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.

arXiv comment13 pages, LaTeX

References in corpus (11)

• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
• Unitary matrix integrals in the framework of Generalized Kontsevich Model. I. Brezin-Gross-Witten Model
• Algebra of differential operators associated with Young diagrams
• Integrability of Hurwitz Partition Functions. I. Summary
• On Equivalence of Topological and Quantum 2d Gravity
• Representation Theory of The $W_{1+\infty }$ Algebra
• Unitary Integrals and Related Matrix Models
• Superintegrability in $\beta$-deformed Gaussian Hermitian matrix model from $W$-operators
• Bilinear character correlators in superintegrable theory
• Virasoro versus superintegrability. Gaussian Hermitian model

Cited by in corpus (24)

• Many-body integrable systems implied by WLZZ models
• Commutative subalgebras from Serre relations
• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions
• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Spectral curves and $W$-representations of matrix models
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• On the status of DELL systems
• Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Summing up perturbation series around superintegrable point
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• On bilinear superintegrability for monomial matrix models in pure phase
• Generalized $\tilde{W}$ algebras
• Supersymmetric polynomials and algebro-combinatorial duality
• $W$-representations of two-matrix models with infinite set of variables
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• Higher order constraints for the ($\beta$-deformed) Hermitian matrix models
• $\beta$-WLZZ models from $\beta$-ensemble integrals directly
• CMM formula as superintegrability property of unitary model
• Phases in WLZZ Matrix Models
• Itzykson-Zuber correlators from character expansion