Baxter Q-operator from quantum K-theory

TL;DR

This paper explores the quantum K-theory of cotangent bundles over Grassmannians, linking quantum multiplication operators to Bethe ansatz equations and quantum groups, and explicitly identifying the Baxter operator.

Contribution

It introduces quantum tautological bundles in quantum K-theory and establishes their connection to integrable systems and quantum groups, providing explicit formulas and relations.

Findings

Spectrum of quantum multiplication operators governed by Bethe ansatz equations.

Identification of Baxter operator with quantum multiplication by tautological bundle.

Explicit universal formula for the Baxter operator in 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.