On matrix models and their $q$-deformations

TL;DR

This paper investigates the connections between classical and $q$-deformed matrix models inspired by BPS/CFT correspondence, providing new correlator results, $W$-algebra representations, and simplified character expectation values.

Contribution

It introduces novel correlators, $W$-algebra representations, and simplified character expectation formulas for matrix models related to supersymmetric gauge theories.

Findings

Derived correlators for $q$-deformed and classical matrix models.

Established $W$-algebra representation of the generating function.

Obtained simple expressions for expectation values of characters.

Abstract

Motivated by the BPS/CFT correspondence, we explore the similarities between the classical $\beta$-deformed Hermitean matrix model and the $q$-deformed matrix models associated to 3d $\mathcal{N}=2$ supersymmetric gauge theories on $D^2\times_{q}S^1$ and $S_b^3$ by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the $W$-algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results.