Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck's dessins d'enfants

TL;DR

This paper proves quasi-polynomiality for monotone orbifold Hurwitz numbers and related combinatorial objects, confirming conjectures and linking these properties to topological recursion via algebraic curve representations.

Contribution

It establishes quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers and connects this property to topological recursion through algebraic curve representations.

Findings

Proves quasi-polynomiality for monotone orbifold Hurwitz numbers.

Confirms conjectures by Do-Karev and Do-Manescu.

Links quasi-polynomiality to topological recursion conditions.

Abstract

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the $n$-point generating function has a natural representation on the $n$-th cartesian powers of a certain algebraic curve. These representations are the necessary conditions for the Chekhov-Eynard-Orantin topological recursion.