On KP-integrable Hurwitz functions

TL;DR

This paper explores the KP-integrable Hurwitz tau-functions, connecting them with matrix models, integrability, and knot theory, and extends the family of related models with new variables and descriptions.

Contribution

It provides a comprehensive framework linking Hurwitz tau-functions with matrix models, integrability, and knot theory, and introduces an extended family of models including the Itsykson-Zuber integral.

Findings

Unified description of Hurwitz tau-functions and matrix models

Multiple representations including W-representations and integral forms

Extension of the model family with new integrability properties

Abstract

There is now a renewed interest to the Hurwitz tau-function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a particular family of Hurwitz tau-functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model tau-functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, several kinds of W-representations (we describe four), two kinds of integral (multi-matrix model) descriptions (of Hermitian and Kontsevich types), Virasoro constraints, character expansion, embedding into generic set of Hurwitz tau-functions and relation to knot theory. When approached in this way, the family of models in the literature has a natural extension, and additional integrability with respect to associated new time-variables. Another member of this extended family is the Itsykson-Zuber integral.