Superintegrability of $q,t$-matrix models and quantum toroidal algebra recursions

TL;DR

This paper explores the superintegrability properties of $q,t$-matrix models linked to quantum toroidal algebras, providing new formulas and insights into their correlation functions and connections to orthogonal polynomials.

Contribution

It introduces a universal algebraic approach to derive superintegrability formulas for $q,t$-matrix models, including new models not previously studied.

Findings

Derived new superintegrability formulas for $q,t$-deformed ensembles

Provided algebraic proofs of known superintegrability results

Connected superintegrability to orthogonal polynomial theory

Abstract

$q,t$-deformed matrix models give rise to representations of the deformed Virasoro algebra and more generally of the quantum toroidal $\mathfrak{gl}_1$ algebra. These representations are described in terms of finite difference equations that induce recursion relations for correlation functions. Under suitable assumptions, these recursions admit unique solutions expressible through "superintegrability" formulas, i.e. explicit closed formulas for averages of Macdonald polynomials. In this paper, we discuss examples arising from localization of 3d $\mathcal{N}=2$ theories, which include $q,t$-deformation of well known classical ensembles: Gaussian, Laguerre and Jacobi. We explain how relations in the quantum toroidal algebra can be used to give a new and universal proof of the known superintegrability formulas, as well as to derive new formulas for models that have not been previously studied in the literature. Finally, we make some remarks regarding the relation between superintegrability and orthogonal polynomials.