Quantum Difference Equations for Grassmannians

TL;DR

This paper develops solutions to quantum difference equations in equivariant quantum K-theory of Grassmannians, linking them to integrable spin chains and deriving Bethe ansatz equations, with extensions to cohomology and nonequivariant cases.

Contribution

It provides explicit solutions to quantum difference equations for Grassmannians and connects quantum K-theory to integrable spin chain models.

Findings

Solutions to QDE for Grassmannians obtained.

Bethe ansatz equations derived asymptotically.

Quantum K-theory ring identified with XXZ spin chain.

Abstract

We consider quantum difference equation (QDE) for equivariant quantum K-theory of the Grassmannian. In this paper we obtain a solution to the QDE and use the solution to asymptotically derive the Bethe ansatz equations. In the limit, we obtain similar results for the cohomological analogue. For both cases, we describe the nonequivariant solutions as well. As an application, we identify the quantum K-theory ring of $\mathrm{Gr}(k,n)$ with a quantum 5 vertex XXZ integrable spin chain.