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 $\mathcal{U}_{\hbar}(\widehat{\mathfrak{sl}}_2)$. 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.