dimensional reduction of the 2d N = (0, 4) fermi multiplet, which on-shell consists of a complex chiral fermion. The flavour symmetry of a 1d N = (0, 4) fermi multiplet can be gauged preserving supersymmetry with a 1d N = (0, 4) vector multiplet. The fields of the 1d N = (0, 4) vector multiplet that couple to the fermi multiplet can be embedded in the 3d N = 4 vector multiplet as follows 19 This embedding makes manifest that the coupling of the fermi multiplet to the bulk N = 4 vector multiplet preserves U (1) C × SU (2) H R-symmetry and the SQM W algebra. 20 The 1d N = 4 SQM fermi multiplet on the defect can be integrated out exactly and it results in the insertion of a supersymmetric Wilson loop (2.5) in the bulk 3d N = 4 theory. This representation 17 The choice of scalar determines an embedding of U (1) C in SU (2) C . 18 When the scalar couples to the loop with constant charge a circular Wilson loop is not supersymmetric in the UV theory. See, however, discussion at the end of this subsection and of circular Wilson loops on S 3 in section 5.1. 19 The Fermi multiplet couples only to a subset of fields in the N = (0, 4) vector multiplet, and those do admit an embedding into the 3d N = 4 vector multiplet. This observation can be applied to the study of supersymmetric surface operators in 4d N = 2, which can preserve either 2d N = (2,2) or N = (0, 4) supersymmetry. We can construct a surface operator by gauging a N = (0, 4) fermi multiplet with a bulk vector multiplet. These surface operators were studied in the context of N = 4 SYM in [22]. 20 The purely 1d N = 4 theory of a gauged fermi multiplet is invariant under SO(4) R-symmetry. The coupling of the fermi multiplet with the bulk through the embedding (2.7) breaks the R-symmetry down to U (1) C × SU (2) H . of a supersymmetric Wilson loop in 4d N = 4 super-Yang-Mills (SYM) as a coupling of a fermi multiplet with bulk fields appeared in [10,11], where the defect field theory was derived from brane intersections in string theory. Inspired by [10,11], a brane realization of Wilson loop defects in 3d N = 4 gauge theories will play a prominent role in section 3, where we will use S-duality of Type IIB string theory to identify the mirror of Wilson loop operators.
As an aside, 1/4-supersymmetric Wilson loops supported on an arbitrary curve γ in R 3 can be defined mimicking the construction in [23] of 1/16-supersymmetric Wilson loops in 4d N = 4 SYM. This requires tuning the coupling of the loop to the three scalars in the vector multiplet. Explicitly, 1/4-supersymmetric Wilson loops are given by and preserve two supercharges: Q H A ≡ Q αAA αA . These Wilson loop operators are in the cohomology of the supercharges Q H A of the Rozansky-Witten twisted theory [24] 21 obtained by twisting spatial rotations with SU (2) C . Half-supersymmetric Higgs branch operators are also in the cohomology of this twisted theory.

Vortex Loop Operators
A supersymmetric line defect in a UV 3d N = 4 theory preserving SQM V can be constructed by coupling the bulk theory to a 1d N = 4 SQM theory with SQM V symmetry. For line defects preserving SQM V , the appropriate 1d N = 4 SQM theories are obtained by dimensionally reducing 4d N = 1 theories (or equivalently 2d N = (2, 2) theories). U (1) H is the R-symmetry already present in 4d while SU (2) V emerges as an R-symmetry in the dimensional reduction down to 1d. Therefore the SQM V invariant 1d N = 4 SQM theories we consider are supersymmetric gauge theories based on the familiar 4d N = 1 vector multiplets and chiral multiplets. The same 4d N = 1 theories dimensionally reduced to 2d define surface operators [26] in 4d N = 2 gauge theories.
We can construct a supersymmetric line defect in a 3d N = 4 gauge theory by gauging flavour symmetries of a 1d N = 4 SQM V invariant theory with bulk vector multiplets. The embedding of the bosonic fields in the 1d vector multiplet (a 3 , σ 1d , d), where σ 1d is a triplet of SU (2) V , in the 3d N = 4 vector multiplet is (see appendix A) This embedding makes manifest that SU (2) V is preserved and that SU (2) H is broken down to U (1) H , as it selects one of the auxiliary fields transforming as a triplet of SU (2) H in the 3d N = 4 vector multiplet, which we have denoted by D. 22 The coupled theory preserves the SQM V algebra. 21 This is the 3d counterpart of the statement in 4d N = 4 SYM that the 1/16-supersymmetric Wilson loop operators in [23] are in the cohomology of a supercharge of the Langlands twist [25]. 22 The choice of auxiliary field determines an embedding of U (1) H in SU (2) H .
We can gauge defect flavour symmetries either with 3d N = 4 fluctuating vector multiplets or background vector multiplets. Background vector multiplets for flavour symmetries are associated to canonical supersymmetric mass deformations in 3d N = 4 and 1d N = 4 theories. 23 Gauging 1d flavour symmetries with background 3d vector multiplets means that 1d and 3d flavour symmetries are identified. 1d N = 4 and 3d N = 4 flavour symmetries are identified by SQM V -preserving defect cubic superpotential couplings between defect chiral multiplets and bulk hypermultiplets 24 where the index I is a 1d gauge index. The indices i, a are simultaneously indices for 1d flavour symmetries and indices for either 3d flavour or gauge symmetries. When a (or i) is a 3d flavour index, the superpotential breaks the (otherwise independent) flavour symmetries acting on chiral multiplets q a (orq i ) and hypermultiplets Q a i to the diagonal flavour symmetry group. 25 The background 3d N = 4 vector multiplet gives the same mass to the 1d chiral multiplets and 3d hypermultiplets that are acted on by the preserved diagonal flavour symmetry group.
The 1d N = 4 gauge theories that appear in the construction of the defects dual to Wilson loops can be encoded in a standard quiver diagram shown in figure 2. 26 An adjoint chiral multiplet may be added to any U (n i ) gauge group factor, an option which we denote by a dashed line. Each adjoint chiral multiplet is coupled to the neighbouring bifundamental chiral multiplets through a cubic superpotential, while nodes without an adjoint chiral multiplet have an associated quartic superpotential coupling the corresponding bifundamental chiral multiplets.
The specific couplings between 1d N = 4 and 3d N = 4 theories can be encoded in a combined 3d/1d quiver diagram (analogous 4d/2d quivers have appeared in [21] (see also [8])).The quiver diagram makes explicit the 1d flavour symmetries which are gauged with bulk dynamical gauge fields and the flavour symmetries which are identified with 3d flavour symmetries, as shown in figure  5. We use the mixed circle and square notation of [8] to denote the 1d flavour symmetries that are gauged with dynamical 3d vector multiplets. This 3d/1d quiver also assigns a defect cubic superpotential coupling between 1d chiral multiplets and 3d hypermultiplets for each triangle that can be formed with these fields.
Demanding that the UV supersymmetric 3d/1d Lagrangian coupling 1d chiral multiplets with a 3d N = 4 vector multiplet is well-defined (finite) requires that [26][27][28] where µ is the moment map for the flavour symmetry acting on the 1d chiral multiplets that is gauged with the bulk (dynamical) vector multiplet and g is its 3d gauge coupling. Therefore, in 23 Obtained by turning on constant commuting values for the three scalars in the 1d and 3d N = 4 vector multiplet. 24 The fields in the 1d N = 4 chiral multiplet can be embedded in the 3d N = 4 hypermultiplet. This embedding (see appendix A), which we denote by Q, allows one to write supersymmetric couplings between defect chiral multiplets and bulk hypermultiplets. 25 When one of the indices is a 3d gauge index the superpotential indeed enforces the gauging of 1d flavour symmetries with a 3d dynamical vector multiplet. 26 These quivers but in a 2d N = (2, 2) were used in [8] to describe M2-brane surface operators. the semiclassical UV description, defect fields induce a singular Vortex field configuration on the 3d gauge fields. This justifies our use of the subscript V to describe this class of line defects, which we refer as Vortex line defects/operators. These UV Vortex line defects flow in the IR to conformal line operators in the SCFT preserving U (1, 1|2) V .
As another aside, we note that just as a 1/4-supersymmetric Wilson loop supported on an arbitrary curve γ in R 3 have been constructed in (2.8), it should be possible to construct a 1/4supersymmetric Vortex loop on an arbitrary curve in R 3 by suitably adjusting the coupling of the 1d N = 4 SQM to the bulk 3d N = 4 theory. Such a Vortex loop would preserve two supercharges: Q C A ≡ Q αAA αA . These Vortex loop operators are in the cohomology of the supercharges Q C A of the other version of the Rozansky-Witten twisted theory, obtained by twisting spatial rotations with SU (2) H . Half-supersymmetric Coulomb branch operators, that is monopole operators, are also in the cohomology of this twisted theory.
Given two UV mirror theories that flow in the IR to the same SCFT, we can construct both classes of line operators in each of the UV theories. How are line operators mapped under mirror symmetry? Since mirror symmetry exchanges SU (2) C with SU (2) H in dual mirror theories, Wilson line operators of one theory are mapped to Vortex line operators in the mirror and viceversa. This can be represented by the following diagram: Our immediate goal is to come up with an algorithm that yields the duality map between Wilson and Vortex loop operators in mirror dual theories.

Brane Realization Of Loop Operators and Mirror Map
In this section we first briefly introduce the Type IIB string theory realization of 3d N = 4 gauge theories of [9] and recall how mirror symmetry gets realized as S-duality in string theory. Central to the main goal of this paper is the brane realization of both types of line defects discussed in the previous section that we put forward in this section. We then devise an explicit algorithm using branes in string theory to identify the map between loop operators in mirror dual theories.
3d N = 4 supersymmetric gauge theories admit an elegant realization as the low-energy limit of brane configurations in Type IIB string theory [9]. This consists of an array of D3, D5 and NS5 branes oriented as shown in table 1. 27 0 1 2 3 4 5 6 7 8 9 D3 X X X X D5 X X X X X X NS5 X X X X X X The gauge theory associated to a brane configuration is constructed by assigning: • A U (N ) vector multiplet to N D3-branes suspended between two NS5-branes • A hypermultiplet in the fundamental representation of U (N ) to a D5-brane intersecting N D3-branes stretched between two NS5-branes • A hypermultiplet in the bifundamental representation of U (N 1 ) × U (N 2 ) to an NS5-brane with N 1 D3-branes ending on its left and N 2 branes ending on its right Depending on whether the x 3 coordinate takes values on the line or is circle valued, the 3d N = 4 gauge theories engineered this way are described either by quiver diagrams of linear topology or circular topology: linear and circular quiver diagrams respectively. The quiver diagrams for linear and circular quiver theories are presented in figure 1. The general brane configuration realizing a linear quiver theory is shown in figure 6. For a circular quiver the x 3 direction is periodic and there are extra D3-branes stretched between the first and last NS5-branes. 28 We are interested in 3d N = 4 gauge theories that flow in the IR to an irreducible, interacting SCFT. A 3d N = 4 gauge theory flows to such a SCFT in the IR if each gauge group factor U (N c ) has a number of fundamental hypermultiplets N f obeying N f ≥ 2N c and i M i ≥ 2. The second condition is automatically obeyed by linear quivers obeying the first condition, but it is an 27 For more details of these brane constructions see [9,[29][30][31]. 28 The number inside a circle denotes the rank of a gauge group factor. The number inside a rectangle denotes the number of hypermultiplets in the fundamental representation of the gauge group factor corresponding to the circle to which attaches. A line between two circles represents a bifundamental hypermultiplet of the two gauge group factors connected by the line. extra requirement for circular quivers. When these conditions are satisfied the gauge group of the quiver can be completely Higgsed [29,32] and there are no monopole operators hitting the unitarity bound [29]. 29 Mirror symmetry is the statement that two different 3d N = 4 gauge theories flow in the IR to the same irreducible SCFT.
The SU (2) C × SU (2) H R-symmetry of such an irreducible SCFT coincides with the SU (2) C × SU (2) H R-symmetry of the UV gauge theory. In the brane construction the R-symmetry is realized geometrically as spacetime rotations: SU (2) C rotates x 789 and SU (2) H rotates x 456 . 3d N = 4 gauge theories admit moduli spaces of vacua: the Coulomb branch and the Higgs branch. 30 These are hyperkähler manifolds invariant under SU (2) H and SU (2) C and acted on by a group of isometries G C and G H respectively. G H is manifest in the UV definition of the SCFT and is realized as the flavour symmetry acting on the hypermultiplets, while only the Cartan subalgebra of G C is manifest in the UV. Each U (1) gauge group factor gives rise to a manifest U (1) global symmetry, known as a topological symmetry, which acts on the Coulomb branch. The abelian symmetry acting on the Coulomb branch can be enhanced to a non-abelian G C symmetry when conserved currents associated to the roots of G C can be constructed with monopoles operators. 31 The non-trivial, irreducible SCFT sits at the intersection of the Higgs and Coulomb branch where the R-symmetry is enhanced to SU (2) C × SU (2) H . The IR SCFT inherits a G C × G H global symmetry. In the brane realization, the Coulomb branch corresponds to the motion of D3-branes along x 789 while the Higgs branch to the motion of D3-branes along x 456 . 32 The brane description of 3d N = 4 UV gauge theories gives an elegant realization of mirror symmetry [9], whereby two different UV gauge theories flow to the same nontrivial SCFT in the IR with the roles of SU (2) C and SU (2) H exchanged. Mirror symmetry is realized as S-duality in Type IIB string theory combined with a spacetime rotation that sends x 456 to x 789 and x 789 to −x 456 , which exchanges SU (2) C with SU (2) H . The combined transformation, which we will refer as S-duality for brevity, maps the class of brane configurations we have discussed to itself. Given the 29 When N f < 2N c or P i=1 M i ≤ 2 for circular quivers the IR theory is believed to contain a decoupled sector. 30 Mixed branches can emerge at submanifolds of the Higgs and Coulomb branch. 31 G C maps to the flavour symmetry acting on the hypermultiplets of the mirror theory. 32 The brane realization makes it clear why P i=1 M i ≥ 2 is required for complete Higgsing in circular quivers. Indeed, unless there are two D5-branes, D3-branes segments cannot be detached from the NS5-branes.
brane configuration corresponding to a UV 3d N = 4 gauge theory, the mirror dual gauge theory is obtained by analyzing the low energy dynamics of the S-dual brane configuration. The mirror UV gauge theory can be read by rearranging the branes along the x 3 direction, possibly using Hanany-Witten moves [9] involving the creation/annihilation of a D3-brane when an NS5-brane crosses a D5-brane, to bring the S-dual brane configuration to a configuration where the low energy gauge theory can be read using the rules summarized above. This transformation preserves the type of quiver, and thus the mirror of a linear quiver is a linear quiver and the mirror of a circular quiver is a circular quiver. 33 . Examples of mirror-dual pairs of quivers are given in figure 7. An N = 4 SCFT in flat space admits canonical relevant deformations preserving 3d N = 4 Poincaré supersymmetry. These deformations are associated to the flavour symmetries G C × G H acting on the Coulomb and Higgs branches of the SCFT. In a UV realization of the SCFT, these deformations couple to a triplet of mass and FI parameters, which transform in the (3, 1) and (1, 3) of SU (2) C × SU (2) H . Mass and FI deformations are obtained by deforming the UV theory with supersymmetric background vector multiplets in the Cartan of G H and supersymmetric background twisted vector multiplets 34 in the Cartan of G C respectively. In the brane realization, these parameters are represented by the positions of five-branes. The position of the i-th D5-brane along x 789 corresponds to a mass deformation m i while the position of the i-th NS5-brane along x 456 corresponds to an FI parameter ξ i . Mass and FI parameters are exchanged between mirror dual theories. Indeed, in the brane realization of mirror symmetry through S-duality the roles of the NS5 and D5-branes are exchanged. The positions of the 5-branes in the x 3 direction are irrelevant in the infrared 3d SCFT. For instance, the separation between two consecutive NS5-branes is inversely proportional to the coupling g 2 YM of the effective low-energy 3d SYM theory living on the D3-branes stretched between the two NS5-branes. In the deep IR, where the Yang-Mills coupling diverges, the dependence on g 2 YM 33 The irreducibility condition of the IR SCFT is preserved under mirror symmetry, except for a circular quiver with a single node, whose mirror dual has a single fundamental hypermultiplet 34 In twisted multiplets the roles of SU (2) C and SU (2) H are exchanged.

disappears.
Linear quivers that flow to irreducible, interacting SCFT's can be labeled by two partitions of N -ρ andρ -and are denoted by T ρ ρ [SU (N )] [29]. Circular quivers flowing to interacting SCFT's are labeled also by two partitions of N and a positive integer L, and can be denoted by C ρ ρ [SU (N ), L] [31]. 35 Under mirror symmetry 2) and the role of the two partitions are exchanged. The Coulomb branch of these theories, and by mirror symmetry the Higgs branch, describe the moduli space of monopoles in the presence of Dirac monopole singularities for linear quivers and the moduli space of instantons on a vector bundle over an ALE space for circular quivers.
In this paper we give a brane realization of both classes of loop operators discussed in section 2 and put forward an algorithm that produces a map between loop operators of mirror dual theories.

Brane Realization Of Wilson Loop Operators
A key ingredient in our derivation of the mirror map of loop operators is identifying a brane realization of Wilson loop operators, which are labeled by a representation R of the gauge group. Inserting a supersymmetric Wilson loop operator in a 3d N = 4 linear or circular quiver gauge theory admits a simple brane interpretation, obtained by enriching the setup in [9]. The construction we propose extends to 3d N = 4 gauge theories the realization of Wilson loops by branes in 4d N = 4 SYM in [10] [11] (see also [33] [34]).
We start with the brane realization of a supersymmetric Wilson loop in the k-th antisymmetric representation of a U (N ) gauge group factor in the quiver, which we denote by A k . Such an operator insertion is realized by adding k F1 strings stretched in the x 9 direction ending at one end on the N D3-branes where the U (N ) gauge group is supported and at the other end on a D5' brane, defined as a D5-brane stretched in the x 045678 directions. 36 The brane configuration realizing such a Wilson loop is given in table 2. The array of fundamental strings ends between the two NS5-branes over which the N D3-branes are suspended. This brane setup is depicted in the example of figure 8-a.
This enriched brane configuration is supersymmetric: it preserves the SQM W subalgebra of the 3d N = 4 super-Poincaré algebra discussed in section 2. Quantization of the open strings stretched 35 In this paper we will not need the explicit mapping between the data of the quivers and ρ,ρ and L, but present it here for completeness. It is based on the linking numbers of the D5-branes l i and NS5-branesl j , which obey  36 Adding the D5'-brane does not break any further symmetries beyond those broken by the F1-strings. 0 1 2 3 4 5 6 7 8 9 D3 X X X X D5 X X X X X X NS5 X X X X X X F1 X X D5' X X X X X X  between the D3 and the D5'-branes gives rise to 1d complex fermions χ localized on the line defect and furnish the dimensional reduction of the 2d N = (0, 4) gauged Fermi multiplet where U = P e i β 0 dtA 0 +σ is the supersymmetric holonomy operator. As it stands there is a global 37 Here we put the system on a circle of length β.
anomaly for the U (1) ⊂ U (N ), since under large gauge transformations Z 0 → −Z 0 . Our brane realization of the Wilson loop, however, engineers a bare supersymmetric Chern-Simons term at level k = 1/2, that is 1/2 (A 0 + σ), which precisely cancels the offending factor det(U) −1/2 , and the brane system is anomaly free. 38 Integrating out the fermions in the presence of the Chern-Simons term therefore yields The presence of k F1-strings stretched between the D3 and D5'-branes is represented in the gauge theory by the insertion of k creation operators for these fermions in the past and k annihilation operators in the future. Physically, these operators insert a charged probe into the gauge theory. Integrating out these fermions inserts a supersymmetric Wilson loop operator in the k-th antisymmetric representation [10] The weights of the k-th antisymmetric representation of U (N ) admit an elegant description in the brane construction. We must distribute k F1-strings among N D3-branes (all k F1-strings terminate at the other end on a single D5'-brane). To a pattern of k F1-strings where k j strings end on the j-th D3-brane (see figure 9) we associate a set of N non-negative integers {k j } obeying k = k 1 + k 2 + · · · + k N , with k j ≥ 0 for all j. However, not all positive integers k j are allowed. There can be at most one F1-string stretched between a D3-brane and a D5'-brane. This is the so-called s-rule [9], and is a constraint that follows from Pauli's exclusion principle [36]. Therefore, the allowed configurations are described by a collection of N non-negative integers {k j } with the constraint that k j ≤ 1. This set of configurations is in one-to-one correspondence with the weights of the k-th antisymmetric representation of U (N ), i.e. of A k .
We now turn to a Wilson loop in the k-th symmetric representation of U (N ), which we denote by S k . Inserting a Wilson loop in the k-th symmetric representation is realized by adding k F1 strings stretched in the x 9 direction ending at one end on the N D3-branes where the U (N ) gauge group is supported and at the other end on a D5-brane stretched in the x 012456 directions and localized in the x 9 direction. The array of fundamental strings ends between the two NS5-branes over which the N D3-branes are suspended. This setup is illustrated in figure 8-b. In this case the charged probe particle inserted by the Wilson loop can be though of as arising from a very heavy hypermultiplet, represented by adding a D5-brane to the theory and then taking the D5-brane far away from the stack, thus giving it a large mass and making the hypermultiplet fields nonrelativistic. Integrating out the heavy hypermultiplet in the presence of k heavy insertions yields a supersymmetric Wilson loop operator in the k-th symmetric representation [10,11]. 39 NS5 F1 D3 NS5 k 1 k 2 k 3 k 4 Figure 9: Configuration with stacks of k j F1-strings, 1 ≤ j ≤ 4, ending on each of the four D3-branes, associated to a weight (k 1 , k 2 , k 3 , k 4 ) of a representation of U (4).
The weights of the k-th symmetric representation of U (N ) admit an elegant description in the brane construction. We again distribute k F1-strings among N D3-branes (all k F1-strings terminate at the other end on a single D5-brane). To a pattern of k F1-strings where k j strings end on the j-th D3-brane (see figure 9) we associate a set of non-negative integers {k j } obeying k = k 1 +k 2 +· · ·+k N , with k j ≥ 0 for all j. In this case an arbitrary number of F1-strings can be stretched between the D5 and a D3-brane. Therefore the set of configurations is in one-to-one correspondence with the weights of the k-th symmetric representation of U (N ), i.e. of S k .
Our brane construction can be easily generalized to Wilson loops in the tensor product of an arbitrary number of symmetric and antisymmetric representations of U (N ): . This requires considering F1-strings stretched between d D5-branes and d D5'-branes and the N D3-branes that support the gauge group. For the above mentioned representation k (a) F1strings must emanate from the a-th D5-brane and l (b) F1-strings from the b-th D5'-brane. Integrating out the massive charged particles produced by this configuration yields a Wilson loop in the desired representation. Furthermore, the set of allowed F1-string configurations, labeled by {k More precisely a configuration is associated to a weight w = (w 1 , w 2 , · · · , w N ) in the orthogonal basis j . Our analysis can be summarized in table 3. In order to describe Wilson loops in the above mentioned representation we have to place the d D5 and d D5'-branes at different positions in the x 3 and x 9 directions. The separation in the x 9 direction is not essential at this stage but plays a role when we identify the S-dual brane configuration and the mirror dual Vortex loop. The separation in the x 3 direction is more crucial: if two D5-branes sit at the same x 3 position, the pattern of F1-strings is more complicated since strings can now break and be stretched between the two D5-branes preserving the same amount of supersymmetry. In this case we expect that the brane configuration would insert a Wilson loop in an irreducible representation of U (N ), as in [10] [11]. An irreducible representation R of U (N ) labeled by a Young diagram with number of D5 number of D5'   In the brane realization we must specify between which two consecutive NS5-branes the F1-strings end. This determines in which gauge group the Wilson loop is inserted. While the F1-strings cannot be moved across an NS5-brane without changing the Wilson loop operator, they can be freely moved across a D5-brane without changing the Wilson loop operator in the IR if the D5-brane has the same number of D3-branes ending on the left and the right, which is the case in the canonical brane configuration in figure 6. The corresponding S-dual statement, that D1-branes can be moved across an NS5-brane with the same number of D3-branes ending on the left and the right but that D1-branes cannot be moved across a D5-brane without changing the IR dynamics, will play an important role in unraveling the mirror map of loop operators.

Mirror Of Wilson Loops From S-duality
After having found a brane realization of Wilson loops, we now make use of the fact that mirror symmetry corresponds to S-duality in the Type IIB brane realization to derive the mirror dual of supersymmetric Wilson loop operators. Here we loosely call S-duality what is really S-duality combined with the rotation that sends x 456 to x 789 and x 789 to −x 456 , so that D5-branes and NS5branes get exchanged. We shall see that the information about the representation of the Wilson loop is encoded in the discrete data of a 1d N = 4 SQM quiver quantum mechanics gauge theory.
We have found that Wilson loop insertions can be realized by a brane configuration with F1strings stretched between the N D3-branes and D5-branes and/or D5'-branes, oriented as in table 2. Under S-duality the D5 and D5'-branes become NS5 and NS5'-branes, while the F1-strings become D1-branes, oriented as in table 4. This brane configuration preserves the SQM V subalgebra of the 3d N = 4 super-Poincaré algebra discussed in section 2. 40 42 The remaining SO(2) 12 − SO(2) 45 isometry (a diagonal SO(2)) is mapped to a flavor symmetry of the SQM theory, which is the commutant of SQM V in the 3d N = 4 supersymmetry algebra. 0 1 2 3 4 5 6 7 8 9 D3 X X X X D5 X X X X X X NS5 X X X X X X D1 X X NS5' X X X X X X We will now exhibit that Wilson loops in a UV 3d N = 4 gauge theory are mirror to loop operators defined by 1d N = 4 SQM V invariant quiver gauge theories coupled to the mirror 3d N = 4 gauge theory. Likewise, the Wilson loops of the mirror theory are mapped to 1d N = 4 quiver gauge theories coupled to the original 3d N = 4 gauge theory. The 1d N = 4 quiver gauge theories and the way they couple to the 3d N = 4 gauge theory are found by identifying the IR gauge theories living on the D1-branes in the S-dual brane configuration. 40 S-duality in Type IIB string theory acts nontrivially on the supersymmetry charges. 41 D1-branes span an interval in the x 6 direction. The low-energy effective theory is thus one-dimensional. 42 For later convenience, we have combined the R-symmetry R H realized by SO(2) 45 with the with the SO(2) 12 − SO(2) 45 commutant to define a new R-symmetry: SO(2) 12 .
How can the mirror operator to a given Wilson loop be constructed? First we perform S-duality on the brane configuration realizing a Wilson loop in a 3d N = 4 gauge theory. In the absence of the F1-strings, the mirror 3d N = 4 theory is found by rearranging the S-dual brane configuration so as to bring it to the canonical frame explained at the beginning of section 3, where the mirror gauge theory can be easily read. After S-duality, the number of D3-branes on the left and on the right of a D5-brane need not be the same while that number is the same for every NS5-brane, as before S-duality the D5-branes had the same number of D3-branes on both sides. In order to bring the brane configuration to one where the gauge theory can be read we must move the D5-branes with an excess of D3-branes in the direction of the excess and make it pass through NS5-branes. Every such move, exchanging a D5-brane with an NS5-brane, results in the creation of a D3-brane [9] on the side of the D5-brane that had a smaller number of D3-branes, thus diminishing the D3-brane excess. See figure 11. This process must be continued until the number of D3-branes on each of the D5-branes is the same on the left and on the right. From this final configuration we can read the mirror dual gauge theory. Insertion of a Wilson loop operator in a gauge group factor requires adding F1-strings ending between the two consecutive NS5-branes where the D3-branes supporting that gauge factor are suspended. After S-duality, we have the same S-dual brane configuration as before but now with extra D1-branes ending between a pair of consecutive D5-branes. It is from this S-dual brane configuration that we read off the 1d N = 4 quiver gauge theory living on the D1-branes and its coupling to the mirror dual gauge theory. In order to read the mirror description of the original Wilson loop we once again move the D5-branes in the direction of D3-brane excess to bring the brane configuration to the canonical one, where the 3d N = 4 mirror gauge theory can be extracted. During this process we do not allow a D5-brane to cross a D1-brane. In the simplest situations, the D5-branes can be moved to the canonical configuration of the mirror dual theory without having to move the D1-branes. Then we can directly read off the 1d N = 4 quiver theory from the final brane configuration with D1-branes (see below). In more complicated examples, we have a situation when a D5-brane must be moved past an NS5-brane while D1-branes stand between them. For definiteness let us imagine that the D5-brane must be moved from the left to the right of an NS5-brane, with the D1-branes in between, as in figure 12-a. Figure 12: a) The D5-brane has to be moved to the right of the NS5-brane to reach the canonical brane configuration.
The D1-branes in between must first be moved across the NS5-brane themselves. b) After moving the D1 and D5branes, stacks of D1-branes end up attached to the NS5-brane and moved away from the main brane configuration along transverse directions (transverse to the picture).
In this situation the D1-branes together with the NS5/NS5'-branes to which they are attached at the other end must be moved first to the right of the NS5-brane. Here we must distinguish two situations: if the number of D3-branes ending on both sides of the NS5-brane are equal, then the D1branes can be moved across it without anything special happening (in the S-dual picture, F1-strings can cross a D5-brane with the same number of D3-branes on both sides in the same trivial way). However the NS5-brane can also have an excess of D3-branes on the right. 43 Then the NS5-brane develops a D3-brane spike [37] and some D1-branes may be moved along the spike so that they end on the NS5-brane, and can then be moved far away from the main stack along transverse directions. If there were k D1-branes in the original stack, they could split into k 1 D1-branes moved from the left to the right of the NS5-brane along the D3-branes and k 2 D1-branes moved along the NS5-brane spike and away from the main stack, with k = k 1 + k 2 . Then the D5-brane can be moved across the NS5-brane as usual, without encountering the D1-branes. This is illustrated in figure 12 The choice of splitting k = k 1 + k 2 may be constrained, but generically several splittings may be allowed. The idea of the mirror map is that the initial Wilson loop is mapped to the sum of the Vortex loops realized by the possible final brane configurations, weighted with coefficients. The precise rules to derive which final brane configurations can be reached (and their weighting coefficients) are not 43 The NS5-brane cannot have an excess of D3-branes on the left, where the D5-brane is coming from. This can be understood as follows. After S-duality of the original configuration, the number of D3-branes on both sides of the NS5-brane are equal. During the brane rearrangement some D5-branes can move across it but only in one direction (because D5-branes do not cross between themselves during the rearrangement). In our example, D5-branes cross the NS5-brane from the left to the right. For each D5-brane crossing the NS5, the Hanany-Witten rule implies that the excess of D3-branes on the right of the NS5 increases by one, so that along the process the NS5 develops an excess of D3-branes on the right. In the other case when D5-branes cross the NS5 from the right to the left, the NS5 develops an excess of D3-branes on the left. obvious. Intuitively the difficulty is related to the fact that the D3-spike is "on the wrong side" of the NS5-branes, so that we cannot directly move the D1-branes along the spikes. We will see shortly that the situation is under much better control starting from the D1-branes configurations realizing Vortex loops and S-dualizing to configurations realizing the mirror Wilson loops.
What are the final brane configurations? In addition to the D3, D5, NS5 system realizing the mirror dual theory, there are D1-branes ending on D3-branes on one side and on NS5 and/or NS5'branes situed far away from the main configuration. Moreover there are also D1-branes ending on NS5-branes in the main stack. The physical interpretation of having q D1-branes ending on an NS5-brane is as a background Wilson loop of charge q for a U (1) global symmetry associated to the NS5-brane, which is a combination of the so-called topological symmetries of the 3d theory. These flavor Wilson loops combine with the Vortex loop realized by the D1-branes ending on D3-branes, which we describe now.
A final brane configuration with D1-branes ending on D3-branes in one end and on NS5 and/or NS5'-branes on the other can be used to give at least two descriptions of the mirror of a Wilson loop. This brane configuration can be thought of as a deformation of two other brane configurations, both of which are conducive to reading off the 1d N = 4 quiver gauge theory living on the D1-branes and its couplings to the bulk 3d N = 4 gauge theory. The D1-branes can be either moved to the nearest NS5-brane to the left or to the nearest NS5-brane to the right. In order to reach this configuration, from which the gauge theory description of the mirror loop operators can be read off, we must again ensure that no D1-strings cross D5-branes. This means that we have to move the D5-branes between the D1-branes and the nearest NS5-brane to the other side of that NS5-brane. In summary, we can realize the mirror loop operator either as: 1. Deformation of the coupled 3d/1d theory realized by the D1-branes when they end on the NS5-brane on the left.
2. Deformation of the coupled 3d/1d theory realized by the D1-branes when they end on the NS5-brane on the right.
These yield two dual descriptions of the same operator.
Once we move the stack of D1-branes so that it ends on a neighbouring NS5-brane, we can read off the 1d N = 4 quiver gauge theory and how it couples to the 3d N = 4 gauge theory. The 1d N = 4 gauge theory associated to a brane configuration is constructed by assigning: • A U (k) vector multiplet to k D1-branes suspended between an NS5-brane and an NS5'-brane.
• A U (k) vector multiplet and an adjoint chiral multiplet 44 to k D1-branes suspended between two NS5-branes or two NS5'-branes.
• Two chiral multiplets in the bifundamental and anti-bifundamental representations of U (k 1 ) × U (k 2 ) 45 to an NS5-brane or NS5'-brane with k 1 D1-branes ending on its left and k 2 D1-branes ending on its right. 44 The adjoint chiral multiplet describes the position of the D1-branes in the x 12 directions (for NS5-branes), or in the x 45 directions (for NS5'-branes). 45 The bifundamental representation of U (k 1 )×U (k 2 ) is (k 1 ,k 2 ) and the anti-bifundamental is its complex conjugate.
• A chiral multiplet in the bifundamental of U (N R ) × U (k) and an chiral multiplet in the bifundamental of U (k) × U (N L ) to k D1-branes ending on an NS5-brane which has N L D3-branes ending on its left and N R D3-branes ending on its right. Figure 13 summarizes the reading of the 1d N = 4 quiver SQM gauge theory from the brane configuration, when moving the D1-strings to the nearest NS5-brane on the left. Figure 13: To read the SQM quiver we move the D1-branes to the closest NS5-brane on the left. The rank of nodes are given by n j = j i=1 q i , where q i is the number of D1-branes emanating from the i-th five-brane. The q i are trivially mapped to the k (a) and l (b) introduced in the text. NS5'-branes are denoted by blue squares, as a opposed to NS5-branes that are blue circles.
The parameters of the 1d N = 4 gauge theory living on the D1-branes admit a brane interpretation. The gauge coupling of the vector multiplet realized on D1-branes stretched between two five-branes (NS5 or NS5') is inversely proportional to the distance in the x 6 direction between the five-branes. The relative position in the x 3 direction between two consecutive five-branes (NS5 or NS5') determines the 1d N = 4 FI parameter for the corresponding gauge group factor. If we denote the position of the NS5-brane in the main stack where the D1-strings originally end by x 3 0 and x 3 i the position of the i-th five-brane away from the main stack, the 1d FI parameters are given by In particular, if the D1-branes are moved to the right of the NS5-brane in the main stack along the D3-branes then the FI parameter for the first 1d gauge group factor is positive, while it is negative if we move the D1-branes to the left along the D3-branes. Therefore, the 3d/1d gauge theories 1) and 2) described above obtained after S-duality should be thought of as a deformation with η 1 > 0 of theory 1) and a deformation with η 1 < 0 of theory 2).
Consider for instance the D1-brane configuration of figure 13 near an NS5-brane in the main stack with k (a) D1-strings emanating from the a-th NS5-brane and l (b) D1-strings from the b-th NS5'-brane and ending on the N R D3-branes to the right of the NS5-brane in the main stack. There are also N L D3-branes to the left of this NS5-brane. To this brane configuration we can associate the representation , where d and d denote the numbers of NS5 and NS5'-branes from which the D1-branes emanate. This brane configuration can be thought of as a deformation of a 1d N = 4 quiver gauge theory in figure 13 with positive FI parameters. There are actually different dual descriptions of the 1d N = 4 theories depending on the relative order of the d NS5 and d NS5'-branes in the x 6 direction. Different relative positions give rise to different dual 1d N = 4 descriptions of the same Vortex loop operator labeled by the representation . Roughly speaking, these dual descriptions are related by a 1d N = 4 version of Seiberg duality [20] (see also [21,38]).
We have described the 1d N = 4 SQM V quiver theory living on the D1-branes. We must now explain how it is coupled to the 3d N = 4 gauge theory. The idea is that the U (N L ) × U (N R ) flavor symmetry of the 1d N = 4 theory is gauged with 3d bulk fields living on the D3-branes ending on the NS5-brane on the main stack. We must, however, distinguish the D3-branes supporting dynamical gauge fields from the non-dynamical D3-branes stretched between the NS5 and a D5-brane. To read the SQM theory we had to move D1-branes to the closest NS5-brane to left (or to the right). Suppose there are n D5 D5-branes standing between this NS5 and the D1-branes, then we have to move them first to the left of the NS5 and, by the Hanany-Witten effect, one D3-brane per D5 is created ending on the left of the NS5-brane. In this case the N L D3-branes ending on the left of the NS5 decompose into N L = n D5 + n D3 , where n D3 is the number of dynamical D3-branes, those supporting a U (n D3 ) 3d N = 4 vector multiplet. On the other side, the N R D3-branes ending on the right of the NS5 support a U (N R ) 3d N = 4 vector multiplet. The 1d N = 4 theory is then coupled to the 3d N = 4 theory by gauging the U (N R ) and U (n D3 ) ⊂ U (N L ) flavor symmetries on the defect with dynamical 3d N = 4 vector multiplets. 46 Furthermore, there is a cubic superpotential as in (2.12) which breaks the U (n D5 ) 3d × U (n D5 ) 1d flavor symmetry to the diagonal U (n D5 ), where U (n D5 ) 1d ⊂ U (N L ) and U (n D5 ) 3d is the 3d flavor symmetry acting on the n D5 hypermultiplets associated to the D5-branes. The 3d/1d coupling that can read from the brane picture is summarized in the example of figure 14.
The 3d/1d coupling for the SQM V theories read from moving the D1s to the nearest NS5 on the right are found similarly, as shown in figure 15.
As mentioned above, it turns out to be simpler to derive the mirror map if we consider the inverse problem of finding the combination of Wilson loops dual to a given Vortex loop. Starting with a configuration of D1-branes realizing a Vortex loop, we can S-dualize to obtain a configuration with F1strings and move the D5-branes to reach the canonical brane configuration of the mirror-dual theory. The crucial difference is that we do not need to move the F1-strings, instead we are allowed to move the D5-branes across the stack of F1-strings if this is necessary to reach the canonical configuration. When we have to move a D5 across the F1-strings the situation is always the same, namely the D5-brane has an excess of D3-branes on the side toward which it is moving. This means that the F1-strings are on the side of the D3-spikes, along which they can be moved without obstruction. When the D5-brane crosses the F1-strings the number of D3-branes at the bottom of the strings decreases, as in figure 16. The initial k strings split into k = k 1 + k 2 strings, where k 1 strings end on the remaining D3-branes, while k 2 strings have been moved along the D3-spikes and end now on the D5, away from the main brane configuration. There are many possible final brane configurations, associated to the various choices of splittings (constrained by the s-rule) as D5-branes are moved across the F1-strings, and each of these final configuration realizes a Wilson loop in a certain node of the mirror theory, combined with flavor Wilson loops inserted by the strings ending on the D5branes. Physically q F1-strings ending on a D5 insert a Wilson loop with charge q under the U (1) flavor symmetry acting on the associated hypermultiplet. The mirror symmetry prediction is then that the initial Vortex loop is mapped to the sum of the Wilson loops realized by the possible final brane configurations. We will provide explicit mirror map predictions using this algorithm in the next section.
The simplest situations arise when no D5-branes must be moved across F1-strings, in which case the Vortex loop is mapped to a single Wilson loop in the mirror theory and both loops are labeled by the same representation R of a certain U (N ) gauge group factor. This is the case for instance in circular quivers with nodes of equal rank, for which mirror symmetry is simply implemented by S-duality on the brane configuration and no D5-brane moves are required to reach the mirror dual configuration (see section 4.4).
In this section we have found a systematic algorithm to construct the mirror map between Wilson loop and Vortex loop operators in mirror dual theories. This can be applied to construct the mirror map for any pair of 3d N = 4 mirror quiver gauge theories of linear or circular type. We will apply this algorithm on several explicit examples below.

Mirror Symmetry and Loop Operators: Examples
In this section we give the explicit mirror map between loop operators using the ideas and tools introduced in the previous section. We provide examples both for linear and circular quivers.
We first discuss the action of mirror symmetry of loop operators for a class of 3d N = 4 linear quivers that are self-mirror: that is T [SU (N )] [29]. T [SU (N )] is encoded by the quiver diagram of Constructing the mirror map for arbitrary loops in T [SU (N )] already incorporates all the subtleties and physical phenomena that emerge in the most general case already discussed in section 3.2. We then provide duality maps for other examples of linear and circular quivers.

T [SU (2)]
Consider a charge k Wilson loop W k in T [SU (2)], a U (1) gauge theory with two fundamental hypermultiplets (see figure 18). The brane construction of T [SU (2)] has a single D3-brane stretched between two NS5-branes which is crossed by two D5-branes. According to the discussion of section 3.1, the insertion of the charge k Wilson loop (which is the same as the S k representation in an abelian theory) is realized by adding k F1-strings stretched between the D3-brane and an extra D5-brane far away from the stack (figure 19-a). The F1-strings (and extra D5-brane) can be moved along the x 3 direction along the D3-brane without changing the infrared 3d theory, so we can without loss of generality place the F1-strings between the two D5-branes.
Acting with S-duality on this enriched brane configuration and moving the D5-branes according Figure 16: a) The D5-brane has to be moved to the right, across F1-strings, to reach the canonical brane configuration.
b) After moving the D5-brane, stacks of F1-strings end on the D5-brane and are moved along transverse directions (transverse to the picture) away from the main brane configuration. to the Hanany-Witten rules we recover the same T [SU (2)] brane configuration but with k D1-branes stretched between the D3-brane and an extra NS5-brane far away from the stack as in figure 19-c. The D1-branes end on the D3-brane between the two D5-branes. We note that this is the final S-dual brane configuration irrespectively of where we decide to place the F1-strings relative to the original D5-branes. The presence of the D1-branes is responsible for the insertion of a supersymmetric one-dimensional defect in the T [SU (2)] theory living on the D3-branes. We denote this defect by V k .
The operator V k can be described by a 1d N = 4 gauge theory coupled to T [SU (2)]. As explained above there are two alternative 3d/1d defect descriptions of V k , that we can read by moving the D1branes on top of the nearest NS5-brane on the left or on the right. It is a non-trivial dynamical statement that these two descriptions of V k do indeed describe the same operator (see section 5).
We start with the description of V k as a deformation of the theory where the D1-branes end on the left NS5-brane. This configuration is reached by first moving the left D5-brane to the left of the NS5-brane, creating a D3-brane by the Hanany-Witten effect, and then moving the D1-branes (together with the extra NS5-brane far away) until they end on the NS5-brane, thus reaching the configuration shown in figure 20. In the IR, the theory supported on the brane configuration is a 3d/1d N = 4 gauge theory coupled to T [SU (2)], which can be read from the rules described in the previous section. It is described by the 1d N = 4 quiver gauge theory of figure 20, a U (k) gauge The way the 1d N = 4 quiver gauge theory couples to T [SU (2)] can be read from the brane configuration. The U (1) − flavor symmetry acting on the anti-fundamental chiral multiplet is gauged with the 3d N = 4 U (1) vector multiplet, and the U (1) + flavor symmetry acting on the fundamental chiral multiplet is identified with the U (1) flavour symmetry acting on the 3d hypermultiplet in T [SU (2)] associated to the left D5-brane. The operator V k is described by the coupled 3d/1d theory summarized by the mixed 3d/1d quiver diagram in 20. As explained in section 2, the identification of 1d with 3d flavour symmetries is implemented with a superpotential coupling (2.12) between the relevant defect chiral multiplets and bulk hypermultiplet. In the initial configuration (figure 19-c) the D1-branes end on the D3-branes instead of the left NS5-brane. The deformation corresponding to moving the D1-branes along the D3-brane to go back to the initial configuration corresponds to turning on an FI term with coupling η ∝ ∆x 3 > 0 in the SQM, where ∆x 3 is the difference of the positions along x 3 between the left NS5-brane and the D1-branes.
Alternatively, we can describe the operator V k as a deformation of a configuration where the D1-branes end on the NS5-brane on the right. Following the same steps as before we reach the brane configuration of figure 21, and we find a description of the infrared loop operator V k as a different coupling of the same 1d N = 4 quiver gauge theory to T [SU (2)]. Now the U (1) + flavor symmetry acting on the fundamental chiral multiplet is gauged with the 3d N = 4 U (1) vector multiplet of  (2)] associated to the right D5-brane. The operator V k is described by the coupled 3d/1d theory summarized by the mixed 3d/1d quiver diagram in 21. In the initial configuration (figure 19-c) the D1-branes end on the D3-branes instead of the right NS5-brane. The deformation corresponding to moving the D1-branes along the D3-brane to go back to the initial configuration corresponds to turning on a negative FI parameter η ∝ ∆x 3 < 0 in the SQM, where ∆x 3 is the difference of the positions along x 3 between the right NS5-brane and the D1-branes. We conclude that there are two different UV descriptions of the operator V k mirror dual to the charge k Wilson loop W k in T [SU (2)]. In section 5.6 we explicitly show that the exact expectation value of V k on S 3 computed using the two different UV definitions define the same operator and in section 5.4.1 that it matches the exact expectation value of W k under the exchange of mass and FI parameters We can also consider the Vortex loop operators realized by D1-branes ending on the D3-brane to the left or to the right of the two D5-branes. The corresponding 3d/1d quivers realizing these Vortex loops can be read by moving the D1-branes on top of the NS5-brane on the right, as shown in figure  22, or on the left. We analyze the mirror of such Vortex loop operators for the general T [SU (N )] theory in section 4.2. They are mirror to flavour Wilson loops (see discussion around equation (4.7)).

Wilson Loops In The
We now turn to Wilson loops in a non-abelian gauge theory. We start with Wilson loop operators in the U (N − 1) node of the self-mirror T [SU (N )] theory. As we shall see, a Wilson loop for this (last) node maps directly to a single Vortex loop operator, described by a specific 3d/1d gauge theory that we construct. Later we show that the expectation value of a Wilson loop operator in another node in T [SU (N )] maps under mirror symmetry to a specific combination of Vortex loop operators.
We start by considering a Wilson loop in the k-symmetric representation of U (N − 1), which we denote by W The brane realization of this Wilson loop insertion comprises the D3-NS5-D5 system realizing the T [SU (N )] theory in figure 17 with k additional F1-strings stretched between the N − 1 D3-branes supporting the U (N − 1) node and an extra D5-brane away from the stack, as depicted in figure 23-a. We are free to place the F1-strings anywhere relative to the N D5-branes that intersect the D3-branes, as this results in the same Wilson loop operator in the IR.
The mirror dual loop operator to W (N −1) S k is obtained by analyzing the S-dual configuration ( figure  23-b), where we have reversed the x 3 direction for convenience. 47 First we move the D5-branes in the direction of D3-brane excess to reach the T [SU (N )] brane configuration, now enriched with k D1-branes stretched between the N − 1 D3-branes associated to the U (N − 1) node and an extra NS5-brane far away from the stack (see figure 23-c). The D1-branes are positioned with one D5 to their left and N −1 D5-branes to their right. This final configuration is the same irrespective of where the original F1-strings were placed relative to the D5-branes. In order to reach this configuration we are careful not to let D5-branes cross the D1-branes in the rearrangement, since this would affect the infrared theory living on the D1-branes. This is S-dual to the statement that we do not let F1-strings move across NS5-branes, as this clearly changes the Wilson loop inserted. The position of the D1-branes relative to the D5-branes is crucial in determining the Vortex loop operator just as the position of the F1-strings relative to the NS5-branes is crucial in determining the Wilson loop operator. On the other hand, we let the D1-branes move freely across an NS5-brane when the number of D3-branes is the same on both sides of the NS5-brane. This is S-dual to the fact that F1-strings can be freely moved across D5-branes when the number of D3-branes is the same on both sides of the D5-branes. We denote the loop operator defined by the S-dual brane configuration by V N −1,S k , where the subscript N − 1 encodes the number of D5-branes to the right of the D1-branes. The coupled 3d/1d theory living on the final brane configuration can be read again in two ways, depending on which of the nearby NS5-branes we let the D1-branes end, as explained in section 3.2. One possibility is to move the N − 1 D5-branes to the right and let the D1-branes end on the NS5-brane on their right as shown in figure 24-a. The associated 1d N = 4 gauge theory is a U (k) vector multiplet, with an adjoint chiral multiplet, N − 1 fundamental and N − 1 anti-fundamental chiral multiplets. This 1d N = 4 theory is captured by the 1d part of quiver diagram in 24-a.
The way the 1d gauge theory couples to the 3d T [SU (N )] theory is also captured by the brane configuration in figure 24-a. The U (N − 1) flavour symmetry rotating the fundamental chirals is gauged with the 3d U (N −1) vector multiplet symmetry and the U (N −1) symmetry rotating the antifundamental chirals is identified with the U (N − 1) flavor symmetry acting on the 3d hypermultiplets associated to the N − 1 D5-branes. In order to describe the actual operator V N −1,S k the 1d N = 4 gauge theory must be deformed with a negative FI parameter. The coupled 3d/1d description of the Vortex loop operator V N −1,S k mirror to W (N −1) S k is captured by the quiver diagram in figure 24-a.
Another description of V N −1,S k can also be read from the configuration where the D1-branes end on the nearest NS5-brane to their left, with the (single) leftmost D5-brane pushed to the left of that NS5-brane. The 1d N = 4 gauge theory living on the D1-branes is the same as before, but the way it couples to T [SU (N )] is rather different. The U (N − 1) flavour symmetry rotating the anti-fundamental chirals is gauged with the 3d U (N − 1) vector multiplet. The U (N − 1) flavour symmetry rotating the fundamental chirals is broken to U (N − 2) × U (1). The U (N − 2) flavour symmetry is gauged with the 3d U (N − 2) vector multiplet and the remaining U (1) flavour symmetry is identified with the U (1) flavor symmetry acting on the 3d hypermultiplet associated to the leftmost D5-brane. Now, the 1d N = 4 gauge theory has to be deformed with a positive FI parameter as the D1-branes end to the right of the reference NS5-brane. The coupled 3d/1d description of the Vortex In summary, we obtain the following mirror map of loop operators in T [SU (N )] theory The situation is almost the same if we start with a Wilson loop in the k-antisymmetric representation of U (N − 1), which we denote by W In this case the Wilson loop in realized with k F1-strings stretched between an extra D5'-brane and the N − 1 D3-branes. After S-duality and brane rearranging we obtain the same brane configuration as in figure 23-c, except that D1-branes end on an NS5'-brane instead of a NS5-brane. The two possible 3d/1d defect theories describing the mirror loop operator, which we denote by V N −1,A k , are given in figure 25. They are the same as in figure 24, except that there is no adjoint chiral multiplet in the SQM. In this case we get the following duality map W In section 5.4 we explicitly show by computing the exact expectation value of these loop operators on S 3 that (4.2) and (4.3) hold under the exchange of mass and FI parameters.
The case of the Wilson loop in a tensor product of symmetric and anti-symmetric representations is analogous. We denote such a Wilson loop by W (N −1) R . As explained in section 3.2, the Wilson loop insertion is realized with a collection of d D5-branes and d D5'-branes, with k (a) F1-strings suspended between the N − 1 D3-branes and the a-th D5-brane, and l (b) F1-strings suspended between the N − 1 D3-branes and the b-th D5'-brane. The case d = d = 2 is displayed in figure 26-a. After S-duality and D5-brane moves, we obtain a configuration with d NS5-branes and d NS5'-branes, with k (a) D1-branes suspended between the N − 1 D3-branes and the a-th NS5-brane, and with l (b) D1-branes suspended between the N − 1 D3-branes and the b-th NS5'-brane, realizing the mirror dual loop operator, which we denote by V N −1,R . In the initial configuration, the relative positions of the D5-branes and D5'-branes in the x 9 direction were irrelevant for the insertion of the Wilson loop operator. In the S-dual picture, the positions of the NS5-branes and NS5'-branes along the x 6 direction are important, since they affect the 1d N = 4 gauge theory living on the D1-branes. The brane picture, however, implies that all the possible 3d/1d defect field theories obtained from picking different orderings of the NS5 and NS5' brane positions along x 6 direction are equivalent/dual descriptions of the loop operator V N −1,R . In section 5.3.2 we provide some explicit evidence that these are indeed dual descriptions of the same operator by computing the exact partition function on S 3 of the 3d/1d defect theories capturing V N −1,R and showing that they are the same.
The 3d/1d defect theories describing the resulting Vortex loop V N −1,R can be read from the rules of section 3.2 by moving the D1-branes to the closest NS5 on the left or on the right. The example of the Vortex loop associated to the representation with a specific ordering of the NS5/NS5'-branes along x 6 is worked out in figure 26 b) S-dual brane configuration with D1-branes ending on the right NS5-brane and associated dual 3d/1d defect theory In summary, we obtain the following mirror map of those loop operators in T [SU (N )] theory: We have found that the mirror of a Wilson loop labeled by a representation of the U (N − 1) node is a Vortex loop operator labeled by the same representation. We now turn to the more involved case of a Wilson loop in a representation of another gauge group factor in T [SU (N )].

The Other T [SU (N )] Loops
We proceed now with constructing the mirror map for the remaining loop operators in T [SU (N )]. This includes the Wilson loop operators in the other nodes of T [SU (N )]. As explained in section 3.2 it is easier to describe the mirror map starting from the brane configuration realizing Vortex loops and going to the S-dual picture with F1-strings realizing the mirror Wilson loops. In the brane realization, defining a Wilson loop requires specifying the location of the F1-strings relative to the NS5-branes and a representation R. Likewise, a Vortex loop is specified by the location of the D1branes relative to the D5-branes and by a choice of representation R of the gauge group supported on the D3-branes where the D1-branes end.
In the T [SU (N )] theory we have Vortex loops labeled by a representation R of the U (N − 1) gauge node, which are realized with D1-strings ending on the N − 1 D3-branes supporting that gauge group factor. As just discussed, the relative position of the D1-branes along the total number of boxes in the Young tableau associated to the representation R, which we also denote by |R|. Finding the mirror loop operator requires performing S-duality on the brane configuration (followed by a left-right reflection). The resulting configuration with F1-strings is shown in figure  (28-b). In order to reach the canonical brane configuration of the mirror T [SU (N )] theory, one has to move the D5-branes until they all lie between the two rightmost NS5-branes. However, except for the configuration realizing V N −1,R studied in the previous section, a D5-brane (the leftmost one) has to cross the F1-strings in this process. After this exchange, the number of D3-branes at the bottom of the F1-stack decreases. As explained earlier, when a D5-brane crosses the stack of k F1-strings, q ≤ k of them can be moved smoothly from ending on the D3-branes to ending on the D5-brane. Intuitively this is possible since the D5-brane has a D3-brane spike and the F1-strings can be moved smoothly far from the main brane configuration along this spike. In the final brane picture shown in figure After these preliminaries we can now keep track of what happens to a single-weight pattern for the Vortex loop V M,R associated to a weight w = (w 1 , w 2 , · · · , w N −1 ), where w j D1-strings end on the j-th D3-brane, as we go to the mirror dual brane configuration. Implementing S-duality and moving the D5-branes as explained, we find the situation where the leftmost D5-brane crosses the stack of F1-strings. A total of N − 1 − M D3-branes at the end of the F1-strings disappear in the process of exchanging the branes, and the F1-strings that are ending on them get attached to the D5-brane. Labelling the D3-branes that disappear by j = M + 1, · · · , N − 1, we find that a total of q = w M +1 + w M +2 + · · · + w N −1 F1-strings get attached to the D5-brane. Therefore, the final brane configuration realizes a charge q U (1) flavor Wilson loop together with aŵ = (w 1 , w 2 , · · · , w M ) single-weight contribution to a Wilson loop associated to the remaining F1-strings ending on the M D3-branes supporting the U (M ) gauge node. Enumerating all the final patterns associated to all the weights w of R, we find that they realize the weights of all the representations (q, R), counted with degeneracies, appearing in the decomposition of the representation R of U (N ) under the subgroup U where ∆ M denotes the set of representations (q, R) in the decomposition of R counted with degeneracies. Our mirror map is therefore R , which is realized with k D1-branes ending on the p D3-branes supporting the U (p) gauge node on one end and on a number of NS5 and NS5'-branes on the other, as shown in figure (29-a). Here k = |R| is again the total number of boxes in the Young tableau associated to the representation R of U (p). In order to find the mirror loop we perform once again S-duality on this brane configuration (together with an x 3 reflection). The resulting configuration with F1-strings is shown in figure (29-b). In order to reach the canonical brane configuration of the mirror T [SU (N )] theory, one has to move the D5-branes. In the process p out the N D5-branes have to cross the stack of F1-strings from the right, after which there are no more D3-branes at the bottom of the initial F1-stack. Therefore, all F1-strings got attached to these p D5-branes and have been moved away from the main brane configuration along them. This means that the final brane configurations are characterized by having the k F1-strings ending on the p D5-branes on one side and on the D5 and/or D5' branes on the other side, as in figure (29-c). We interpret this configuration as realizing a non-abelian Wilson loop in the representation R of the U (p) flavor symmetry acting on the p hypermultiplets associated to the p D5-branes, which we denote W fl U (p),R . Therefore, the mirror map is (4.10) Inverting these relations one obtain It is tempting to try to derive these relation directly from the brane picture. This would necessitate new rules constraining the brane moves which are rather ad hoc. It could be interesting to pursue this.
We have found an explicit map relating Vortex loop operators to gauge and flavor Wilson loop operators. We will check these predictions by exact computations of the expectation value of the loop operators on S 3 in section 5.4.

Loops And Mirror Map In Circular Quivers
Our prescription to derive the mirror map of loop operators described in section 3 is used here to find the mirror map for loop operators in 3d N = 4 gauge theories described by circular quivers with equal ranks. The brane realization of this class of quivers is special: after performing S-duality, the branes are already in the canonical brane configuration and no Hanany-Witten moves are required to derive the mirror quiver gauge theory. This immediately implies that a Wilson loop labeled by a representation R of the gauge group directly maps to a Vortex loop labeled by the same representation R, and not to a sum of Vortex loop operators.
For concreteness, let us consider the circular quiver in figure 30-a, which has two U (N ) gauge group nodes U (N ) a × U (N ) b and L ≥ 2 fundamental hypermultiplets of U (N ) b . 50 The mirror dual quiver is easily found by acting with S-duality and is shown in figure 30 R . This is realized by the brane configuration in figure 31-a, where F1-strings emanate from extra D5 and/or D5'-branes far away from the main stack and end on the N D3-branes supporting the U (N ) a gauge node. After S-duality the brane configuration is already in its canonical form to read the mirror quiver theory. The F1-strings become D1-branes lying between the two D5-branes and 50 L ≥ 2 is required for the UV gauge theory to flow to an irreducible SCFT in the IR. can be described by a coupled 3d/1d theory, which when obtained by moving the D1-branes on top of the nearest NS5-brane to their left, is summarized by the mixed 3d/1d quiver diagram in figure 31-b (for m = 2). We leave as an exercise for the interested reader to identify the the 3d/1d quiver theories describing V In summary, we find the following mirror maps (4.13) The mirror map between Vortex loops of the initial quiver in figure 31-a and Wilson loops of the mirror quiver in 31-b can be found in the same way. The two-node quiver has Vortex loops V (4.14) We find once again that some Vortex loops have inequivalent descriptions, predicting hopping dualities for the associated 3d/1d theories.
Other circular quivers with nodes of equal rank and arbitrary number of fundamental hypermultiplets in each node can be treated similarly, leading to explicit maps between loop operators labeled by the same representation R. For circular quivers with varying ranks, one has to rely on the more elaborate analysis presented in section 3.2, which still results in explicit mirror maps between loop operators very similar to those in section 4.3.

Loops In SQCD with 2N Quarks And Its Mirror
For our final example we consider 3d N = 4 SQCD, with U (N ) gauge group and 2N fundamental hypermultiplets, which is the smallest number of fundamental hypermultiplets (N f ≥ 2N c ) required for SQCD to flow in the IR to an irreducible SCFT. The quiver diagram and brane realization of this theory is given in figure 33-a.  We now derive the mirror maps using the brane picture. Consider first the Wilson loop W R of the SQCD theory. It is realized with F1-strings ending on the N D3-branes supporting the U (N ) gauge group, placed among the 2N D5-branes at our convenience. We choose to place the strings with N D5-branes on each side. After S-duality and D5-brane rearrangements, we end up with D1-branes ending on the N D3-branes supporting the central U (N ) gauge node of the mirror theory, placed between the two D5-branes. This is summarized in figure 35. This configuration realizes the vortex loop V 1,R , which is labeled by the same representation R of U (N ) as the initial Wilson loop. The mirror map is then simply (4.17) The only loops we did not discuss thus far are the Vortex loops V where U (n) F is the flavor group acting on the n hypermultiplets associated to the n leftmost D5branes, U (n) F is the flavor group acting on the n hypermultiplets associated to the n rightmost D5-branes and W fl U (n),R denotes a flavor Wilson loop in the representation R of U (n). This completes the derivation of the mirror map between the loops of the U (N ) theory with 2N fundamental hypermultiplets and its mirror dual. The analysis generalizes easily to the U (N ) theory with N f ≥ 2N fundamental hypermultiplets.

Loop Operators In 3d N = 4 Theories On S 3
In this section we provide explicit quantitative evidence for our proposed action of mirror symmetry on loop operators by computing the exact expectation value of supersymmetric loop operators on S 3 in 3d N = 4 gauge theories. We confirm our proposal for the action of mirror symmetry on loop operators by showing that the expectation value of mirror loop operators match perfectly and indeed give alternative UV descriptions of superconformal line defects in the IR SCFT.
Our first task is to define these observables in N = 4 theories on S 3 , where exact computations can be performed by supersymmetric localization. Recall that in R 3 the UV Lagrangian definition of a 3d N = 4 SCFT is invariant under those supercharges in OSp(4|4) that close into the isometries of flat space, which generate the 3d N = 4 super-Poincaré algebra. 51 While an N = 4 SCFT can be placed canonically on the round S 3 by stereographic projection, its UV Lagrangian description likewise breaks OSp(4|4) to a subalgebra, which we now identify. On S 3 , the supersymmetries of a generic UV (massive) 3d N = 4 theory are generated by the supercharges in OSp(4|4) that close into 51 Super-Euclidean in R 3 . the SU (2) l × SU (2) r isometries of S 3 and project out the remaining SO(4, 1) conformal generators on S 3 . Since we allow arbitrary massive deformations, and mass and FI parameters transform in the (3, 1) and (1, 3) representations of SU (2) C × SU (2) H , the R-symmetry in OSp(4|4) is broken down to its Cartan subalgebra on S 3 . Under the SU (2) l × SU (2) r × U (1) C × U (1) H embedding in SO(4, 1)×SU (2) C ×SU (2) H , the supercharges generating OSp(4|4), which transform in the (4, 2, 2), decompose as (2, 1) The supersymmetries preserved by a UV 3d N = 4 theory on S 3 are (2, 1) ++ ⊕ (2, 1) −− ⊕ (1, 2) +− ⊕ (1, 2) −+ . These supercharges generate the following supergroup 52 where R C and R H are the Cartan generators of SU (2) C and SU (2) H respectively. The isometries of S 3 are SU (2) l × SU (2) r . Explicitly, the anticommutators of the SU (2|1) l × SU (2|1) r supercharges on an S 3 of radius L are where J l a and J r a generate SU (2) l and SU (2) r respectively. In the flat space L → ∞ limit, the SU (2|1) l × SU (2|1) r symmetry on the S 3 contracts to the 3d N = 4 super-Poincaré symmetry. Indeed, writing J l a = 1 2 (J a + LP a ) and J r a = 1 2 (J a − LP a ), and taking the flat space limit we get where P a and J a are the translation and rotation generators of R 3 . Comparing with the 3d N = 4 super-Poincaré supercharges Q αAA in (2.1) we find the identification: The SU (2|1) l supersymmetry transformations are generated by Killing spinors l and¯ l on S 3 obeying [17] 53 while SU (2|1) r transformations are generated by Killing spinors r and¯ r on S 3 which satisfy [39] From the viewpoint of off-shell supersymmetric supergravity backgrounds [40], these Killing spinor equations arise from 3d N = 4 supergravity in the presence of a background auxiliary field. 54 The superisometry of this supergravity background is SU (2|1) l × SU (2|1) r .
We take the left-invariant frame e a = Lµ a , a = 1, 2, 3, where the left-invariant one-forms of SU (2) are defined by g −1 dg = iµ a τ a with g = z 1 z 2 −z 2z1 so that det g = 1. In this frame l and¯ l are constant while r and¯ r are given by r = g −1 0 ,¯ r = g −1 0 , with 0 , 0 constant. 55 The explicit 3d N = 4 supersymmetric Lagrangians on S 3 can be written down using the formulae for 3d N = 2 gauge theories on S 3 in [17,39,[42][43][44], which are invariant under SU (2|1) l × SU (2) r . We first decompose the N = 4 vector multiplet and hypermultiplet into N = 2 vector and chiral multiplets. By assigning U (1) l charge 1 and 1/2 to the chiral multiplets inside an N = 4 vector multiplet and hypermultiplet respectively, the supersymmetry of such an N = 2 theory on S 3 is enhanced with extra four supercharges [39], which generate the remaining SU (2|1) r . Just as in flat space, 3d N = 4 theories on S 3 admit canonical relevant deformations associated to the G C × G H symmetries of the IR SCFT. These are introduced by turning on SU (2|1) l × SU (2|1) r invariant background vector and twisted vector multiplets on S 3 for G H and G C respectively. Supersymmetric backgrounds for 3d N = 4 vector and twisted vector multiples on S 3 allow a single scalar in the multiplet to be turned on, instead of the triplet of parameters in flat space. Therefore on S 3 there is a single mass parameter and a single FI for each Cartan generator in G H and G C respectively.

Wilson Loops On
By a choice of parametrization of the loop γ ∈ S 3 we can always make |ẋ| = 1. Since in the SU (2) l left-invariant frame the Killing spinors l and¯ l are constant, imposing that the Wilson loop preserves half of the supercharges in SU (2|1) l requires thaṫ x µ = n a e µ a , (5.11) where n a is a constant unit three-vector n a n a = 1. A Wilson loop wrapping the curveẋ µ = n a e µ a on S 3 preserves the following half of the SU (2|1) l supersymmetries γ a n a l = l γ a n a¯ l = −¯ l . Without loss of generality we take n a = (0, 0, 1). The Hopf circle preserves J l 3 generating U (1) l ⊂ SU (2) l and the SU (2|1) l supersymetries The Hopf circle, which is labeled by a point on S 2 , also preserves U (1) r ⊂ SU (2) r . A Wilson loop supported on this Hopf circle preserves the following supersymmetries in SU (2|1) r γ 3 r = r γ 3¯ r = −¯ r , (5.14) where r and¯ r are non-constant spinors in the left-invariant frame. We can without loss of generality take the loop at the North or South pole of S 2 , in which case U (1) r is generated by J r 3 . For another Hopf circle at a different point in S 2 , which can be obtained by the action of an isometry, the preserved U (1) r and supersymmetry generators are obtained from those at the poles by conjugating by the action of the isometry.
In summary, the Wilson loop (5.8) wrapping the Hopf circle at the North pole of S 2 is halfsupersymmetric in a 3d N = 4 theory on S 3 . It preserves an SU (1|1) l × SU (1|1) r inside the SU (2|1) l × SU (2|1) r supersymmetry of a 3d N = 4 theory on S 3 . The explicit supersymmetry algebra of the Wilson loop is

Vortex Loops On S 3
Another family of half-supersymmetric line operators can be defined in 3d N = 4 gauge theories on S 3 . These loop operators preserve a different SU (1|1) l × SU (1|1) r inside the SU (2|1) l × SU (2|1) r supersymmetry on S 3 . They correspond to Vortex loop operators. A Vortex loop operator supported on a Hopf circle at the North or South pole of S 3 preserves the following supersymmetries 57 Indeed the supersymmetry algebra preserved by a Vortex loop is different to the one preserved by a Wilson loop (cf. (5.13)(5.14)). We record in table 5 the quantum numbers of the preserved supercharges. We note that J l 3 − 1 2 (R C + R H ) and J r 3 + 1 2 (R C − R H ) commute with all four supercharges, a fact will make good use of shortly. The explicit SU (1|1) l × SU (1|1) r anticommutators of the preserved supercharges by a Vortex loop operator are where the sphere generators J l 3 = 1 2 (J 12 +LH) and J r 3 = 1 2 (J 12 −LH) become the translation generator H and transverse rotation generator J 12 in flat space.
Coupling a 1d N = 4 SQM V gauge theory on the loop to a 3d N = 4 theory on S 3 requires turning on background fields in the 1d N = 4 theory in order to make the 3d/1d theory invariant under the SU (1|1) l × SU (1|1) r supersymmetry algebra of the Vortex loop. This is not surprising. Placing a 3d N = 4 gauge theory on S 3 deforms the flat space supersymmetry transformations and action. Both modifications can be interpreted as arising from background 3d N = 4 supergravity fields [40]. We now show that the 1d N = 4 supersymmetry transformations and action can be deformed in such a way as to yield a supersymmetric Vortex loop operator on S 3 .
The starting point is the SQM V supersymmetry algebra discussed in section 2 In order to couple the 1d N = 4 theory on S 3 supersymmetrically we must deform the 1d N = 4 theory in such a way that the supersymmetry algebra becomes the SU (1|1) l × SU (1|1) r algebra of the Vortex loop on S 3 This can be accomplished in two steps: • Turning on a fixed background gauge field for the R-symmetry J − ≡ J l 3 + J r 3 − R C • Turning on a fixed imaginary mass parameter for the flavour symmetry G F ≡ J l 3 + J r 3 − R H G F does indeed commute with all four supercharges, as it is the sum of the bosonic generators in (5.19) and (5.20), both of which are central elements. The charge J − commutes with q l andq l but not with q r andq r . We also use in an important way that the 1d N = 4 theory lives on an S 1 and non-trivial Wilson lines can be turned on around the circle. Deforming the 1d N = 4 theory by these backgrounds allows to couple a 1d N = 4 theory on a Hopf circle to a 3d N = 4 theory on S 3 while preserving SU (1|1) l × SU (1|1) r .
The SQM V supersymmetry algebra (5.23) gets deformed in the presence of a background 1d N = 4 vector multiplet for a flavour symmetry G. We consider a background gauge field a on S 1 and a background scalar field in the vector multiplet corresponding to a real mass m. 58 The deformed SQM V algebra becomes 59 H is the generator of translations on the circle, which we take to have length β = 2πL. 58 Turning on a complex mass associated to the other two scalars in the vector multiplet deforms other commutators. 59 This can be easily understood by dimensional reduction of the 4d N = 1 supersymmetry algebra.
Turning on a background gauge field for the R-symmetry J − generically breaks the supersymmetries associated to the supercharged Q − andQ − charged under J − . However for quantized values of the gauge field a − = n L , n ∈ Z, supersymmetry remains unbroken with Q − andQ − generated by the non-constant Killing spinors − = e i nτ L and¯ − = e −i nτ L , with τ ∈ [0, 2πL] parametrizing S 1 . 60 For these quantized holonomies, the supersymmetry algebra is deformed in the same way as if it were a flavor symmetry. In conclusion, turning on the following background fields for J − and G F the SQM V algebra (5.23) becomes the SU (1|1) l × SU (1|1) r supersymmetry preserved by a line defect in a 3d N = 4 theory on S 3 . We can identify the generators as follows: 61 Consistently with the deformation by the R-symmetry gauge field a − = − 1 L , we find that the Killing spinors generating q r andq r behave at the position of the loop as¯ − = e iτ L and − = e − iτ L , respectively. Note that these position dependent Killing spinors are nevertheless periodic along the Hopf circle.
The background fields (5.27) ensure that the 1d N = 4 theory living on a maximal circle of S 3 can be coupled supersymmetrically to the bulk 3d N = 4 theory. This can also be understood by looking at the 3d N = 4 supersymmetry transformations of the vector multiplet and hypermultiplet on S 3 generated by the four Killing spinors preserving the Vortex loop. These transformations decompose into SQM V supersymmetry transformations deformed by the advertised background. To couple the bulk theory to SQM V , preserving the four supercharges, it is then required to turn on this background in the SQM V supersymmetry transformations and Lagrangian. The derivation of how the supersymmetry transformations on S 3 decompose is presented in appendix A.
The presence of these background fields can be recast, upon field redefinition, into twisted boundary conditions for the fields in the 1d N = 4 theory around S 1 , thus affecting the partition function (or supersymmetric index) of the 1d N = 4 theory. This will play a crucial role in the evaluation of the expectation value of Vortex loop operators in section 5.3.
Having defined the half-supersymmetric Wilson and Vortex loop operators on S 3 , we now propose to evaluate their expectation values exactly, using the results of supersymmetric localization and to test the mirror symmetry predictions of section 4.
60 This can be understood as follows. The constant gauge field a − can be absorbed into a redefinition of the fields φ = e −ir−a−τ φ, with r − the J − -charge of φ. The non-standard periodicity of the fields around S 1 are then preserved by the supersymmetry transformations generated with − = e ia−τ and¯ − = e −ia−τ . These spinors are globally defined only for a − = n L , with n ∈ Z. 61 In the flat space limit, the R-symmetry generators J ± are identified with ∓J 12 − R C . They are related to the generators J ± presented in section 2 by a shift by the flavor symmetry generator J 12 − R H , which is the flat space limit of G F .

Exact Partition Function Of 3d/1d Theories On S 3
In this section we identify the matrix integral representation of the exact expectation value of a half-supersymmetric Vortex (and Wilson) loop operator in an N = 4 theory on S 3 , which takes into account the coupling of 1d N = 4 SQM on S 1 to the N = 4 theory on S 3 described in the previous section. This matrix model model is obtained by combining in an interesting way the matrix integral computing the S 3 partition function of N = 4 theory on S 3 [17] and the matrix model representation of the supersymmetric index of 1d N = 4 SQM found in [18,19].

S 3 Partition Function And Wilson loops
A powerful probe of the dynamics of a strongly coupled 3d N = 4 IR SCFT emerging at the endpoint of an RG flow from a UV 3d N = 4 supersymmetric gauge theory is the partition function of the gauge theory on S 3 . The S 3 partition function is a renormalization group invariant observable, and the computation performed in the UV exactly captures the partition function of the IR SCFT.
The S 3 partition function of 3d N = 4 gauge theories can be localized to a finite dimensional matrix integral [17]. The matrix integral is defined by integrating over the Cartan subalgebra of the gauge group the product of the classical and one-loop contributions in the (exact) saddle point analysis [17] where W is the Weyl group and C is the contour of integration. The classical contribution depends on the FI parameter η for each abelian gauge group factor Z class = e 2πiηTrσ . (5.31) The 3d N = 4 vector multiplet contribution is while a hypermultiplet in a representation R of the gauge group with mass m yields , (5.33) where α are the roots of the Lie algebra and w the weights for the representation. Throughout we use the following short-hand notation (borrowed from [15]) By combining these building blocks the exact S 3 partition function of an arbitrary 3d N = 4 gauge theory acquires an elegant matrix model representation.
The contour of integration C of the matrix model that arises from the localization computation is the real axis. The condition that the matrix integral over the real axis is convergent [14] is precisely the same as the criterion for the quiver theory to be "good" or "ugly" [29], that is that for each U (N ) gauge group factor the number of fundamental hypermultiplets N f obeys N f ≥ 2N − 1. In this paper we have considered gauge theories that flow in the IR to an irreducible SCFT, so the matrix models (in the absence of Wilson loops) are convergent. We note that the matrix integrals defined over the real contour capturing the expectation value of Wilson loops can become divergent when the charge of the Wilson loop is sufficiently large. For these superficially divergent matrix models we regulate the divergent integrals by deforming the contour of integration C of the matrix model away from the real axis. The results that we find using the deformed contours are completely consistent with our mirror symmetry predictions. It may be possible to justify these deformed contours of integration for the divergent matrix integrals by carefully performing the localization computation.
More precisely, for positive FI parameter η > 0, we deform the contour of integration of each eigenvalue so that it encloses the poles of the matrix model integrand in the upper half-plane, as in figure 37. This is then the same as closing the real line with a semi-circle going through +i∞. Conversely if η < 0 we close the contour so that it encloses the poles in the lower half-plane. These choices of contour lead to finite results for arbitrary large values of the Wilson loop charge, as long as η = 0. For each matrix eigenvalue the Wilson loop charge q combines with the FI parameter η in a complex parameter η + iq and the evaluation of the integral with our prescription coincides with the analytic continuation to the complex value of the FI parameter.

1d N = 4 Supersymmetric Quantum Mechanics Partition Function
In this section we introduce and evaluate the partition function of the 1d N = 4 SQM quiver gauge theories on a circle relevant for the description of Vortex loop operators in 3d N = 4 gauge theories. The 1d N = 4 SQM partition function admits a Hilbert space interpretation as an index with fugacities (or chemical potentials). It is defined as a trace over the Hilbert space of the theory [45] I(z, µ) = Tr H (−1) F e 2πizJ − e 2πiµΠ , (5.37) where F is the fermion number, z is a chemical potential for the U (1) − R-symmetry with generator J − and µ = {µ j } is a collection of chemical potentials for flavor symmetries with Cartan generators Π j . In the path integral representation, the fugacities are associated to twisted boundary conditions for the fields, which can be undone by turning on background vector multiplets for the relevant symmetries. z is associated to the presence of a U (1) − R-symmetry gauge field, with the relation z = −La − , where 2πL is the length of S 1 . A chemical potential µ j is associated to the presence of a U (1) j flavor symmetry gauge field a j and a real mass parameter m j , through the relation µ j = −L(a j + im j ).
As explained in section 5.2, coupling the 1d N = 4 theory on a maximal S 1 ⊂ S 3 to a 3d N = 4 theory requires turning on a constant R-symmetry gauge field corresponding to z = 1 and a mass deformation corresponding to a chemical potential µ = 1 for the flavor symmetry generated by Of course, the fugacities for all other flavour symmetries are arbitrary. The 1d N = 4 SQM theories are deformed by FI terms with parameters ζ ≡ ζ, taking real values in the center of the gauge algebra. The partition function or index is well-defined when all FI parameters ζ a are non-zero. When the FI parameters vanish important subtleties arise as the theory may develop a continuous spectrum.
Using supersymmetric localization, the exact index of a 1d N = 4 SQM theory is given by the explicit formula [18,19] where k is the rank of the gauge group G, |W | is the order of the Weyl group W of G and JK-Res denotes the sum over Jeffrey-Kirwan residues [46] of a meromorphic k-form with complex variables u = {u I }. The factors g vec , g chi are the localization one-loop determinants associated to 1d N = 4 vector and chiral multiplets, , (5.40) where α runs over the roots of the gauge algebra, w runs over the weights of the chiral multiplet representation, R is the chiral multiplet R-charge under 2J − and q = {q j } are the chiral multiplet charges under the flavor symmetry generators Π = {Π j }.
We now explain briefly how to compute the Jeffrey-Kirwan residues of the meromorphic k-form with simple poles and with FI parameters ζ. A given set pole u * = {u * I } arises at the intersection of k hyperplanes in C k , defined by the equations w (I) ·u * +q (I) ·µ+ R (I) 2 z = 0, 62 with I = 1, · · · , k, appearing in the denominator of the chiral multiplet one-loop determinant. In principle we should also consider poles in the vector multiplet one-loop determinant but it happens that one cannot find a collection of k hyperplanes intersecting at a point u * if one of the hyperplanes is described by α · u − z = 0. 63 Thus each pole u * is associated with a set of k weights {w (I) }, each weight appearing in some chiral multiplet factor. A set of weights {w (I) } defines a cone C(w (I) ) = { k I c I w (I) | c I > 0} ⊂ R k . The Jeffrey-Kirwan residue at u * is then given by where Res[u * ]g(z, µ, u) denotes the usual residue at the pole u = u * and ζ is understood as a kcomponent vector.
The total index (5.38) is obtained by summing over all JK-residues at poles u * We will give here the result of the evaluation of the index for the class of 1d N = 4 SQM quivers entering our construction of Vortex loop operators mirror to Wilson loops, relegating part of the details of the residue computations to appendix B.
The class of 1d N = 4 SQM quiver gauge theories we consider is described in figure 2. They are linear quivers connected by pairs of bifundamental chiral multiplets, with N L fundamental and N R anti-fundamental chiral multiplets in the terminating node, which we mean to be the node on the right of the quiver in 2 (this is the one that is closest to the 3d quiver in the combined 3d/1d quiver). The gauge group is then G = P p=1 U (n p ), where U (n P ) denotes the terminating node. Moreover each node has either zero or one adjoint chiral multiplet and we denote c adj p ∈ {0, 1} the number of adjoint chiral multiplets in the U (n p ) node. As explained in section 3.2, the 1d FI parameters of all nodes in the quiver are taken negative if the 1d N = 4 theory is read from the brane configuration with D1-strings moved to the closest NS5-brane on the right. Conversely they are all taken positive if the 1d N = 4 theory is read from the brane configuration with D1-strings moved to the closest NS5-brane on the left. 64 Superpotentials: The 1d N = 4 SQM theories have superpotentials constraining the R-charges of the chiral multiplets: cubic superpotential couplings between adjoint and bifundamental chiral multiplets and quartic superpotential couplings between bifundamental chiral multiplets for the nodes without adjoint chiral multiplet. The N fundamental and M anti-fundamental chiral multiplets in the terminating U (N P ) node do not enter into such cubic and quartic superpotentials. Instead they are coupled to a 3d hypermultiplet through a cubic superpotential as described in section 3.2. This 3d/1d coupling is responsible for the identification of bulk and defect flavor symmetries.
A cubic superpotential imposes the constraint R adj + R bif1 + R bif2 = 2 on the R-charges of the fields with respect to the generator 2J − . A quartic superpotential imposes the constraint R bif1 + R bif2 + R bif3 + R bif4 = 2. The cubic superpotential which couples the fundamental and anti-fundamental chirals to the 3d hypermultiplet imposes the constraints R fund + R a−fund = 2, since the bulk hypermultiplet is not charged under J − = J 3 l + J 3 r − R C . In order to perform the computation of the Jeffrey-Kirwan residues, we only impose that the R-charges obey these superpotential constraints, but otherwise leave them arbitrary, to avoid having to deal with higher order poles. It turns out that the final results depend only on the R-charges of the adjoint chiral multiplets, which we need to specify at the end of the computation.
The R-charges for the adjoint chiral multiplets can be read from the flat space brane realization of the quiver (see section 3.2). The complex scalar degrees of freedom of an adjoint chiral living on D1-branes stretched between two NS5-branes correspond to displacements of the D1-branes along the plane x 1 − x 2 . The scalar transforms as a vector under SO(2) 12 and is uncharged under J 78 . The adjoint chiral then has charge 2 under 2(J 12 − J 78 ) which is identified with the 1d N = 4 SQM generator 2J − in the flat space limit. On the other hand the complex scalar of adjoint chiral living on D1-branes stretched between two NS5'-branes are associated with displacements of the D1-branes along the plane x 4 − x 5 . It transforms as a vector under SO(2) 45 and is uncharged under J 12 and J 78 . The adjoint chiral is then uncharged under 2J − .
The R-charges (under 2J − ) of the adjoint chiral multiplets are then fixed to be either 2 or 0, depending on whether they arise from a brane construction with NS5 or NS5' branes. In the 1d 64 Less restrictive conditions could be imposed on the signs of the FI parameters in non-terminating nodes. For instance in a two-node quiver with FI parameter ζ 2 > 0 in the terminating node, we could allow for a FI parameter ζ 1 > −ζ 2 , with ζ 1 = 0, in the other node. This constraint follows from a careful analysis of the positions of the various NS5-branes in the brane realization of the loop operator. We checked with a few explicit computations that the index does not depend on ζ 1 taking values in this range. At ζ 1 = 0, corresponding to aligned NS5-branes, we do not know how to evaluate the 1d partition function, but we expect a dramatic change in the result (see discussion around 3.1). To simplify computations we require the FI parameters of all the nodes in a quiver to have the same sign (ζ 1 , ζ 2 > 0 in the example). N = 4 quiver gauge theory description, this prescription is re-expressed as follows. Define the integers c p = P − p − P q=p c adj q . An adjoint chiral multiplet in the U (n p ) node has R-charge R adj, p given by Similarly one can derive the charge of the adjoint chiral multiplets under the flavor symmetry G F , which is identified in the flat space limit with the 10d rotation generator J 12 − J 45 . The charges of the multiplets under the symmetries relevant for the computation of the index and the constraints they obey are summarized in the following table.
fund. chiral The last entry in the table, q + + q − = 1/2, follows from the superpotential constraint coupling bulk and defect fields and the fact that the 3d hypermultiplet entering in this superpotential has charge −1/2 under G F = J 3 l + J 3 r − R H . As explained in section 3.2 , each 1d N = 4 quiver gauge theory obtained by moving the D1branes to the closest NS5-brane on the right is associated to a tensor product representation R of U (N ), with N the number of fundamental chiral multiplets, which depends on the ranks of the nodes n p and on the distribution of adjoint chiral multiplets among the nodes. The representation of U (N ) associated to the quiver is R = ⊗ P p=1 R kp with k p = n p − n p−1 (with n 0 = 0) and R kp is given by S kp and A kp denote the k p -symmetric representation and k p -antisymmetric representation respectively.
Once the representation R is identified we can perform the evaluation of the 1d N = 4 SQM index. As we will explain shortly, we first compute the index at arbitrary values of z and then we take the analytic continuation to z → 1 in the combined 3d/1d partition function. 65 This turns out to be the correct prescription to compute the final exact partition function of the 3d/1d theory describing a Vortex loop operator. The G F chemical potential µ can be readily set to µ = 1.
The details of the computations are given in appendix B, for single node and two nodes quivers. We use the explicit results of our computations to conjecture the index formula for the general class 65 Poles of the 3d integrand cross the integration contour as z → 1, which is why we analytically continue to z = 1.
of quivers relevant in this paper. Our computations in explicit examples gives us confidence in our formulas (5.45)(5.47) below and ultimately result in the matching between mirror and Vortex loops extended to an arbitrary number of nodes.
Let us denote I r and I l the partition functions, or supersymmetric indices, of the 1d N = 4 gauge theory realized by moving the D1-strings on the closest NS5-brane on the right and on the left respectively. For the 'right' SQM V theory, which has N fundamentals with mass −σ j and M anti-fundamentals of mass m a in the terminating node, we find that the index I r at arbitrary z is given by a sum of contributions labeled by the weights of the representation R of U (N ): , (5.45) where F(σ i , z) is a factor that will not play any role in our final computation of the expectation value of Vortex loop operators. Let us just mention here that it is a product of the form where α, κ take real values. In the analytic continuation to z = 1, this factor will not affect the evaluation of the coupled 3d/1d matrix model.
For the 1d N = 4 quiver theory obtained by moving the D1-branes to the closest NS5-brane on the left the computation is similar, except for the fact that the FI parameters of the nodes are all positive. This affects very significantly the index. The answer is now written as a sum over weights of a representation R of U (N ), where now N is the number of anti-fundamental chiral multiplets in the terminating node and R is given by the same tensor product of symmetric and anti-symmetric representation described above. Denoting by − m a the mass of the M fundamental chiral multiplets and σ j the mass of the N anti-fundamental chiral multiplets, the index is then given by , (5.47) where F(σ i , z) is again a factor that trivializes when we consider the full 3d/1d matrix model in the analytic continuation to z = 1.
The formulae (5.45) and (5.47) are compatible with the expectations from 1d Seiberg duality. This can be understood as follows. The computation shows that the partition function depends only on the representation R and on the masses of the fundamental and anti-fundamental chiral multiplets in the terminating node. Multiple 1d quivers (except for single node quivers) lead to the same tensor product representation R and hence have equal partition functions, namely the quivers read from all possible orderings of the NS5 and NS5'-branes in the x 6 direction. These reorderings affect the rank of the gauge nodes and the distribution of adjoint chiral multiplets among them but not the associated representation R. These Seiberg-dual quivers are expected to realize the same Vortex loop in the infrared limit and we find indeed that their partition functions match (at least when z → 1).

Matrix Model Computing Vortex Loops
Our aim is to compute the exact expectation value of Vortex loops in 3d N = 4 gauge theories on S 3 . We now explain how this quantity is captured by a matrix model that combines in an interesting way the exact S 3 partition of 3d N = 4 theories and the exact index of the 1d N = 4 theories defining Vortex loops, both of which we have already introduced.
A Vortex loop on S 3 is defined by a supersymmetric coupling of a 3d N = 4 gauge theory on S 3 to a 1d N = 4 gauge theory on an S 1 ⊂ S 3 . Recall that in order for the 3d/1d defect theory to be supersymmetric, invariant under SU (1|1) l × SU (1|1) r , we had to turn on very specific background fields in the SQM. We can compute the exact partition function of the combined 3d/1d theory by choosing the supercharge q l in SU (1|1) l to localize the full functional integral. In order to localize the combined 3d/1d partition function, we must add suitable deformation terms for both the 3d N = 4 theory and the 1d N = 4 theory. Importantly, the saddle points for the 3d and 1d fields are not modified. The way the combined 3d/1d partition function encodes the specific couplings between the 3d and 1d theories is by taking into account that the flavour symmetries of the 1d N = 4 theory are gauged with 3d N = 4 vector multiplets. At the level of the combined matrix model this means that the 1d mass parameters that are gauged are replaced by the scalar in the corresponding 3d N = 4 vector multiplet. This scalar can be either dynamical, in which case it corresponds to an eigenvalue in the 3d matrix integral that needs to be integrated over, or it is identified with a mass parameter in the 3d theory. Therefore, the combined 3d/1d partition function is found by convolving the "gauged" 1d partition function with the 3d partition function.
The expectation value of a Vortex loop V wrapping a maximal circle of S 3 is the partition function of the 3d N = 4 theory coupled to the 1d N = 4 SQM quiver, normalized by the partition function of the 3d theory. It takes the form ] is the matrix model integrand of a 3d N = 4 theory with eigenvalues σ, hypermultiplet masses m and FI parameters ξ, while I[σ, m, z] is the index of the 1d N = 4 SQM V defect theory that realizes the Vortex loop. We note that the flavour fugacities of the 1d theory are identified either with 3d matrix eigenvalues σ or with 3d mass parameters m, depending on whether the corresponding 1d flavour symmetry is gauged by dynamical or background 3d vector multiplets.
The factor W fl takes the form of an abelian background Wilson loop that we add to the matrix model. It follows from the analysis of the brane realization of the Vortex loop and appears as a necessary ingredient to check successfully mirror maps with Wilson loops. In the brane picture each five-brane is associated to a deformation parameter of the theory (see section 3), masses m a for D5-branes and "FI" parameters ξ a for NS5-branes. When q F1-strings end on a D5 with parameter m, they realize a Wilson loop of charge q for the flavor symmetry associated to the D5-brane, and the corresponding matrix model factor is e 2πqm . Similarly when q D1-branes end on an NS5-brane with parameter ξ, they realize a Wilson loop of charge q for the global symmetry associated to the NS5-brane, and the corresponding matrix model factor is e 2πqξ . Thus when we read the 3d/1d quiver for a Vortex loop by moving the D1-branes on top of the closest NS5-brane to the left, resp. to the right, we propose to add to the matrix model the background Wilson loop factor left: W fl = e 2π|R| ξ L , right: W fl = e 2π|R| ξ R , (5.49) where ξ L , resp. ξ R , is the parameter associated to the NS5-brane on the left, resp. on the right, and |R| is the total number of boxes in the Young tableau of the representation R labeling the Vortex loop.
|R| coincides with the total number of D1-branes ending on the NS5-brane. It should be noticed that this Wilson loop factor cannot be associated to a global symmetry acting on the fields of the theory, since the parameter ξ L/R is not a combination of the true FI parameters ξ a −ξ a+1 . With this addition, the matrix model computing the Vortex loop depends on all ξ a parameters. Similarly the matrix model computing the Wilson loop depends on all the masses m a of fundamental hypermultiplets, although only the parameters m a − m a+1 are associated to actual flavors symmetries. 66 In the presence of a loop operator, the theory seems to admit one extra deformation parameter, mass or FI parameter, associated to a "hidden" symmetry, which act on the fields trivially.
M,R coupled to a U (N ) node of the 3d N = 4 quiver theory is labeled by a representation R of U (N ) and by a splitting K = M +(K −M ) of the K fundamental hypermultiplets of this U (N ) gauge node. Let us consider the Vortex loop realization furnished by coupling the 3d N = 4 theory to the 1d N = 4 SQM quiver obtained by moving the D1-branes to the NS5-brane on the right, illustrated in figure 5. In this case the SQM is labeled by a certain representation R of U (N 1 ) and a splitting M = M 2 , K −M = M 1 . In the terminating node, the SQM has N 1 fundamental chiral multiplets with masses −σ j , j = 1, · · · , N 1 and M 2 + N 2 anti-fundamental chiral multiplets with masses m a , a = 1, · · · , M 2 and σ k , k = 1, · · · , N 2 . The couplings to the bulk theory identify σ j with the eigenvalues of the 3d U (N 1 ) gauge node, σ k with the eigenvalues of the 3d U (N 2 ) gauge node and m a with the real masses of the M 2 fundamental bulk hypermultiplets. The identification of bulk and defect parameters works similarly for the SQM obtained by moving the D1-branes to the left NS5-brane.
We must also explain what we mean by lim z→1 . The evaluation of (5.48) at imaginary values of z defines a function that admits an analytic continuation to the complex plane. We can thus evaluate the integral for z ∈ iR and consider the analytic continuation to z = 1. 67 This is how we extract our final result. This recipe is motivated by the observation that plugging z = 1 directly in the matrix model integrand would trivialize the index (reduce it to be just one) and the coupling to the SQM quiver would not affect at all the total partition function. Instead, the analytic continuation that we propose takes into account the matrix model contributions from residues of poles crossing the σ i -integration contours, as we vary z continuously from zero to one. The fact that these residues should be included should follow from a more detailed analysis of the localization computation.
We can now explain our claim that the factor F(σ i , z) appearing in the evaluation of the SQM index (5.45) does not contribute to V (N ) M,R . We noticed that F(σ i , z) is a product of terms with the generic form (5.46). In taking the analytic continuation z → 1 these factors introduce residues from poles crossing the integration contours proportional to sin(±πz). In the limit z = 1 these extra residues all vanish, implying that the limit z = 1 can be taken directly in the integrand of (5.48) for this factor F(σ i , z). Noting that F(σ i , z = 1) = ±1, we can simply drop this factor from our computations. Our results will be valid only up to an overall sign.
The same argument would not work for the other factors in (5.45) because the numerators ch(m a − σ j ) are exactly canceled by inverse factors from the 3d hypermultiplets contributions! This means that the residues of the poles crossing the integration contour as z → 1 due to these factors are non-vanishing. The discussion is the same for the left index (5.47).

Loops in T [SU (N )]
In this section we perform the explicit computation of the exact expectation value of Wilson and Vortex loop operators in the T [SU (N )] theory. We show precise agreement with all of our branebased predictions in section 4.3.
We will use the notations  [15,47] and is given by where S N is the group of permutations of N elements and (−1) τ is the signature of the permutation τ . T [SU (N )] has the property of being self-mirror. This is realized by virtue of the partition function (5.51) being invariant under the exchange m j ↔ ξ j , except for the phase e −2iπξ N j m j , which is unphysical, in the sense that it can be removed by a local counterterm constructed from the background fields (namely mixed background Chern-Simons terms) defining the UV partition function [48,49].
For the explicit mirror symmetry maps, it will prove useful to define the mirror symmetric quantity which can be understood as the partition function expressed in another renormalization scheme and which is manifestly invariant under the exchange of mass and FI parameters m j ↔ ξ j .
All our results are nicely expressed in terms of two shift operators S q and S q acting on the partition function Z ≡ Z T [SU (N )] of the theory. These operators act by shifting respectively the FI parameters and the masses by imaginary terms, where q = (q 1 , · · · , q N ) is a N -component vector.

T[SU(2)] loops
Let us start our analysis with the abelian theory T [SU (2)]. The partition function is given by the matrix model One can evaluate this integral by closing the contour of integration by a semi-circle going through i∞ (or −i∞) and summing over the residues inside the contour. This yields which agrees with (5.51) for N = 2.
The matrix model computing the vev of a Wilson loop of charge q ∈ Z is given by where we evaluated the integral at non-zero q by the analytical continuation ξ 1 → ξ 1 − iq, following from our choice of deformed contour integral (see discussion after (5.36)). 69 Let us turn now to the computation of Vortex loops. The T [SU (2)] theory has two fundamental hypermultiplets, which give rise to three possible splitting 2 = i + (2 − i), i = 0, 1, 2, defining three possible Vortex loops V i,q . Using the result for the Vortex loop factor (5.45), computed from its right SQM quiver realization (and ignoring the factor F for the reasons mentioned before), inserted in (5.48) and with extra Wilson line (5.49), the matrix models computing the Vortex loop vevs are 69 As explained before, divergent Wilson loops are computed by deforming the contour of integration.
found to be 70 The mirror symmetry prediction (4.1) is recovered by our exact computations, namely The explicit results also show that the other Vortex loops V 0,q , V 2,q are mapped to the flavor Wilson loops of the mirror theory, as predicted in section 4.3.

Wilson loops in T[SU(N)]
We turn now to the non-abelian theories by considering Wilson  We compute now the matrix model W U (p) (w) associated to a single weight w = (w 1 , w 2 , ..., w p ). To simplify expressions we omit the factors depending on (and the integrals over) the eigenvalues σ (a) j with a > p in the matrix model because they do not affect the computation. With this simplification the matrix model is given by .

(5.65)
For each term in the sum over S p the factor e 2π p j w τ (j) σ j can be reabsorbed as a shift of the parameters ξ j by −iw τ (j) , for j = 1, ..., p, at the cost of an extra factor (−1) (p−1)(w 1 +w 2 +...+wp) . The matrix model without these imaginary shifts is exactly Z (this can be seen by noticing that Z is the matrix model obtained when w = 0). The evaluation of the integrals (with deformed integration contours as explained after (5.36)) then coincides with an analytical continuation of Z to complex FI parameters. We obtain with w tot = p j=1 w j and w τ = (w τ (1) , w τ (2) , · · · , w τ (p) , 0, · · · , 0 N −p ). We need to sum over these single weight contributions to get the final result (5.62). Recognizing S p as the Weyl group W of U (p), we can simplify the result using the property This leads to our final result where we also used the property that for any weight w, p j w j = |R|, the number of boxes in the Young tableau of R, and we replaced trivially Z by Z defined in (5.52) .

Vortex loops in T[SU(N)]
We now evaluate the matrix models computing the exact expectation value of Vortex loops in T [SU (N )]. The Vortex loops are labeled by a representation R of a U (p) node and, for the U (N − 1) node, by a splitting N = M + (N − M ) of the fundamental hypermultiplets (see section 4.3). We denote by V (p) R , 1 ≤ p ≤ N − 2, the U (p) Vortex loops and by V M,R the U (N − 1) Vortex loops. As for the Wilson loops, the matrix model computing a Vortex loop V R , which is labeled (in particular) by a representation R, decomposes into a sum of contributions labeled by the weights w of R This decomposition in a sum over weights follows from the evaluation of the SQM index (5.45), (5.47). We will be computing the matrix model V (w) associated to a single weight w = (w 1 , w 2 , ..., w p ) for each Vortex loop.
Let us first discuss Vortex loops in the U (p) nodes with 1 ≤ p ≤ N − 2. The prediction from the brane picture of section 4.3 is that such a Vortex loop must evaluate to a background Wilson loop for a global symmetry, depending on the parameters ξ j , 1 ≤ j ≤ p. In the special case of the T [SU (N )] theory, the brane picture predicts that the Vortex loop V (p) R will evaluate to a background Wilson loop transforming in a representation R of the topological symmetry subgroup U (p) ⊂ U (N ) J = G C acting on the Coulomb branch, associated to the deformation parameters ξ j , 1 ≤ j ≤ p.
To compute the vev of this Vortex loop, we choose to consider its left-SQM quiver realization. The matrix model is given by (5.48) with the defect contribution (5.47) and additional background loop (5.49). We simplify the matrix model by replacing the factors depending on the eigenvalues σ (a) j with a ≥ p, which do not affect the computation, by [· · · ]. This gives for the contribution of a single weight w = (w 1 , w 2 , · · · , w p ) to the Vortex loop where w tot = p k=1 w k = |R|. We remind that lim z→1 means that we compute the matrix model for z ∈ iR and analytically continue the result to z = 1. For z ∈ iR, we recognize in the integrand the matrix model computing the partition function of T [SU (p)] with FI parameters ξ (p) = (ξ 1 , ξ 2 , · · · , ξ p ) and shifted mass parameters σ (p) − iwz = (σ (p) p − iw p z). We can use (5.51) to evaluate it and easily perform the analytic continuation to z = 1 71 .

(5.71)
Note that the background factor e 2πwtot ξp canceled in the limit z = 1. One can now pull out the sum over permutations τ ∈ S p and play with eigenvalues relabelings σ (knowing that the hidden factors in [· · · ] are invariant under such relabelings). After several operations we can pull out of the integral the factors depending on the weight w, The full Vortex loop vev is obtained by summing over all the single weight contributions. Using (5.67) we obtain where |R|(= w tot ) is the total number of boxes in the Young tableau associated to the representation R. Up to the overall sign factor, V R evaluates to a background Wilson loop in the representation R of the U (p) ⊂ U (N ) J subgroup of the topological symmetry acting on the Coulomb branch. This is precisely the prediction (4.7) derived from the brane picture. 71 The analytic continuation z → 1 can be directly taken in the integrand for the factor p j<k sh −1 (σ k − iw j z +iw k z), as no pole crosses the contour of integration when z → 1. This is because the rest of the matrix integrand has p j<k sh(σ k ) factors, which kill the poles of the first factor. We will rely on this property on several other occasions. Enforcing z = 1 simplifies the factor to (−1) (p−1)wtot p j<k sh −1 (σ We now turn to the evaluation of the vevs of the Vortex loops V M,R of the U (N − 1) node.
Note that the Vortex loop vev V N,R , associated to the splitting of hypermultiplets N = N + 0, can be evaluated in exactly the same fashion as the vevs of the V (p) R loops, leading to the result (5.74) with p = N − 1 and reproduces our prediction (4.8).
In order to evaluate the V M,R vevs, we are going to consider their "right" SQM realization, namely we choose to insert in the 3d matrix model the right SQM index (5.45) with the additional background loop (5.49). Again we start by considering a single weight contribution V M (w), with w = (w 1 , · · · , w N −1 ), and we simplify the matrix model by replacing the factors irrelevant to the computation by [· · · ], where σ j denote the eigenvalues of the U (N − 1) node and we defined N ≡ N − 1 for convenience. The cancellation between numerator and denominator factors leads to We compute this matrix model by using a generalized Cauchy determinant formula. Let us remind the Cauchy determinant formula .

(5.77)
A generalized version of this formula, for We use this formula in the matrix model to replace the factor depending on the weight w, with We may now plug this result into the matrix model and pull out the sum over permutation τ ∈ S N V M (w) = lim We can simplify the result by taking directly setting z = 1 in the factors in the first line of (5.79), which just produces a sign. This is justified because these factors do not have poles crossing the integration contours as z goes from 0 to 1. We obtain It is now possible to relabel the eigenvalues σ τ (j) ↔ σ j in each integral and recognize each term in the sum as the same matrix model with shifted masses m j → m j − iw τ (j) z, for 1 ≤ j ≤ p. The shifted matrix model is simply the bare partition function Z, as can be understood by taking w j = 0 for all j. We have precisely . Summing over the weights w of the representation R and using (5.67) yields the final result where w M = (w 1 , w 2 , · · · , w M , 0, · · · , 0 N −M ). To check mirror symmetry, it is useful to re-express the result in terms of Z, which is manifestly invariant under the exchange of mass and FI parameters, where we have reabsorbed (−1) factors into an imaginary shift of ξ N withξ N = ξ N + i N −1 2 . We remind the reader that we did not keep track of the overall sign in the evaluation of the SQM index, so that our evaluation of Vortex loops are only valid up to an overall sign.
This completes our evaluation of the matrix models computing the vevs of the T [SU (N )] Vortex loop operators.

Mirror Map
where we used N −1 j=1 w j = w tot = |R|. Recalling the vev of the Wilson loop in the U (N − 1) node given by (5.68),  This completes successfully the checks of mirror symmetry for the T [SU (N )] theory. We have found that T [SU (N )] Wilson loops and Vortex loops can be expressed as operators acting on the partition function by imaginary shifts of the FI or mass parameters (see also [50]). More generally, in all T ρ ρ [SU (N )] linear quiver theories it seems possible to express Wilson loops / Vortex loops as operators acting on the partition function by imaginary shifts of a "generalized" set of FI parameters/ mass parameters. (we will not provide a proof of this result in this paper).

Loops In SQCD
As our final example, we consider loops in the U (N ) theory with 2N fundamental and its mirror dual theory, discussed in section 4.5. We focus on the prediction (4.15), relating the Wilson loop W R of the U (N ) theory to the Vortex loop V (N ) 1,R of the mirror theory. The other mirror maps (4.16) can be easily checked with the explicit matrix models computing the vevs of the operators, by computations essentially identical to those presented above for the T [SU (N )] loops.
We denote ξ 1 − ξ 2 the FI parameter and m a , a = 1, · · · , 2N , the masses of fundamental hypermultiplets in the U (N ) theory, ξ a − ξ a+1 , a = 1, · · · , 2N − 1, the FI parameters and m 1 , m 2 the masses of fundamental hypermultiplets in the mirror theory. The matrix models computing the S 3 partition function of the U (N ) theory and its mirror dual are given by where we used to fact that the mirror theory can be decomposed into three pieces: two T [SU (N )] theories (left and right parts of the quiver in figure 33-b) whose SU (N ) hypermultiplet flavor symmetries are gauged with the U (N ) gauge symmetry of the central node (which has two more hypermultiplets by itself). The matrix model is then a combination of these three pieces and can be expressed using two T [SU (N )] partition functions with mass parameters identified with the U (N ) node eigenvalues σ j and FI parameters as indicated. We have also added background Chern-Simons terms given by This result will follow from our computations.
The matrix model computing the vev of a Wilson loop W R in the U (N ) theory is given by Using twice the Cauchy determinant formula (5.77) we find To compute the integrals we use the identity analytically continued to β ∈ C. After simplifications we obtain We now turn to the mirror dual operator which should be the Vortex loop V Plugging the explicit values (5.51) leads to . (5.101) As we did already several times, we can plug directly z = 1 in the factors on the second lign, since these factors do not have poles as z goes from 0 to 1. The ratio of sh factors then simplifies to (−1) (N −1)wtot . The remaining integrals can be performed using (5.97) and the final result can be analytically continued to z = 1. After simplification, we obtain This matches precisely (5.98) upon identifying FI and mass parameters (ξ 1 , ξ 2 , m a ) = ( m 1 , m 2 , ξ a ), up to a sign, which was not carefully analyzed in the computations leading to the matrix models. This confirms the mirror symmetry prediction (4.15) We notice that the partition functions Z and Z can be found by setting to zero the weight w and removing the normalization factors Z −1 and Z −1 in the formula for W (w) and V (w) respectively. We observe then that Z and Z are exactly mapped under the identification (ξ 1 , ξ 2 , m a ) = ( m 1 , m 2 , ξ a ). This would not have happened if we did not add the unphysical phases e φ , e φ to the matrix models.

Hopping Duality
We have been claiming several times that each Vortex loop can be realized (at least) by two different 3d/1d defect theories, which are read from the brane realization by moving the D1-branes to the closest NS5 on the left or on the right. The equivalence between the two defect theories is called hopping duality, in analogy with [21] (see also [8]). We can show that the S 3 partition function of the two defect theories indeed match.

Consider a Vortex loop V (N )
M,R in a certain 3d quiver theory, labeled by a representation R of a U (N ) node and a splitting K = M + (K − M ) of the K fundamental hypermultiplets of that node. It is realized by a brane configuration of figure 38-a, with |R| D1-branes ending on N D3-branes with K − M D5-branes on the left and M D5-branes on the right. The associated left and right 3d/1d theories are shown in 38-b.
where [ · · · ] indicates the matrix model associated to the other nodes of the 3d quiver, which play no role in the check of the hopping duality, ξ l and ξ r are the "FI parameters" associated to the left and right NS5-branes (ξ l − ξ r is the FI parameter of the U (N ) node), m a are the masses of the fundamental hypermultiplets and σ l j and σ r k are the eigenvalues of the U (N l ) and U (N r ) nodes standing respectively on the left and on the right of the U (N ) node in the quiver diagram. This simplifies to The meaning of the z → 1 limit, as explained after equation 5.48, is to take the analytical continuation of the matrix model computed with iz ∈ R. We can thus perform the change of variable σ j → σ j − iw j z, leading to The analytical continuation z → 1 can be taken directly in the integrand for the factors in the numerator, because they do not have poles for z ∈ C. The expression then simplifies again and matches the matrix model computing V

(N )
M,R from the left 3d/1d theory realization: where we have used sh(x + in) = (−1) n sh(x), for n ∈ Z. This is indeed the matrix model associated to the left 3d/1d theory given by 5.48 with the 1d index 5.47, after simplification of some factors of ch. Note that the additional background Wilson loop is important to get a precise match.
The hopping duality also explains the equivalence of Vortex loops labeled by representations of different nodes. This occurs for instance for the Vortex loops of circular quivers with nodes of equal ranks described in section 4.4 (see (4.13), (4.14)).
Consider the quiver in figure 39-a with two adjacent nodes U (N 1 ) and U (N 2 ), with K 1 and K 2 fundamental hypermultiplets respectively and assume N 1 ≥ N 2 . The vortex loop V (N 1 ) 0,R 1 , with R 1 a representation of U (N 1 ) and the subscript 0 indicating the splitting K 1 = 0 + (K 1 − 0), can be a) computed from the matrix model associated to the right 3d/1d theory: where we indicated only the matrix factors inserted by the 1d defect and part of sh factors of the two nodes, which will play a role in the check of hopping duality. Moreover the ch factors in the numerator of (5.45) have been canceled by the matrix factor of the U (N 1 ) × U (N 2 ) bifundamental hypermultiplet. This Vortex loop is realized by the brane configuration of figure 39-a with |R 1 | D1-branes ending on the N 1 D3-branes supporting the U (N − 1) node and standing to the right of the K 1 D5-branes.
In the simplest case when N 1 = N 2 , we have argued that the D1-branes can be moved to the right of the NS5-brane without changing the loop operator inserted. When D1-branes stand on the right of the NS5-branes, they realize a Vortex loop V (N 2 ) K 2 ,R 2 with R 2 a representation of U (N 2 ) and the subscript K 2 indicating the splitting of U (N 2 ) fundamental hypermultiplets K 2 = K 2 + (K 2 − K 2 ). The left 3d/1d theory associated to V (N 2 ) K 2 ,R 2 is actually the same as the right 3d/1d theory associated to V (N 1 ) 0,R 1 , except for one important difference, which is that the FI parameter of the terminating SQM node (the node with N 1 fundamental and N 2 anti-fundamental chiral multiplets) is positive for V (N 2 ) K 2 ,R 2 and negative for V (N 1 ) 0,R 1 . The matrix model computing V (N 2 ) K 2 ,R 2 from the left 3d/1d theory is given by 0,R , as the D1-branes realizing the loops can be moved freely across the NS5. Let us consider this case first. Starting from the matrix model (5.108) computing V (N 1 ) 0,R , with N 1 = N , we can use the same trick as above and make the replacement in the integrand. This allows us to use the Cauchy determinant formula (5.77) with σ j → σ j + iw j , leading to Relabeling σ j → σ τ −1 (j) and σ j → σ τ (j) , we obtain .
where we have used the Cauchy identity to obtain the second equality and we have again enforced z = 1 in the sh factors of the integrand to reach the matrix model (5.109) computing V This shows that the Vortex loops realized by D1-branes ending on the left or on the right of an NS5 with equal numbers of D3s on both sides are equivalent. Combining this property with the hopping duality between the left and right 3d/1d quiver realization of a Vortex loop, we prove the equivalence of Vortex loops in circular quivers with nodes of equal rank (4.13), (4.14). This is illustrated in figure 39 When N 1 > N 2 the map of Vortex loops is more complicated. It can be found from the brane picture in the same way as we found mirror maps between loop operators. We consider the brane realization of V (N 1 ) 0,R 1 with |R 1 | D1-strings ending on the N 1 D3-branes as in figure 39-a. As N 1 > N 2 , the NS5 has a D3-spike on its left side. We then move the D1-branes to the left, across the NS5-brane. Some D1-branes can be moved along the D3-spike and get attached to the NS5-brane, realizing flavor Wilson loops for the U (1) global symmetry associated to the NS5-brane. The other D1-branes end on the N 2 D3-branes on the right of the NS5 and realize a Vortex loop V (N 2 ) K 2 ,R 2 . Recycling the ideas of section 3.2, the precise prediction from the brane picture is found to be where ∆ is the set of representations (q s , R s ) appearing in the decomposition of R 1 under the This map can be checked by explicit computations, using the generalized Cauchy formula (5.78). It implies that the Vortex loops V (N 1 ) 0,R 1 and V (N 2 ) K 2 ,R 2 are redundant and that in order to describe the mirror map with Wilson loops of the mirror theory, it is sufficient to consider only the loops V In general for each pair of consecutive D5-branes in the brane picture, we need only to consider the Vortex loops realized with D1-branes placed between the two D5-branes, with a fixed number of NS5-branes on their left and on their right. This is the mirror statement to the fact that between two consecutive NS5-branes, we need only consider Wilson loops realized with F1-strings placed between the two NS5s, with a fixed number of D5-branes on their left and on their right. In the left-invariant frame, l and¯ l are constant, while r and¯ r have a spatial dependent phase e iτ /L and e −iτ /L respectively, where τ ∈ [0, 2πL] is the coordinate along the loop. We recall that the equations solved by the Killing spinors are where L is the radius of S 3 . The supersymmetry transformations under the supercharges in deformed 72 We derived these transformations by "covariantizing" the 3d N = 2 transformations of [4].
These correspond to the supersymmetry transformations of a 1d N = 4 SQM V abelian vector multiplet (see for instance [18]) but in the presence of a background gauge field a − = − 1 L for the U (1) − R-symmetry, generated by J − . The presence of the background gauge field a − = − 1 L affects the covariant derivative of Φ,Φ in the transformations above. The identification of the fields in the 3d N = 4 vector multiplet with the fields of the 1d N = 2 vector and adjoint chiral multiplet that furnish a 1d N = 4 SQM V vector multiplet is We now turn to the 3d N = 4 hypermultiplet supersymmetry transformations. These do not admit an off-shell formulation, so we provide the on-shell supersymmetry transformations: (A.8) 73 We do not give the supersymmetry transformations of the auxiliary fields and transverse gauge fields, which combine into another 1d N = 4 chiral multiplet.
74 U (1) − is a global symmetry from the point of view of the 1d N = 2 subalgebra, so that each N = 2 multiplet comes with a U (1) − charge.
Restricting to the same four supercharges generating deformed SQM V as above, we obtain This matches the on-shell deformed supersymmetry transformations of two 1d N = 4 chiral multiplets (φ, ψ + , ψ − ) q F ∼ (φ 1 , (ψ 1 ) 1 , (ψ 2 ) 1 ) − 1 2 and (φ 2 , (ψ 1 ) 2 , (ψ 2 ) 2 )1 2 . The shift of Φ 3 in the supersymmetry transformations is identified with a real mass deformation with complex parameter m F = i L for a flavor symmetry G F , such that the 1d N = 4 chiral multiplets have G F charge q F = ∓ 1 2 . The term (D 1 + iD 2 )φ 1 in the on-shell 1d N = 4 supersymmetry transformations arises from a superpotential coupling. In the off-shell transformations it gets replaced with a complex auxiliary field.
We note that the mass deformation obtained by giving a background to the G F flavour symmetry is not visible anywhere in the transformations A.4, which means that the adjoint chiral multiplet with bottom component Φ = Φ 1 + iΦ 2 is not charged under G F . This allows us to identify this flavor symmetry with Moreover the background gauge field a − = − 1 L for J − is not visible in the transformations (A.9) of the chiral multiplets with bottom components φ 1 and φ 2 , which means that these multiplets are not charged under J − . We then have the identification This identification of generators allows us to match the deformed SQM W supersymmetry algebra with the SU (1|1) l × SU (1|1) r subalgebra preserved by the defect, as explained in section 5.2.

B Evaluations of the SQM Index
In this appendix we compute the 1d N = 4 SQM index, or partition function on S 1 , for some quiver gauge theories with generic U (1) − R-symmetry background z and U (1) F flavor chemical potential µ = 1, as defined in section 5.3.2. The R-symmetry and flavor symmetry charges are summarized in table 6 and the specific adjoint R-charges are given in equation (5.43).
B.1 U (k) Theory With N Fundamental And M Anti-fundamental Chirals We consider a 1d N = 4 SQM gauge theory with U (k) gauge group and N fundamental chiral multiplets with real masses −σ j , R-charges r + and G F flavor charge q + , and M anti-fundamental chiral multiplets with masses m a , R-charges r − and G F flavor charge q − . Mass parameters are in units of the inverse S 1 radius. As explained in the main text the charges obey the superpotential constraints r − +r + = 2 and q − +q + = 1/2. We introduce the complex parametersσ j = σ j +iq + +i r + 2 z andm a = m a − iq − − i r − 2 z. Importantly we take a negative FI parameter ζ < 0. This corresponds to the choice of FI parameter when the 1d N = 4 theory is read by moving the D1-branes to the right NS5-brane.
The partition function is given by where the integration contour is defined so that it picks the residues at the poles from the fundamental chiral multiplet factors, namely factors in the product N j=1 . In particular it does not pick the poles from factors in the product k I =J and M a=1 . This follows from the definition of JK − Res ζ for ζ < 0, reviewed in the main text. This integral has poles at u * I = iσ j , j = 1, · · · , N , however, due to the sin[−π(u I −u J )] factors, we get a non-zero residue only when each u * I hits a different iσ j , in particular we have non-zero residues only when k ≤ N . A non-vanishing residue at u * = {u * I } 1≤I≤k is then associated to a decomposition k = N k=1 k j , with k 1 , · · · , k N ∈ {0, 1}, where k j = 1 if u * I = iσ j for a given I and k j = 0 otherwise. This residue contribution appears with a multiplicity k!, corresponding to permutations of the u * I . The partition function is then expressed as The partition function vanishes when k > N , consistent with the fact that there are no supersymmetry vacua in that range.

(B.5)
This result matches the formula (5.45) giving the index as a sum over the weights of the representation A k associated to the 1d N = 4 gauge theory.
B.2 U (k) Theory With N Fundamental, M Anti-fundamental Chirals And One Adjoint Chiral We consider a 1d N = 4 SQM gauge theory with U (k) gauge group and N fundamental chiral multiplets with real masses −σ j , R-charges r + and G F flavor charge q + , M anti-fundamental chiral multiplets with masses m a , R-charges r − and G F flavor charge q − and an adjoint chiral multiplet with R-charge R adj = 2 and flavor charge q adj = 1. Mass parameters are in units of the inverse S 1 radius. The charges obey the superpotential constraints r − + r + = 2 and q − + q + = 1/2. We introduce the complex parametersσ j = σ j + iq + + i r + 2 z andm a = m a − iq − − i r − 2 z. Importantly we have a negative FI parameter ζ < 0.
In order to avoid higher order poles in the computation we keep R adj generic and only set it to 2 at the end of the computation. The partition function or index is given by where the integration contour is defined by picking the residues of the poles from the fundamental chiral mutliplets factors, namely the factors in the product N j=1 and from "half" of the adjoint chiral multiplet factors k I,J according to the JK − Res ζ prescription with ζ < 0. Concretely the integral is a sum over residues, each contribution corresponding to a pole u * = {u * I } 1≤I≤k described by taking a decomposition of k into N non-negative integers, k = N i=1 k i , k i ≥ 0, and picking u * I → u * i,s i = iσ i − s i R adj 2 z with s i = 0, · · · , k i − 1. The arrow → indicates a mapping between the index I into the index (i, s i ). The residue contribution coming from a given pole u * = {u * i,s i } arises with k! degeneracy, associated to permutations of the u * I . The partition function is then given by a sum over the residue contributions I (k i ) associated to possible decompositions k = N i=1 k i , counted with k! degeneracy: The explicit evaluation of the residues leads to If we plug the adjoint R-charge R adj = 2 then the result simplifies to where we have introduced a regulating mass parameter for vanishing factors. 75 Let us write I (z) = N i=1 sh[ ] sh[ik i z+ ] . In the limit z → 1 (at finite ) this factor become trivial. Using the contraints r + + r − = 2 and q + + q − = 1/2, we obtain Here again the result matches the formula (5.45) giving the index as a sum over the weights of the representation S k associated to the 1d N = 4 theory.

B.3 Two-Node Quiver
We consider a U (N 1 ) × U (N 2 ) quiver gauge theory with bifundamental chiral multiplets of R-charges r, r and G F -charges q, q. In addition we have N U (N 2 )-fundamental chiral multiplets with real masses −σ j , R-charges r + and G F -charge q + , M U (N 2 )-anti-fundamental chiral multiplets with masses m a , R-charges r − and G F -charge q − and an adjoint chiral multiplet with R-charge R adj = 0 and G Fcharge q adj = −1. The charges obey the superpotential constraints r − + r + = 2, q − + q + = 1/2, r + r + R adj = 2 and q + q = 1. The FI parameters are both taken negative ζ 1 , ζ 2 < 0. We also assume  Figure 40: Brane realization of a Vortex loop associated with the representation A k1 ⊗ A k2 and SQM two-node quiver read by moving the D1-branes to the NS5-brane on the right. Here N 1 = k 1 and N 2 = k 1 + k 2 .
N 2 > N 1 . This setup corresponds to the 1d N = 4 theory obtained by moving the D1-branes to right NS5-brane, with k 1 = N 1 D1-branes ending on an extra NS5'-brane and k 2 = N 2 − N 1 D1-branes ending on a second extra NS5'-brane (see figure 40). The representation associated to this quiver is We introduce the complex parametersσ j = σ j + iq + + i r + 2 z andm a = m a − iq − − i r − 2 z. in order to avoid higher order poles in the computation we keep R adj generic and only set it to 0 at the end. The partition function or index is given by sin[−π(v J − u I + q + r 2 z − z)] sin[π(v J − u I + q + r 2 z)] .
The integration contour picks the residues at the poles selected by the JK-recipe as explained in the main text. The recipe allows to take poles from the the fundamental chiral mulitplet factors, as well as from "half" of the bifundamental and adjoint chiral multiplet factors. However one can realize, for instance, that picking a pole u * I = u * J − R adj 2 z from the adjoint factor and a pole u * J = v * K − q − r 2 z from the bifundamental factor, leads to u * I = v * K + q + r 2 z − z − 1, where we have used the superpotential constraints, and in this case u * I has an extra zero from the bifundamental factor, canceling the pole from the adjoint factor. This kind of reasoning leads to the conclusion that we cannot take a pole from the adjoint factor, as it yields a vanishing contribution.
• With k 1 = N 1 and k 2 = N 2 − N 1 , a pole {u * I , v * J } is characterized by a choice of decomposition • The explicit single-integral poles are given by with → denoting a mapping between the relevant indices. The range of the s j is such that it correctly gives the N 1 u * I -poles and the N 2 v * I -poles. • Each residue contribution to the index appears with the degeneracy k 1 !(k 1 + k 2 )! = N 1 !N 2 !.
The evaluation of the index gives where we have used the superpotential constraints. F(σ i , z) is a complicated expression which will not be relevant for our analysis of mirror symmetry. It is expressed as a product of the form F(σ i , z) = α,κ N i =j sin[−π(iσ i − iσ j + αz ± z + κ/2)] sin[π(iσ i − iσ j + αz + κ/2)] , (B.14) where α, κ take real values. As in (B.10), there are terms which require a regularization by a small mass deformation and which evaluates to ±1 as z → 1.
Plugging the adjoint R-charge R adj = 0, the result simplifies to I (k (1) ,k (2) ) = F(σ i , z) This result reproduces correctly (5.45) as a sum over the weights of the representation A k 1 ⊗ A k 2 .