Wall-crossing formulae and strong piecewise polynomiality for mixed Grothendieck dessins d'enfant, monotone, and simple double Hurwitz numbers

TL;DR

This paper derives explicit generating series formulae for mixed Hurwitz numbers using semi-infinite wedge formalism, demonstrating strong piecewise polynomiality and wall-crossing phenomena, thus generalizing and providing new proofs for known results.

Contribution

It introduces explicit formulae for mixed Hurwitz numbers' generating series and establishes their piecewise polynomiality and wall-crossing behavior, extending previous results to a broader mixed setting.

Findings

Explicit generating series formulae for mixed Hurwitz numbers

Proof of strong piecewise polynomiality in the mixed case

Wall-crossing formulae for the mixed Hurwitz numbers

Abstract

We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden-Jackson-Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair.