On KP-integrable Hurwitz functions

Understand

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.