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 τ-functions to description of a wide family of KP-integrable skew Hurwitz τ-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 β-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∞). 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.