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 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians

We study a correspondence between 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 topologically twisted Chern-Simons-matter theories on S1× Σg and quantum K -theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter β associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with β = −1 is isomorphic to the (equivariant) small quantum K -ring of the Grassmannian, and the Frobenius algebra with β = 0 is isomorphic to the equivariant small quantum cohomology of the Grassmannian. We apply supersymmetric localization formulas to the correlation functions of supersymmetric Wilson loops in the Chern-Simons-matter theory and show that the algebra of Wilson loops is isomorphic to the Frobenius algebra with β = −1. This allows us to identify the algebra of Wilson loops with the quantum K - ring of the Grassmannian. We also show that correlation functions of Wilson loops on S1× Σg satisfy the axiom of 2d TQFT. For β = 0, we show the correspondence between an A-twisted GLSM, the Frobenius algebra for β = 0, and the quantum cohomology of the Grassmannian. We also discuss deformations of Verlinde algebras, omega-deformations, and the K -theoretic I -functions of Grassmannians with level structures.

1 Introduction and summary N = (2, 2) supersymmetric gauged linear sigma models (GLSM) in two dimensions [1] play important roles in connection with quantum cohomology, Gromov-Witten invariants, and mirror symmetry. Correlation functions of the vector multiplet scalars of abelian A-twisted GLSM on the two-sphere S 2 was studied in [2]. When the Higgs branch of N = (2, 2) GLSM is a toric Fano manifold, the three-point correlation functions give the three-point genus-zero Gromov-Witten invariants of that manifold. When the Higgs branch is a Calabi-Yau 3-fold, the identification of the three-point functions with the B-model Yukawa couplings of the mirror is known as toric residue mirror symmetry [3][4][5][6][7]. Threepoint functions of non-abelian A-twisted GLSMs on S 2 , possibly with the Ω-background, are computed in [8] using supersymmetric localization. It was suggested in [9] that the JHEP08(2020)157 resulting formulas give genus-zero quasimap invariants of the space of Higgs branch vacua of the GLSM, and proved mathematically in [10] for Grassmannians and their Calabi-Yau complete intersections defined by equivariant vector bundles. It follows that the twisted chiral ring of 2d A-twisted GLSM, whose Higgs branch is the Grassmannian, agrees with the small quantum cohomology ring of the Grassmannian.
When a supersymmetric quantum field theory on a manifold M is related to a cohomological object, an S 1 uplift of the theory, i.e., a supersymmetric quantum field theory on S 1 × M , is often related to a K-theoretic object. A typical example is Nekrasov's instanton partition functions; roughly speaking, instanton partition functions on R 4 [11] are integrals of cohomology classes over instanton moduli spaces [12], whereas five-dimensional instanton partition functions on S 1 × R 4 [13] are indices of instanton moduli spaces (K-theoretic instanton partition functions) [14]. Hence it is natural to expect the existence of 3d (topologically twisted) supersymmetric gauge theories relevant to quantum K-theory [15], which is a K-theoretic analogue of quantum cohomology.
In this paper, we establish a relation between certain 3d topologically twisted supersymmetric gauge theories and the quantum K-rings of Grassmannians. Although there are earlier works in this direction, starting with [16] which pointed out a similarity between the small K-theoretic J-function of P M −1 and the vortex partition function of a certain 3d U(1) gauge theory in the context of open BPS state counting, to our best knowledge, an identification of the small quantum K-ring of Grassmannians (or even projective spaces) with the level of precision of this paper is new.
To find the precise relation between the 3d supersymmetric theories and the quantum K-ring of Grassmannians, we focus on Bethe/Gauge correspondence. It was pointed out in [17,18] that the saddle point equations of the mass deformations of a 2d N = (4, 4) U(N ) GLSM coincide with the Bethe ansatz equation of the spin- 1 2 XXX spin chain. The relation between the twisted chiral ring and the quantum cohomology mentioned above implies that the Bethe ansatz equation is closely related to the relations of the equivariant quantum cohomology ring of the cotangent bundle of the Grassmannian, which appears as the Higgs branch of the N = (4, 4) GLSM. See e.g. [19] for a mathematical formulation of this observation.
For the quantum cohomology of the Grassmannian, a description in terms of the algebraic Bethe ansatz of a five-vertex model is given by . In a subsequent work [21], they introduced a finite-dimensional commutative Frobenius algebra depending on a parameter β. They showed that the Frobenius algebra with β = −1 is isomorphic to the small quantum K-ring of Grassmannian 1 and the Frobenius algebra with β = 0 reproduces their previous result on the quantum cohomology in [20].
By using supersymmetric localization formulas, we show that the saddle point equations of the 3d U(N ) Chern-Simons-matter theory with M chiral multiplets coincide with the Bethe ansatz equations for β = −1. This implies that the operator product expansions (OPEs) of Wilson loop operators W λ can be identified with the quantum products of the JHEP08(2020)157 equivariant quantum K-theory ring QK T (Gr(N, M )) of the Grassmannian; Here is the quantum product in QK T (Gr(N, M )) and [O λ ]'s are the K-theory classes of Schubert varieties of the Grassmannian. We also revisit the relation between A-twisted 2d GLSM and quantum cohomology from the viewpoint of Bethe/Gauge correspondence, and derive an isomorphism of the twisted chiral ring with the equivariant small quantum cohomology of the Grassmannians. Generalizations of the supersymmetric localization formula from S 1 × S 2 (resp. S 2 ) to higher genus cases S 1 × Σ g (resp. Σ g ) were studied in [23,24]. We show that the correlation functions at higher genus can be identified with the correlation functions of the 2d topological quantum field theories (TQFTs) associated with the Frobenius algebra.
Besides the correspondence between the algebra of Wilson loops in the topologically twisted CS-matter theories and the quantum K-rings of Grassmannians, we discuss several other aspects of CS-matter theories; CS-matter theories with Ω-background, K-theoretic I-functions of Grassmannians with level structures, supersymmetric index on S 1 × D 2 , and deformations of Verlinde algebra in terms of indices over moduli stacks of G-bundles on Riemann surfaces.
This article is organized as follows. In section 2.1, we recall results by Gorbounov-Korff which will be used later to show the relation between quantum K-ring (resp. quantum cohomology) and CS-matter theory (resp. GLSM). In section 2.2, we compute the correlation functions of 2d TQFT on a genus g surface associated with the Frobenius algebra. In section 2.3, we evaluate the inner product of the on-shell Bethe vectors by the Izergin-Korepin method. In section 3, we show that the genus g correlation function of the 2d TQFT associated with the Frobenius algebra with β = 0 agrees with the genus g correlation functions of the A-twisted GLSM. In section 4, which is the main part of our paper, we study the correspondence between CS-matter theories and the quantum K-theory of Grassmannians. In section 4.1, we show that the supersymmetric localization formulas of the correlation functions of Wilson loops on S 1 × Σ g agree with the genus g correlation functions of 2d TQFT associated with the Frobenius algebra with β = −1. These equalities lead to the isomorphism between the algebras of Wilson loops and the quantum K-ring of Grassmannians. In section 4.2, we turn on the Ω-background and show that the partition function of topologically twisted CS-matter theory with specific choices of CS levels factorizes to a pair of K-theoretic I-functions, and discuss the relation between CS levels and level structures of quantum K-theory. In section 4.3, we show that K-theoretic I-function derived in section 4.2 can be derived from a supersymmetric index on S 1 × D 2 . In section 5, we discuss connections between CS-matter theories and deformations of Verlinde algebra.

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2 Quantum K-ring, quantum cohomology and algebraic Bethe ansatz  defined a Frobenius algebra depending on a parameter β in terms of algebraic Bethe ansatz. The elements of the Frobenius algebra are defined by the N excitations (particle states) in the Hilbert space of the quantum integrable system with M -sites. The Frobenius bilinear form is defined in terms of the inner product of states and dual of states in the Hilbert spaces. They showed that the structure constants in the spin basis agree with the structure constants of the equivariant quantum K-ring of Grassmannian QK T (Gr(N, M )) for β = −1 and the equivariant quantum cohomology of Grassmannian QH T (Gr(N, M )) for β = 0.
In section 2.1, we summarize results from [21] which will be useful later. In particular, we recall the definition and properties of the Frobenius algebra introduced in [21]. In section 2.2, we describe the correlation functions of the 2d TQFT associated with the Frobenius algebra. In section 2.3, we evaluate the on-shell square norm of Bethe vectors and dual Bethe vectors.

Results of Gorbounov-Korff
First we introduce the R-matrix and L-operator which characterize the quantum integrable model in [21].
where is defined in appendix A. The R-matrix satisfies the following Yang-Baxter equation: Here I 2 is the identity matrix of size two and P is a permutation matrix defined by
also satisfies the Yang-Baxter type relation (RTT relation) Here L(x|t i ) non-trivially acts on C 2 (x) ⊗ C 2 (t i ). (2.10) gives the sixteen relations between four 2 M × 2 M matrices A(x|t), B(x|t), C(x|t) and D(x|t) listed in appendix B. The twisted transfer matrix is defined by the trace taken over the auxiliary space as The periodic boundary condition is given by q = 1. Note that the twist parameter q will be identified with the quantum parameter of the quantum K-ring for β = −1 and the quantum parameter of the quantum cohomology for β = 0. From the relation (2.10), the transfer matrices commute with each other: [T(y|t), T(y |t)] = 0. (2.12) The coefficients of y l for l = 0, 1, · · · in T(y|t) give conserved charges. The states and dual states on which A, B, C, D act are defined as follows. First we introduce a basis of the 2 where v 0 and v 1 are defined by and we also introduce the dual space spanned by in one-to-one correspondence with the elements of a set of partitions P N,M defined by (2.16)

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We define a vector |v λ by ⊗ M i=1 v i which correspond to a partition λ ∈ P N,M . We call {|v λ } λ∈P N,M as the spin basis of V N,M . The dual spin basis { v λ |} λ∈P N,M for the dual space of V N,M is defined in a similar way and the inner products satisfy v λ |v µ = δ λµ . We define ∅ by ∅ := (0, · · · , 0) ∈ P N,M .
The pseudo vacuum |0 and the dual pseudo vacuum 0| are defined by The A, B, C, D act on the pseudo vacuum as is an eigen vector of the transfer matrix (2.11), then {y 1 , · · · , y N } have to satisfy the following system of equations called the Bethe ansatz equation: A root {y a } N a=1 of the Bethe ansatz equation is called as a Bethe root. When {y a } N a=1 is a Bethe root, N a=1 B(y a )|0 (resp. 0| N a=1 C(y a )) is called an on-shell Bethe vector, (resp. on-shell dual Bethe vector), vice versa. When {y a } N a=1 is indeterminate, Bethe vectors are called off-shell.
(2.26) Π(y), Π(t λ ), t, t and (y b |t) k are defined in appendix A . The partition λ ∨ associated to λ = (λ 1 , · · · , λ N ) ∈ P N,M is defined by Note that G λ (y|t) for β = 0 agrees with a factorial Schur polynomial given by . (2.28) When t i = 0, a factorial Schur polynomial s λ (y|t) is reduced to a Schur polynomial s λ (y). Here the inner product is called on-shell (resp. off-shell), when {y a } N a=1 are a Bethe root (resp. indeterminate). As we will see (2.24) and (2.25) are useful to show the equality of the equivalence between the partition function of 2d TQFT on Σ g and the topologically twisted index (for simplicity we call partition function) of a Chern-Simons-matter theory on S 1 ×Σ g for β = −1 and also the partition function of a 2d A-twisted GLSM on Σ g for β = 0.
The definition and properties of the Frobenius algebra. Now we explain the definition and properties of the Frobenius algebra in [21]. Let {Y λ } λ∈P N,M be a basis of V N,M defined by The associative product * :

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The identity element is given by |v . (2.32) Here the coefficient ring is defined by R := Z[[q]] ⊗ R(t 1 , t 2 , · · · , t M ) and R is the ring of rational function of β regular at β = 0, −1. η is non-degenerate, since {|y λ } λ∈P N,M is a basis. (V N,M , * , η) is a finite dimensional commutative Frobenius algebra. For simplicity we call V N,M as the Frobenius algebra. Next we explain the relations between the Frobenius algebra, quantum K-ring and quantum cohomology. Let {C λ µν (q, t, β)} λ,µ,ν∈P N,M be the structure constants of V N,M in the spin basis {|v λ } λ∈V N,M : The transformation from the on-shell Bethe vectors to the spin basis, C λ µν (q, t, β) is written as Gorbounov-Korff showed that the structure constants (2.34) for β = −1 agree with the structure constants of the K-theory classes of structure sheaves of the T := (C × ) M equivariant quantum K-ring of Grassmannian QK T (Gr(N, M )), i.e., Therefore the isomorphism between V N,M /(β) and QH T (Gr(N, M )) is given by We define η µν ∈ R as the elements of the Frobenius bilinear form η in the spin basis: We define the genus zero three point function C Note that η(|v λ , |v µ * |v ν ) = η(|v λ * |v µ , |v ν ). C Here λ is the conjugate (transpose) partition of λ. We will see the level-rank duality leads to a Seiberg-like (level-rank) duality between correlation function of U(N ) Chern-Simonsmatter theory and a U(M − N ) Chern-Simons-matter theory on S 1 × Σ g and also leads to Seiberg-like duality of two A-twisted GLSMs on Σ g .

Frobenius algebra and genus g n-point correlation functions in 2d TQFT
In order to show the correspondence between the quantum K-rings and CS-matter theories on S 1 × Σ g and also the correspondence between the quantum cohomologies and A-twisted GLSMs on Σ g , it is useful to introduce a pictorial description [27] (see also [28] for a more rigorous treatment) of the finite dimensional commutative Frobenius algebras in terms of 2d TQFTs. After explaining the 2d TQFT aspect of the Frobenius algebra, we will calculate quantities associated to the surface Σ g,n with genus g and n in-boundaries which will be JHEP08(2020)157   (1)ν λ corresponds to gluing two genus one surface with an in-boundary and out-boundary, which is a genus two surface with an in-boundary and out-boundary and is expressed by C identified with genus g correlation function of topologically twisted gauge theories in two and three dimensions.
First we introduce η µν defined by Then η µν is the inverse of η µν , since these satisfy the relation ν∈P N,M η µν η νρ = δ ρ µ which follows from the formula (2.25). We introduce the pictorial expression of the product * , the bilinear form η, and identity |v ∅ as depicted in figure 1. The inverse of η is associated to an annulus with two out-boundaries. A contraction of an upper index and a lower index corresponds to a gluing of an in-boundary and an out-boundary (See figure 3b).

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Note that C (g)µ 1 ···µ l λ 1 ···λn satisfies the following properties Here we used (2.24). C (g)µ λ is calculated by contracting upper and lower indices of C The genus g Riemann surface corresponds to the contraction µ∈P N,M C (g)µ µ given by of the choices of bases. Z(Σ g ) is sometimes called the genus g partition function of 2d TQFT. In section 3, we will show Z(Σ g ) for β = 0 agrees with a partition function of A-twisted GLSM on Σ g and, in section 4.1, we will show Z(Σ g ) for β = −1 agrees with the partition function (topologically twisted index) of a CS-matter theory on S 1 × Σ g . Next we express C (g) µ 1 µ 2 ···µn in terms of Grothendieck polynomials and the on-shell square norms. By using the relation (2.24), we obtain
The level-rank duality (2.39) and (2.40) leads to the equality of C (g)

Determinant representation of the on-shell square norm
In order to relate the quantum K-ring with the inner product (2.32) to correlation functions and the algebra of supersymmetric Wilson loops in a supersymmetric Chern-Simons-matter theory, we have to express the on-shell inner product e(y λ , y λ ) in (2.38) as a nice function of the Bethe roots {y λ } λ∈P N,M and the inhomogeneous parameters It was shown in [29,30] that the partition functions of U(N )/U(N ) gauged Wess-Zumino-Witten (WZW) model and its deformation (called gauged WZW-matter model) 3 are written in terms of the determinant type formulas of the on-shell norms for the phase model and q-boson, respectively. Moreover the partition functions of the gauged WZW model and the gauged WZW-matter model on the Riemann surfaces agree with the partition functions of 2d TQFTs associated to the Frobenius algebras in [32,33]. Motivated by these agreements between gauge theories and 2d TQFTs observed in [29,30,34,35], we apply Izergin-Korepin method [36] and express the on-shell square norm by the determinant type formula. See also [37] for a review of derivation of the norm for the spin- 1 2 XXZ spin chain. The derivation of determinant type formula for e(y λ , y λ ) is parallel to that for the XXZ spin chain.
First we study the off-shell inner product between the Bethe vector and the dual Bethe vector for indeterminate variables {x} := {x 1 , · · · , x N } and {y} = {y 1 , · · · , y N }. Recall that the off-shell inner product is defined by (2.53) From (B.8) and (2.18), the most general form of (2.53) is expressed as
is expressed in terms of the leading and conjugate leading coefficients as follows.
Here g(x, y) is a coefficient in the RTT relation given by Then, it is enough to evaluate the leading and the conjugate leading coefficients to evaluate the off-shell inner product (2.54). Let us consider to evaluate the leading and the conjugate leading coefficients. From (2.58), we obtain (2.60) Here the ellipses denote the terms which are irrelevant to the evaluation of the leading coefficient. Multiplying (2.60) by 0| N −1 i=1 C(x i ) , we find the leading coefficients satisfy the following recurrence relation.
In a similar way, the recurrence relation for the conjugate leading coefficients is given by (2.62)
(2.66) det j,k denotes the determinant with respect to indices j, k. We show that (2.64) satisfies the recurrence relation (2.61) as follows. By adding the j-th row of the matrix t(x j , y k ) multiplied by u j /u N with j = 1, · · · , N − 1 to the N -th row of t(x j , y k ), we obtain the following relation between determinants for t(x j , y k ): then G l satisfies the following identity: To show (2.69), we consider the integral: . (2.70) Here the contour encloses all the poles of the integrand; z = ∞, x 1 , · · · , x N . The residues at z = ∞ and z = x 1 , · · · , x N are evaluated as (2.71) Since I = 0, we obtain (2.69).

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From (2.67) with the identity (2.69), we find that (2.64) satisfies the recurrence relation (2.61). Thus the right hand side of (2.64) is the leading coefficient. In a similar way, we can show that (2.65) satisfies the recurrence relation forK N .
Substituting the solutions of the recurrence relations to (2.54), we have P α is the parity of the partition (α 1 , · · · , α n ,ᾱ 1 , · · · ,ᾱ N −n ), defined as 0 (resp. 1) when (α 1 , · · · ,ᾱ N −n ) is an even (resp. odd) permutation of (1, 2, · · · , N ). Now assume that {y} in (2.72) is a Bethe root. By using the Bethe ansatz equation and the identity we obtain the following expression of the square norm after some calculation: where Finally, by taking limit x a → y a , we obtain the determinant formula of the on-shell square norm given by (2.77) We are interseted in the expression of e(y, y) with slices β = −1, 0. When β = 0, the inverse of e(y, y) is given by (2.78) When β = −1, the inverse of e(y, y) is given by

2d A-twisted GLSM and quantum coholomogy of Grassmannian
In this section we study a 2d A-twisted GLSM on the genus g Riemann surface Σ g . The gauge group of GLSM is taken to be G = U(N ) and the M chiral multiplets are in the fundamental representation of U(N ) with M > N . If we put the GLSM on R 2 , the Higgs branch vacua is a Grassmannian Gr(N, M ) in the positive Fayet-Iliopoulos (FI) parameter region. The motivation of this section is threefold. The first is to establish Bethe/Gauge correspondence between the algebraic Bethe ansatz for β = 0 and the 2d U(N ) GLSM with M fundamental chiral multiplets. As far as we know, Bethe/Gauge correspondence for Atwisted GLSM for M fundamental chiral multiplet has not been studied. The second is to explain how the expression (2.38) of three-point Gromov-Witten invariants appears in Atwisted 2d GLSM on S 2 . 4 The third is to show that the correlation functions of A-twisted 2d GLSM on higher-genus Riemann surfaces agree with those of 2d TQFT associated with the Frobenius algebra. This suggests that these correlation functions are quasimap invariants with fixed domains.

A-twisted GLSM on S 2 and genus zero quasimap invariants
First we consider the genus zero case. In terms of supersymmetric localization formula [8], the n-point correlation functions of gauge invariant functions O 2d.l of the U(N ) vector multiplet scalar in the A-twisted GLSM on S 2 is written as the following form: Here the residues in (3.1) are evaluated at σ a = m ia with a = 1, · · · , N and i a = 1, · · · , M . {σ a } N a=1 is the saddle point value of u(N ) ⊗ C valued scalar σ in the U(N ) vector multiplet: {k a } N a=1 is magnetic charges of gauge fields for the maximal torus of U(N ). Z FI is the contribution from the saddle point value of the FI term and theta term which is given by where S FI is the actions of FI-term and theta-term. log q = 2πiθ − ξ is the holomorphic combination of the FI parameter ξ and the theta angle θ. q is identified with the JHEP08(2020)157 quantum parameter or equivalently the parameter of twisted boundary condition in the algebraic Bethe ansatz.
2d.chi (σ, k) are the one-loop determinant of the 2d A-twisted super Yang-Mills action on S 2 and the one-loop determinant of the M chiral multiplets on S 2 given by (3.5) Here m i is the twisted mass of the i-th chiral multiplet and r ∈ Z ≥0 is a vector type U(1) R-charge. For any given symmetric polynomial f (σ 1 , · · · , σ N ) of σ 1 , · · · , σ N , there exists a gauge invariant operator of σ whose saddle value is given by f (σ 1 , · · · , σ N ). We may defineŝ λ by the gauge invariant operator of σ including m i , whose saddle point is given by a factorial Schur polynomial s λ (σ|m). It is enough to consider the correlation functions ofŝ λ , since {s λ (σ|m)} λ∈P N,M generate the symmetric polynomials of σ 1 , · · · , σ N .
The algebra of scalars in the vector multiplet (twisted chiral ring) is given by Here ellipses denote the arbitrary insertions of gauge invariant combination of vector multiplet scalar and N k ij 's are the coefficients of OPEs. We remark on properties of the algebra of scalars. The Q-exact terms in the correlation functions vanish and do not contribute to the OPEs. If there exist a basis of the algebra of vector multiplet scalars such that the coefficients N k ij agree with the structure constants C λ µν of the quantum cohomology, then the algebra of vector multiplet scalar is isomorphic to the quantum cohomology of Grassmannian Gr(N, M ). When we regard A-twisted GLSM as a Frobenius algebra, Taking the expectation values of O 2d.l correspond to a Frobenius bilinear form and a choice of R-charges r corresponds to a choice of Frobenius bilinear form.

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Although the expectation values depend on the R-charge, The coefficients of OPE of O 2d.l , i.e., the structure constants are independent of r, we will show this later. We will choose a particular integer of R-charge r to give identification with the Frobenius algebra with β = 0.
It is shown in [10] that (3.1) with r = 0 is the generating function of genus zero degree-k (k := k 1 + · · · + k N ) quasimap invariants, which imply that the three point function correlation functions of factorial Schur polynomials agrees with the genus zero three points T -equivariant Gromov-Witten invariants of Schubert classes [X λ ], [X µ ], and [X ν ] of Gr(N, M ): We show (3.8) as follows. We first perform sums of {k a } N a=1 in (3.1) and deform the integration contours. Then we obtain the following expression of n-point correlation function: Here W 2d.eff is the effective twisted superpotential defined by (3.10) From the first to the second line in (3.9), the residues are evaluated at roots of the saddle point equations of the twisted superpotential; exp (∂ σa W 2d.eff ) = 1, a = 1, · · · , N .
(σ a − m i ) = (−1) N −1 q, a = 1, · · · , N. ( Since the saddle point equations is independent of the R-charge r, the structure constants of the algebra ofÔ 2d in the correlation functions are independent of the R-charge. This can been seen as follows. From the saddle point equation (3.11), the genus zero n-correlation function with r is related to that with r = 0 as Here · · · S 2 ,r denotes the correlation functions on S 2 with R-charge r. Then we find that the R-charge dependence is canceled out in the structure constants C λ µν = η λρ C (0) ρµν .

A-twisted GLSM on the general Riemann surfaces and 2d TQFT
Next we consider A-twisted GLSM on Σ g and show that the correlation functions of Atwisted GLSM agree with (2.51). In terms of the supersymmetric localization computa-JHEP08(2020)157 tion [23], an n-point correlation function on Σ g is written as Here the one-loop determinants for genus g case are given by . (3.20) When g = 0, the formula is same as the localization formula on S 2 (3.1). We comment on the contour integrals in (3.1), (3.18). When g = 0, in the order by order residue computations in the power of q, the contour integrals are determined by Jeffrey-Kirwan (JK) residues [39]. As we have seen in the previous section, we have two expressions (3.14) and (3.15) of the genus zero correlation functions in the A-twisted GLSM, both of them give the same result. In the derivation of the JK residue in supersymmetric localization, it is required that the singular hyperplane arrangement is projective, for example see [40]. When g > 1, the one-loop determinant of the vector multiplet of non-abelian gauge group (N ≥ 2) violates the projective condition and the order by order residue operations are not well-defined. 5 We first perform the sums and obtain the following expression of the correlation function n l=1ŝ .
(3.23) 5 The violation of the projective condition implies that the order of the sums and the integrals do not commute. A prescription to make sense of JK residue operation is proposed in [40] that regulator mass for vector multiplet is introduced to satisfy the condition and the regular mass is taken to zero after the JK residue operations. A different choice of contour was proposed in [24], but we do not have any agreement between the contour prescription in [24] and the higher genus partition functions constructed by the axiom of 2d TQFT.

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Again, if we choose r = 0, (3.23) perfectly agrees with the correlation function on a genus g correlation function obtained from the axiom of 2d TQFT; n l=1ŝ λ l Σg = C (g) λ 1 λ 2 ···λn (q, t, β = 0) for r = 0. (3.24) In particular the partition function of the A-twisted GLSM with r = 0 on Σ g agrees with the genus g partition function of 2 TQFT with β = 0;  In this section we introduce topologically twisted 3d N = 2 Chern-Simons (CS) theories coupled to chiral multiplets (CS-matter theory) on S 1 × Σ g . We consider the gauge group G = U(N ). The M chiral multiplets are in the fundamental representation of the gauge group U(N ). We will show the correspondence between the algebra of supersymmetric Wilson loops and the equivariant small quantum K-theory ring of Grassmannian Gr(N, M ). For the moment we take supersymmetric gauge, flavor, R-symmetry mixed Chern-Simons terms with the generic CS levels: Here A is the U(N ) dynamical gauge field. B is the background gauge field which couples to the topological U(1)-current. A (R) is the background gauge field for U(1) R-symmetry.

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C (i) is the background gauge field for a U(1) flavor symmetry non-trivially acting on the i-th chiral multiplet for i = 1, · · · , M . The ellipses denote supersymmetric completion of the CS-terms which includes 3d N = 2 super partners of the dynamical and background gauge fields. κ (1) , κ (2) , κ i κ (4) , κ (5) i , κ (6) ij denote integer or half integer valued CS levels κ (1) , κ (2) are the gauge CS levels, κ i 's are the gauge-flavor mixed CS levels, κ (4) is the gauge-R-symmetry mixed CS level, κ (5) i 's are the flavor-R-symmetry mixed CS levels, and κ (6) ij 's are the flavor CS levels. The CS levels in the Lagrangian have to satisfy the condition for the absence of anomaly. The values of CS levels will be determined later.
By supersymmetric localization formula [23,24,44] (see also an earlier work [45]), the path integrals of the topologically twisted 3d N = 2 CS-matter theory on S 1 × Σ g is reduced to N -dimensional multi-contour integrals and infinite magnetic sums. An n-point correlation function of supersymmetric gauge and flavor Wilson loops is given by (

4.2)
Here the integrand is given as follows.
σ is the adjoint scalar in the U(N ) vector multiplet. The algebra of Wilson loop is defined by the following OPE Here ellipses denote the arbitrary insertions of Wilson loop operators and N k ij is the coefficient of OPE (structure constants). If there exist a basis of the algebra of Wilson loops JHEP08(2020)157 such that the coefficients N k ij agree with the structure constants C λ µν of the quantum Kring, then the algebra of Wilson loops is isomorphic to the quantum K-ring. Note that the correlation functions and the coefficients of OPE do not depend on the coordinates of S 1 × Σ g , since topologically twisted CS-matter theories are topological field theories. Z (g) CS is the saddle point value of the mixed CS terms (4.1) given by Here q = e S 1 (iBτ −σ B )dτ is the supersymmetric Wilson loop for the background gauge field B. σ B is U(1) scalar belonging to the same background supermultiplet of B, which is the FI-parameter in three dimensions. k a and n i are magnetic charges (the first Chern numbers) of the gauge field of the a-th diagonal U(1) ⊂ U(N ) and a U(1) background gauge field C (i) in the Riemann surface direction, respectively. Z .

(4.8)
r ∈ Z ≥0 is an integer valued U(1) R-charge of the 3d chiral multiplets. W 3d.eff is the effective twisted superpotential given by The last two terms in (4.9) do not contribute to the computation of the correlation functions on S 1 × Σ g . As explained in the section 3.2, when g = 0, 1, the singular hyperplane arrangements are projective and the integration contours in (4.2), order by order in powers of q are determined by the JK residue operations. The contours are chosen to enclose x a = z i for all a = 1, · · · , N and i = 1, · · · , M . On the other hand, when g ≥ 2 and N ≥ 2, JHEP08(2020)157 the singular hyperplane arrangements including the one-loop determinant of the super Yang-Mills action are not projective. We expect the ordering of the sums and the contour integrals does not commute in general. When the sums of k a for a = 1, · · · , N are performed before the integration, we obtain the following expression of the correlation function. z, κ, r). (4.11) Here we introduced H(x, z, κ, r) to shorten the formulas by (4.12) The Hessian of the 3d effective twisted superpotential is written as In (4.10), the residues are evaluated at the roots of the saddle point equations of the 3d twisted superpotential exp (∂ log xa W 3d.eff ) = 1, a = 1, · · · , N with x a = x b for 1 ≤ a = b ≤ N . The sum x runs over the solution of the saddle point equation of twisted superpotential; (4.14) Note that the expression of the correlation functions in terms of the roots of the saddle point equation (the Bethe roots) (4.11) is useful to show the isomorphism between the Wilson loop algebra and the quantum K-ring QK T (Gr(N, M )) for general N, M , but it is not easy to compute correlation functions explicitly from it, since we do not have simple closed expressions of the Bethe roots for β = −1 and N ≥ 2. 6 On the other hand (4.2) JHEP08(2020)157 for g = 0 gives an explicit procedure to compute genus zero correlation functions, order by order in the q-expansions in terms of JK residues, which will be useful to compute structure constants of quantum K-rings. Now we verify the correspondence between the algebra of Wilson loops and the quantum K-ring. We choose the gauge CS levels and the gauge-flavor mixed CS levels as When we choose gauge, gauge-flavor mixed CS levels as (4.15), the saddle point equation (4.14) becomes We find that the saddle point equation (4.16) equals to the Bethe ansatz equation (2.20) for β = −1 with the following identification of variables: where the x a and z i are a root of the twisted superpotentials and a flavor Wilson loop in the CS-matter theory and y a and t i are a Bethe root and an inhomogeneous parameter in the algebraic Bethe ansatz. For U(1) gauge theory, it is easy to see the agreement between the algebra of Wilson loops and the quantum K-ring of projective space as follows. In this case, the saddle point value of a U(1) gauge Wilson loop is an element of C[x, x −1 ], where x := x 1 and the saddle point equation is written as From (4.11), the following relation holds in the correlation function Here we express (polynomials of) U(1) Wilson loops asx and O 3d (x, z i ) such that the saddle point values are given by x and O 3d (x, z i ), respectively. O 3d (x, z i ) is an arbitrary JHEP08(2020)157 Then the algebra of Wilson loop for U(1) theory with M chiral multiplets is identified with 7 If x is identified with K-theory class of a line bundle O(−1) of P M −1 , we find that the algebra of Wilson loops (4.20) reproduces the equivariant quantum K-ring QK T (P M −1 ). Note that the coefficients of OPEs of Wilson loops or equivalently the structure constants are independent of choices of κ (4) , κ i , κ ij , n i and r. A choice of κ (4) , κ i , κ ij , n i and r fixes the value of the Wilson loop correlation functions (4.11) which corresponds to a choice of Frobenius bilinear form.
For general N , it is not easy to find the ring relation of the quotient of Laurent polynomial ring for the Wilson loop algebra. But, to show the isomorphism between the Wilson loop algebra and the quantum K-ring, it is enough to see the equality between the three point correlation functions of basis of Wilson loop algebra on S 1 × S 2 and C (0) λµν for β = −1 since genus zero three point functions fix the ring structure.
To show the agreement between the quantum K-ring and the CS-matter theory, we define an operator W λ in the algebra of the Wilson loops such that the saddle point value of W λ equals to a Grothendieck polynomial G λ (y| t) for β = −1 with the identification of variables (4.17): (4.21) For example, an operator W (1,0 N −1 ) with (1, 0 N −1 ) := (1, 0, · · · , 0) ∈ P N,M corresponds to the following combination gauge and flavor Wilson loops where W ∧ N is the gauge Wilson loop in the N -th anti-symmetric products of the fundamental representation of U(N ). Since Grothendieck polynomials {G λ (y| t)} λ∈P N,M generate the symmetric polynomials of a Bethe root y 1 , · · · , y N [21], the arbitrary functions of Wilson loops are generated by {W λ } λ∈P N,M . Here the structure constants is determined by the Bethe ansatz equation. We choose the R-charge, flavor charges, the gauge-R-symmetry mixed CS level, flavor-R-symmetry mixed CS levels as On the other hand, if we consider parameters to agree with a Frobenius bilinear form defined in [46], the correlation functions are polynomials with respect to q, for example see (4.28) and (4.29).

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From the determinant formula of the on-shell square norm evaluated in the section 2.3, we find that H equals to the inverse of the on-shell square norm of Bethe vectors (2.79) for β = −1 under the choice (4.15), (4.23) and the identification of variables (4.17): Therefore the arbitrary n-point correlation functions of W λ l for l = 1, · · · , n on S 1 × Σ g perfectly agree with C (g) λ 1 ,··· ,λn obtained by the contractions of structure constants of QK T (Gr(N, M )), η µν and η µν : In particular, (4.25) includes the agreement between the genus zero three point functions. Therefore the algebra of Wilson loops is isomorphic to the quantum K-ring of Grassmannian QK T (Gr(N, M )) with the Frobenius bilinear form in [21]. We comment on properties of the algebra of Wilson loops. If we choose different values of κ (4) , κ (5) , κ (6) ij , n i , and r, then correlation functions correspond to different Frobenius bilinear forms. For example, a choice of parameters agrees with a Frobenius bilinear form of the quantum K-ring in [46] (see also sentences below (5.42) in [21]). If (4.15) is satisfied, both of (4.23) and (4.26) gives the structure constants of quantum K-ring. Namely the OPEs of Wilson loop have the same form of the quantum products in the quantum K-ring; Here C λ µν are the structure constants of QK T (Gr(N, M )). If we choose as (4.23), W λ W µ W ν S 1 ×S 2 is not polynomial with respect to the quantum parameter q. On the other hand, if we choose (4.26), W λ W µ W ν S 1 ×S 2 is polynomials with respect to the quantum parameter q. For example we explicitly calculate the genus zero two point functions and the structure constants for z i = 1, N = 2 and M = 4 . Let us (1) and so on. We also define C abc by O a O b O c S 1 ×S 2 with the choices (4.15) and (4.23) and define C abc by O a O b O c S 1 ×S 2 with the choices (4.15) and (4.26). From (4.2) for g = 0, we can compute the two point correlation functions C 1ab and C 1ab . In matrix notation (C 1 ) ab := C 1ab = O a O b S 1 ×S 2 , the two point functions are JHEP08(2020)157 given by (4.28) The matrix notation of two point functions (C 1 ) ab := C 1ab is given by In a similar way we compute all the C abc and C abc and obtain the same structure constants C a bc = 6 d=1 (C −1 1 ) ad C dbc = 6 d=1 (C −1 1 ) ad C dbc which reproduce the structure constants of QK (Gr(2, 4)) in [46]: (1,1) , (4.30) Here we write [O λ ] as O λ to shorten the expression. We explicitly evaluate the structure constants for QK T (Gr(N, M )) for small N, M in terms of supersymmetric localization formula. The values of structure constants for QK T (P 1 ), QK T (P 2 ) QK T (Gr(2, 3)), QK T (P 3 ) QK T (Gr(3, 4)), and also a non-equivariant case QK(Gr(2, 5)) QK(Gr (3,5)) are listed in appendix C.
The correlation function (3.1) of the 2d A-twisted GLSM on S 2 gives the generating function of genus-zero quasimap invariants on S 2 . Similarly, in three dimensions, the JHEP08(2020)157 (4.31) of the CS-matter theory on S 1 × S 2 with the choices (4.15) and (4.23) gives the generating function of indices of quasimap spaces. For example, when N = 1, the degree k quasimap space is given by P (k+1)M −1 , and (4.31) can be written as the equivariant Euler characteristics; the expectation value of O 3d =x l is given by Then (2.52) and (4.25) lead to the following equality of correlation functions between two CS-matter theories: under the identification of flavor Wilson loops z i = z −1 M −i+1 for i = 1, · · · , M following from (4.33). · · · U(N ),S 1 ×Σg denotes correlation functions of U(N ) CS-matter theory on S 1 × Σ g with the CS levels and parameters taken as (4.15) and (4.23). · · · U(M −N ),S 1 ×Σg denotes correlation functions of U(M − N ) CS-matter theory on S 1 × Σ g with the CS levels and parameters taken as JHEP08(2020)157

CS-matter theory with Ω-background and K-theoretic I-function
In the previous section we studied the correspondence between the algebra of Wilson loops in topologically twisted CS-matter theories without Ω-background and the quantum Kring of Grassmannians. When g = 0, i.e., S 1 × S 2 , a fugacity q (log q is proportional to the Ω-background parameter) for the rotation of S 2 can be included in the partition function (supersymmetric index) or the correlation functions of supersymmetric Wilson loops. The supersymmetric localization computation was performed in [44]. In this section we study the geometric aspects of topologically twisted CS-matter theory with the Ω-background and relate it to the K-theoretic I-function of Grassmannians.
For 2d A-twisted GLSM on S 2 with the Ω-background parameter (Ω-deformed Atwisted GLSM, also called equivariant A-twist), the supersymmetric localization computation was performed in [8]. The supersymmetric localization formula of 2d Ω-deformed A-twisted GLSMs has a nice mathematical interpretations; it was shown in [9,10] that the partition function of Ω-deformed A-twisted GLSM on S 2 factorizes to a pair of Givental I-function [47] for the Grassmannians 8 [48], and also shown in [10] that supersymmetric localization fromula is equivalent to the generating function of integral over the graph spaces of Higgs branch vacuum manifold. 9 We will show a similar result in three dimensions; the partition function of topologically twisted CS-matter theory with Ω-background factorizes to a pair of functions, which are related to the small K-theoretic I-function of Grassmannian. 10 We choose an R-charge and flavor charges same as before; r = n i = 0 . On the other hand we take generic CS levels κ (1) , κ (2) , κ (3) i , κ (4) and κ (5) i in order to compare physical results with K-theoretic I-functions with level structures [53][54][55]. It is convienient to introduceκ (1) ,κ (2) ,κ (3) i ,κ (4) andκ (5) i to express the factorization defined bỹ i := κ = 0 corresponds to the U(N ) CS-matter theory in which the Wilson loop algebra is isomorphic to the quantum K-ring. 8 For the Grassmannians, the I-function is same as the J-function. 9 This factorization in mathematical literature is known as Givental's double construction lemma. See, e.g., [49,Lemma 11.2.12]. 10 The factorization formulas for 3d A-type linear quiver CS-matter theories were shown in [50]. The physical reason why the factorization occurs is that, by choosing a Q-exact term appropriately, point-like vortices and anti-vortices exist on two antipodal points of S 2 that lead to a pair of 3d vortex partition functions [51,52]. The 3d vortex partition function coincides with the specialization of the K-theoretic I-function for xa = zi a .

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In the presence of Ω-background, the one-loop determinants of the vector and chiral multiplets are modified to (4.38) When Ω-background is turned off, i.e., q = 1, the one-loop determinants is reduced to the one-loop determinant without Ω-background (4.7) and (4.8) for g = 0, n i = 0 and r = 0. The CS-terms are independent of q and given by (4.6) for g = 0. Then the partition function with the Ω-background 1 S 1 ×S 2 Ω is rewritten as (4.39) To show the factorization, we introduce k (1) a , k (2) a and x a for a = 1, · · · , N by Then we obtain the factorization of the partition function (4.39) given by I(x , z, q, q,κ)I(x , z, q, q −1 ,κ), (4.42)
We obtained two equivalent expressions (4.43) and (4.44) of I associated to two expressions of the one-loop determinants (4.37) and (4.38). Ifκ (1) andκ (2) are integers, all the square roots of x a 's in I are canceled out. If CS levels are chosen to reproduce quantum K-ring, i.e.κ (1) i =κ (4) = 0, the function I becomes I Gr(N,M ) (x, z, q, q) := I(x, z, q, q,κ)|κ (1) . When x −1 a , (resp. x a ) for a = 1, · · · N is identified with the K-theoretic Chern roots of universal subbundle S (resp. it dual S ∨ ) of the Grassmannian Gr(N, M ), we find that I Gr(N,M ) reproduces the K-theoretic I-function of Gr(N, M ) up to a overall factor (1−q). 11 i = 0, (4.44) reproduces K-theoretic I-function with level structures. For example, when N = 1, (4.44) is written as .

(4.46)
This agrees with the equivariant K-theoretic I-function of P M −1 with level structures [53]. We comment on generalizations of the factorization formula (4.42) to the correlation functions with the Ω-background. In the presence of the Ω-background, the operators can be inserted only on the fixed points of Ω-rotation of S 2 which we call the north and south JHEP08(2020)157 poles of S 2 , respectively. If we insert a Wilson loop O 3d , ( O 3d ) on the north (south) pole, we obtain the following expressions .
Here | N/S denotes the operator insertion on the north/south pole, respectively. O 3d (x, z),

Comparison with K-theoretic I-function from supersymmetric index on
In [57], 12 a supersymmetric index Z S 1 ×D 2 for 3d N ≥ 2 CS-matter theories on S 1 × D 2 was defined and also evaluated in terms of supersymmetric localization techniques. When the Higss branch is P M −1 , it was found in [57] that Z S 1 ×D 2 factorizes to products of Ktheoretic I-function of P M −1 and q-pochhammer symbol which is regarded as q-deformation of Gamma class. 13 We expect a similar story also holds for Gr(N, M ) and study a relation between Z S 1 ×D 2 and I Gr(N,M ) . The supersymmetric localization formula of Z S 1 ×D 2 for the CS-matter theory treated in section 4.1 is written as (4.49) 12 The supersymmetric index in [57] is different from the supersymmetric index (holomorphic block) studied in [58]. For examples, the vector multiplet contribution in these two indices are different. And also the definitions of the two indices themselves are different; the index in [57] is defined by the trace with the insertion of (−1) F , where F is the fermion number. On the other hand, the holomorphic block in [58] is defined by the trace with the insertion of (−1) R , where R is the R-charge. Therefore the chiral multiplet contributions are also different in general. The supersymmetric index in [57] for 3d N = 4 gauge theory with a N = (2, 2) boundary condition reproduces vertex functions (z-solutions) in [59] with appropriate identification of global symmetry parameters. 13 Properties of I-functions of P M −1 evaluated by Z S 1 ×D 2 were studied in [60].

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Here the contributions from the CS-terms, the FI-term, and the one-loop determinants of vector and chiral multiplets on S 1 × D 2 are given by Here (a; q) ∞ : In the presence of the boundary, CS-terms are not invariant under gauge transformations in general. It was observed in [57] that the cancellation of the logarithmic terms arising from the CS-terms and the ones from the one-loop determinants are correlated with the gauge anomaly cancellation. If we choose the gauge and gauge-flavor and gauge-Rsymmetry mixed CS levels as (4.15) and (4.23), all the log(x a ) and log(x a ) log(x b ) terms in Z CS , Z vec and Z chi.N are canceled out. Moreover, if we choose κ (6) ij = − N 2 δ ij , log(z i ) log(z j ) terms in Z S 1 ×D 2 also vanish, which is the signal of the zero effective flavor CS levels. Up to overall q-factors, Z S 1 ×D 2 (z, q, q) can be written as (4.55) The residues are evaluated at x a = z ia q −ka with i a = 1, · · · , M and k a ∈ N ≥0 . We obtain 14 q and log(q)/ log(q) in this section correspond to q 2 and 2πrζ in [57], respectively. JHEP08(2020)157 the following expression of the supersymmetric index ond S 1 × D 2 ; 1 (x a z −1 i ; q) ∞ · I Gr(N,M ) (x, z, q, q −1 ). (4.56) Therefore we find the K-theoretic I-function of the Grassmannians can be derived from the supersymmetric index on S 1 × D 2 . In other choices of CS levels, we have to introduce the elliptic genus of 2d N = (0, 2) multiplets on the boundary of S 1 × D 2 to cancel the gauge anomaly by anomaly inflow mechanism.

Levels in CS-matter theory and deformations of U(N ) Verlinde algebra
The correlation functions of Wilson loops in G Chern-Simons theory on S 1 × Σ g , described by the Verlinde formula [61,62], are indices of line bundles on moduli stacks of G C -bundles on Σ g . Similarly, we expect that correlation functions in CS-matter theories are indices of vector bundles on moduli stacks of G C -bundles with sections. In this section, we briefly discuss choices of CS levels in U(N ) CS-matter theories with M fundamental chiral multiplets in connection with variants of Verlinde algebra. Precise relations between CS-matter theories and indices on moduli stacks will be studied elsewhere.

CS-matter theory and quantum cohomology of Grassmannian
First we study the connection between U(N ) CS-matter theories with M fundamental chiral multiplets and a deformation of Verlinde algebra, which is isomorphic to quantum cohomology of Grassmannian. It was shown in [31]

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The correlation function (4.11) of CS-matter theory on S 1 × Σ g is written as where the sum is taken over the disctinct roots of the 3d saddle point equations We find that correlation functions in the CS-matter theory on S 1 ×Σ g agree with correlation functions in A-twisted GLSM (up to overall sign) on Σ g with r = 0; Therefore we obtain the CS-matter theory in which the algebra of Wilson loop has the same structure constant of U(N ) Verlinde algebra with level M − N , but a different Frobenius bilinear form.

CS-matter theory and Telemann-Woodward index
For a line bundle L on a Riemann surface Σ, an admissible line bundle L on the moduli stack M of G C -bundle on Σ, and representations V 1 , . . . , V M , U of G, the index formula of Telemen-Woodward [63] gives where ψ p is the p-th Adams operation and ρ runs over the set of weights of V i . It was shown in [64,65] that the index (5.6) for M = 1 and V = g C agrees with the correlation functions of Wilson loops in G CS-matter theory with a chiral multiplet in the adjoint representation and R-charge r = deg(L). When G = U(N ) and V i is the fundamental representation of G for all i, we have θ t (f t ) = with respect to ξ a and (l 1 , l 2 ) is the level of L. The sum in (5.6) runs over the saddle points of (5.9). If one has q = 1, κ M , and M i=1 z i = 1, then (5.9) matches the 3d twisted superpotential by setting e ξa = x a , t i = z −1 i , l 1 + N = κ (1) + M 2 , and l 2 = κ (2) , and the index (5.6) agrees with the genus g correlation function of the CS-matter theory. QK T (P 2 ) QK T (Gr(2, 3)). The structure sheaves of the Schurbert varieties of JHEP08(2020)157 QK T (P 3 ) QK T (Gr(3, 4)). The structure sheaves of the Schurbert varieties of P 3 :