On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations

TL;DR

This paper extends the formalism of cut-and-join operators to describe a broad class of KP-integrable skew Hurwitz tau-functions, including new interpolating WLZZ models, with detailed proofs, generalizations, and beta-deformations.

Contribution

It introduces a comprehensive framework for skew Hurwitz tau-functions, generalizes to multi-matrix models, and proposes beta-deformations, connecting various integrable models via W-representations.

Findings

Identified a family of KP-integrable skew Hurwitz tau-functions.

Connected WLZZ models to matrix models and Calogero-Sutherland Hamiltonians.

Proposed beta-deformation of matrix models for these tau-functions.

Abstract

We extend the old formalism of cut-and-join operators in the theory of Hurwitz $\tau$-functions to description of a wide family of KP-integrable {\it skew} Hurwitz $\tau$-functions, which include, in particular, the newly discovered interpolating WLZZ models. Recently, the simplest of them was related to a superintegrable two-matrix model with two potentials and one external matrix field. Now we provide detailed proofs, and a generalization to a multi-matrix representation, and propose the $\beta$ deformation of the matrix model as well. The general interpolating WLZZ model is generated by a $W$-representation given by a sum of operators from a one-parametric commutative sub-family (a commutative subalgebra of $w_\infty$). Different commutative families are related by cut-and-join rotations. Two of these sub-families (`vertical' and `45-degree') turn out to be nothing but the trigonometric and rational Calogero-Sutherland Hamiltonians, the `horizontal' family is represented by simple derivatives. Other families require an additional analysis.