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arXiv · 1610.08376

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

Reinier Kramer·Danilo Lewanski·Sergey Shadrin
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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.

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Reinier Kramer, Danilo Lewanski, Sergey Shadrin. 2016-10-26. Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck's dessins d'enfants. https://doi.org/10.25537/dm.2019v24.857-898

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