Topologically Twisted SUSY Gauge Theory, Gauge-Bethe Correspondence and Quantum Cohomology

We calculate partition function and correlation functions in A-twisted 2d $\mathcal{N}=(2,2)$ theories and topologically twisted 3d $\mathcal{N}=2$ theories containing adjoint chiral multiplet with particular choices of $R$-charges and the magnetic fluxes for flavor symmetries. According to Gauge-Bethe correspondence, they correspond to Heisenberg XXX and XXZ spin chain models. We identify the partition function as the inverse of the norm of the Bethe eigenstates. Correlation functions are identified as the coefficients of the expectation value of Baxter $Q$-operators. In addition, we consider correlation functions of 2d $\mathcal{N}=(2,2)^*$ theory and their relation to equivariant quantum cohomology and equivariant integration of cotangent bundle of Grassmann manifolds. Also, we study the ring relations of supersymmetric Wilson loops in 3d $\mathcal{N}=2^*$ theory and Bethe subalgebra of XXZ spin chain model.


Introduction
The Gauge-Bethe correspondence states that quantum integrable models correspond to the supersymmetric gauge theories. The XXX Heisenberg spin chain model was considered as one of primary examples of Gauge-Bethe correspondence in the original papers [1,2]. It was argued that the supersymmetric vacua of softly broken 2d N = (4, 4) theory by the mass of adjoint chiral multiplet, usually called as 2d N = (2, 2) * theory, is naturally identified with Bethe ansatz equation for XXX spin chain model. Also, twisted superpotential was identified with Yang-Yang potential.
In this paper, we study 2d N = (2, 2) theories and 3d N = 2 theories containing adjoint chiral multiplet with two different choices of of R-charges and background magnetic fluxes but with same gauge group and matter contents. We calculate the partition functions of A-twisted 2d N = (2, 2) theory on S 2 and the partition function of topologically twisted 3d N = 2 theory on S 1 × S 2 with all equivariant parameters associated to flavor symmetries but the one associated to rotational symmetry on S 2 turned off. We match them with the inverse of the norm of Bethe eigenstates by choosing a particular R-charges and background fluxes for flavor symmetries. The gauge invariant operators forms chiral ring and expectation values of them provide the coefficient of the expectation value of Baxter Q-operator. Thus with proper choice of coefficients the expectation value of the gauge invariant operators provide the expectation value of conserved charges of corresponding spin chain model.
We also explicitly calculate the correlation functions of A-twisted 2d N = (2, 2) * theory whose target space (in nonlinear sigma model limit) is cotangent bundle of Grassmannians for several examples. On the other hand, we evaluate the equivariant integration of equivariant cohomology for cotangent bundle of Grassmannians [9] where they showed that Bethe subalgebra of XXX spin chain model is isomorphic to the equivariant quantum cohomology ring 1 . We provide a way to calculate the equivariant integration, and by checking several examples we see that the result is consistent with the correlation functions. In the calculation, we also use Seiberg-like duality.
It was shown in [10] that Bethe subalgebra of XXZ spin chain model is given by certain generators and relations analogous in [9]. 2 With Gauge-Bethe correspondence in mind, we see that Wilson loop algebra agree with Bethe subalgebra by checking several examples. Also, we consider Seiberg-like duality of 3d N = 2 * theory in the context of Bethe subalgebra.
In the final section, we conclude with summary of our results and discuss some future directions.
Note added: While this work is being completed, we heard that a paper [11], which has a partial overlap with ours, will appear.

Gauge/Bethe Correspondence and Bethe Norm
Given 2d N = (2, 2) gauge theories with the effective twisted superpotential W eff (σ), the condition for supersymmetric vacua is given by and according to Gauge/Bethe correspondence [1,2,12], 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 istotropic SU (2) Heisenberg spin chain model XXX 1/2 , similarly for anisotropic XXZ 1/2 spin chain model, with spin-1/2 degree of freedom at each sites, the twisted mass parameters for flavor symmetries are related to the parameters for the position of the lattice sites with respect to symmetric round lattice configuration.
In this section, we relate the norm of the Bethe eigenstates of XXX 1/2 and XXZ 1/2 spin chain model to the partition function of topologically twisted 2d N = (2, 2) theory and 3d N = 2 theory, respectively. We also discuss the relation between correlation functions and the coefficients of the expectation value of Baxter Q-operator as well as conserved charges.

2.1
The norm of Bethe eigenstates in XXX 1/2 and XXZ 1/2 spin chain model We are interested in inhomogeneous XXX 1/2 and XXZ 1/2 spin chain model with M lattice sites. The monodromy matrix, T(λ), of XXX 1/2 and XXZ 1/2 model takes a form of 2 × 2 matrix acting on the 2-dimensional auxiliary space where λ is a spectral parameter. Therefore, the transfer matrix τ obtained by the trace of monodromy matrix, where S a = 1 2 σ (a) is the generators at a-th site, the transfer matrix is given by A(λ)+e iϑ D(λ) [1,13].
The R-matrix of XXX 1/2 and XXZ 1/2 model is for XXX 1/2 model where c is an auxiliary parameter, and for XXZ 1/2 model where η is related to the anisotropy parameter. The R-matrix satisfies Yang-Baxter equation where ⊗ denotes tensor product of two matrices. 3 The Yang-Baxter equation implies R(λ, µ) (T(λ) ⊗ T(µ)) = (T(µ) ⊗ T(λ)) R(λ, µ), (2.8) and this provides the commutation relations of the matrix elements of monodromy matrix T(λ). Also, from (2.8) and due to trace identities, one can show that transfer matrix commutes with Hamiltonian, [τ (λ) , H] = 0 so τ (λ) is a generating function of conserved charges. As they commute, the eigenfunction of the transfer matrix is also the eigenfunction of Hamiltonian. The pseudo-vacuum |0 satisfies the following condition where a(λ) and d(λ) are called vacuum eigenvalues. For Heisenberg spin chain model, the pseudo-vacuum |0 is given by the state where spins are up at all sites.

The Bethe Eigenstates
Consider a state which is obtained by acting B operator on the pseudo-vacuum, where N is the number of particles or excitations. This state becomes the eigenvector of transfer matrix when spectral parameters, λ a , satisfy the Bethe ansatz equation, and the eigenvector is called Bethe eigenstate. The dual vector of |Ψ N (λ) is defined by The vacuum eigenvalues, a(λ) and d(λ), of inhomogeneous XXX 1/2 and XXZ 1/2 model are and for the quasi-periodic boundary condition.
The Norm of Bethe Eigenstate for XXX 1/2 Model The norm of Bethe eigenstate [14] is given by and we obtain (2.20) Therefore, the inverse of the norm of Bethe eigenstate is given by Here P XXX is the set of the independent solutions of (λ) = (λ 1 , · · · , λ N ) satisfying the Bethe ansatz (2.15) with the quasi-periodic boundary condition.

The Norm of Bethe Eigenstate for XXZ 1/2 Model
The norm of Bethe eigenstates for XXZ 1/2 model [14] can be obtained similarly as in the case of XXX 1/2 model and it is In terms of ϕ ′ , the inverse of the norm of Bethe eigenstates is given by and P XXZ is the set of the independent solutions of (λ) = (λ 1 , · · · , λ N ) satisfying the Bethe ansatz equation (2.15) with the quasi-periodic boundary condition.
2.2 Correlation functions in 2d N = (2, 2) gauge theory and XXX 1/2 spin chain model We consider a topologically twisted N = (2, 2) U (N c ) gauge theory. The matter chiral multiplets are an adjoint chiral multiplet Φ, N f fundamental chiral multiplets Q i a and antifundamental chiral multiplet Q a i , a = 1, . . . , N c , i = 1, . . . , N f . The flavor symmetry group The charge assignment is specified in the Table 1.
The partition function of 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 graviphoton associated on S 2 .
The one-loop contributions from chiral, anti-chiral, and adjoint chiral are given by and one-loop contribution from vector multiplet is We denote σ = diag(σ 1 , · · · , σ Nc ) as a constant configuration of the scalar in the vector multiplet. The partition function of A-twisted gauged linear sigma mode on two sphere is given by Here the choice of contour is specified by Jeffrey-Kirwan residue prescription. 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 the covector η, for example, to be η < 0, the residues are taken from anti-chiral multiplets or part of them with part of factors from adjoint chiral multiplet. Poles from anti-chiral multiplets exist when k a < n i − 1 2 l − r 2 + 1. Summing over k a < K first for sufficiently large positive integer K, the partition function is expressed as (2.32) 4 In our notation, the background flux l is an even number.

Comparison and Matching
The following identification give the agreement between Bethe ansatz equation (2.14) for XXX 1/2 spin chain model and the condition for supersymmetric vacua (2.40) of 2d N = (2, 2) gauge theory. Moreover, with identification the partition function of A-twisted 2d N = (2, 2) gauge theory 5 and the inverse of the norm of the Bethe eigenstates (2.21) agree; This type of relation was first studied for U (N )/U (N ) gauged WZW model on genus-g Riemann surfaces Σ g in [15] where the corresponding integrable model is phase model. This correspondence was generalized to one-parameter deformation U (N )/U (N ) gauged WZW model and q-boson model in [16]. The relation between the deformation of gauged WZW model and Chern-Simons matter theories on S 1 × Σ g was discussed in [17,18]. See also [19].

Correlation Functions and Conserved Charges
We have identified the partition functions and the norm of the inverse of Bethe eigenstates. We can also consider the correlation functions of A-twisted 2d N = (2, 2) * theory in the context of Gauge-Bethe correspondence.
In A-twisted 2d N = (2, 2) theory, the correlation functions of the gauge invariant operators O(σ) are given by (2.45) 5 For example, we can choose all the background fluxes and R-charges to be zero. However the canonical assignment of R-charge is not allowed if we want to match the A-twisted partition function and the inverse of the norm of the Bethe eigenstates. Indeed if we sum three conditions in (2.42), we obtain However, as the 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, 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, Bethe ansatz equation are same whatever R-charges and background magnetic fluxes are.
This can also be written as The operator O(σ) is provided by gauge invariant polynomials of Cartan of the scalar component σ of 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 gives l-th elementary symmetric polynomial of σ a . Meanwhile, in integrable models there is a fundamental quantity known as Baxter Qoperator Q(x) whose eigenvalue is actually (2.47) with N c identified with the number of particles N and σ a identified with spectral parameters λ a . Thus, we see that the expectation value of Baxter Q-operator provides the generating function of correlation function of gauge invariant operators in 2d N = (2, 2) * theory, i.e. . (2.48) The eigenvalue of transfer matrix τ (µ) for XXX 1/2 model is given by Therefore, the eigenvalue θ (µ, {λ a }) is generating function of symmetric polynomial of λ a . As discussed above, the eigenvalue of transfer matrix is actually generating function of mutually commuting conserved charges (or Hamiltonians). Accordingly, we can identify the expectation value of conserved charges of XXX 1/2 spin chain model with the twisted GLSM correlators with appropriate coefficients.

Correlation functions in 3d N = 2 theory and XXZ 1/2 spin chain model
We consider a topologically twisted 3d N = 2 U (N c ) gauge theory with an adjoint chiral multiplet Φ and N f chiral and anti-chiral multiplet Q ai , Q ai , a = 1, . . . , N c , i = 1, . . . , N f , respectively, where we use same notation as 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 Wilson loop for a-th diagonal U (1) of gauge group U (N c ). We take, for example, η = (−1, −, 1, . . . , −1) to choose a contour so that it picks the poles from anti-fundamental chiral multiplets or from part of them with part of factors from adjoint chiral. Poles exist for m a < n i − l 2 − r 2 + 1, and we resum over m i < K for sufficiently large positive integer K. With f i = −n i − l 2 − r 1 + 1, f i = n i − l 2 − r 1 + 1, and h = l − R + 1, above expression can be organized to and summing over all fluxes for m i < K, we get Due to (ζe iBa(x) ) K with large K in numerator, 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 as where we removed solutions which are same up to permutation of (x 1 , · · · , x Nc ). With the contour integral gives Also, upon (2.56), the condition for supersymmetric vacua, ζe iBa(x) = 1, i.e.
is exactly same as the Bethe ansatz for XXZ 1/2 spin chain (2.15) If we choose R-charges and magnetic fluxes in such a way that holds, then 3d topologically twisted index (2.57) and the inverse of the norm of Bethe eigenstate of XXZ 1/2 spin chain model agree up to overall constants.
We can also consider correlation functions and conserved charges in 3d N = 2 theory and XXZ 1/2 spin chain model as in section 2.3. The eigenvalue Q(u) of the Baxter Qoperator Q(u) in XXZ 1/2 model is given by where Y is a Young tableaux for the representation R of U (N c ). When R is totally antisymmetric representation Y = 1 r , r = 1, . . . , N c , Schur polynomial is given by elementary symmetric polynomial, s 1 r (x 1 , . . . , x Nc ) = e r (x 1 , . . . , x Nc ). Therefore, upon (2.56), 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 XXZ 1/2 model is also (2.49) with (2.13), we can identify the expectation value of conserved charges of XXZ 1/2 model with the expectation value of Wilson loop with appropriate coefficients.

Equivariant Quantum Cohomology, GLSM, and Integrable Model
In the previous section, we studied the relation between an A-twisted N = (2, 2) GLSM and correlation functions of XXX 1/2 spin chain. It is known that correlation functions of an A-twisted N = (2, 2) GLSM with a particular choice of R-charge give the integration of quantum cohomology classes of Fano manifolds, which are Higgs branch vacua of the GLSM [20] [add references]. We expect that a similar story holds for N = (4, 4) GLSM where Higgs branch vacua is now hyperKähler manifold. We turn on all the possible twisted mass parameters including the one associated to N = (2, 2) adjoint chiral multiplet 6 , In this section, we consider correlation functions of A-twisted N = (2, 2) * GLSM on S 2 and study its relation to equivariant quantum cohomology of cotangent bundle of Grassmannians.

Equivariant quantum cohomology and Equivariant Integration
Firstly, we summarize the equivariant quantum cohomology of the cotangent bundle of Grassmannians T * Gr(r, n) [9]. A Grassmannian Gr(r, n) are specified by the chains of subspaces, with dim F 1 = r. It is convenient to define (λ 1 , λ 2 ) := (r, n − r). We would like to consider the cotangent bundle T * Gr(r, n) of Grassmannian Gr(r, n).
There is a torus action (C * ) n ⊂ GL n (C) on C n , accordingly on Gr(r, n). In addition, there is also C * action on the fiber direction of T * Gr(r, n). With these actions, one can consider GL n (C) × C * equivariant cohomology ring. We denote Γ i = {γ i,1 , · · · , γ i,λ i } with i = 1, 2 as the set of Chern roots of bundles on Gr(r, n) with fiber F i /F i−1 , z = {z 1 ; · · · ; z n } as Chern roots corresponding to each factors of (C * ) n action, and h as the Chern root corresponding to C * action. 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 the n coefficients of a degree n − 1 polynomial of u: The equivariant quantum cohomology ring of cotangent bundle of Grassmannian is given by ] is the ring of formal series of q, which is the quantum parameter. The ideal I q is generated by the n coefficients p l defined by The coefficients p l are degree l polynomials of each Γ and z and invariant under the action of S n ×S λ 1 ×S λ 2 . Meanwhile, the authors of [21] constructed the Yangian acting on equivariant cohomology and identified equivariant quantum cohomology ring as Bethe subalgebra. The cotangent bundle of Grassmannian is a typical example of [21].
The equivariant integration of a 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 subsets of I = {1, · · · , n} with |I r | = r and I n−r is the compliment of I r in I. The factor f (z I , z; h) in 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 subset in I with fixed r.
As will be discussed with explicit examples in section 3.2, by considering the case of Fano target space and also from the comparison with the GLSM results, we expect that the equivariant integration of the element [f (Γ, z, h; q)] of equivariant quantum cohomology ring QH * GLn(C)×C * (T * Gr(r, n); C) is also calculated by using the formula (3.6). 7 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 generically depends on the parameter q. The GLSM correlation function of the operator corresponding to a given original ring element is expected to match with 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 A-twisted 2d N = (2, 2) * GLSM and equivariant integration for quantum cohomology. The gauge group and the matter content are same as section 2.2, but we choose different R-charge from the previous case in such a way that we have now the superpotential Then the Higgs branch vacua is T * Gr(N c , N f ) in positive FI-parameter regions. The su- We turn off all the background flux for flavor symmetry group. We denote the twisted mass parameters for SU (N f ) flavor symmetry by m i and the twisted masses parameter for U (1) D flavor symmetry by m z . Table 3. The charge assignment of GLSM 7 As we will see, we expect that it works for T * Gr(r, n) when r ≤ n−r, and we use ideal when calculating the case r ≥ n − r.
The correlation function of gauge invariant operator O(σ) constructed from σ = diag(σ 1 , · · · , σ Nc ) is Here we take the charge vector in Jeffrey-Kirwan reisdues as Re q < 1. Then the residue is 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

Bethe ansatz equation and matching of parameters
Dividing (3.10) by (3.11), we get Then the identification gives the condition for supersymmetric vacua (2.40) for N = (2, 2) * U (N c ) GLSM with N f fundamental hypermultiplets.

Quantum cohomology of CP n−1 and correlation functions of A-twisted GLSM
Let us briefly recall the well known relation between N = (2, 2) U(1) GLSM with n charge +1 chiral multiplets and quantum cohomology of CP n−1 . This GLSM flows to N = (2, 2) non-linear sigma model with target space CP n−1 [22] [cite references]. The quantum cohomology of CP n−1 is given by 14) The equivariant integration of γ a 1,1 ∈ QH * (CP n−1 ; C), which we denote as γ a 1,1 CP n−1 , is written as follows. If a < n, γ l 1,1 CP n−1 is same as integration of cohomology class γ a 1,1 ∈ H * (CP n−1 ; C) and is given by If l = mn + a with a < n, we reduce the degree by using the relation γ n 1,1 − q = 0 to γ an+l 1,1 = q a γ 1,1 and obtain γ an+l On the other hand, the expectation value of σ m is evaluated by supersymmetric localization as Therefore we have We expect that similar way of calculation for equivariant integration as above also works for our case.

T * CP n−1
We would like to relate the expectation value of σ l in GLSM to the equivariant integration of quantum cohomology when target space is T * CP n−1 .
T * CP 1 From (3.12), the following relation holds This relation is same as chiral ring relation via (3.13). By using (3.20), γ l 1,1 can be uniquely expressed as From (3.21), we expect that equivariant integration of γ l 1,1 on equivariant quantum cohomology is given by is expected to be related to equivariant integral on T * CP 1 as with the parameter identification (3.13). Here, we directly compute σ l Nc=1,N f =2 A-twist from localization formula (3.8), also γ l 1,1 T * CP 1 by using (3.22), and check their relation (3.23).
On the other hand , we evaluated 1 Nc=1,N f =2 A-twist in order by order of q, which suggest the q-correction vanish and given by Here we fixed the overall sign by requiring to reproduce the equivariant integration T * CP 1 [1]. This shows We computed σ l Nc=1,N f =2 A-twist , l = 2, 3, 4, 5 perturbatively and have agreement of (3.23).
We conjecture 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 This relation is same as chiral ring relation via (3.13). From (3.32), 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 integrals of γ l 1,1 on quantum cohomology, which means We also checked several examples, n = 3, 4, also several higher power of σ and found agreement.
3.2.2 T * Gr(r, n) with r ≤ n − r Here we consider the case r ≤ n − r. The case r > n − r will be discussed in 3.2.3. We expect that the way of calculation for equivariant integration discussed previous cases can be applied to the non-Abelian cases with r ≤ n − r. For non-Abelian cases, we use the ideal of quantum cohomology to reduce the degree of symmetric polynomials of γ 1,1 , · · · , γ 1,λ 1 , which correspond to gauge invariant operators in GLSM side, differently from the abelian case where the Bethe ansatz equation was enough.

T * Gr(r, n) with r > n − r and Seiberg-like duality
From the ideal p 1 = 0 for T * Gr(r, n), we have In section 3.2.2, we expect, for example, i.e. the equivariant integration on quantum cohomology r a=1 γ 1,a T * Gr(r,n) does not have any q-correction and is simply given by the equivariant integration T * Gr(r,n) r a=1 γ 1,a . From the relation (3.38), this implies that λ 2 a=1 γ 2,a T * Gr(r,n) for r > n − r receives q-correction and differs from classical equivariant integration. To clarify the above implication, it is useful to study Seiberg-like duality for twisted chiral ring elements.
Next we find the relations between ideals, the equivariant quantum cohomology of T * Gr(n − r, n) is given by where we use tilde to distinguish the notations from QH * GLn(C)×C * (T * Gr(r, n); C). The idealĨ q is generated by the n polynomialsp l defined by n l=1p l (z,Γ,h,q)u n−l : The ideal quantum cohomology of T * Gr(r, n) is same as that of T * Gr(n − r, n) with the following parameter identification This identification was given in [23]. When the equivariant parameters are turned off, γ 1,a andγ 2,a are exchanged 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(r, n); C) with U (n − r) GLSM. We substitute u =γ 1,c andγ 1,c + h into n l=1p l (z,Γ,h,q)u n−l = 0. Then we obtaiñ This gives the map between the σ in U (r) and U (n − r) GLSMs.
Therefore we obtain r a=1 σ a Nc=r, N f =n We evaluate the left hand side of (3.50) in |q| < 1. From q =q −1 , the right hand side is evaluated in the region |q| > 1 in which the one-loop determinant of anti-chiral multiplets contribute to Jeffrey-Kirwan residues. Note that with |q| > 1 is same as n−r a=1σ a Nc=n−r, N f =n A-twist with |q| < 1 by parameter changeq −1 →q, but from (3.39), n−r a=1σ a Nc=n−r, N f =n A-twist for r > n − r does not haveq-correction. Thus we have where r > n − r and |q| > 1. On the other hand, the correspondence between A-twisted GLSM and quantum cohomology gives We computed r a=1 σ a Nc=r, N f =n A-twist in |q| < 1 for (N c , N f ) = (2, 3), (2,4), (3,4) in several order of q and confirm the equality.
4 Wilson loops in 3d N = 2 * theory and Bethe subalgebra of XXZ model In previous section, we saw that the twisted chiral ring relation of GLSM is given by equivariant quantum cohomology ring, which corresponds to Bethe subalgebra of XXX spin chain model, by calculating correlation functions of gauge invariant operators. Therefore we can do similar calculations and checks for the chiral ring relation of 3d N = 2 * theory and Bethe subalgebra of XXZ spin chain model [10]. In 3d N = 2 * theory on S 1 × S 2 , chiral ring is generated by Wilson loops wrapped on S 1 . In 3d N = 2 * theory, which is obtained by adjoint mass deformation of 3d N = 4 theory, there is a superpotential Also, we turn off all the background magnetic fluxes for flavor symmetries. We denote the fugacity for SU (N f ) flavor symmetry by y i and the one for U (1) D flavor symmetry by z. The expectation values of supersymmetric Wilson loop in the representation R is given by Here we absorbed (−1) Nc−1 into the definition of ζ. 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.

Bethe ansatz equation and match of parameters
We consider how the generators of K q are identified with the variables in 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 We find that (4.8) is same as the relation ζe iBa = 1 of the supersymmetric Wilson loop with the following identification

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 the following relation Here we used the parameter identification (4.9). By using (4.8), the higher order correlation functions W l Nc=1,N f for l ≥ n is expressed in terms of W k , (k = 0, 1 · · · , n − 1) as In the 2d N = (2, 2) * theory with N f = n flavors, we found σ l Nc=1,N f =n A-twist , (l ≤ n − 1) do not have q-correction. We have a similar property in 3d N = 2 * theory. For 0 ≤ l ≤ n − 1, W l Nc=1,N f =n does not have ζ-correction and is given by the zero magnetic charge sector as We have check (4.10) and (4.12) for N f = 2, 3, 4 in several orders of ζ.

Non-Abelian cases
In two dimension, we observed that the partition function does not receive any q-correction and is given by the residue at the zero magnetic charge sector. Similarly we observed that the partition function (index) of 3d N = 2 * theory does not receive ζ-correction neither and is given by In two dimensions, 1 Nc=r,N f =n A-twist has a geometrical interpretation as equivariant integration of [1] ∈ H * (T * Gr(r, n); C). (4.13) also have geometrical interpretation as follows. If we identify 3d parameter z i and h as z i = e z i and h = e −h , (4.13) is sinh uplift of the equivariant integration which can be interpreted as equivariant Dirac index.
For 2 < n − 2, we also observed that the expectation values of x ±1 (4.14) On the other hand, the properties of the correlation function (x 1 + x 2 ) 2 , (x 1 + x 2 )(x 1 x 2 ), and (x 1 x 2 ) 2 are different from the two dimensions. In the 2d N = (2, 2) * theory with N c < N f − N c , we expected that the correlation functions of symmetric polynomial of σ Nc a=1 e la a (σ) with Nc a=1 l a ≤ N c , (4.15) don't have q-dependence. This may be interpreted as that the degree of polynomial is not reducible in the polynomial ring by the ideal. For example, (σ 1 + σ 2 ) l (σ 1 σ 2 ) k Nc=2,N f with k+l = 2 agrees the residues at zero magnetic charges sector. do not have q-dependence. For examples (σ 1 + σ 2 ) l (σ 1 σ 2 ) k Nc=2,N f =4 with k + l = 2 agrees the residues at zero magnetic charges sector. 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 the following relation 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 equation and (x 1 + x 2 ) l (x 1 x 2 ) k has ζ dependence.

Wilson loops and Seiberg-like duality in 3d N = 2 * theory
From the expression of (4.13), the partition functions (index) agree in the U (N c ) and To find the map of Wilson loops between U (r) and U (n − r) gauge theory, let us consider polynomial P (Γ,z,h,q) = 0 for (λ 1 ,λ 2 ) = (n − r, n) In the following identification (4.19) is identical to P (Γ, z, h, q). (4.20) give isomorphism between Bethe subalgebras with (λ 1 , λ 2 ) = (r, n − r) and (λ 1 ,λ 2 ) = (n − r, n). Substitution u =γ 1,a ,γ 1,ah −1 into gives the relation which is again same asζe iBa = 1 with the following identification Here we take tilde for the variables in U (n − r) gauge theory to distinguish from that of U (r) gauge theory. From (4.9), (4.20) and (4.22), we have the maps of the variables between U (r) and U (n − r) 3d N = 2 * gauge theories: From now on, we do not distinguish y i , z and z i fromỹ i ,z andz i . We write down the Seiberg-like duality for supersymmetric Wilson loops. The coefficient of u −n+1 of P = 0 gives By using the relations between Bethe subalgebra and fugacities in 3d index, (4.24) leads to Seiberg-like duality for Wilson loop in fundamental representation W F = Nc a=1 x a . ( and the Seiberg-like duality for expectation values is given by For example, the evaluation of the right hand side in the region ζ < 1, (ζ > 1) means that the left hand side is evaluated in the regionζ = ζ −1 > 1, (ζ = ζ −1 < 1) in which negative (positive) magnetic charge contribute to the Jeffrey-Kirwan residue operations, vice versa. We evaluated the left and right hand side independently and have agreement in several (r, n).
Next we derive the Seiberg-like duality for second anti-symmetric representation. The derivation is parallel to that in two dimensions. 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 anx a as (1 − h −2 q)e 2 (x) = −(1 − h 2 q)e 2 (xh −1 ) (e 1 (xh −1 )) 2 + (1 − q) (1 − h −1 q) e 1 (z)e 1 (xh −1 ) + e 2 (z) (4.28) Therefore we obtain the Seiberg-like duality for the second anti-symmetric representation W A 2 = a<b x a x b as Nc=r,N f =n In the similar way, we can obtain Seiberg duality for Wilson loops in other representations.

Conclusion and Future Directions
In this paper, we discussed the relation between the partition function in A-twisted 2d N = (2, 2) and topologically twisted 3d N = 2 gauge theories and the inverse of the norm of Bethe eigenstates for XXX 1/2 and XXZ 1/2 spin chain model with a particular choice of R-charges and background magnetic fluxes for flavor symmetries in gauge theory sides. The correlation functions in gauge theories could be understood as the coefficients of the expectation values of Baxter Q-operators as well as conserved charges with appropriate coefficient.
We also studied the relation between correlation functions in A-twisted 2d N = (2, 2) * gauge theories and equivariant integration of equivariant quantum cohomology for the cotangent bundle of Grassmannians. A method of calculating the equivariant integration was described in Section 3, which we expect to hold. We also considered Seiberg-like duality of 2d N = (2, 2) * to calculate the equivariant integration. We have checked several examples and found that the equivariant integration via the method described above are matched with the correlation functions.
From the 2d N = (2, 2) * calculations where chiral ring relation is identified with Bethe subalgebra of XXX spin chain model, we did similar calculation for 3d N = 2 * . We calculated correlation functions of Wilson loops and checked that they agree with Bethe subalgebra of XXZ spin chain model.
There are several interesting directions. Firstly, it will be interesting to find the analogue of the equivariant integration in equivariant quantum K-theory and match them with the correlation functions of Wilson loops in topologically twisted 3d N = 2 * theory.
Another interesting direction is to study relations between Bethe ansatz and finite dimensional commutative Frobenius algebra. In [24], a finite dimensional commutative Frobenius algebra was constructed in terms of Bethe ansatz for q-boson model. It is known that finite dimensional commutative Frobenius algebra is essentially same as 2d topological quantum field theory (TQFT) and 2d partition functions on genus g Riemann surface Σ g corresponding to q-boson can be written as [16] Z(Σ g ) = (λ)∈P q-boson Ψ(λ)|Ψ(λ) g−1 (5.1) Here |Ψ(λ) is n eigen vector of q-boson determined by Bethe root (λ). We obtained same type formulas for XXX and also XXZ spin chains and corresponding TQFT are topologically twisted 2d N = (2, 2) and 3d N = 2 theories. By using a recent result [25], the partition function of 2d N = (2, 2) and 3d N = 2 theories studied in this paper can be generalized to the Riemann surfaces as These formulas are very similarly to q-boson case and implies that there exist dimensional commutative Frobenius algebras associated with Bethe ansatz for XXX and also XXZ spin chains. It would be interesting to construct Frobenius algebras in terms of XXX and XXZ spin chains.