Toroidal q-Opers

TL;DR

This paper introduces and analyzes the space of $q$-opers linked to Bethe equations for XXZ-type integrable models with quantum toroidal symmetry, connecting algebraic, geometric, and analytic perspectives.

Contribution

It defines $(ar{GL}( infty),q)$-opers with regular singularities and derives Bethe equations for toroidal $q$-opers through analytic conditions.

Findings

Definition of $q$-opers related to Bethe equations

Construction of $(ar{GL}( infty),q)$-opers with singularities

Derivation of Bethe equations from analytic conditions

Abstract

We define and study the space of $q$-opers associated with Bethe equations for integrable models of XXZ type with quantum toroidal algebra symmetry. Our construction is suggested by the study of the enumerative geometry of cyclic quiver varieties, in particular, the ADHM moduli spaces. We define $(\overline{GL}(\infty),q)$-opers with regular singularities and then, by imposing various analytic conditions on singularities, arrive at the desired Bethe equations for toroidal $q$-opers.