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

• DOI: 10.1016/j.physletb.2023.137805
• Publication Date: 2023-02-28T13:29:30Z
• Authors: A. Mironov; V. Mishnyakov; A. Morozov; A. Popolitov; Wei-Zhong Zhao
• Keywords: Matrix (chemical analysis)

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