Expanding the Bethe/Gauge dictionary

Journal of High Energy Physics, Nov 2017

We expand the Bethe/Gauge dictionary between the XXX Heisenberg spin chain and 2d \( \mathcal{N} \) = (2, 2) supersymmetric gauge theories to include aspects of the algebraic Bethe ansatz. We construct the wave functions of off-shell Bethe states as orbifold defects in the A-twisted supersymmetric gauge theory and study their correlation functions. We also present an alternative description of off-shell Bethe states as boundary conditions in an effective \( \mathcal{N} \) = 4 supersymmetric quantum mechanics. Finally, we interpret spin chain R-matrices as correlation functions of Janus interfaces for mass parameters in the supersymmetric quantum mechanics.

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Expanding the Bethe/Gauge dictionary

HJE Expanding the Bethe/Gauge dictionary Mathew Bullimore 0 1 2 5 Hee-Cheol Kim 0 1 2 3 4 Tomasz Lukowski 0 1 2 5 Andrew Wiles Building 0 1 2 Radcli e Observatory Quarter 0 1 2 0 in Lower Dimensions , Topological Field Theories 1 Cambridge , MA 02138 , U.S.A 2 Woodstock Road , Oxford, OX2 6GG , U.K 3 Je erson Physical Laboratory, Harvard University 4 Department of Physics , Postech 5 Mathematical Institute, University of Oxford We expand the Bethe/Gauge dictionary between the XXX Heisenberg spin chain and 2d N = (2; 2) supersymmetric gauge theories to include aspects of the algebraic Bethe ansatz. We construct the wave functions of o -shell Bethe states as orbifold defects in the A-twisted supersymmetric gauge theory and study their correlation functions. We also present an alternative description of o -shell Bethe states as boundary conditions = 4 supersymmetric quantum mechanics. Finally, we interpret spin chain R-matrices as correlation functions of Janus interfaces for mass parameters in the supersymmetric quantum mechanics. Supersymmetric Gauge Theory; Supersymmetry and Duality; Field Theories - in an e ective N 1 Introduction 2 Spin chain primer 3 Setup Heisenberg spin chain R-matrices Bethe states The model Sphere partition function Vortex partition function Factorization 4 Defect operators in 2d Abelian theories Orbifold construction Bethe wavefunctions 5 Quantum mechanical description 6 The R-matrix 7 Discussion A Conventions N = 4 quantum mechanics Empty boundary condition Stable boundary conditions Thimble boundary conditions Orthonormality of stable basis R-matrix from Janus interface Yang-Baxter equation A.1 2d N = (2; 2) supersymmetry A.2 N = 4 quantum mechanics 2.1 2.2 2.3 3.1 3.2 3.3 3.4 4.1 4.2 4.3 5.1 5.2 5.3 5.4 6.1 6.2 6.3 1 Introduction The aim of this paper is to extend the dictionary between quantum integrable systems and supersymmetric gauge theories introduced and studied in [1{3], the so-called Bethe/Gauge correspondence. We focus on an elementary example of this phenomenon: the correspondence between the XXX 1 Heisenberg spin chain and a family of 2d N = (2; 2) super2 symmetric gauge theories. Some basic aspects of this correspondence are summarized in gure 1. { 1 { U (k) gauge group N flavours periodic boundary condition and ~ is Planck's constant. On the supersymmetric side of the correspondence, mj and ~ are complex mass parameters associated to avour symmetries of the gauge theory in gure 1, while q is the exponential of a complexi ed FI parameter. For generic values of the mass parameters, the theory has a low energy description as a U(1)k gauge theory with an e ective twisted superpotential Wf( ) depending on complex vectormultiplet scalars a. This function is determined exactly by a one-loop calculation. The equations for supersymmetric vacua, exp a = 1; : : : ; k ; (1.2) coincide with the Bethe equations for the spin chain (1.1). The ring generated by gaugeinvariant functions of the vectormultiplet scalars 1; : : : ; k modulo the relations (1.2) is the twisted chiral ring of the supersymmetric gauge theory. The twisted superpotential itself can be identi ed with Yang-Yang function of the spin chain. A powerful approach to computing a wide range of observables in quantum integrable systems is the algebraic Bethe ansatz, as explained in [5]. In this paper, we will understand how elements of this approach arise in the Bethe/Gauge correspondence. For this purpose, we will perform exact computations in the original supersymmetric gauge theory shown in gure 1, rather than the e ective abelian description. In particular, we will interpret aspects of the algebraic Bethe ansatz in terms of correlation functions in the A-type topological twist of the supersymmetric gauge theory, using techniques from supersymmetric localization [6, 7]. Investigations of the Bethe/Gauge correspondence in this context have appeared in [8, 9]. The remainder of the introduction is dedicated to summarizing our results. An important part of the algebraic Bethe ansatz is the construction of o -shell Bethe states j 1; : : : ; ki, which are elements of the spin chain Hilbert space depending on auxiliary parameters 1 : : : ; k. The inner product hf j 1; : : : ; ki with another state jf i is { 2 { hf |gi = functions in the A-twisted supersymmetric gauge theory on CP1. a symmetric function f ( 1; : : : ; k) of the auxiliary parameters, which can be identi ed with a gauge-invariant function of the vectormultiplet scalar in the supersymmetric gauge theory in gure 1. The correlation functions of such operators in the A-type topological twist depend only on the class [f ( 1; : : : ; k)] of the function modulo the twisted chiral ring relations (1.2). The map jf i ! [f ( 1; : : : ; k)] ; then sets up a correspondence between states in the spin chain Hilbert space and invariant functions of 1; : : : ; a modulo relations, such that the inner product hf jgi on the spin chain Hilbert space coincides with the two-point correlation function of f ( 1; : : : ; k) and g( 1; : : : ; k) in the A-twisted theory on CP1. This is illustrated in gure 2. In order to investigate this relation, it is convenient to introduce an orthonormal `updown' basis for the spin chain Hilbert space. The basis elements are labelled by subsets I = fi1; : : : ; ikg f1; : : : ; N g such that jIi is the state with spin " at positions i1; : : : ; ik and spin # everywhere else. We can then introduce the wavefunctions of o -shell Bethe states in this basis, SI ( 1; : : : ; k) / hIj 1; : : : ; ki ; which provide a set of generators for the twisted chiral ring. Nekrasov has proposed a physical de nition of the corresponding twisted chiral ring elements as `orbifold defects' [10]. In this paper, we explain how to implement this orbifold construction in the A-twisted supersymmetric gauge theory to compute correlation functions of the twisted chiral operators (1.4). We furthermore demonstrate that these operators are orthonormal with respect to the A-model two-point functions, corresponding to the fact that hIjJ i = I;J in the spin chain Hilbert space. In the algebraic Bethe ansatz, the eigenstates of the spin chain Hamiltonian are obtained by evaluating the o -shell Bethe state j 1; : : : ; ki on a solution of the Bethe equations (1.1). The functions SI ( 1; : : : ; k) evaluated on solutions of the Bethe equations are therefore the wavefunctions of the eigenstates in the `up-down' basis jIi. We will show that this wavefunction can be obtained directly from the supersymmetric gauge theory by computing the A-model in a cigar geometry with a vacuum corresponding to I at in nity. More precisely, we rst introduce an -background and then compute a normalized correlation function that is nite in the limit ! 0. This is shown in gure 3. (1.3) (1.4) { 3 { SI ( ) ✏ vJ f ( ) g( ) Bf Bg equations, the steps in the algebraic Bethe ansatz are independent of the parameter q. It is therefore su cient to understand these aspects in the limit q ! 0, which corresponds to discarding instanton corrections in the A-twisted supersymmetric gauge theory. In this limit, correlation functions can be understood in a nite-dimensional N = 4 supersymmetric quantum mechanics with boundary conditions preserving the same pair of supercharges as the A-twist. In particular, each twisted chiral operator generates a boundary condition in the supersymmetric quantum mechanics, and two-point functions are computed by interval partition functions | as shown in gure 4. In particular, we will provide two independent constructions of the boundary conditions generated by the operators SI ( 1; : : : ; k), either by coupling Neumann boundary conditions to additional degrees of freedom or as `thimble' boundary conditions. The setup described above is compatible with turning on background holonomies for avour symmetries. In the supersymmetric quantum mechanics description, background avour holonomies around the circle become `real mass parameters' for avour symmetries. The ordering of the holonomy eigenvalues or real masses can be identi ed with an ordering of sites on the spin chain. It is therefore natural to consider `Janus' interfaces which permute the ordering of the masses. We will show that A-model correlation functions of such Janus interfaces in between the elements SI ( 1; : : : ; k) reproduce matrix elements of the spin chain R-matrix. The Yang-Baxter relation is interpreted as the statement that a given permutation of real mass parameters can be decomposed in a number of ways into elementary Janus interfaces permuting a pair of real mass parameters. { 4 { HJEP1(207)5 Finally, the Bethe/Gauge correspondence provides a physical realization of a parallel developments in geometry and representation theory, and many of the objects we consider here have already appeared in this context. The starting point is the statement that the twisted chiral ring is the equivariant quantum cohomology ring of the vacuum manifold of the supersymmetric gauge theory, T G(k; N ). The A-model correlation functions considered here can be formulated in the language of quasi-maps to the vacuum manifold. In particular, the functions SI ( 1; : : : ; k) were introduced in work of Maulik and Okounkov [11] as the `stable basis' in the quantum equivariant cohomology (see [12] for connections to Bethe wavefunctions). This paper is largely motivated by understanding these mathematical constructions in the language of supersymmetric gauge theory. The paper is organized as follows. In section 2 we collect some relevant properties of the Heisenberg spin chain. In section 3 we describe 2d N = (2; 2) supersymmetric gauge theory and explain how to calculate their A-twisted sphere and cigar partition functions using supersymmetric localization. Section 4 focuses on the de nition of the distinguished twisted chiral operators and their correlation functions. In section 5 we reduce the problem to a 1d quantum mechanics and explain how the twisted chiral operators can arise from appropriate boundary conditions in this quantum mechanics. Finally, in section 6 we show how the spin chain R-matrices can be obtained as correlation functions of twisted chiral operators in the supersymmetric gauge theory. Our conventions and more technical details of the calculation are postponed to the appendix. 2 Spin chain primer In this section we collect some basic information on the Heisenberg XXX 1 spin chain, where all spins transform in the fundamental representation of su(2). Many of the statements we present here and in subsequent sections have a natural generalization to spin chains with higher representations, as well as to higher rank algebras. Our notation is designed to match that of supersymmetric gauge theory and therefore di ers from standard 2 integrability conventions. 2.1 Heisenberg spin chain In order to de ne the Heisenberg spin chain we need to specify a Hamiltonian and a Hilbert space on which it acts. The Hilbert space of the spin chain is the N -fold tensor product of the fundamental representation of su(2), We introduce standard basis elements j " i and j # i for each spin chain site C2. There is then a natural basis for V that is labelled by subsets I = fI1; : : : ; Ikg f1; : : : ; N g such that V = C 2 | C 2 N {tizmes : : : C2 : } jIi = j # : : : " : : : " : : : #i : |{I1z} |{Ikz} { 5 { (2.1) (2.2) We de ne an inner product by demanding that the basis vectors at each site are orthonormal h # j # i = h " j " i = 1; h # j " i = h " j # i = 0 ; a 2N and naturally extending this de nition to V. Any operator A : V ! V can be represented 2N matrix of its expectation values between tensor products of j " i and j # i. The (twisted, homogeneous) Heisenberg spin chain is de ned by the Hamiltonian, with the twisted boundary condition, Later on, we will also introduce inhomogeneities for each spin chain site. The Hamiltonian commutes with the operator counting up spins and therefore the Hilbert space can be decomposed into a direct sum of spaces with xed number of excitations, sion relation The spectrum of the Heisenberg spin chain can be then found using the celebrated Bethe ansatz. In particular, the eigenvalues of the Hamiltonian H are obtained from the dispera a H = include inhomogeneities mi at each site of the spin chain.2 In that case we denote the spin chain Hilbert space as and the Bethe equations turn into mi + ~~2 = q Y 2 where we sum over the rapidities1 of excitations a, a = 1; : : : ; k, which are solutions to the Bethe equations a + ~2 !N ~ 2 = q Y a For a given number of excitations k, there are Nk solutions distinct solutions of the Bethe equations. The solutions aI can be labelled by a subset I = fI1; : : : ; Ikg that, expanding around q ! 0, the solutions are of the form aI = mIa ~2 + O(q). f1; : : : ; N g such 1It is common to use the letter u to denote rapidities. In this paper we use the letter instead in order to make connection with the gauge theory side of our story in the following sections. 2It is common to use the letters vi to denote inhomogeneities. In this paper we use the letters mi instead in order to make connection with the gauge theory side of our story in the following sections. { 6 { (a) Yang-Baxter equation. (b) Unitarity of the R-matrix. most powerful method is the algebraic Bethe ansatz, which is based on the construction of an R-matrix. For the inhomogeneous spin chain, this is an operator acting on two sites, Rij (mj mi) : C2mi 2 Cmj ! Cmj 2 C2mi : It has rational dependence on mj mi and satis es the regularity property Rij (0) Pij where Pij is the permutation operator, together with the Yang-Baxter equation (shown graphically in gure 5a) R12(m2 m1)R13(m3 m1)R23(m3 m2) = R23(m3 m2)R13(m3 m1)R12(m2 where Rij acts non-trivially only on C2mi and C2mj . An explicit form of the R-matrix is (2.11) m1) ; (2.12) (2.13) (2.14) (2.15) (2.16) where each matrix element is an operator acting on the Hilbert space (2.9). Namely, each matrix element of M ( ) in the auxiliary space can itself be represented by a 2N 2N matrix. M ( ) = A( ) B( ) C( ) D( ) ! ; { 7 { 1 mji + ~ Rij (mji) = (mji Iij + ~ Pij ) ; R12(m)R21( m) = I : where mji = mj mi and we xed the normalisation by the unitarity condition (shown graphically in gure 5b) 2.3 Bethe states odromy matrix We now introduce an auxiliary space C 2 with spectral parameter and de ne the monM ( ) = R10( m1)R20( m2) : : : RN0( mN ) ; where each Ri0( 2 2 matrix in the auxiliary space mi) acts non-trivially only on the auxiliary space and C2mi . This is a For a given k, we de ne an o -shell Bethe state by j 1; : : : ; ki = B( 1) : : : B( k)j i ; where j i = j # : : : #i. Additionally, for given subset I = fI1; : : : ; Ikg of f1; : : : ; N g, we de ne the functions SI ( ) as the overlaps of o -shell Bethe state j 1; : : : ; ki with the basis vectors: SI ( ) = ( 1)jIjN ( )hIj 1; : : : ; ki : Here we have introduced a normalization factor HJEP1(207)5 N ( ) = ( 1) 2 k(k 1) +kN Qa;i( a Qa;b( a mi + ~2 ) ; b + ~) which is independent of I. These functions can be computed explicitly with the result, SI ( ) = Sym k Q a=1 Ia 1 Q ( a i=1 It can be shown that the states j 1; : : : ; ki become eigenstates of the spin chain Hamiltonian provided a are evaluated on a solution aJ of the Bethe equations. The functions SI ( J ) are then (up to normalization) the wavefunctions of the Bethe eigenstates in the position basis jJ i. In the following sections, we will explain how to construct such wavefunctions in the Bethe/Gauge correspondence. 3 Setup In this section, we review the computation of correlation functions in A-twisted supersymmetric gauge theories on CP1 and a cigar. We review two approaches to computing such correlation functions using supersymmetric localization. The rst leads to a contour integral in the complex Cartan subalgebra of the gauge group. The second is via equivariant localization on the moduli space of quasi-maps into the vacuum manifold. This will provide a foundation for the results presented in the following sections. 3.1 The model We consider 2d N = (2; 2) supersymmetric gauge theories with R-symmetry U(1)V that ow to sigma models onto cotangent bundles to complex Grassmannians, T G(k; N ). Such a theory has gauge symmetry G = U(k) and avour symmetry Gf = PSU(N ) The eld content is depicted in gure 6 and can be summarized as follows: U(1)A U(1)~. A vectormultiplet containing bosonic elds (A ; ; D) transforming in the adjoint representation of U(k), where A is the gauge eld, is a complex scalar and D is an auxiliary scalar. { 8 { (2.17) (2.18) (2.19) (2.20) Chiral multiplets ( ; X; Y ) transforming as shown in the table below. X Y U(k) adj U(1)~ 1 and theta angle . Our conventions are summarized in appendix A. space of solutions the vacuum equations For positive FI parameter, r > 0, the theory ows to a sigma model onto the moduli modulo constant gauge transformations. We de ne R to be the moment map for the gauge symmetry. It can be shown that solutions require = 0 and = 0, and that the remaining equations reproduce the hyper-Kahler quotient construction of T G(k; N ) where r is the Kahler parameter of the base Grassmannian G(k; N ). We refer to this as the vacuum manifold V. It is useful to provide an algebraic description of the vacuum manifold. For r > 0, we can replace the D-term equation (3.1) by the stability condition that the matrix X has maximal rank and divide by complex gauge transformations, V = fX; Y jX Y = 0; rk(X) = kg=GL(k; C) = T G(k; N ) : (3.5) From this perspective, X de nes a k-plane in CN corresponding to a point in the base Grassmannian G(k; N ). For example, in the case k = 1, we have T CPN 1 with homogeneous coordinates [X1; : : : ; XN ] on the base. For negative FI parameter r < 0, the roles of X and Y would be interchanged. We can also introduce complex mass parameters for the avour symmetry Gf by coupling to a background vectormultiplet and introducing non-zero vacuum expectation { 9 { (3.1) (3.2) (3.3) (3.4) . {+} { } values (m1; : : : ; mN ; ~) to the complex scalar in the vectormultiplet in a Cartan subalgebra equations (3.3) are replaced by j mj = 0. In the presence of complex masses, a mj + ( 1; : : : ; k) denote the eigenvalues of the vectormultiplet scalar . For generic values of the complex masses (m1; : : : ; mN ; ~), the vacuum manifold V is lifted, leaving behind Xaj = p r j;Ia ; Y j a = 0 ; ab = 0 ; (3.7) labelled by subsets I = fI1; : : : ; Ikg f1; : : : ; N g of size jIj = k. The massive vacua can be identi ed with the xed points of the in nitesimal Tf action on the vacuum manifold V = T G(k; N ) generated by (m1; : : : ; mN ; ~) and correspond geometrically to the coordinate hyperplanes in the base G(k; N ). 3.2 Sphere partition function We will now consider correlation functions in the -deformed A-model on C = CP1, introduced in [6, 7]. We introduce homogeneous coordinates [z : w] on CP1 and de ne a U(1)J isometry that transforms the homogeneous coordinates by (z; w) ! (e =2z; e =2w) with xed points f+g = f z = 0g and f g = fw = 0g, as shown in gure 7. The background preserves a pair of supercharges Q , Q+ that commute with the combination U(1) := U(1)J + U(1)V . 3.2.1 Contour integral Partition functions in the -deformed A-model can be computed exactly using supersymmetric localization for the supercharge Q = Q + Q+ [6, 7]. This reduces the path integral to a contour integral over the complex Cartan subalgebra of G parametrized by the eigenvalues ( 1; : : : ; N ) of the vectormultiplet scalar . In order to express the contributions to the integrand of the contour integral from the various multiplets, it is convenient to introduce the following function Z(n)( ) = 1( is Barnes' gamma function. Due to the functional equation 1(x + ) = x 1(x), this ratio of Barnes' gamma functions is in fact a rational function of . The contribution to the integrand from a chiral multiplet of charge r under the U(1)V vector R-symmetry and charge qf under a U(1)f avour symmetry is Z(qf n r)(qf ), where is the vectormultiplet scalar and n 2 Z is the quantized ux through CP1. Coming back to the model introduced in section 3.1, partition functions are expressed as a contour integral over the complex vectormultiplet scalar ~ = ( 1; : : : ; k) together with a summation over the ux ~n = (n1; : : : ; nk) 2 Zk. The contributions to the integrand from the chiral multiplets are k Y a;b=1 N N where we introduce a shorthand notation ab = a b and nab = na nb. There is an additional contribution from the vectormultiplet V Z(~n)(~ ) = Y Z ( nab 2)( ab) a6=b a<b = Y( 1)nab+1 ( a b) 2 (na nb)2 : The partition function is then given by where (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) the contributions from the chiral multiplets [6] and (k + N + 1) P combines the contributions from the vector and chiral multiplets. We include an additional sign ( 1)P with P = k2 + (k + N + 1) P a na, where the factor k2 xes a sign ambiguity in a na is an additional sign that can be absorbed into the de nition of q. The contour is given by the Je rey-Kirwan prescription, which reduces for r > 0 to the contour surrounding poles at a = mi ~ 2 na 2 ` ; scalar. As explained in [6, 7], there are then additional contributions f+g f g : : f ~ f ~ + ~n 2 ~n 2 ; ; notation n := P a na. multiplets Xai in the fundamental representation of the gauge group. The summation over uxes can therefore be restricted to the region ~n 2 Z 0 k . We will often use the shorthand The partition function is enriched by inserting twisted chiral operators annihilated by . We will consider gauge-invariant functions f (~ ) of the vectormultiplet to the integrand in equation (3.12). We denote a correlation function with f (~ ) inserted at f+g and g(~ ) inserted at f g by hf (~ ); g(~ )iS2 = X ~n2Zk 0 qPi ni Z d k k! ZN(~n;)k(~ )f ~ ~n 2 ~n 2 g ~ + : (3.16) Importantly, the contributions from na > 0 vanish unless certain conditions are satised. For example, in the abelian case instanton corrections to hf ( ); g( )iS2 vanish unless the combined degree of the polynomials is greater than or equal to 2N 1. This follows from the fact that for n > 0 the only potential pole outside of the contour is at which only exists if deg(f ) + deg(g) 2N 1. This phenomenon can be understood as the condition to cancel the U(1)A axial anomaly. In all cases, the partition function h1iS2 receives contributions only from ~n = 0 and is therefore independent of q. ! 1, Moreover, correlation functions involving particular combinations of twisted chiral operators vanish, re ecting the structure of the twisted chiral ring. For example, in the abelian case (3.14) (3.15) (3.17) (3.18) hf ( ) Y N j=1 hf ( ); g( ) Y N j=1 mj + N Y j=1 ~ 2 mj + q f ( ~ 2 mj + ~ 2 for any f ( ) and g( ). In the limit ! 0, we recover the twisted chiral ring relations, N ) Y j=1 q g( + ) Y N j=1 N q Y j=1 ; g( )iS2 = 0 ; ~ 2 iS2 = 0 ; ~ 2 mj ~ 2 mj = 0 ; which coincide with the equivariant quantum cohomology ring of the vacuum manifold V = T CPN 1 . that hold inside correlation functions are ! 0, the general twisted chiral ring relations of a non-abelian theory (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) together with R r1 = modulo gauge transformations. The solutions of such `generalized vortex equations' are known as freckled instantons [17, 18]. The moduli space of solutions has an algebraic description by dropping the D-term equation (3.20) in favour of a stability condition and dividing by complex gauge transformations. This leads to a description in terms of stable `quasi-maps' from C = CP1 into the vacuum moduli space V = T G(k; N ). The moduli space of solutions decomposes into a union of components labelled by the vortex number or ux n 2 Z through CP1, which coincides with the degree of the quasi-map. YN ( a j=1 ( a mj + ~2 ) mj ~ ) 2 = q Y a This coincides with the quantum equivariant cohomology ring of V = T G(k; N ) and the Bethe equations (2.10) for an inhomogeneous XXX 1 spin chain of length N with quasiperiodic boundary conditions. We now consider alternative approach to computing correlation functions in A-twisted gauged linear sigma models introduced in [ 13, 14 ] in terms of vortex counting. This approach was derived rigorously from supersymmetric localization in [6]. The mathe matical formulation of this construction involves equivariant integrals over holomorphic `quasi-maps' to the vacuum manifold V, which may be computed by equivariant localization [15, 16]. This provides the link with recent mathematical work on the Bethe/Gauge correspondence [12]. In this approach, we rst set the complex mass parameters (m1; : : : ; mN ; ~) and the to zero, and consider con gurations preserving both Q and Q+. Such con gurations are given by b + ~ b ~ ; 2 M = Mn ; [ n2Z Now turning on the mass parameters (m1; : : : ; mN ; ~) and the -deformation parameter deforms the equations (3.23){(3.24) that determine by replacing ! + m + ~ + LV ; where (m1; : : : ; mN ; ~) are understood to mean the in nitesimal Tf avour transformation generated by these parameters and LV is the Lie derivative along the vector eld V generating U(1) rotations. This restricts the system to the xed points of the corresponding Tf action on the moduli space M. This can be understood as working equivariantly with respect to the action of Tf U(1) on the moduli space M with equivariant di erential Q = Q + Q+. In particular, localization of the path integral to Gaussian uctuations around Q , Q+-invariant con gurations is equivalent to computing the following sum of equivariant integrals where [Mn]vir is the virtual fundamental class. The correlation functions hf (~ ); g(~ )iS2 correspond to computing the equivariant integrals of certain virtual equivariant cohomology classes [f ] and [g] on Mn. We will rst explain how to compute the partition function in this manner in the abelian case, before considering the general case. Abelian case. We rst set the mass parameters (m1; : : : ; mN ; ~) and -deformation to vanish. Assuming r > 0, we then have = = 0 and the remaining equations become (3.26) (3.27) (3.28) (3.29) (3.30) Solutions are labelled by the ux N X(jXj2 j=1 jYj j2) r = 2i through C = CP1 and we denote the corresponding moduli space by Mn. It is convenient to introduce the following algebraic description of the moduli space Mn. We rst remove the D-term equation (3.28) and replace it for r > 0 by the stability condition that Xj 6= 0 for all j = 1; : : : ; N except at a nite number of points on CP1. In addition, we divide by complex gauge transformations that leave the remaining equations (3.29) invariant. A point in Mn is now speci ed by N holomorphic sections (Xj ; Yj ) of O(n) O( n), such that P straightforward to compute the moduli spaces explicitly: j Xj Yj = 0 and the sections Xj are not all zero. It is now If n < 0, the moduli space is empty Mn = ;. If n = 0, we recover the algebraic description of the vacuum manifold M0 = V = T CPN 1 . If n > 0, we have Yj = 0 and the moduli space is parametrized by N holomorphic sections Xj of O(n). Using a complex gauge transformation to set Az = 0, the holomorphic sections are homogeneous polynomials Xj (z; w) = X xj;rzn rwr : n r=0 (3.31) HJEP1(207)5 The moduli space is therefore parametrized by the N (n + 1) coordinates xj;r that are not all zero, modulo residual constant C gauge transformations preserving Az = 0. We therefore nd that Mn = CPN(n+1) 1. We now consider the uctuations around a point on the moduli space Mn for n uctuations decompose into chiral and Fermi multiplets with respect to the supersymmetry algebra generated by Q and Q+. A 2d N = (2; 2) chiral multiplet of U(1)V charge r transforming as a section of a line bundle L contributes: 1. Chiral multiplets: H0(C; KCr=2 2. Fermi multiplets: H1(C; KCr=2 L). L). Here, KC is the canonical bundle of the Riemann surface C. For us, KC = O( 2). This can be summarized by the statement that the contribute H (C; KCr=2 uctuations of a 2d N = (2; 2) chiral multiplet L) to the `virtual tangent bundle' of the moduli space. Turning on (m1; : : : ; mN ; ~) and corresponds to working equivariantly with respect to the action of Tf U(1) on Mn. Let us consider the uctuations from each chiral multiplet in turn for n > 0, leaving the special case n = 0 until the end. The uctuations from each Xj transform in H (C; O(n)). There are therefore N (n+1) chiral multiplets corresponding to uctuations of the coordinates xj;r in equation (3.31) and no Fermi multiplets. Under a G Gf U(1) transformation generated by parameters ( ; m1; : : : ; mN ; ~; ), they transform with weight s ; j = 1; : : : ; N ; s = 0; : : : ; n : (3.32) The uctuations from each Yj transform in H (C; O( n)). There are therefore no chiral multiplets and N (n 1) Fermi multiplets corresponding to fermion zero modes in the vortex background. They transform with weight mj + + ~ 2 n 2 ~ 2 n 2 2 s ; j = 1; : : : ; N ; s = 0; : : : ; n 2 : (3.33) The uctuations from transform in H (C; O( 2)). There is therefore a single Fermi multiplet transforming with weight ~. T vir Mn = X N " n X e 1 comes from the vectormultiplet. The moduli space Mn = CPN(n+1) 1 has isolated xed points under a generic Tf U(1) transformation generated by (m1; : : : ; mN ; ~; ), which correspond to the N (n + 1) coordinate lines. We can label the xed points by the pair (i; r) with i = 1 : : : ; N and r = 0; : : : ; n. The xed points correspond to sections In addition there is a contribution H (O) from the vectormultiplet. Combining these contributions, the equivariant index of the virtual tangent bundle is The contribution to the partition function from uctuations around each xed point of Mn is encompassed in the virtual localization formula Z [Mn]vir whose transformation under (m1; : : : ; mN ; ~; ) is compensated by a gauge transformation r) . The equivariant index at the xed point (i; r), by = ji;r := mi T(vii;rr)Mn = ~ 2 N X j=1 s=0 ( n 2 n by the replacement ! ji;r. X emi mj+(r s) X e mi+mj+~ (r s) e ~ 1 ; (3.36) ! Z [M0]vir d ( ~) Y N j=1 ( 1 mj + ~2 )( + mj + ~2 ) ; (3.35) (3.37) (3.38) ji;r = (3.39) where we have introduced the replacement rule e : Pi niewi ! Qi wini to compute the equivariant Euler character. This result is most neatly expressed as the following contour integral Z d ( ~) Y s=1 N j=1 s s ; where the contour surrounds the poles corresponding to the xed points = mi ~ 2 2 r) . This exactly reproduces the coe cient of qn for n > 0 in the contour integral formula (3.12). Note that the Je rey-Kirwan residue corresponds to computing residues at poles of the integrand corresponding to xed points of Mn. Let us now consider the special case n = 0. The moduli space now corresponds to constant maps to the vacuum manifold V = T G(k; N ) with bosonic uctuations from both X and Y . As above, the equivariant localization expression is neatly expressed as a contour integral where the contour surrounds the poles at = mj ~2 from the contribution of Xj . This is a regular equivariant integral of 1 over the vacuum manifold V = T G(k; N ). The extension to include twisted chiral operators inserted at f g will be discussed in detail in section 4. Non-Abelian case. With gauge group U(k), we again pass to an algebraic description of the moduli space M of solutions to equations (3.20){(3.22) by removing the D-term equation in favour of a stability condition and dividing by complex gauge transformations. We therefore consider only with the stability condition that the k N matrix X has maximal rank away from isolated points on C = CP1 and modulo complex GL(k; C) gauge transformations. A point in Mn is then speci ed by: A holomorphic GL(k; C) bundle V . W ' C of the PSU(N ) avour symmetry. Constraints X Y = 0. Holomorphic sections X and Y of associated vector bundles V W and V W where CN is a trivial vector bundle associated to the fundamental representation Stability condition that rk(X) = k except at isolated points. According to a theorem of Grothendieck, on C = CP1 we can decompose V = O(n1) O(nk) X na = n ; a such that Xa become sections of O(na) W and Ya become sections of O( na) W . This leads to a strati cation of the moduli space for ux n 2 Z into components labelled by integers (n1; : : : ; nk) 2 Z k with Pa na = n. The moduli space is empty if na < 0 for any a = 1; : : : ; k. We therefore restrict attention to the region na 2 Z 0 . Fluctuations around a point on the moduli space Mn decompose into chiral and Fermi multiplets with respect to the superalgebra generated by Q and Q+. Following the discussion above, the contributions can be summarized by the equivariant index T vir = H V W 1 W 2 ~ 1 W 2 ~ + H V W + H (KC V V W 1 ~ ) H (V where we have introduced yet another trivial line bundle W~ ' C C associated to the fundamental representation of the avour symmetry U(1)~. The rst three contributions arise from the uctuations of the chiral multiplets X, Y and respectively. The nal contribution H (V V ) is the contribution from the vectormultiplet. (3.41) V ) ; (3.42) The equivariant index is straightforward to write down explicitly for any ~n 2 Zk 0. Let us write the formula for the case when na 1 for all a = 1; : : : ; k, and na 6= nb for a 6= b: T vir Mn = X e b mj+ ~2 +( n2b s) X e b+mj+ ~2 ( n2b s 1) # e bc ~+( n2bc s 1) X e cb ~+( n2bc s) # (3.43) X e bc ( n2bc s) e cb ( n2bc s 1) # : ~ 2 nb 2 s=0 nbc s=0 The moduli spaces themselves for k > 1 are singular and do not admit an explicit description as in the abelian case. It is nevertheless possible to compute the equivariant xed points in terms of the algebraic data and apply the virtual localization formula to compute the partition function. The xed points are labelled by a decomposition ~n = fn1; : : : ; nkg, a choice of vacuum I = fI1; : : : ; Ikg f1; : : : ; N g and a vector ~s = fs1; : : : ; skg where sa 2 f0; 1; : : : ; nag. The vectormultiplet scalar takes the following value at this point and the virtual localization formula is a = ajI;~s mIa Z [Mn]vir 1 = X X j~nj=n (I;~s) e(T(v~nir;I;~s)Mn) na 2 1 sa : ; (3.44) (3.45) This reproduces coe cient of qn for n > 0 in the contour integral formula (3.12) where the data f~n; I; ~sg enumerate poles of the integrand chosen by the Je rey-Kirwan description. The case n = 0 should again be treated separately and reproduces a regular equivariant integral over the vacuum manifold V = T G(k; N ). 3.3 Vortex partition function We will also consider the vortex partition function or `cigar' partition function with a xed vacuum vI at in nity. We can equivalently view this as a sphere with the boundary condition that the system sits in the vacuum vI at f g, as shown in gure 8. In section 4.3, this partition function will be used to construct the wavefunctions of spin chain Bethe eigenstates. As above, we present the partition function both as a contour integral over the complex Cartan subalgebra of the gauge group and its interpretation in terms of counting quasi-maps that are `based' at f g Let us rst consider the abelian case. The partition function with the vacuum vi at f g can be expressed as a contour integral in the vectormultiplet scalar , h1ivi = 1 1(~) Z i d q~ N Y j=1 1 ~ 2 mj + 1 ; (3.46) ~ 2 U (1)J vI at in nity. This can also be viewed as a sphere with a xed vacuum at f g where q~ = ( 1)N q. The integrand has poles at = mj ~ 2 ` for all j = 1; : : : ; N and ` 2 Z 0. The contour i selects only those poles with j = i arising from the 1-loop determinant for the chiral multiplet Xi that has a non-zero expectation value in the vacuum vi. The classical and 1-loop contributions can be factored out by normalizing by the value of the partition function at q ! 0 HJEP1(207)5 !0 h1ivI lim hf (~ )ivI = f (~ I (q)) ; The result, h1ivi h1ivi jq!0 = Vi(q) : Vi(q) = 1 X q n=0 N n n Y Y mi j=1 `=1 mj mi ~ mj (` ` 1) ; is the vortex partition function with vacuum vi at in nity. This can be generalized to the non-abelian case with vacuum vI , h1ivI = Z d k I k! q~ P a a= Q a6=b k Q a;b=1 1( ab + ) 1( ab +~) a=1 j=1 k n Y Y ~ 2 1 a mj + 1 a +mj + ; where q~ = ( 1)N q. The same integrand appears in the computation of the hemisphere partition function with the boundary condition supported on the whole of V = T G(k; N ) [19, 20]. However, the contour I surrounds only the poles arising from the 1-loop determinant of the chiral multiplets Xi for all i 2 I. As above, we can extract the corresponding vortex partition function VI (q), which we will not write down explicitly. We will denote the correlation function of a twisted chiral operator f (~ ) at f+g in the background with a supersymmetric vacuum vI at in nity f g by hf (~ )ivI . In the limit that we remove the -deformation, asymptotic behavior ! 0, such correlation functions have the common hf (~ )ivI ! e 1 Wf(~ I (q)) + : : : ; where Wf(~ ) is the e ective twisted superpotential and particular solution of the Bethe equations (2.10) associated to the aI(q) = mIa ~2 + O(q) is the xed vacuum vI at in nity. Therefore, normalizing by the vortex partition function, we nd that (3.47) (3.48) ~ 2 (3.49) (3.50) (3.51) We now perform an orbifold construction at either of the xed points f+g or f g by choosing a subgroup ZN U(1)J and turning on a discrete holonomy g = ! i 1 for the U(1)f avour symmetry for some choice of i = 0; : : : ; N 1 where !N = 1. The introduction of this orbifold is implemented at the level of the equivariant index by replacing j ! j + (i ; ! N ; and then averaging over the transformation xed. Applying this operation to the equivariant indices (4.22) in 1) < N we nd that eqf j+ if 0 < qf (i 1) < N ; e(z) = <>`~ 1 ` > P e~ z >:`~ 0 ` + `~ if 8> P e~ z + `~ if 0 < qf (i 1) < N ; 1) 0 ; Therefore, the partition function remains unchanged for g. Otherwise, the partition function is multiplied by an additional factor qf j . In this case, the orbifold construction is then equivalent to inserting the twisted chiral operator qf . Let us now explain this procedure by implementing the orbifold construction directly on the mode expansion. We focus on the xed point f+g and set the coordinate w = 1 with the understanding that transforms with weight j+ = formations. This is natural because j+ is the vacuum expectation value of the complex scalar at f+g. With this understanding, the chiral eld is expanded n under U(1)f 2 avour trans(z) = X ` 0 + ` ` z : e(z) = X ` 0 `+zqf (i 1)+` : In the presence of the orbifold, the chiral eld should transform under ZN transformations as (z) ! gqf (!z), where g = !i 1. It is therefore convenient to de ne a deformed eld ~(z) := zqf (i 1) (z), which absorbs the e ect of the discrete holonomy and transforms in the standard way ~(z) ! ~(!z). This has an expansion z ! z1=N , it is straightforward to see that The modes that are invariant under the ZN action are `+ such that qf (i some `~ 2 Z. Projecting onto ZN -invariant modes and rede ning the complex coordinate 1) + ` = `~N for (4.24) (4.25) (4.26) (4.27) (4.28) where we have de ned e+~ := ` expansion is trivial if orbifold has removed the mode + `~N qf (i 1) partition function by the corresponding equivariant weight qf j+. 0. However, in the region 0 < qf (i 1) < N the 0+ and we should therefore multiply the integrand of the . Therefore the e ect of the orbifold on the mode We now consider the supersymmetric gauge theory introduced in section 3.1 and introduce a ZN orbifold with discrete holonomy that breaks the U(k) gauge and PSU(N ) gauge and avour symmetry to a maximal torus [10]. The construction depends on the following data: HJEP1(207)5 A permutation symmetry, of f1; : : : ; N g, which speci es the discrete holonomy for the avour (gF )ij = ! (i) 1 ij : (gG)ab = !I (a) 1 ab : An ordered subset I = fI1; : : : ; Ikg f1; : : : ; N g with Ia < Ib for a < b, which speci es the discrete holonomy for the gauge symmetry, and de ne SI (~ ) := Sf1;:::;Ng(~ ). I We rst perform the orbifold construction at the point f+g. We will denote the twisted chiral operator introduced by this orbifold construction at f+g by SI( )(~ ). In the following, in order to simplify our notation, we write formulae for the unit permutation = f1; : : : ; N g The starting point for the computation is the equivariant index for contributions X X e aj+ mi+ ~2 + e aj++mi+ ~2 + e abj+ ~+ SI (~ ) = Sym k Q a=1 Ia 1 Q ( a i=1 ( N b)( a b ~) Q ( a a<b (4.29) (4.30) (4.31) (4.32) (4.34) 3 X e abj+ 5 : a6=b ! ; ! The orbifold construction is implemented by shifting the complex avour and gauge parameters according to the discrete holonomy, aj ! ! mi ! mi + (i 1) N aj + (Ia 1) N ! N ; and averaging over the transformations + 2 is for s = 0; : : : ; N 1. This operation leads to a modi cation of the equivariant index by I+ ! I+ k X Ia 1 X e aj+ mi+ ~2 + and therefore to an insertion of k X a;b=1 X a<b e aj++mi+ ~2 e abj+ +e abj+ ~ ; (4.33) ? ? ? ? in the integrand of the partition function. In writing this expression, we have symmetrized over 1; : : : ; k as the operator is inserted inside a contour integral that is symmetric in these parameters. Note that in the abelian case k = 1 with I = fig, the above expression reduces to the abelian formula (4.14) considered above. This formula can be understood graphically as explained in gure 12. In order to read o the numerator of (4.34) one draws a k N table and indicates the positions of fI1; : : : ; Ikg by ?. Then one lls in the cells to the right (light grey) with equivariant weights weights a a + mi + ~2 corresponding to elds Y ia, and the cells to the left (dark grey) with mi + ~2 corresponding to elds Xai. Finally, one multiplies all the weights in the table. The denominator of (4.34) is universal for all I for a given k. permutation inserts the operator SI( )(~ ) given by the expression Performing the orbifold construction at f+g with a holonomy labelled by a general SI( )(~ ) = S 1(I)(~ )jmj7!m (j) : (4.35) an insertion of the operator SI re ection permutation. Finally, performing the same orbifold construction at the other xed point f g leads to ( )(~ ) where : f1; : : : ; N g ! fN; : : : ; 1g is the longest or We now perform the same computation at f+g using a zero mode analysis, highlighting the additional features that appear compared with a single chiral multiplet. For simplicity, we restrict ourselves here to the zero ux sector. The computation in the general ux sector is obtained by replacing a ! aj+ = a orbifold action. na everywhere with j+ xed under the 2 In absence of the orbifold defect, we can expand the elds X, Y and around the 1 `=0 point f+g as holomorphic functions of z, Xai(z) = X(x`)ai z` ; Y ia(z) = eld starts at O(z) in the twisted theory. we can introduce the deformed elds Note that since the adjoint chiral multiplet has U(1)V charge +2, the expansion of this In the presence of the ZN orbifold with discrete holonomies (4.29) and (4.30) at f+g 1 X(y`)ia z` ; `=0 1 `=1 ab(z) = X( `)ab z` : (4.36) 1 `=0 1 `=1 Xe ai(z) = zIa i X(x`)ai z` ; Ye ia(z) = zi Ia X(x`)ai z` ; eab(z) = zIa Ib X( `)ab z` ; 1 `=0 (4.37) HJEP1(207)5 8 1 > P ( e`)ai z` ; a < b > : `=1 > P ( e`)ai z` ; a > b i > Ye ia(z) = < `=0 e 1 8 1 > P (y`)ai z` ; Ia : `=1 e > P (y`)ai z` ; Ia < i > i ; near f+g. The orbifold defect has completely changed the zero mode structure at f+g, as one can see from the above expansions. Note that before orbifolding, each component of X and Y had a zero mode at f+g, while some of the uctuations of X and Y at f+g with equivariant weights vanished there. However, the orbifold has eliminated k Y a=1 Ia 1 Y i=1 a mi + now project onto modes in the expansion that are invariant under ZN . For example, the invariant modes in the expansion of Xe ai are parametrized by (x`)a i with Ia for some `~ 2 Z. Replacing z ! z1=N and relabelling the coe cients, the invariant parts of i + ` = `~N the expansions are ; i=Ia+1 Y( a a<b b ~) 1 : Y( a a<b b) 1 ; while introducing additional zero modes for with equivariant weights Furthermore, the orbifold breaks the gauge symmetry U(k) ! U(1)k at f+g and the broken generators develop additional zero modes in the defect background parametrizing the complete ag variety Fk = U(k)=U(1)k. This leads to an additional contribution corresponding to the equivariant weight at a xed point of Fk. This combines with the contribution from to form the equivariant weight of the cotangent bundle T Fk. The symmetrization over 1; : : : ; k together with these denominator factors can be interpreted as an equivariant integral over the moduli space T Fk of the defect. The appearance of a hyper-Kahler moduli space is expected since in the absence of the mass parameter ~ the defect preserves a N = (0; 4) supersymmetry. Combining these contributions reproduces the function SI ( ) obtained in equation (4.34). 4.3 Bethe wavefunctions The functions SI ( ) are up to normalization the wavefunctions of the o -shell Bethe states for the spin chain in the up-down basis, hIj 1; : : : ; ki. The Bethe eigenstates themselves (4.38) (4.39) (4.40) (4.41) f ( ) g( ) Bf Bg are obtained by evaluating the auxiliary variables on a solution aJ of the Bethe equations, j J i = j 1J ; : : : ; kJ i. The wavefunctions of the Bethe eigenstates in the up-down basis are then hIj J i = ( 1)jIjN ( J ) 1SI ( J ) where the normalization factor N ( ) is de ned in equation (2.19). These wavefunctions arise in the supersymmetric gauge theory from expectation value of the stable basis elements SI ( ) in the cigar background with vacuum in equation (3.51). More precisely, we compute the limit function with SI ( ) inserted at the tip of cigar and vacuum vJ at in nity, ! 0 of a normalized correlation J at in nity, as !0 h1ivJ lim hSI (~ )ivJ = SI ( J ) : (4.42) We have already mentioned that this evaluates the function SI ( ) at the solution to Bethe equations (2.10) associated to the vacuum vJ , with expansion emphasize that the Bethe wavefunctions SI ( J ) can be found directly from the gauge aJ = mJa ~2 + O(q). We theory computation without solving any Bethe equations. 5 Quantum mechanical description In this section, we provide an alternative viewpoint on A-model computations on CP1 in terms of supersymmetric quantum mechanics. We replace CP1 by a long cylinder capped o by A-twisted cigars. In the cylindrical region, the theory preserves 2d N = (2; 2) supersymmetry and we can reduce on S1 to obtain an e ective N = 4 supersymmetric quantum mechanics. The capped regions become boundary conditions in the supersymmetric quantum mechanics preserving the supercharges Q ; Q+. This setup is summarized in gure 13. In principle, we can nd a nite dimensional N = 4 supersymmetric quantum mechanics for each to the zero ux sector n 2 Z individually. Here we restrict ourselves to a description ux sector n = 0, or equivalently the limit q ! 0. As we have emphasized in the introduction, much of the representation theoretic apparatus of the algebraic Bethe ansatz does not depend on the choice of quasi-periodic boundary condition speci ed by q, and should therefore have a description in this supersymmetric quantum mechanics. After a description of the N = 4 supersymmetric quantum mechanics and a general description of how to translate insertions of twisted chiral operators to boundary conditions, we will provide two explicit constructions of the boundary conditions that arise from insertions SI( )(~ ). The rst involves a combination of Neumann or Dirichlet boundary conditions coupled to boundary degrees of freedom. The second involves the notion of a thimble boundary condition. Let us rst consider the N = 4 supersymmetric quantum mechanics obtained from the ux sector n = 0 of our 2d N = (2; 2) theory, which is obtained by plain dimensional reduction of section 3.1 on a circle. In the absence of mass deformations, the supersymmetric quantum mechanics has U(1)V SU(2)A R-symmetry with U(1)A SU(2)A as the Cartan subalgebra. We denote the euclidean coordinate of the supersymmetric quantum mechanics by . We refer the reader to appendix A.2 for further details on our conventions for supersymmetric quantum mechanics. The U(k) vectormultiplet now contains scalar elds A transforming in a triplet of the SU(2)A R-symmetry. The complex combination = 1 + i 2 is the two-dimensional scalar and 3 arises from the integral of the two-dimensional gauge eld around S1. The fermion now transform as a spinor of SU(2)A. The lagrangian for the vectormultiplet Lvec = 1 A denote the SU(2)A gamma matrices and we have suppressed contractions over SU(2)A spinor indices. A chiral multiplet consists of a complex scalar and fermions transforming as a spinor of SU(2)A with lagrangian Lchiral = jD 2 D A j2 + D + i + i + jF j2 : (5.2) We have N fundamental chiral multiplets X, N anti-fundamental chiral multiplets Y , and an adjoint chiral multiplet . There is also a cubic superpotential W = Tr( XY ). The supersymmetric quantum mechanics has a vacuum manifold determined by solutions to the equations R A which coincide with con gurations annihilated by all four supercharges. As in the twodimensional theory, with r > 0 solutions are forced to have A = 0 and the vacuum manifold is V = T G(k; N ). avour symmetry around the S1. The supersymmetric quantum mechanics has avour symmetry Gf = PSU(N ) U(1)~. We can turn on SU(2)A triplets of mass parameters (m1A; : : : ; mAN ; ~A) of mass parameters by coupling to a background vectormultiplet with a vacuum expectation value for A in Tf . The components of these mass parameters are the complex masses (m1; : : : ; mN ; ~) introduced in two dimensions. In section 5.4, we will also want to turn on real mass parameters (m31; : : : ; m3N ) which arise from turning on background holonomy for the PSU(N ) In the presence of such mass parameters, equations (5.5) are deformed to ~ 2 a mj + ~ 2 mj + ) ab = 0 : (5.6) (5.7) For generic values of the mass parameters, this lifts the vacuum manifold to the isolated xed points of the Tf action on V. Empty boundary condition Before constructing boundary conditions in the supersymmetric quantum mechanics that correspond to the inserting SI( )( ) at the tip of the cigar, we rst consider the empty boundary condition associated to a cigar without any insertion. Our discussion of such boundary conditions has much in common with the description of B-type boundary conditions in 2d N = (2; 2) gauge theories [19, 20, 25]. The appropriate boundary condition for the vectormultiplet can be determined as follows. First, as we are considering the sector with vanishing ux and 3 impose the Dirichlet boundary conditions 3j = 0. The remaining boundary conditions, RS1 A, we should A j = 0 ; 3j = 0 ; D j = 0 ; 1j = 1j = 0 ; are determined by preserving the 0d N Q1 = Q and Q1 = Q+ at the boundary. Let us now consider the boundary conditions for the chiral multiplets. A 1d N = 4 chiral multiplet can be decomposed into a chiral multiplet ( ; 2) and a Fermi multiplet 1 with superpotential E ( ) = D in terms of the boundary N = (0; 2) supersymmetry algebra generated by Q and Q+. A basic boundary condition therefore involves Neumann for the chiral component and Dirichlet for the Fermi component, = (0; 2) supersymmetry algebra generated by D = 0j ; 1j = 0 ; D 2j = 0 : The components ( ; 2) transform as a N = (0; 2) chiral multiplet at the boundary. We call this a `Neumann' boundary condition. In the presence of a bulk superpotential, this boundary condition must be supplemented by a choice of matrix factorization. The boundary condition corresponding to the empty cigar can now be described as follows. We rst impose Neumann boundary conditions for X, Y and . We then couple to a 0d Fermi multiplet with the same charges as and boundary superpotentials E = j ; J = X Y j : This provides a matrix factorization of the bulk superpotential, W j = E J . Note that the boundary superpotential E = j e ectively modi es the boundary condition for from Neumann to Dirichlet. This is compatible with the zero mode analysis in section 4.2. In particular, the chiral elds X(z) and Y (z) were non-zero at z = 0 corresponding to a Neumann boundary condition, whereas (0) = 0 reproducing a Dirichlet boundary condition. (5.8) (5.9) (5.10) The `empty' boundary condition is compatible with any vacuum con guration in V. It will therefore ow to Neumann boundary conditions in the supersymmetric sigma model supported on the whole of V. In particular, the partition function on an interval with the empty boundary condition at each end reproduces the n = 0 contribution to the CP1 partition function, h1i1d = ( 1)k2 Z d k Q ( a k! k N Q Q ( a k b) Q ( a a;b=1 mi + ~2 )( ; (5.11) provide the factor ( 1)k2 Qa;b( a following the Je rey-Kirwan prescription. Stable boundary conditions which is the equivariant integral of `1' over V. From this perspective, the denominators arise from the N = (0; 2) chiral multiplets X; Y . The contributions coming from two Fermi multiplets at two boundaries cancel the contribution from the chiral multiplet and ~) in the numerator. The remaining numerator b) is the vector multiplet contribution. The contour is chosen by We now consider the class of boundary conditions that arise from inserting SI ( ) at the tip of the cigar. Flowing to a supersymmetric sigma model to the vacuum manifold V, such boundary conditions are supported on holomorphic lagrangian submanifolds in V that are xed by Tf . In this section, we provide an elementary description of these boundary conditions in the supersymmetric gauge theory by coupling the empty boundary condition to additional boundary degrees of freedom. The construction is similar in spirit to the exceptional Dirichlet boundary conditions introduced in [26]. Let us rst consider the abelian case. Following the arguments of section 4.1, let us rst consider the boundary condition obtained by inserting the homogeneous polynomial mj + ~2 at the tip of the cigar, corresponding to the equivariant weight of the coordinate Xj . In the vanishing ux sector, this is a polynomial representative of the equivariant cohomology class Poincare dual to fXj = 0g T CPN 1 . This boundary condition is described in the supersymmetric quantum mechanics by coupling the empty boundary condition to a 0d N = (0; 2) Fermi multiplet j with the same charges as Xj with boundary superpotentials E j = Xj ; J j = 0 : The 0d N = (0; 2) chiral multiplet part of Xj will receive a mass from the superpotential at the boundary modifying the boundary condition for Xj from Neumann to Dirichlet. This boundary condition is therefore supported on the subspace fXj = 0 g of the vacuum manifold. In the computation of partition functions, the boundary Fermi T CPN 1 multiplet j provides an additional contribution mj + ~ 2 ; (5.12) (5.13) in the numerator of the integrand. This reproduces the e ect of inserting the twisted chiral operator f ( ) = mj + ~2 at the tip of the cigar. It is now straightforward to write down the boundary condition corresponding to inserting the twisted chiral operator SI ( ) with I = fig in abelian theories. We deform the empty boundary condition by introducing (N with the boundary superpotentials 1) boundary Fermi multiplets j for j 6= i E j = (Xj if j < i Yj if j > i ; J i = 0 : The charges of the Fermi multiplets are uniquely determined by their superpotentials. The boundary condition is now supported on the subspace HJEP1(207)5 Xj = 0 for j < I ; Yj = 0 for j > I ; of the vacuum manifold V. A partition function with this boundary condition will include an additional contribution compared to the empty boundary condition, I 1 Y i=1 mi + ~ 2 N Y i=I+1 + mi + ; ~ 2 from the additional contributions of the Fermi multiplets j . Therefore, this boundary condition reproduced the computation with SI ( ) inserted. Note that due to the symmetry ( j ; Ej ; Jj ) $ ( j ; Jj ; Ej ), we could equivalently have coupled to boundary fermi multiplets with J -type superpotentials. This boundary condition can be easily generalized to non-abelian theories in a manner consistent with the zero mode analysis of section 4.2. The boundary condition for SI (~ ) with I = fI1; : : : ; Ikg should be supported on the subspace of V de ned by Xaj = 0 for j < Ia ; Y j a = 0 for j > Ia : These constraints can be implemented by introducing k(N 1) Fermi multiplets denoted by aj for j < Ia and j a for j > Ia with superpotentials (E )aj = Xaj for j < Ia ; (E )ja = Y j a for j > Ia ; together with J = 0. Recalling that Ia Ib if and only if a b, these constraints imply that the complex moment map for the gauge symmetry is upper triangular at the boundary, (X Y )abj = < 8 > >:0 ; X Ia<j<Ib Xaj Y j bj ; if a b ; if a > b : Therefore the matrix factorization requires that we introduce only k(k+1) Fermi multiplets at the boundary, with components ab for a 2 b, with boundary superpotentials (5.14) (5.15) (5.16) (5.17) (5.18) (5.19) SI ( ) = k Q Q ( a i=1 mi + ~2 ) IaQ+1( N ! Here, the numerator factors are the contributions from the boundary Fermi multiplets a and j a. The denominator factors arise from additional bosonic zero modes of a < b and from the breaking of the gauge symmetry. As mentioned in section 4.2, the denominator factors combine to form an equivariant weight of the tangent bundle at a xed point of T Fk, and symmetrizing over 1; : : : ; k, the result can be interpreted as an equivariant integral over this boundary moduli space. Thimble boundary conditions In this section, we turn on real mass parameters (m31; : : : ; m3N ) valued in the Cartan subalgebra of the PSU(N ) avour symmetry. These parameters arise from a background avour holonomy around S1 in the two-dimensional setup. In the presence of real mass parameters, there is a natural class of boundary conditions for the supersymmetric quantum mechanics preserving Q , Q+, which are analogous to thimble boundary conditions in 2d N = (2; 2) theories [27{29]. We will need a slight generalization of the standard notion appropriate for theories with multiple isolated vacua connected by gradient ows [26]. We will adapt this construction here to the context of 1d N = 4 supersymmetric quantum mechanics. We expect this to provide a physical counterpart to the construction of stable envelopes [11]. In the presence of real mass parameters fm31; : : : ; m3N g, the con gurations of the supersymmetric quantum mechanics preserving Q , Q+ are (J )ab = X Y j and (E )ab = boundary from the components ab with a < b. abj. This leads to additional bosonic uctuations at the In addition, the boundary condition breaks complex gauge transformations GC = GL(k) to the Borel subgroup B of upper triangular transformations. There are therefore additional bosonic uctuations at the boundary parametrizing Fk = GC=B, which is complex version of the breaking of U(k) to its maximal torus described in section 4.2. As mentioned there, these uctuations combine with those of a bj for a < b to form the cotangent bundle T Fk, whose hyper-Kahler structure is a re ection of the fact that the boundary condition preserves a N = (0; 4) supersymmetry in the limit ~ ! 0. In the computation of interval partition functions, this boundary condition leads to an additional contribution compared to the empty boundary condition, R r1 = D X = Xy D X = 3 X + X m3 ; D Y = Y = These equations can be reformulated as the gradient ow equations (5.20) j a bj for (5.21) (5.22) where h = R;m ; (5.23) and R and R;m denote the moment maps for the gauge and avour symmetry respectively. The notation X refers to one of the elds fX; Y; ; 3 g and gXXy is the inverse metric on the space of elds inherited from the Lagrangian. Note that inside of gauge invariant combinations of the elds X and Y , we have (5.24) and therefore gauge invariant combinations of X and Y will grow or decay along the direction according to their charge under the avour transformation generated by m3. This is gradient ow on the vacuum manifold V for the Morse function hm = m3 A preliminary de nition of a left thimble boundary condition BI on be given as follows: it is a boundary that is equivalent for the computation of correlation functions preserving Q , Q+ to the placing theory on 1 with a xed isolated vacuum vI at vacuum vI at the submanifold of points in V that can be reached by an in nite gradient ow from the parameters. Suppose that the real masses are ordered as m3(1) < < m3(N) for some 1. This submanifold clearly depend on the ordering of the real mass 1. The support of such a boundary condition is in the rst instance 0 can now permutation , then we denote this submanifold by V 1 to However, in passing from = 0, we will need to allow for a sequence of domain walls preserving Q , Q+ that interpolate between di erent isolated vacua connected by gradient ows. In order to formalize this notion, we can introduce a partial ordering on the set of isolated vacua fvI g depending on the permutation , by the requirement (I ). there exists a -gradient ow vI ! vJ vI < vJ ; and extending transitively, namely if vI < vJ and vJ < vK then also vI < vK . Allowing for sequences of domain walls interpolating between vacua, the support of a left thimble boundary condition BI is V(BI) = [ vI vJ V (J ) : We claim that the thimble boundary condition BI in the presence of mass parameters corresponds to boundary condition constructed in section 5.3 ordered by the permutation associated to inserting SI( )( ). Let us illustrate this construction with an abelian example: k = 1 and N = 2 with V = T CP1 | see table 1. Introducing a real mass parameter m3 for the U(1)m symmetry, we have hm = m3( jX1j2 + jX2j2 + jY1j2 jY2j2) : Note that in the graphical representation of V = T CP1 in gure 10, the Morse function is proportional to the coordinate along the horizontal axis. The gradient ows are therefore straightforward to understand in this graphical representation. (5.25) (5.26) avour (5.27) V{B22,1} (5.28) (5.29) HJEP1(207)5 = f1; 2g corresponds to m3 < 0 and the permutation = f2; 1g corresponds to m3 > 0. In the graphical representation, the direction of ow for increasing is from left to right for = f1; 2g and right to left for = f2; 1g. The vacua are therefore ordered such that v1 < v2 for straightforward to see that = f1; 2g and v2 < v1 for = f2; 1g. First, it is Vf11;2g = CP1 Vf21;2g = F2 ; fv2g ; Vf12;1g = F1 ; Vf22;1g = CP1 fv1g ; where Fi denotes the ber of V = T CP1 at the xed point vi. We therefore generate thimble boundary conditions with support VfB11;2g = Vf11;2g [ Vf21;2g = CP1 [ F2 ; VfB21;1g = Vf22;1g = F1 ; VfB12;2g = Vf21;2g = F2 ; VfB22;1g = Vf22;1g [ Vf12;1g = CP1 [ F1 ; = f1; 2g, and Sf2;1g( ) and Sf2;1g( ) for 1 2 = f2; 1g. which are illustrated in gure 14. The supports of these boundary conditions clearly coincide with those obtained from the cigar with insertions of S1f1;2g( ) and Sf1;2g( ) for 2 Finally, let us attempt to make a general statement. We expect that the left thimble boundary condition BI generated by real mass parameters m3(1) < m3(N) is equivalent for the purpose of computing correlation functions preserving Q , Q+ to the boundary condition in section 5.3 corresponding to S ow equations in the same way as mj3 ! mj3, the right thimble boundary condition for the same mass parameters reproduces the function SI ( ), where : f1; : : : ; N g ! fN; : : : ; 1g ( ) I( )( ). Since ! transforms the gradient is the longest permutation. In this section we return to studying the Heisenberg spin chain and the question of how the R-matrix arises in the study of supersymmetric gauge theory. In particular, we will examine the two-point functions of stable basis elements S I( )(~ ) in the A-twisted supersymmetric gauge theory on the sphere. These correlation functions are in fact independent of q and can therefore be interpreted as the partition function of a supersymmetric quantum mechanics of section 5 on a interval with thimble boundary conditions at either end. Orthonormality of stable basis and S Let us rst consider the two-point correlation functions of basis elements SI( )(~ ) at f+g ( )(~ ) at f g where : f1; : : : ; N g ! fN; : : : ; 1g is the longest permutation. This J correlation functions are independent of q and evaluate to hSI( )(~ )SJ( )(~ )iS2 = I;J : The appearance of the re ection here is natural from the orbifold construction. The orbifold construction at f+g producing SI( )(~ ) has avour holonomy (gF )ij = ! (i) 1 ij . This is compatible with turning on form ei i ij such that (1) < avour holonomy in a neighbourhood of f+g of the (N). Translating to f g without allowing these holonomy eigenvalues to cross, the compatible orbifold construction leads to functions SI Similarly in the supersymmetric quantum mechanics, introducing constant real masses ordered such that m3(1) < < m3(N) leads to left thimble boundary conditions generating ( )(~ ). SI( )(~ ) and right thimble boundary conditions generating S I ( )(~ ). This observation motivates to de ne an inner product, (6.1) (6.2) (6.3) (6.4) where the conjugation sends a ! a; mi ! therefore the stable basis elements for a given permutation are orthonormal, mi. In particular, SI( )(~ ) = S I ( )(~ ) and hf (~ ); g(~ )i = hf (~ ); g(~ )iS2 ; hSI( )(~ ); SJ( )(~ )i = I;J : By construction, this inner product depends only on the functions f (~ ) and g(~ ) modulo the twisted chiral ring relations. Under the correspondence outlined in the introduction 1, this corresponds to the inner product on the spin chain Hilbert space,3 V ( ) = Cm (1) 2 2 Cm (2) : : : 2 Cm (N) ; with sites ordered according to the permutation . In particular, equation (6.3) corresponds to the orthonormality of the up-down basis of the spin chain, hIjJ i = I;J . 3In the supersymmetric gauge theory, we have allowed the parameters ( 1; : : : ; k), and (m1; : : : ; mN ; ~) to be complex. In order to recover the honest inner product on the spin chain Hilbert space, we would need to specify certain reality conditions on these parameters such that they map exactly onto the corresponding spin chain parameters. 6.2 The natural next step is to consider the inner product of stable basis elements for di erent permutations and 0. These correlation functions are again independent of q. Let us therefore consider this problem from the perspective of the supersymmetric quantum mechanics of section 5. (m31; : : : ; m3N ) vary as a function of Consistency of such a correlation function requires that the mass parameters across an interval, such that they are ordered by the permutation at the left boundary and by 0 at the right boundary. This is an exact deformation and therefore correlation functions do not depend on the particular pro le HJEP1(207)5 of this variation. In particular, we can say that such a correlation function requires the presence of a `Janus interface' J ; 0 for the real mass parameters. The inner product hSI( )(~ ); SJ( 0)(~ )i ; One can immediately recognize this table as the matrix elements of the spin chain R2 matrix (2.13) acting on Cm1 sets of correlation functions C2m2 , up to a sign. Summarizing this example, there are two which can be represented graphically as in gure 15. The rst line consists of correlation functions consistent with constant real masses ordered by the permutation f1; 2g, whereas the second line contains correlation functions consistent with the presence of a Janus interface Jf2;1g;f1;2g. is computing the partition function of the supersymmetric quantum mechanics on a interval with the interface J ; 0 between thimble boundary conditions generated by the vacua vI on the left and vJ on the right. In order to develop the connection between such correlation functions and spin chain quantities, let us consider the simplest example corresponding to a spin chain of length N = 2. In that case we have two distinct permutations, f1; 2g and f2; 1g, and three supersymmetric theories with k = 0, 1 and 2. The correlation functions hSIf2;1g(~ ); SJf1;2g(~ )i ; are straightforward to evaluate explicitly. The result is summarized in the following table Sf2;1g fg Sff12g;1g Sff22g;1g Sff12;;21gg Sf1;2g fg Sff11g;2g Sff21g;2g Sff11;;22gg 1 0 0 0 m1 m1 + ~ m1 + ~ 0 m1 + ~ m2 m1 + ~ 0 0 0 1 hSIf1;2g(~ ); SJf1;2g(~ )i = IJ ; hSIf2;1g(~ ); SJf1;2g(~ )i RIJ (m1 m2) ; (6.5) (6.6) (6.8) : (6.7) 1 2 1 2 a spin chain of length N = 2. In order to extend this correspondence to supersymmetric gauge theories with N > 2, we will need to introduce the notion of a Weyl R-matrix depending on a pair of permutations , sitions. functions where In order to de ne it, let decompose the + 1 = (i1 j1) (i2 j2) : : : (iL jL) : R( +; )(m1; : : : ; mN ) := R (i1) (j1) m : : : As a consequence of the Yang-Baxter equation (2.12) and the unitarity condition (2.14) the Weyl R-matrix is independent of the choice of decomposition into elementary transpoWe have found that the components of the Weyl R-matrix coincide with the correlation RI(J+; )(m1; : : : ; mN ) = NIJ ( +; ) h I S( +)(~ ); SJ( )(~ )i ; NIJ ( +; ) = ( 1)j +1(I)j+j 1(J)j ; is a sign. We have performed extensive checks of this relation in numerous examples. In the supersymmetric quantum mechanics setup of section 5, the matrix elements of the Weyl R-matrix are therefore identi ed with the partition function of a Janus interface J +; between thimble boundary conditions generated by the vacua vI and vJ . We end this section with an example of a Weyl R-matrix. Let us x the spin chain length N = 5 and choose permutations decompose the permutation 1 = (25) (34) (24) (23) and therefore de ne + = f1; 4; 3; 5; 2g and = f1; 2; 3; 4; 5g. We can R(f1;4;3;5;2g;f1;2;3;4;5g)(m1; : : : ; m5) = R25(m52)R34(m43)R24(m42)R23(m32) ; (6.14) which can be straightforwardly computed from the matrix elements of the elementary Rmatrices, Rij (mji). This R-matrix can be depicted as in gure 16. Equivalently, the same matrix elements can be computed from the correlation functions RI(fJ1;4;3;5;2g;f1;2;3;4;5g)(m1; : : : ; m5) = NIJ hSIf1;4;3;5;2g(~ ); SJf1;2;3;4;5g(~ )i ; (6.15) 1 as a convolution of elementary transpositions : : : R (iL) (jL) m (6.9) (6.10) (6.12) (6.13) 1 2 5 Weyl R-matrix is independent of the decomposition as a consequence of the Yang-Baxter equation ( gure 5a) and unitarity ( gure 5b). which can be evaluated by directly performing the contour integral in (3.16). Both of these computations give the same result. Yang-Baxter equation Janus interface J +; the real masses mi3 and m3, j As explained in the previous section, we have performed extensive checks that the Weyl R-matrix RIJ ( +; )(m1; : : : ; mN ) corresponds to matrix elements of a Janus interface J +; in the supersymmetric quantum mechanics setup of section 5. In the same manner that the Weyl R-matrix RIJ elementary R-matrices according to a decomposition ( +; )(m1; : : : ; mN ) is constructed from 1 = (i1 j1) : : : (iL jL), the is a composition of elementary Janus interfaces Jij that interchanges J +; = Ji1j1 : : : JiLjL : (6.16) This can be understood since deformations of the pro le m31( ); : : : ; m3N ( ) for the real mass parameters are exact in Q , Q+. We are therefore free to choose a pro le consisting of a sequence of `jumps' where pairs of mass parameters mi3 and mj3 are interchanged. Each of these jumps can be regarded as an elementary Janus interface Jij . Inserting the complete set of states provided by the stable basis SI( )(~ ) in between each elementary Janus interface then reproduces the decomposition (6.11) of the Weyl R-matrix. Equation (6.16) can therefore be understood as a basis-independent statement of this decomposition. The fact that the Weyl R-matrix is independent of the choice of decomposition into elementary transpositions followed from the Yang-Baxter equation (2.12) and the unitarity condition (2.14). From the perspective of supersymmetric quantum mechanics, this property is guaranteed since di erent decompositions of a pro le m31( ); : : : ; m3N ( ) into elementary jumps are related by exact deformations. In particular, we have and where I is an identity interface preserving the order of the real mass parameters. This is a basis-independent statement of the Yang-Baxter equation and unitarity relation. The standard equations for R-matrices are recovered by inserting the complete set of states provided by the stable basis SI( )(~ ) in between each elementary Janus interface. Jij Jik Jjk = Jjk Jik Jij ; Jij Jji = I ; (6.17) (6.18) Discussion In this paper, we have investigated aspects of the correspondence between XXX 1 Heisenberg spin chains and 2d N = (2; 2) supersymmetric gauge theories. We have focussed 2 on reproducing components of the algebraic Bethe ansatz for spin chains from correlation functions in A-twisted supersymmetric gauge theory and their reduction to partition functions in N = 4 supersymmetric quantum mechanics. In particular, we have provided a concrete construction of the wavefunctions of o -shell Bethe states as orbifold defects in A-twisted supersymmetric gauge theory, and as thimble boundary conditions in supersymmetric quantum mechanics. We have also developed a new interpretation of the spin chain R-matrix as the matrix elements of Janus interfaces for mass parameters, leading to a novel and basis-independent presentation of the Yang-Baxter equations. Let us conclude with some directions for further research: First, there are some important components of the algebraic Bethe ansatz that we have omitted from our presentation. One example is the generators of the Yangian symmetry of the spin chain. Unlike the R-matrix, Yangian generators have non-vanishing matrix elements between spin chain states with di erent number of excitations. On the supersymmetric side of the correspondence, this will correspond to correlation functions of interfaces that change the rank of the gauge group, U(k) ! U(k0). It is straightforward to construct such interfaces in the supersymmetric quantum mechanics description of section 5, following methods introduced in [30]. However, we expect a complete discussion of Yangian representation theory and the algebraic Bethe ansatz will arise from `tri-partite' interfaces in supersymmetric quantum mechanics relating theories (k; N ), (k0; N 0) and (k00; N 00) with di erent gauge and avour symmetries [31]. Secondly, in this paper we have considered only su(2) spin chains with the fundamental representation at each site. It would be interesting to extend the results presented here to more general groups and representations, by studying more general supersymmetric quiver gauge theories. Finally, it would be interesting to extend our results to trigonometric or elliptic spin chains, corresponding to three and four dimensional supersymmetric gauge theories. The corresponding localization techniques for correlation functions of twisted theories on S2 S1 or S2 T 2 have been developed in [7, 32, 33]. Acknowledgments We would like to thank Stefano Cremonesi for useful discussions, and the organizers of the Pollica summer workshop for kind hospitality while this paper was being completed. MB and TL are supported by ERC STG grant 306260. HK would like to thank Mathematical Institute at University of Oxford and the 2017 Summer Workshop at the Simons Center for Geometry and Physics for their hospitality and support during di erent stages of this work. The research of HK is supported in part by NSF grant PHY-1067976. Az = Az = 2 2 i i( + i( + D = +Dz + i 2 +[ ; +); Dz 2 1 2 1 2 1 2 1 2 + = i + D 2iFzz + [ ; ] + 2 Dz ; = i D + 2iFzz [ ; ] 2 +Dz ; + = i + D + 2iFzz + [ ; ] + 2 Dz ; D 2iFzz [ ; ] 2 +Dz ; Conventions 2d N = (2; 2) supersymmetry We consider two-dimensional N = (2; 2) supersymmetric theory on a at space with Euclidean coordinates (x1; x2). We will also introduce a complex coordinate z = x1 + ix2. Our conventions are taken directly from appendix A of [25] with x0 = The supersymmetry transformations of a vectormultiplet are (A.2) (A.4) (A.6) (A.7) (A.8) (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) (A.17) +Dz + + Dz 2 +[ ; = + + ; + ; + = 2i Dz + +F = 2i +Dz + F + Dz + +F 2i +Dz + F + ; ; ; ; + = F = F = 2i 2i + ( + 2i ( + Dz Dz + 2i +Dz + + 2i +Dz + +) + i( + ) + i ( ) ; ) ; For a chiral multiplet transforming in a unitary representation of the gauge group G, the supersymmetry transformations are where D = @ + iA and it is understood that vectormultiplet elds act in the appropriate representation of G. LV = Tr d Z 1 4 and FI term 1 2e2 The chiral multiplet Lagrangian is L = Tr Z d + total derivative = Tr 2Dz Dz 2Dz Dz +i and superpotential term 2Dz Dz 2Dz Dz +2i Dz 2i +Dz + +4Fz2z +D2 (A.18) LF I = Re Z d2 ~( t ) = rD 2 Fzz : HJEP1(207)5 i + +i +i + + + [ ; ] i +Dz + + D +jF j2 j j 2 1 2 Dz Z LW = Re d 2 W ( ) : (A.19) (A.21) (A.22) (A.23) (A.24) (A.25) (A.26) Writing a general supersymmetry transformation as we nd that = i( +Q Q+ fQ ; Q g = 2iDz fQ ; Q+g = fQ+; Q g = 0 Q2 = 0 fQ+; Q+g = fQ+; Q g = fQ+; Q g = 0 Q 2 = 0 : 2iDz The charges of the supersymmetry generators under U(1)J rotations and the axial U(1)A and vector U(1)V R-symmetries are shown below: Q+ Q+ Q Q 1 1 +1 +1 1 1 +1 +1 U(1)V U(1)A 1 +1 +1 1 U(1)0J 2 0 0 +2 In order to write the N = (2; 2) supersymmetry algebra as a 1d N = 4 supersymmetry algebra, we compactify a spatial direction on a circle x1 x1 + 2 R and rename = x2. We can then organize the supercharges into spinors Q Q+ Q Q Q+ Q combining supercharges of U(1)V charge 1 and +1 respectively. Note that the top (resp. bottom) components of both spinors have charge +1 (resp. 1) under U(1)A. With this notation, the supersymmetry algebra with Z = 0 can be re-expressed as follows where ( I ) mation is given by are the Pauli matrices. In terms of these elds, the supersymmetry transforwhere fQ ; Q g = fQ ; Q g = 0 ; D + Z 0 1 0 1 Z U(1)V This takes the form of an N = 4 supersymmetric quantum mechanics with R-symmetry SU(2)A R-symmetry. States with KK momentum in the x1 direction clearly break SU(2)A to the U(1)A axial R-symmetry in two dimensions. We now write the supersymmetry transformations of the elds in SU(2)A covariant notation. We need to choose a convention for raising and lowering indices and will choose 12 = 21 = 1 with transformations in SU(2)A covariant notation as and . 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Mathew Bullimore, Hee-Cheol Kim, Tomasz Lukowski. Expanding the Bethe/Gauge dictionary, Journal of High Energy Physics, 2017, 55, DOI: 10.1007/JHEP11(2017)055