Generating functions for weighted Hurwitz numbers

TL;DR

This paper introduces a unifying framework using generating functions and hypergeometric tau-functions to enumerate weighted Hurwitz numbers, including new quantum variants linked to statistical mechanics and Bosonic gases.

Contribution

It develops a 1-parameter family of generating functions for weighted Hurwitz numbers, encompassing classical cases and introducing quantum deformations via the quantum dilogarithm.

Findings

Unified generating functions for classical and quantum Hurwitz numbers.

Explicit formulas for four classical weighted Hurwitz number cases.

Introduction of quantum Hurwitz numbers depending on a deformation parameter q.

Abstract

Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating function. A uniquely determined $1$-parameter family of 2D Toda $\tau$-functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov's double Hurwitz numbers, for which the ramification is simple at all but two specified branch points; the case of Belyi curves, with three branch points, two with specified profiles; the general case, with a specified number of branch points, two with fixed profiles, the rest constrained only by the genus; and the signed enumeration case, with sign determined by the parity of the number of branch points. Using the exponentiated quantum dilogarithm function as weight generator, three new types of weighted enumerations are introduced. These determine {\em quantum} Hurwitz numbers depending on a deformation parameter $q$. By suitable interpretation of $q$, the statistical mechanics of quantum weighted branched covers may be related to that of Bosonic gases. The standard double Hurwitz numbers are recovered in the classical limit.