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

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.

31 pages. Version 2: simplified the computation in section 4.2, also clarifying the deduction of the diagonal part of the A-operators

References in corpus (2)

• Equivariant Gromov-Witten theory of one dimensional stacks
• Rationality in map and hypermap enumeration by genus

Cited by in corpus (5)

• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• Loop equations and a proof of Zvonkine's $qr$-ELSV formula
• Towards an orbifold generalization of Zvonkine's $r$-ELSV formula
• Cut-and-join equation for monotone Hurwitz numbers revisited
• Wall-crossing formulae and strong piecewise polynomiality for mixed Grothendieck dessins d'enfant, monotone, and simple double Hurwitz numbers