Geometric realizations of the Bethe ansatz equations

TL;DR

This paper explores the geometric aspects of quantum integrable systems, focusing on quantum K-theory of Nakajima quiver varieties and a q-deformation of the oper-Gaudin correspondence, advancing understanding of Bethe ansatz solutions.

Contribution

It introduces new geometric realizations of Bethe ansatz equations via quantum K-theory and a novel q-deformation of oper-Gaudin models, linking geometry and integrable systems.

Findings

Quantum K-theory provides a geometric framework for Bethe ansatz equations.

A new q-deformation relates oper connections to Gaudin models.

The notes synthesize recent advances in geometric methods for quantum integrable systems.

Abstract

These lecture notes are devoted to the recent progress in the geometric aspects of quantum integrable systems based on quantum groups solved using the Bethe ansatz technique. One part is devoted to their enumerative geometry realization through the quantum K-theory of Nakajima quiver varieties. The other part describes a recently studied $q$-deformation of the correspondence between oper connections and Gaudin models. The notes are based on a minicourse at C.I.M.E. Summer School ``Enumerative geometry, quantisation and moduli spaces," September 04-08, 2023.