Constellations and $\tau$-functions for rationally weighted Hurwitz numbers

TL;DR

This paper generalizes the combinatorial representation of weighted Hurwitz numbers using constellations by incorporating rational weight generating functions, leading to a new expression for the associated $ au$-functions.

Contribution

It introduces a novel framework for weighted constellations with rational weights, expanding the combinatorial interpretation of hypergeometric $ au$-functions for Hurwitz numbers.

Findings

Generalization to rational weight generating functions.

Expression of $ au$-functions as sums over doubly labelled constellations.

Introduction of colour and flavour indices for branch point labelling.

Abstract

Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $\tau$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $\tau$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated $\tau$-function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as "colour" and "flavour" indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.