Superintegrability for some $(q,t)$-deformed matrix models

TL;DR

This paper develops a method to prove superintegrability relations in $(q,t)$-deformed matrix models using hypergeometric function constraints, confirming a conjecture for the refined Chern-Simons model.

Contribution

It introduces a concise proof technique for superintegrability in $(q,t)$-deformed models and constructs a general $(q,t)$-deformed matrix model with parameter constraints.

Findings

Proved the uniqueness of solutions to hypergeometric function constraints.

Established superintegrability relations for $(q,t)$-deformed integrals.

Confirmed a conjectured superintegrability relation in the literature.

Abstract

We analyze the Macdonald's $(q,t)$-deformed hypergeometric functions with one and two set variables and present their constraints. We prove the uniqueness to the solutions of these constraints. We propose a concise method to prove the superintegrability relations for $(q,t)$-deformed matrix models, where the constraints of hypergeometric functions play a crucial role. A conjectured superintegrability relation in the literature for the refined Chern-Simons model can be easily proved by our method. Moreover, we construct a general $(q,t)$-deformed matrix model. We give the constraint conditions for parameters in the integral. The superintegrability relations for the $(q,t)$-deformed integrals with allowed parameters are derived from the hypergeometric constraints.