On KP-integrable Hurwitz functions

Abstract

There is now a renewed interest [1]-[4] to a Hurwitz τ -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 τ -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 τ -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 τ -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.

Ward identities and combinatorics of rainbow tensor models

“…its complex matrix model representation found in [140] being slightly different from Z C {t}. The factorization property (2.46) also survives, with a simple modification:…”

“…W -representation [136][137][138][139][140] provides a simple "dual" formula for Z{t}, expressing it through differentiation rather than integration:…”

“…However, there is another possibility to relate the Gaussian Hermitian model with the Toda lattice τ -function [163]: one can note that it is equal to the concrete model from the big family of Hurwitz partition functions considered in [140],…”

“…The W -representation for the rectangular complex model can be read off from formulas of [140] upon its identification with Z (1,2) :…”

Checks of integrality properties in topological strings

“…This expansion results in the appearance of Hurwitz-Tau function when substituted in the Ooguri-Vafa partition function Z K OV (A, q,p) [110][111][112][113][114]. Here the sum goes over the Young diagrams ∆ with l(∆) lines of lengths δ i and the number of boxes |∆| = l(∆) i δ i , while ϕ R (∆) are proportional to the characters of symmetric groups ψ R (∆) at |R| = |∆|: ψ R (∆) = z ∆ d R ϕ R (∆), and continued to |R| > |∆| as in [110, eq.…”