Superintegrability in $\beta$-deformed Gaussian Hermitian matrix model from $W$-operators

TL;DR

This paper develops a method based on Virasoro constraints and $W$-representation to prove superintegrability formulas for character averages, specifically applying it to Jack functions in the $eta$-deformed Gaussian Hermitian matrix model.

Contribution

It introduces a first-principles proof technique for superintegrability formulas using $W$-operators and Virasoro constraints, applied to $eta$-deformed matrix models.

Findings

Proved the formula for Jack functions averages in the $eta$-deformed model.

Developed a method to derive superintegrability formulas from fundamental principles.

Outlined the construction of $W$-operators from Calogero-Ruijsenaars Hamiltonians.

Abstract

This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and $W$-representation. We apply it to prove the formula for the Jack functions averages - appropriate analogue of characters for the $\beta$-deformed Hermitian Gaussian matrix model. We also sketch the construction of $W$-operators from Calogero-Ruijsenaars Hamiltonians.