Remarks on $q$-difference opers arising from quantum toroidal algebras

TL;DR

This paper explores a conjectural link between the spectra of Bethe algebras for quantum toroidal algebras and $q$-hypergeometric opers, introducing apparent singularities for $q$-difference operators and discussing generalizations.

Contribution

It introduces a new conjectural correspondence and the concept of apparent singularities for $q$-difference operators, extending to $rak{gl}_n$.

Findings

Proposes a conjectural spectrum correspondence.

Defines apparent singularities for $q$-difference operators.

Discusses potential generalizations to $rak{gl}_n$.

Abstract

We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal $\mathfrak{gl}_2$ algebra on relaxed Verma modules, and $q$-hypergeometric opers with apparent singularities. We introduce alongside the notion of apparent singularities for linear $q$-difference operators and discuss some of their properties. We also touch on a generalization to $\mathfrak{gl}_n$.