Wilson loop algebras and quantum K-theory for Grassmannians

We study the algebra of Wilson line operators in three-dimensional N=2 supersymmetric U(M) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M,N), and its connection to K-theoretic Gromov-Witten invariants for Gr(M,N). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M,N), isomorphic to the Verlinde algebra for U(M), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.


Introduction and outline
The relation [1] between the Verlinde algebra [2] and quantum cohomology of Grassmannians [3,4,5] has been derived in ref. [6,7,8] by studying the small radius limit in an S 1 -compactification of a 3d N = 2 supersymmetric gauge theory. This setup fits into the general framework of refs. [9,10], which connects 3d supersymmetric gauge theories to quantum K-theory, as opposed to quantum cohomology. An explicit 3d gauge theory/quantum K-theory correspondence between the N = 2 supersymmetric gauge theory and Givental's equivariant quantum K-theory [11] was proposed and studied in ref. [12]. For non-trivial Chern-Simons level, it involves the generalization to non-zero level defined in ref. [13]. A concrete relation between quantum K-theoretic correlators and the Verlinde numbers was proposed in [14].
In this note we carefully reconsider the relation between the Verlinde algebra, quantum cohomology and quantum K-theory of Grassmannians by studying the algebra of 3d Wilson line operators. The 3d Wilson loop algebra depends on the choice of Chern-Simons (CS) levels κ and the matter spectrum. In a certain window for κ, the dimension of this 3d algebra is fixed and equal to dim(H * (Gr(M, N))), but the quantum corrections to the structure constants of the ring of chiral operators associated with the loop algebra depend on the values of κ. The family of quantum algebras for different κ contains the Verlinde algebra and quantum cohomology, the ordinary quantum K-theory studied by Buch and Mihalcea [15], and closely related algebras. The dependence on κ drops out in the 2d scaling limit, where the quantum algebras become isomorphic to quantum cohomology, as predicted by the arguments of refs. [1,7].

Generalized I-functions
In the following we determine the Wilson line algebra by mapping Wilson line operators in a representation R of the 3d N = 2 U(M) gauge theory to difference operators acting on Givental's I-function in quantum K-theory [11]. The I-function is the generating function of K-theoretic correlators with at least one insertion, defined as an Euler number on an appropriate compactification of the moduli space of maps from a curve Σ to the target space X. The basic difference operators p a shift the exponentiated, complexified Kähler parameters Q a as p a Q a = qQ a , a = 1, . . . , h 1,1 (X). Here q is the weight for the twisting of the world-volume geometry C × q S 1 of the 3d theory, where the Riemann surface C is topologically a two-sphere S 2 or a disk D 2 . These twisted world-volume geometries relate to the Omega background of refs. [6,16]. In Givental's formalism, Q a represent Novikov variables and q is the weight for the C * action in equivariant Gromov-Witten theory [17]. A relation between Wilson line operators and difference operators has been first studied in refs. [6,18,19]. A difference module structure in the variables Q a in quantum K-theory has been described in ref. [20]. For complete flag manifolds, the difference module had been related to the difference Toda lattice already in ref. [21]. Such a connection between gauge theory and integrable systems holds much more generally, as shown in the context of Bethe/gauge correspondence [10].

Generalized I-functions of Gr(M, N)
The 3d gauge theory/quantum K-theory correspondence of ref. [12] equates the Ifunction of permutation symmetric quantum K-theory of ref. [11] with the vortex sum of a supersymmetric partition function on D 2 × q S 1 . There are by now several methods to determine this partition function [22,23,19,24]. In this note we study the partition function for 3d N = 2 U(M) gauge theory with N chiral matter multiplets in the fundamental representation, which realizes (for N > M) the complex Grassmannian Gr(M, N) as its target space. The details of the computation of the partition function will be given elsewhere [25]; the result is 1 .
Here c 0 is a normalization factor chosen such that I SQK Gr(M,N ) | Q=0 = (1 − q). The parameters y α arise as real masses of the N matter multiplets and define the equivariant I-function with respect to global flavor symmetry. Moreoverd ab =d a −d b and d a = d a − ǫ a , where the parameters ǫ a are related to the Chern roots x a of the dual S * of the tautological U(M) bundle S over Gr(M, N) by q ǫa = e βxa . Here β is the radius of S 1 , which will be set to β = 1 in most of the following. For vanishing real masses y α = 1, the I-function takes values in the topological K-group K(Gr(M, N)) -abbreviated in the following by K(Gr). 2 Using the Chern isomorphism, a basis of dim(K(Gr)) = N M elements is provided by the Schur polynomials σ µ (x K ) in the variables x K a = 1 − e −xa , where µ is a Young tableau in the set of tableaus fitting in a M × (N − M) box B N . The expansion represents the overlap of the groundstate of the 3d theory with boundary states defined by a 3d generalization of B-branes [26,12]. The choice of the variables x K a is motivated by the 2d limit β → 0, where the basis {σ µ (x K )} reduces to the cohomological basis dual to the Schubert cycles, σ µ (x K ) β→0 −→ β |µ| (σ µ (x) + O(β)). 3 The effective CS levels in the 3d gauge theory are encoded in the term 4 (2. 3) The parametersκ S andκ A specify the levels for the SU(M) and U(1) subgroups of the gauge group U(M), respectively, whileκ R is a level for the mixed gauge/Rsymmetry CS term. The effective CS levelsκ i fulfill 5 1 To simplify some of the following formulas, a sign factor (−1) M has been absorbed in the definition of Q relative to the conventions used in the mathematical literature. 2 More precisely, the I-function takes values in the equivariant K-group, and we obtain the Kgroup K(Gr(M, N )) in the non-equivariant limit y α = 1. In the following we mostly consider the non-equivariant limit. 3 The connection of the K-theory basis {σ µ (x K )} to the basis of Grothendieck polynomials defined in ref. [27] will be discussed in sect. 4. 4 For a matrix σ = diag(σ 1 , . . . , σ M ) the trace symbols are defined as tr U(M) (σ 2 ) = a σ 2 a , tr U(1) (σ 2 ) = 1 M ( a σ a ) 2 , tr SU(M) (σ 2 ) = tr U(M) (σ 2 ) − tr U(1) (σ 2 ) and tr R (σ) = a σ a . 5 The quantization relation linkingκ S andκ A is a consequence of the global structure of the Lie group U (M ) ≃ (SU (M ) × U (1)) /Z M . and the relation to the bare bulk CS terms κ i in the Lagrangian are [25] Following the arguments of ref. [12], the level zero quantum K-theory of ref. [11] corresponds to zero effective CS levelsκ i = 0. For other levelsκ i , one obtains a three parameter family of generalized I-functions. One can check that on the oneparameter slices (κ S ,κ A ,κ R ) = (ℓ , ℓ , −ℓ /2) and (κ S ,κ A ,κ R ) = (0, Mℓ det , −ℓ det /2) the above result respectively reproduces the I-functions at level ℓ in the fundamental representation of U(M) and at level ℓ det in the determinantal representation of U(M) in quantum K-theory with level structure, as derived in ref. [28] (see also ref. [14]). 6 To study the action of Wilson line operators in a representation R of U(M) we will also need an abelianized version of the path integral for the maximal torus U(1) M ⊂ U(M). 7 To this end we consider a modified sumÎ(Q a ) depending on M weights Q a ,Î where the coefficients c d are defined by the requirementÎ(Q a ) Qa=Q = I SQK Gr(M,N ) (Q). The sumÎ satisfies the relation: which can be viewed as a K-theoretic version of the Hori-Vafa formula [31]. Here p a = q θa , θ a = Q a ∂ Qa , and The constants are The last factor in eq. (2.8) involves the coefficients of a generalized I-function I α,β P N−1 = d≥0 I d,ǫ,α,β P N−1 (Q) for P N −1 obtained from a U(1) gauge theory with CS levels (α, β), which reads (2.10) 6 Different K-theoretic I-functions for complex Grassmannians have also appeared in refs. [29,30], however we did not find a suitable choice of CS terms (2.5) such that these match with the field theory result (2.1). 7 Abelianization of Grassmannian sigma models has been used many times before in physics and mathematics; for some literature closely related to the present context see refs. [5,31,32,28].
The cross terms ∼ q γdad b in eq. (2.8) obstruct a complete factorization into a product of M P N −1 factors, except for ∆ κ = 0. This condition is however incompatible with the canonical levelsκ i = 0 relevant for the level zero quantum K-theory.

A window for the Chern-Simons levels
The 3d gauge theory/quantum K-theory correspondence equates the vortex sum of the 3d gauge theory with the I-function for the permutation symmetric K-theory of refs. [11,13]. That is, the I-function I SQK Gr(N,M ) (δt) in eq. (2.1) has in general a non-zero permutation symmetric input δt, determined by the expansion which separates the perturbations δt from the correlator part I corr (δt).
] is the symplectic loop space, and K ± the Lagrangian subspaces defined as [20,33,13] where R(q) denotes the field of rational functions in the variable q. The permutation symmetric input corresponds to a 3d partition function perturbed by multi-trace operators [12]. To compute the chiral ring one wants to study the family of 3d gauge theories perturbed only by single trace operators, corresponding to the ordinary quantum K-theory. Hence, one has to shift the multi-trace perturbations to zero, and subsequently perturb in the single trace directions, There is a certain window of CS levels, for which the I-function (2.1) is already at zero input, δt = 0, and coincides with the unperturbed I-function I QK Gr(M,N ) (0) of ordinary quantum K-theory. Inspection of eq. (2.10) shows that this requires the CS levels to lie in the window As discussed in the next section, the same condition assures that the Z[[Q]]-module of Wilson loops has N M generators -in the following referred to as the dimension of the Wilson loop algebra -which is the number of generators of the K-theory group K(Gr). There are extra operators outside the window H κ .

Wilson loops
In the 3d gauge theory one can consider more generally expectation values W R of Wilson loop operators W R = tr R P exp(i S 1 (A θ − iσ)dθ) wrapping S 1 , where σ is the real scalar field in the 3d vector multiplet and R is a representation of the gauge group. 8 The insertion of a charge minus one Wilson line in the a-th U(1) factor in the path integral produces an extra factor qd a in the sector with vortex number d a . The vacuum expectation value of a U(1) M Wilson line W n = a W na a is then (up to a normalization factor) equal to a vortex sum where ∆ is the operator defined in eq. (2.7) andc d are the coefficients of the I-functioñ The extra factor a qd ana can be generated by acting with a difference operator Classically, a product of two Wilson line operators W n and W m corresponds to the insertion of a qd a(ma+na) in the sum, or the action with D n · D m = D m+ n onÎ.
In the quantum theory, there are non-trivial relations between the Wilson lines for different n. To this end, one notices that the sumĨ satisfies the following difference equations: These equations are inherited from the difference equations of the P N −1 factors, taking into account the additional CS terms. The O(ǫ N a ) terms are set to zero in H * (Gr(M, N)). The difference equations allow to replace a difference operator p D a acting onĨ by an operator of lower degree in p a . Here D is the degree in p a of the polynomial obtained by clearing denominators in the operator L a : This reduces the algebra of difference operators acting onĨ to D n (p)/(p D a ). Taking into account the extra operator ∆ inÎ, which has degree M − 1 in each variable p a , the algebra of difference operators acting onÎ is generated as a Z[[Q]]-module by the operators {D n , 0 ≤ n a ≤ D − M}. In view of the relations (3.3), it is useful to also define shifted Wilson line operatorsŴ a = 1 − W a , corresponding to the action of the difference operator δ a = 1 − p a .
In the U(M) theory, one has to restrict to permutation symmetric combinations of U(1) M Wilson lines, which can be represented by Schur polynomials σ µ (labeled by Young tableaus µ) either in the operators δ a or the operators p a , i.e., For explicitness we focus now on the operatorsD µ , which satisfy the algebrâ where the structure constants C λ µν are the Littlewood-Richardson coefficients. Classically the ideal of relations is This yields the number of representation-theoretic independent Wilson linesŴ λ , which is given by For ∆ κ = 0 one recovers the 3d Witten index computed already in refs. [35,36]. 9 After the reduction by the ideal (3.3), the structure constants depend on Q, q and the CS levels, namelyD Here the structure constants c λ µν (Q, q, κ i ) are obtained from the action of the D µ onÎ and taking the Q a = Q limit. One has c λ µν Q=0 = C λ µν . The reduced algebra of difference operators is not equal to the Wilson loop algebra, due to extra terms generated by the action of a difference operator on the Q-dependent terms in the relations (3.3). Since these terms are necessarily of O(1 − q) and can be eliminated by setting q = 1. The algebra of Wilson line operators is thereforê 3) are related to the vacuum equations of ref. [10], by first replacing q θa by a commuting variable p a and subsequently taking the q → 1 limit.
The discussion is also similar to the treatment in ref. [7], where the algebra of Wilson loops has been computed from an ideal of relations for a S 3 -partition function. 10 It has been argued there, that the algebra should be independent of the manifold, on which the 3d theory is compactified. A comparison of the above results with those in ref. [7] will be made below.

Diff. equations for the I-function
As discussed around eq. (2.13), the permutation symmetric I-function (2.1) coincides with the unperturbed I-function of the ordinary quantum K-theory inside the level window H κ . The latter can be perturbed by operators Φ i ∈ K(Gr), introducing a dependence on N M additional variables t i . The correlators of the perturbed theory are captured by the Givental J-function 11 The sum over β runs over the degrees of rational curves and n counts the number of insertions of the perturbations (3.12) The correlator part lies in K − and is defined as a holomorphic Euler characteristic on the moduli space of stable maps [38,39]. The J-function of Gr(M, N) relates to a fundamental solution T ∈ End(K(Gr)) of the flatness equations [38,40] ( where * denotes the quantum product, A ∈ End(K(Gr)) and we suppress the Q, t, qdependence of the endomorphisms. In terms of T one has

14)
10 See also refs. [35,36]. 11 The I-function studied before, and the J-function below, are in general connected by a nontrivial transformation, the 3d mirror map. At zero perturbation, this transformation is trivial for Gr(M, N ) with canonical CS levels, i.e., J(0) = I(0). For non-zero level, the correlators and the inner product in the following two equations have to be defined as in ref. [13].
where Φ 0 = 1 denotes the unit. The systems of first order equations (3.13) imply differential equations for the J-function. For concreteness and simplicity we consider the small deformation space spanned by the element Φ 1 = 1 − e c 1 (S) and suppress the index on t. Suppose there is a relation in the quantum K-theoretic algebra of the form for some integer d ≤ N M , where a k are some (Q, q)-dependent coefficients. Then where the second step uses that the basis elements Φ i are constant. This implies the differential equation Similarly, a relation k b k (Aq θ ) k Φ 0 = 0 in the quantum algebra implies the difference equation In this way, relations in the quantum K-theory algebra translate to quantum differential and difference equations for the J-function and vice versa.

Factorized case
The simplest subset of theories arises for the choice of CS levels with ∆ κ = 0, i.e., equal bare CS levels in the SU(N) and U(1) factors, but unequal effective levels, κ S =κ A . In this case, the relations in eq. This is, up to our sign convention, the same relation as in quantum cohomology [3,4,5], and thus will give an isomorphic algebra after projecting to U(M). The operators generating these algebras are 3d Wilson linesŴ µ in the quantum K-theoretic ring or the corresponding operators related to the cohomology classes [σ µ ] ∈ H * (Gr(M, N)) in the 2d case [8].

2d limit
In the 3d gauge theory compactified on an S 1 of radius β, one can consider a simple small radius limit, in which the quantum K-theoretic I-function reduces to the cohomological I-function. To this end one sets q = e − β , and rescales the operators and the FI parameter by factors of β [12]. For any CS levels, the 2d limit of the relations which is again the relation in quantum cohomology, after a renormalization Q a → Q a e N ln β . The different K-theoretic algebras of Wilson loops distinguished by different CS levels therefore reduce all to the quantum cohomology algebra, in agreement with the arguments of refs. [1,7].

Chiral ring at level zero
The canonical case from the point of quantum K-theory is however at zero effective CS level, related the ordinary K-theory of refs. [38,39]. In the following we study the chiral ring from the algebra of difference operators and show that the result agrees with the results in the mathematical literature on the quantum product of Schubert structure sheaves [15] . The canonical case lies inside the level window H κ if N > M, which is required for the theory to have a supersymmetric vacuum, see e.g., ref. [7]. The chiral ring can therefore be studied directly from the I-function  The appearance of a denominator on the r.h.s. is harmless, as the operators p a are invertible. Due to the non-factorized form, the general polynomial reduction defined by the ideal (4.4) is not entirely trivial, but straightforward to perform for given values of M and N.
As an illustrative example we discuss the first non-trivial case Gr (2,4). Performing the polynomial reduction of the difference operators one obtains, up to order O(1 − q) terms, the multiplication table Here σ µ stands for eitherŴ µ orD µ (δ) and we used the abbreviations In the classical limit Q = 0, the difference operators σ µ (δ) become Schur polynomials σ µ (x K a ) in the variables x K a = 1 − e −xa . The polynomials σ µ (x K ) represent Chern characters of a basis {φ µ } for K(Gr). The pairing χ(φ µ , φ ν ) on the basis {φ µ } has already determinant one, but is non-minimal in the sense that the entries are not all 0 or 1. The polynomials σ µ (x K ) are related by a linear transformation of determinant one to the Grothendieck polynomials O µ (x K ) of ref. [27]: The inner product (3.12) on the new basis is The O µ are the Chern characters of the the structure sheaves of the Schubert cycles. After the basis change, we obtain for the quantum multiplication of the structure These multiplications agree with the result of ref. [15], which has been obtained by quite different methods. At zero deformations t = 0, as detailed in ref. [25], the K-theoretic quantum product can be also obtained from the critical locus of the 1-loop effective N = 2 twisted superpotential W [10]. In the example, there are two relations obtained from e d W = 1, which can be written as One can check that, similiar to the cohomological case [3,4,5], the K-theoretic quantum product (4.8) is isomorphic to a polynomial ring in two generators O 1 , O 1,1 , divided by the ideal of relations r a = 0 The Grassmannians Gr(M, N) and Gr(N − M, N) are related by the well-known classical geometric duality, and this duality persists at the level of quantum K-theory. Under the duality a Young tableau is exchanged with its transposed Young tableau. In the self-dual example Gr (2,4), the duality in the quantum theory is reflected in the symmetry of the quantum multiplication (4.8) under the exchange of O 2 and O 1,1 . In the app. A.2 we discuss the less trivial example of Gr(2, 5) ≃ Gr(3, 5) of dual quantum K-theories and its relation to Seiberg like dualities of ref. [41].

Perturbed theory for Gr(2, 4)
The above results for the I-function and the chiral ring hold at zero perturbation t = 0. Methods to reconstruct the I-function at non-zero t from the I-function at t = 0 have been described in refs. [40,33,11], using the difference module structure in quantum K-theory, first established in ref. [20] for the ordinary theory.
These reconstruction theorems use the difference operators (3.13) to deform the I-function and are therefore limited to the directions in K(X) generated by repeated action of A on the unit Φ 0 . In the case of the Grassmannian Gr(M, N), this reconstructs the subspace of the deformations t = i t i (O 1 ) i generated by the difference operator associated with the line bundle e c 1 (S) = 1 − O 1 .
From the view point of the 3d gauge theory, the full deformation space can be accessed by adding single trace deformations in any representation of the gauge group [12], and this is not limited to the subspace generated by powers of O 1 . It suffices to know the action of Wilson line operatorsŴ µ on the I-function for any representation µ. This action has been already reconstructed in sect. 3 in terms of the difference operatorsD µ acting on the generalized I-functionÎ(Q a ) depending on M Novikov variables Q a , and setting Q a = Q at the end.
The general reconstruction for Gr(M, N) will be discussed in ref. [25]. Here we briefly discuss the simplest case of the level zero quantum K-theory for Gr (2,4) at non-zero t, where one can use duality to reconstruct the I-function from the reconstruction methods of refs. [40,33]. Classically, the Grassmannian Gr(2, 4) can be described via the Plücker embedding as a quadratic hypersurface P 5 [2] in P 5 . One can verify that this relation extends to a duality of the quantum K-theory at t = 0, by establishing the equality of the level zero I-functions (4.11) The easiest way to show this equation is to check that the r.h.s. satisfies the same difference equation from Tab. 1 and to compare the first terms in the Q-expansion to fix the normalization.
One can check that the r.h.s. agrees with the structure constants at t = 0 computed from I P 5 [2] (0).
One way to reconstruct the perturbed theory is to integrate the flatness equations (3.13), as described in ref. [40]. This has the advantage of getting easily all order expressions in the general perturbation 12 T = 4 k=1 T k Φ k at fixed power of Q. Restricting to the direction t = T 1 , which is the integrable deformation in the 2d limit, the perturbed structure constants at order Q 1 obtained in this way are In the perturbed theory, the structure constants get contributions also from higher order in Q. Some higher order terms are given in eq. (A.1).
(A.7) The inner product on the basis of structure sheaves is Taking acount a sign change Q → −Q, which is due to our sign convention explained in footnote 1, the quantum multiplication table for Gr (3,5) computed by polynomial reduction is equal to the one obtained from (A.4) by transposing Young tableaus labelling the Grothendieck polynomials, i.e., O µ → O µ T . Moreover, the difference operator for Gr (3,5) coincides with that in eq. (A.6), which implies that the expansion coefficients (2.2) of the I-functions agree up to linear combination. From the point of the underlying 3d field theory, the agreement of the level zero quantum K-theory for Gr (2,5) and Gr (3,5) is expected in view of the Seiberg type dualities studied in ref. [41]. The agreement of the I-functions is the statement, that the D 2 × S 1 partition functions for these 3d theories agree upon appropriate identification of the vacua on both sides. From the algebraic point, the isomorphism between the Wilson line algebras is less obvious, and the vacuum equations e d W = 1 of the Bethe/gauge correspondence of ref. [10] are different in the two theories. Nontrivial equivalences of algebras of this type have been discussed ref. [7].