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

Abstract

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 $$\beta $$ β -deformed Hermitian Gaussian matrix model. We also sketch the construction of W-operators from Calogero–Ruijsenaars Hamiltonians.