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

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.

13 pages, LaTeX

Cited by in corpus (6)

• Many-body integrable systems implied by WLZZ models
• Summing up perturbation series around superintegrable point
• Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems
• $W$-representations of two-matrix models with infinite set of variables
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• CMM formula as superintegrability property of unitary model