Baxter Q-operator from quantum K-theory

Abstract

We define and study the quantum equivariant K-theory of cotangent bundles over Grassmannians. For every tautological bundle in the K-theory we define its one-parametric deformation, referred to as quantum tautological bundle. We prove that the spectrum of operators of quantum multiplication by these quantum classes is governed by the Bethe ansatz equations for the inhomogeneous XXZ spin chain. In addition, we prove that each such operator corresponds to the universal elements of quantum group U ( sl 2 ). In particular, we identify the Baxter operator for the XXZ spin chain with the operator of quantum multiplication by the exterior algebra tautological bundle. The explicit universal combinatorial formula for this operator is found. The relation between quantum line bundles and quantum dynamical Weyl group is shown.

Elliptic and K-theoretic stable envelopes and Newton polytopes

“…Conjecturally, the associated Bethe algebra of that action is the algebra of quantum multiplication on the equivariant K-theory algebra, cf. [OS,RTV2,PSZ]. For the cotangent bundle T * F λ of a partial flag variety F λ the constructions of this paper and [OS,RTV2] identify the Bethe algebra of the cotangent bundle with the image of the Bethe algebra B ′ q ′ in the endomorphism algebra of (C N ) ⊗n λ .…”

Twisted indices of 3d $$ \mathcal{N} $$ = 4 gauge theories and enumerative geometry of quasi-maps

“…As mentioned, the additional supercharges do not commute with U (1) t and therefore we consider the limit t → 1 of the twisted index. In this limit, the virtual χ t -genus greatly 9 Notice that there are interesting additional terms that could be added to the action. For example, This forces ϕ and ϕ † to be covariantly constant on Σ, not just covariantly holomorphic.…”

“…The moduli space of quasi-maps and their enumerative geometry have been discussed in various contexts, e.g.,[8][9][10][11][12][13].…”

Elliptic stable envelope for Hilbert scheme of points in the plane

“…and t 1 , t 2 are the corresponding coordinates (equivariant parameters) . Similarly, by (30) we have…”

“…Even before the term "stable envelope" was coined, this object already manifested itself under different names in sometimes unrelated areas of mathematics. For instance, the theory of canonical bases, the so called off-shell Bethe vectors [3,30,16] (which are the main object of investigation in the theory of the quantum integrable systems), weight functions for integral solutions of qKZ equations [33] are now understood as different incarnations of stable envelopes. Stable envelopes also appear as partition functions of integrable lattice models [13] and play important role in enumerative geometry [26].…”