Topologically twisted SUSY gauge theory, gauge-Bethe correspondence and quantum cohomology

We calculate the partition function and correlation functions in A-twisted 2d N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = (2, 2) U(N) gauge theories and topologically twisted 3d N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 U(N) gauge theories containing an adjoint chiral multiplet with particular choices of R-charges and the magnetic fluxes for flavor symmetries. According to the Gauge-Bethe correspondence, they correspond to the Heisenberg XXX1/2 and XXZ1/2 spin chain models, respectively. We identify the partition function with the inverse of the norm of the Bethe eigenstate. Correlation functions are identified to coefficients of the expectation value of Baxter Q-operator. In addition, we consider correlation functions of 2d N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = (2, 2)* theories and their relations to the equivariant integration of the equivariant quantum cohomology classes of the cotangent bundle of Grassmann manifolds and the equivariant quantum cohomology ring. Also, we study the twisted chiral ring relations of supersymmetric Wilson loops in 3d N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2* theories and the Bethe subalgebra of the XXZ1/2 spin chain models.


Introduction
The Gauge-Bethe correspondence states that quantum integrable models correspond to supersymmetric gauge theories. The XXX Heisenberg spin chain model was considered as one of the primary examples of the Gauge-Bethe correspondence in the original papers [1,2]. It was argued that the supersymmetric vacua of the softly broken 2d N = (4, 4) U(N ) gauge theory by the mass of the adjoint chiral multiplet, usually called 2d N = (2, 2) * U(N ) gauge theory, is naturally identified with the Bethe ansatz equation for the XXX 1/2 spin chain model. Also, the twisted superpotential was identified with the Yang-Yang potential.
In this paper, we study 2d N = (2, 2) and 3d N = 2 theories containing an adjoint chiral multiplet with two different choices of R-charges and background magnetic fluxes but with same gauge group and matter contents. We calculate partition functions of A-twisted 2d N = (2, 2) theories on S 2 and partition functions of topologically twisted 3d N = 2 theories on S 1 × S 2 with all the equivariant parameters associated to flavor symmetries turned on but with the equivariant parameter associated to the rotational symmetry on S 2 turned off. We match them with the inverse of the norm of Bethe eigenstates by choosing particular R-charges and background fluxes for flavor symmetries. The gauge invariant operators form a twisted chiral ring and expectation values of them provide the coefficient of the expectation value of the Baxter Q-operator. Thus, with a proper choice of coefficients, the expectation value of gauge invariant operators provide the expectation value of conserved charges of the corresponding spin chain model.
We also calculate correlation functions of the A-twisted 2d N = (2, 2) * theory whose target space (in the nonlinear sigma model limit) is the cotangent bundle of the Grassmannian for several examples. We calculate the equivariant integration by using the results in [9] where they showed that the Bethe subalgebra of the XXX spin chain model is isomorphic to the equivariant quantum cohomology ring, 1 and check that the result is consistent with correlation functions of the A-twisted 2d N = (2, 2) * theory and also with the Seiberg-like duality.
It was shown in [10] that the Bethe subalgebra of the XXZ spin chain model is given by certain generators and relations analogous to the equivariant quantum cohomology ring in [9]. 2 With the Gauge-Bethe correspondence in mind, we see that the Wilson loop algebra agrees with the Bethe subalgebra of the XXZ 1/2 model by checking several examples. Also, we consider the Seiberg-like duality of the 3d N = 2 * theory in the context of the Bethe subalgebra of the XXZ 1/2 model.
In the final section, we conclude with a summary of our results and discuss some future directions.
2 The gauge-Bethe correspondence and the Bethe norm Given 2d N = (2, 2) gauge theories, the condition for the supersymmetric vacua is given by exp 2πi ∂ W eff (σ) ∂σ a = 1 , (2.1) 1 They considered general partial flag manifolds and the Grassmannian is a part of them. 2 The Bethe subalgebra of the XXZ spin chain model was conjectured to be identical to the equivariant quantum K-theory ring [10].

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where W eff (σ) is the effective twisted superpotential. According to the Gauge-Bethe correspondence [1,2,11], it is identified with the Bethe ansatz equation of a certain integrable model. Also, the twisted superpotential W eff (σ) of 2d N = (2, 2) theories corresponds to the Yang-Yang potential of the corresponding integrable model. For the isotropic SU(2) Heisenberg XXX 1/2 spin chain model and similarly for the anisotropic XXZ 1/2 spin chain model where spin-1/2 degree of freedom of SU(2) is attached to each sites, twisted mass parameters for flavor symmetries are related to parameters for the displacement of lattice sites with respect to the symmetric round lattice configuration.
In this section, we relate the norm of the Bethe eigenstates of the XXX 1/2 and the XXZ 1/2 spin chain model to the partition function of a certain topologically twisted 2d N = (2, 2) and 3d N = 2 theory, respectively. We also discuss coefficients of the expectation value of the Baxter Q-operator and conserved charges in terms of correlation functions.

2.1
The norm of the Bethe eigenstate in the XXX 1/2 and the XXZ 1/2 spin chain model We are interested in the inhomogeneous XXX 1/2 and XXZ 1/2 spin chain model with M lattice sites. 3 The monodromy matrix, T(λ), of the XXX 1/2 and the XXZ 1/2 model takes a form of a 2 × 2 matrix acting on the 2-dimensional auxiliary space V where λ is a spectral parameter. Therefore the transfer matrix, τ , which is given by the trace of monodromy matrix is τ (λ) = A(λ)+D(λ). With the quasi-periodic boundary condition S M +1 = e where S a = 1 2 σ (a) are generators at the a-th site and σ are the Pauli matrices, the transfer matrix is given by A(λ) + e iϑ D(λ) [1,14].
The R-matrix of the XXX 1/2 and the XXZ 1/2 model is for the XXX 1/2 model where c is an auxiliary parameter, and for the XXZ 1/2 model where η is related to the anisotropy parameter ∆ = cos 2η, 0 < 2η ≤ π.

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The R-matrix satisfies the Yang-Baxter equation acting on the auxiliary space where T a acts on the auxiliary space V a , and this provides the commutation relations of matrix elements of the monodromy matrix T(λ). Also, from (2.8) and due to the trace identities, one can show that the transfer matrix, τ (λ), commutes with the Hamiltonian, Therefore τ (λ) is a generating function of conserved charges. As they commute, eigenfunctions of the transfer matrix are also eigenfunctions of the Hamiltonian. The pseudo-vacuum |0 satisfies the following conditions where a(λ) and d(λ) are called the vacuum eigenvalues. For the Heisenberg spin chain model, the pseudo-vacuum |0 is given by the state with spins being all up or all down.
The Bethe eigenstate. Consider a state that is obtained by acting an operator B on the pseudo-vacuum where N is the number of particles or excitations. This state becomes the eigenvector of the transfer matrix when spectral parameters, λ a , satisfy the Bethe ansatz equation, and the eigenvector is called the Bethe eigenstate. The dual vector of |Ψ N (λ) is defined by The vacuum eigenvalues, a(λ) and d(λ), of the inhomogeneous XXX 1/2 and XXZ 1/2 model are and respectively, and the Bethe ansatz equation is respectively, for the quasi-periodic boundary condition.
The norm of the Bethe eigenstate for XXX 1/2 model. The norm of the Bethe eigenstate [15] is given by .
For the inhomogeneous XXX 1 2 spin chain model, and we obtain (2.21) Therefore, the inverse of the norm of the Bethe eigenstate is given by Here P XXX is a set of independent solutions of (λ) := (λ 1 , · · · , λ N ) satisfying the Bethe ansatz (2.15) with the quasi-periodic boundary condition.
The norm of the Bethe eigenstate for XXZ 1/2 model. The norm of the Bethe eigenstate for the XXZ 1/2 model [15] can be obtained similarly as in the case of the XXX 1/2 JHEP02(2019)052 (2.24) In terms of ϕ ′ , the inverse of the norm of the Bethe eigenstate is given by and P XXZ is a set of independent solutions of (λ) := (λ 1 , · · · , λ N ) satisfying the Bethe ansatz equation (2.16) with the quasi-periodic boundary condition.

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The partition function of the A-type topologically twisted theory can be calculated by using the formula in [3]. In the following calculation, we turn off the background value of the graviphoton associated to S 2 .
The one-loop contributions from the chiral, anti-chiral, and adjoint chiral multiplets are given by and the one-loop contribution from the vector multiplet is We denote a constant configuration of the scalar in the vector multiplet as σ = diag(σ 1 , · · · , σ Nc ). The partition function of A-twisted gauged linear sigma models on S 2 is given by Here the choice of the contour is specified by the Jeffrey-Kirwan residue prescription, which depends on the choice of the covector η ∈ R Nc . The parameter q is the exponential of the complexified FI-parameter q := exp 2πiτ = exp 2πi θ 2π + iξ and Z 1-loop total (k) is By choosing a covector η, for example, to be η = (−1, · · · , −1), the one-loop determinants of anti-chiral multiplets and an adjoint chiral multiplet contribute to Jeffrey-Kirwan residues. Poles from anti-chiral multiplets exist when k a < n i − 1 2 l − r 2 + 1. Summing over k a < K first for a sufficiently large positive integer K, the partition function is expressed as where W eff is the effective twisted superpotential Due to the factor exp(2πi∂ σa W eff ) K with large K in numerator, there are no poles at −σ a + m y i − 1 2 m z = 0 and σ a − σ b + m z = 0 and only poles at exp(2πi∂ σa W eff ) − 1 = 0 contribute. Then, dependence on K disappears and we obtain where P 2d := {(σ 1 , · · · , σ Nc ) | exp(2πi∂ σa W eff ) = 1 for all a = 1, . . . , N c }/S Nc (2.37) (2.40) In P 2d , we identify solutions which are the same up to Weyl permutations, S Nc , of (σ 1 , · · · , σ Nc ). And the condition for supersymmetric vacua, exp(2πi∂ σa W eff ) = 1, is given by the partition function of the A-twisted 2d N = (2, 2) gauge theory 5 and the inverse of the norm of the Bethe eigenstate (2.22) agree up to an overall factor. This type of relation was first studied for the U(N )/U(N ) gauged WZW model on genus-g Riemann surfaces Σ g in [16] where the corresponding integrable model is the phase model. See also [17][18][19][20].
Correlation functions, the Baxter Q-operator, and conserved charges. We have identified the partition function and the norm of the inverse of the Bethe eigenstate. We can also consider correlation functions of the A-twisted 2d N = (2, 2) theory discussed in section 2.2 in the context of the Gauge-Bethe correspondence.
In the A-twisted 2d N = (2, 2) theory, correlation functions of gauge invariant operators O(σ) are given by This can also be written as (2.47) 5 For example, we can choose all background fluxes and R-charges to be zero and don't include the superpotential QΦQ in the theory. The canonical assignment of the R-charge for superpotential QΦQ is not allowed if we want to match the A-twisted partition function and the inverse of the norm of the Bethe eigenstate. Indeed if we sum three conditions in (2.43), we obtain However, as flavor symmetries are SU(N f ) instead of U(N f ), we have r1 + r2 + R = 0. Therefore, the canonical assignment of the R-charge such as r1 = r2 = 0 and R = 2 is not allowed for the match with the inverse of the norm. Also note that, given same matter contents, the Bethe ansatz equation is same whatever the R-charges and background magnetic fluxes are.

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The operator O(σ) is provided by gauge invariant polynomials of the Cartan of the scalar component σ of the vector multiplet, which is a symmetric function of σ a , a = 1, · · · , N c . Thus it can be written in terms of the elementary symmetric polynomials. We denote the polynomial Q(x) as then the coefficients of x Nc−l provide the l-th elementary symmetric polynomial of σ a . Meanwhile, in integrable models there is a fundamental quantity known as the Baxter Q-operator Q(x) whose eigenvalue is actually (2.48) with N c identified with the number of particles N and σ a with spectral parameters λ a . Thus, we see that the expectation value of the Baxter Q-operator provides the generating function of correlation functions of gauge invariant operators in the 2d N = (2, 2) theory in section 2.2, i.e. .
The eigenvalue of the transfer matrix τ (µ) for the XXX 1/2 model is given by Therefore, the eigenvalue θ (µ, {λ a }) is expressed in terms of symmetric polynomials of λ a . As discussed above, the eigenvalue of the transfer matrix is actually a generating function of mutually commuting conserved charges (or Hamiltonians). Accordingly, we can identify the expectation value of conserved charges of the XXX 1/2 spin chain model with the twisted GLSM correlators with appropriate coefficients.

Correlation functions in the 3d N = 2 theory and the XXZ 1/2 spin chain model
We consider topologically twisted 3d N = 2 U(N c ) gauge theories with an adjoint chiral multiplet Φ and N f chiral and anti-chiral multiplet Q a i , Q i a , a = 1, . . . , N c , i = 1, . . . , N f , respectively, where we use the same notation as in the 2d case. There are flavor symmetries SU(N f ) Q , SU(N f ) Q , and U(1) D . In addition, there is a U(1) T topological symmetry in three dimensions. The matter contents and charge assignment are specified in table 2.
We denote fugacities and magnetic fluxes of the Cartan part of global symmetries as follows; Then the topologically twisted index of the 3d N = 2 theory is given by Nc a,b=1 Here x a is a constant value of the Wilson loop for the a-th diagonal U(1) of the gauge group U(N c ). We take, for example, η = (−1, −1, . . . , −1) to choose a contour so that it picks poles from anti-chiral multiplets and the ajdoint chiral multiplet. Poles exist when m a < n i − l 2 − r 2 + 1, and we resum over m i < K for a sufficiently large positive integer Summing over all fluxes for m i < K in (2.53), we get where B a (x) is given by Due to (ζe iBa(x) ) K factor in the numerator with a sufficiently large K, poles at x a = 0, 1 − x −1 a y i z −1/2 = 0, and x a − zx b = 0 are not available and only relevant poles come from ζe iBa(x) = 1 for all a. We denote the solution for this equation by where solutions that are related by Weyl permutations S Nc of (x 1 , · · · , x Nc ) are identified. With the contour integral becomes (2.60) Also, upon (2.57), the condition for supersymmetric vacua, ζe iBa(x) = 1, i.e.
is exactly same as the Bethe ansatz for the XXZ 1/2 spin chain (2.16) If we choose R-charges and magnetic fluxes in such a way that hold, then the 3d topologically twisted index (2.58) and the inverse of the norm of the Bethe eigenstate of the XXZ 1/2 spin chain model agree up to overall constants. We can also consider correlation functions and conserved charges in the 3d N = 2 theory and the XXZ 1/2 spin chain model as in section 2.2. The eigenvalue Q(u) of the Baxter Q-operator Q(u) in the XXZ 1/2 model is given by Meanwhile, the Wilson loop in 3d N = 2 theories is given by the Schur polynomial where Y is the Young diagram for the representation R of U(N c ). When R is a totally antisymmetric representation Y = 1 r , r = 1, . . . , N c , the Schur polynomial is given by the elementary symmetric polynomials, s 1 r (x 1 , . . . , x Nc ) = e r (x 1 , . . . , x Nc ). Therefore, with the identifications (2.57), the expectation value of Wilson loop operators is proportional to the coefficient of the eigenvalue of the Baxter Q-operator. Also, as the eigenvalue of the transfer matrix τ (µ) for the XXZ 1/2 model is given by (2.50) with (2.14), we can identify the expectation value of conserved charges of the XXZ 1/2 model with the expectation value of Wilson loops with appropriate coefficients.

Equivariant quantum cohomology, GLSM, and integrable model
In the previous section, we studied the relation between the A-twisted N = (2, 2) GLSM and the XXX 1/2 spin chain. It was shown in [21] that integrations of cohomology classes of toric Fano manifolds can be interpreted as correlation functions of σ of the corresponding A-twisted N = (2, 2) GLSM where the cup product of cohomology classes are deformed by using three point Gromov-Witten invariants (quantum cup product). We may expect that such a relation holds for N = (4, 4) GLSM where the target space is a hyperKähler manifold. We turn on all the possible twisted mass parameters including the one for the N = (2, 2) adjoint chiral multiplet. 6 In this section, we consider correlation functions of the A-twisted N = (2, 2) * GLSM on S 2 and study its relation to the equivariant quantum cohomology of the cotangent bundle of the Grassmannian.

Equivariant quantum cohomology and equivariant integration
Firstly, we summarize the equivariant quantum cohomology of the cotangent bundle of the Grassmannian T * Gr(r, n) [9]. The Grassmannian Gr(r, n) is specified by the chains of subspaces, We would like to consider the cotangent bundle T * Gr(r, n) of the Grassmannian Gr(r, n). We sometimes denote (r, n − r) by (λ 1 , λ 2 ) := (r, n − r) below.
There is a torus action (C * ) n ⊂ GL n (C) on C n , accordingly on Gr(r, n). In addition, there is also a C * action on the fiber direction of T * Gr(r, n). With these actions, one can consider a GL n (C) × C * equivariant cohomology ring. The set of the Chern roots of bundles on Gr(r, n) with fiber Also, the Chern root corresponding to each factors of (C * ) n action and C * action is denoted by z = {z 1 ; · · · ; z n } and h, respectively. Then the equivariant cohomology ring is given by where S n , S λ 1 and S λ 2 denote the symmetrization of variables {z 1 , · · · , z n }, {γ 1,1 , · · · , γ 1,λ 1 } and {γ 2,1 , · · · , γ 2,λ 2 }, respectively. The ideal I is generated by n coefficients of a degree n − 1 polynomial of u, The equivariant quantum cohomology ring of the cotangent bundle of the Grassmannian is given by ] is a ring of formal series of the quantum parameter q. The ideal I q is generated by n coefficients, p l , defined by n l=1 p l (z, Γ, h, q)u n−l : (3.5) The coefficients p l are degree l polynomials of each Γ and z, and are invariant under the action of S n × S λ 1 × S λ 2 . Meanwhile, in [22] the Yangian acting on the equivariant cohomology was constructed and the equivariant quantum cohomology ring was identified with the Bethe subalgebra of the integrable model. The cotangent bundle of the Grassmannian is a typical example of [22]. The equivariant integration of the cohomology class [f (Γ, z, h)] ∈ H * GLn(C)×C * (T * Gr(r, n); C) is calculated by the formula T * Gr(r,n) where I r is a subset of I = {1, · · · , n} with |I r | = r and I n−r is the complement of I r in I. The factor f (z I , z; h) in the numerator is defined by the substitution Γ = (Γ 1 , Γ 2 ) → (z Ir , z I n−r ) in f (Γ, z, h). Summation Ir⊂I in (3.6) runs for all the possible subsets in I with fixed r.
In section 3.2 we calculate the equivariant integration of the elements [f (Γ, z, h; q)] in the equivariant quantum cohomology ring QH * GLn(C)×C * (T * Gr(r, n); C) for several examples by using the formula (3.6) and check that they match with the corresponding GLSM correlators. More specifically, given a ring element, we reduce the degree of the ring element by using the ideal I q whenever it is possible and then apply the formula (3.6) to the resulting ring element, which depends on the parameter q in general. 7 The GLSM correlation function of the operator corresponding to a given original ring element before reducing is expected to match with the result of the equivariant integration obtained in a way we have just described.

Correlation functions of A-twisted GLSM and equivariant integration of equivariant quantum cohomology
We study the relation between correlation functions of the A-twisted 2d N = (2, 2) * GLSM and the equivariant integration for the equivariant quantum cohomology classes in the cotangent bundle of the Grassmannian. The gauge group and the matter contents are the same as in section 2.2, but we choose different R-charges from the previous case in such a way that we now have the superpotential In the positive FI-parameter region, the target space of the non-linear sigma model limit of the theory is T * Gr(N c , N f ) where the base space Gr(N c , N f ) is parametrized by Q b i . On the other hand, in the negative FI-parameter region, the target is again T * Gr(N c , N f ) but the base space is parametrized by Q i a . The superpotential breaks SU We turn off all the background fluxes for flavor symmetry groups. The twisted mass parameters for the SU(N f ) flavor symmetry are denoted by m i and the twisted mass parameter for U(1) D flavor symmetry by m z .

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The correlation function of the gauge invariant operator O(σ) constructed from σ = diag(σ 1 , · · · , σ Nc ) is (3.8) Here we take the charge vector in the Jeffrey-Kirwan reisdue formula as Re q < 1. Then residues are evaluated at the poles (σ a − m i − m z 2 ) −(ka+1) and it is easy to show that poles coming from (σ a − σ b + m z ) −(ka−k b −1) do not contribute to the residues. The overall sign ambiguity will be fixed below. we obtain two equations −q Dividing (3.10) by (3.11), we get Upon the identifications

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Quantum cohomology of CP n−1 and correlation functions of the A-twisted GLSM. We briefly recall the well-known relation between the N = (2, 2) U(1) GLSM with n charge +1 chiral multiplets and the quantum cohomology of CP n−1 . This GLSM flows to the N = (2, 2) non-linear sigma model with target space CP n−1 [21,23]. The quantum cohomology of CP n−1 is given by (3.14) The equivariant integration of γ l 1,1 ∈ QH * (CP n−1 ; C), which we denote as γ l 1,1 CP n−1 , is obtained as follows. If a < n, γ a 1,1 CP n−1 is the same as the integral of the cohomology class γ a 1,1 ∈ H * (CP n−1 ; C) and is given by For γ mn+a 1,1 CP n−1 with a < n, we reduce the degree by using the relation γ n 1,1 − q = 0 to γ mn+a 1,1 = q m γ a 1,1 and obtain γ mn+a On the other hand, the expectation value of σ l is obtained by supersymmetric localization We perform a similar calculation for the cotangent bundle of the Grassmannian.

T * CP n−1
We would like to relate the expectation value of σ l in the GLSM to the equivariant integration of equivariant quantum cohomology classes when the target space is T * CP n−1 .

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From (3.21), the equivariant integration of γ l 1,1 is given by The correlation function σ l Nc=1,N f =2 A-twist is expected to be related to the equivariant integral on via the identification of parameters (3.13). We can check this explicitly. For example, when l ≤ 1, the equivariant integration γ l 1,1 T * CP 1 gives for several orders of q and see that there are no q corrections, . (3.27) Here we fixed the overall sign in order to have an agreement with the equivariant integration T * CP 1 [1]. Therefore, we checked that We also computed σ l Nc=1,N f =2 A-twist perturbatively and γ l 1,1 T * CP 1 exactly by using (3.22) for l = 2, 3, 4, 5, and checked agreement (3.23).
• T * CP n−1 . We expect that the expectation value of σ l agrees with the integration of γ l 1,1 ∈ QH GLn(C)×C * (T * CP n−1 ; C), From the ideal, we obtain the following relation

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This relation is the same as the twisted chiral ring relation of the corresponding GLSM via (3.13). From (3.31), γ l 1,1 with l > n − 1 is uniquely expressed as With the identification σ = γ 1,1 , we expect that σ l Nc=1,N f =n A-twist agrees with the equivariant integration of the equivariant quantum cohomology class γ l 1,1 , We also checked this for n = 3, 4 with several higher powers of σ and found agreement.
(3.36) We have checked (3.34) and (3.35) for k + l ≤ 3 perturbatively. The detailed calculation of the reduction (3.35) is available in appendix A as an example. We also checked the cases of k + l = 4 and some of k + l = 5 for T * Gr(2, 5) and found agreement. We expect to have agreement for general r ≤ n − r.

T * Gr(r, n) with r > n − r and the Seiberg-like duality
From the ideal p 1 = 0 for the equivariant quantum cohomology of T * Gr(r, n), we have

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where λ 1 = r and λ 2 = n − r. In section 3.2.2, we expected, for example, r a=1 σ a Nc=r,N f =n A-twist = T * Gr(r,n) r a=1 γ 1,a for r ≤ n − r , (3.38) i.e. r a=1 γ 1,a T * Gr(r,n) does not have any q corrections and can be computed by using (3.6). From the relation (3.37), it is expected that r a=1 γ 1,a T * Gr(r,n) with r > n − r receives q corrections and differs from the result directly obtained by the classical equivariant integration (3.6). In order to calculate the equivariant integration properly for the case r > n − r, it is useful to study the isomorphism T * Gr(r, n) ≃ T * Gr(n − r, n), which corresponds to the Seiberg-like duality [24] between A-twisted N = (2, 2) * GLSM's with gauge groups U(r) and U(n − r).
For this purpose, we consider the relation between ideals of the equivariant quantum cohomology of T * Gr(r, n) and T * Gr(n − r, n). The latter is given by where we use tilde to distinguish the notations for QH * GLn(C)×C * (T * Gr(r, n); C). The ideal I q is generated by n polynomialsp l defined by The ideals of the quantum cohomology of T * Gr(r, n) and of T * Gr(n − r, n) are the same upon the following parameter identification 8 When equivariant parameters are turned off, γ 1,a andγ 2,a are exchanged with each other under T * Gr(r, n) ↔ T * Gr(n − r, n). This is consistent with the fact that vector bundles with fibers F 1 and F 2 /F 1 are exchanged vice versa under Gr(r, n) ↔ Gr(n − r, n).
Next we identify the variables in QH * GLn(C)×C * (T * Gr(n−r, n); C) with those in U(n−r) GLSM. By substituting u =γ 1,c andγ 1,c + h into n l=1p l (z,Γ,h,q)u n−l = 0, we obtaiñ With the identifications We begin with the simplest case, which corresponds to the partition function. From (3.6), we obtain T * Gr(r,n) [1] = T * Gr(n−r,n) [1] (3.45) and this implies 1 We computed each side of (3.46) for (N c , N f ) = (1, 3), (1,4), (2,5) in several orders of q and checked the agreement. Next, with the identification (3.41), we have There is another way of identification, but considering the Seiberg-like duality above identification is more appropriate.
4 Wilson loops in the 3d N = 2 * theory and the Bethe subalgebra of the XXZ 1/2 model In the previous section, we saw that the twisted chiral ring relation of the GLSM agrees with the equivariant quantum cohomology ring of the cotangent bundle of the Grassmannian, which corresponds to the Bethe subalgebra of the XXX 1/2 spin chain model. Therefore we can do similar calculations and checks for the S 1 uplift of the twisted chiral ring relation of the 3d N = 2 * theory and the Bethe subalgebra of the XXZ 1/2 spin chain model [10]. The S 1 uplift of the twisted chiral ring in the 3d N = 2 * theory on S 1 × S 2 is generated by Wilson loops wrapped on S 1 .
In the 3d N = 2 * theory, which is obtained by the adjoint mass deformation of the 3d N = 4 theory, there is a superpotential Table 4. Matter contents of the 3d N = 2 * theory.

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Here, we turn off all the background magnetic fluxes for flavor symmetries. Then the expectation value of supersymmetric Wilson loops in the representation R is given by where we absorbed (−1) Nc−1 into the definition of the fugacity ζ for U(1) T . Here the fugacity for SU(N f ) flavor symmetry is denoted by y i and the one for U(1) D flavor symmetry by z. When the representation R is the l-th anti-symmetric representation A l , Tr A l (x) is given by the l-th elementary symmetric polynomial of (x) = diag(x 1 , · · · , x Nc ) Note that any product of supersymmetric Wilson loops is a symmetric function of (x), which is also expressed in terms of the elementary symmetric polynomials.

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Bethe ansatz equation and match of parameters. We consider the identification between generators of K q and variables in the topologically twisted 3d N = 2 * supersymmetric theory by deriving the Bethe ansatz equation from (4.5). By substituting u = γ 1,a and γ 1,a h −1 into P (Γ, z, h, q) = 0, we obtain, respectively, Dividing (4.6) by (4.7), we get the Bethe ansatz equations, which is the SUSY vacua condition ζe iBa = 1 of the 3d N = 2 * theory with the following identifications Abelian cases. From (4.8), which is equivalent to ζe iBa = 1, we expect that the supersymmetric Wilson loop W = x for U(1) gauge theories satisfy with the parameter identification (4.9). Also by using (4.8), the higher order correlation functions W l Nc=1,N f for l ≥ n are expressed in terms of W k , (k = 0, 1 · · · , n − 1) as (4.11) In the 2d N = (2, 2) * theory with N f = n flavors, we found σ l Nc=1,N f =n A-twist with l ≤ n − 1 do not have q corrections. There is a similar property in the 3d N = 2 * theory. For 0 ≤ l ≤ n − 1, W l Nc=1,N f =n does not have ζ corrections and is given by the zero magnetic charge sector We have checked (4.10) and (4.12) for N f = 2, 3, 4 in several orders of ζ.

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Non-Abelian cases. In two dimensions, we observed that the partition function 1 A-twist does not receive any q corrections and is given by residues at the zero magnetic charge sector. Similarly we observed that the partition function (index) of the topologically twisted 3d N = 2 * theory on S 1 × S 2 does not receive ζ corrections neither and is given by (4.13) In two dimensions, 1 Nc=r,N f =n A-twist has a geometrical interpretation as the equivariant integration of [1] ∈ H * (T * Gr(r, n); C). The index (4.13) also has a geometrical interpretation. If we identify 3d parameters z i and h as z i = e z i and h = e −h , respectively, (4.13) is the sinh uplift of the equivariant integration, which can be interpreted as the equivariant Dirac index.
For 2 ≤ n − 2 (r = 2), we also observe that the expectation values of x ±1 (4.14) However, the properties of correlation functions (x 1 + x 2 ) 2 , (x 1 + x 2 )(x 1 x 2 ), and (x 1 x 2 ) 2 are different from the 2d case. In the 2d N = (2, 2) * theory with N c ≤ N f − N c , we expected that correlation functions of symmetric polynomials of σ, Nc a=1 e la a (σ) with Nc a=1 l a ≤ N c , don't have q dependence. This may be because the degree of the polynomial cannot be reduced to a lower degree in the polynomial ring by the ideal. For example, (σ 1 + σ 2 ) l (σ 1 σ 2 ) k Nc=2, N f =4 with k + l = 2 agrees with the residues at zero magnetic flux sector and do not have q dependence. On the other hand, if we eliminate γ 2,1 + γ 2,2 and γ 2,1 γ 2,2 from the ideal of K q , we obtain Then we find that the degree of (x 1 + x 2 ) l (x 1 x 2 ) k Nc=2,N f =4 with k + l = 2 is reduced by the above equations and that (x 1 + x 2 ) l (x 1 x 2 ) k has ζ dependence. We have checked (4.15) and (4.16) hold for several orders of ζ in terms of expectation values of Wilson loops.

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With the identifications λ 1 =λ 2 , λ 2 =λ 1 , γ 1,a =γ 2,ah , γ 2,a =γ 1,ah −1 , z i =z i , h =h, q =q −1 , (4.18) (4.17) is identical to P (Γ, z, h, q) for (λ 1 , λ 2 ) = (r, n − r). Thus, with (4.18) the Bethe subalgebra for (λ 1 , λ 2 ) = (r, n − r) and the one for (λ 1 ,λ 2 ) = (n − r, n) are isomorphic. By substituting u =γ 1,a ,γ 1,ah −1 into (4.17), we obtain which is again the same with the SUSY vacua conditionζe iBa = 1 of the U(n − r) gauge theory with the identifications From (4.9), (4.18) and (4.20), we have maps between parameters in U(r) and U(n − r) 3d N = 2 * gauge theories, So from now on, we don't distinguish y i , z, and z i fromỹ i ,z, andz i , respectively. From (4.13), we have at the level of the partition function (or index). We have checked this for several N f and N c . For the fundamental representation, we consider the coefficient of u −n+1 of P = 0, which becomes by using relations between two sets of parameters, (4.9), (4.18) and (4.20). Therefore, this indicates that the Wilson loop in the fundamental representation W F = Nc a=1 x a in the U(N c ) gauge theory with N c > N f − N c is provided by where W F = n−r a=1x a is the Wilson loop in the fundamental representation in the U(N f − N c ) gauge theory.

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When calculating the index, the evaluation of the l.h.s. in the region ζ < 1 (resp. ζ > 1) means that the r.h.s. is evaluated in the regionζ = ζ −1 > 1 (resp.ζ = ζ −1 < 1) where the negative (resp. positive) magnetic fluxes contribute to the Jeffrey-Kirwan residue operations. We evaluated the l.h.s. and the r.h.s. separately and have agreement for several r and n.
Next we consider the second antisymmetric representation. We eliminate e 1 (γ 1 ) from the coefficient of u −2 in P = 0. Then we obtain the relation This can be written in terms of x a andx a as Therefore, this suggests that the expectation value of the second antisymmetric representation W A 2 = a<b x a x b is given by Nc=r,N f =n (4.28) 2 −Nc ζ, and h = z −1 =z −1 . We checked this for several N c and N f .
In a similar way, we can have the Seiberg-like duality for Wilson loops in other representations from the ideal with identification of parameters (4.9), (4.18), and (4.20). Also, as done in the 2d case, we can have the S 1 -uplift of twisted chiral rings by eliminating symmetric polynomials of γ a,2 in (4.5) with the identification of parameters (2.57).

Conclusion and future directions
In this paper, we discussed the relation between the partition function in the A-twisted 2d N = (2, 2) theory (resp. the topologically twisted 3d N = 2 gauge theory) and the inverse of the norm of the Bethe eigenstate for the XXX 1/2 (resp. XXZ 1/2 ) spin chain model with a particular choice of R-charges and background magnetic fluxes for flavor symmetries in the gauge theory side. Coefficients of the expectation value of the Baxter Q-operator and the conserved charges were understood in terms of correlation functions in gauge theories.
We also studied the relation between correlation functions in the A-twisted 2d N = (2, 2) * U(N c ) gauge theories and the equivariant integration of equivariant quantum cohomology classes for the cotangent bundle of the Grassmannian. We calculated each of them for several examples, checked that they agree, and expect that the relation holds for general cases. For the case N c > N f − N c , we used the isomorphism of Grassmannians to calculate JHEP02(2019)052 the equivariant integration where such isomorphism corresponds to the Seiberg-like duality in the GLSM side.
As the twisted chiral ring of the 2d N = (2, 2) * theory is identified with the Bethe subalgebra of the XXX 1/2 spin chain model, we were able to make a similar identification for the 3d N = 2 * theory. We calculated correlation functions of Wilson loops and checked that they agree with the Bethe subalgebra of the XXZ 1/2 spin chain model.
There are several interesting directions. Firstly, it will be interesting to find the analogue of the equivariant integration in the equivariant quantum K-theory and match them with the correlation functions of Wilson loops in the topologically twisted 3d N = 2 * theory.
Another interesting direction is to study relations between the Bethe ansatz and the finite-dimensional commutative Frobenius algebra. In [26], a finite-dimensional commutative Frobenius algebra was constructed in terms of the Bethe ansatz for the q-boson model. It is known that the finite-dimensional commutative Frobenius algebra is essentially same as the 2d topological quantum field theory (TQFT) and the 2d partition function on genus g Riemann surface Σ g corresponding to q-boson can be written as [17] Z(Σ g ) = (λ)∈P q-boson Ψ(λ)|Ψ(λ) g−1 . (5.1) Here |Ψ(λ) is the eigenvector of the q-boson determined by the Bethe root (λ). We obtained the same type of formula for the XXX 1/2 spin chain model where the corresponding TQFT is the topologically twisted 2d N = (2, 2) theory and also for the XXZ 1/2 model that corresponds to the 3d N = 2 theory with the partial topological twist along S 2 . By using recent results [27,28], the partition function of the 2d N = (2, 2) and the 3d N = 2 theories studied in this paper can be generalized to Riemann surfaces of genus g as Z(Σ g ) = These formulas are similar to the q-boson case and imply that there exist finite-dimensional commutative Frobenius algebras associated with the Bethe ansatz for the XXX 1/2 and also the XXZ 1/2 spin chain models. It would be interesting to construct the Frobenius algebras in terms of the XXX 1/2 and the XXZ 1/2 spin chain models.
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