On KP-integrable skew Hurwitz Τ-functions and their β-deformations

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 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 infinity). Different commutative families are related by cut-and-join rotations. Two of these sub-families ('vertical' and '45degree') 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. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/). Funded by SCOAP3.