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

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.

arXiv comment12 pages

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• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• $\beta$-WLZZ models from $\beta$-ensemble integrals directly
• $\beta$-ensembles and higher genera Catalan numbers
• Proving superintegrability in $\beta$-deformed eigenvalue models