Kramer, Reinier; Lewański, Danilo; Shadrin, Sergey

Quasi-Polynomiality of Monotone Orbifold Hurwitz Numbers and Grothendieck's Dessins d'Enfants

Doc. Math. 24, 857-898 (2019)
DOI: 10.25537/dm.2019v24.857-898

Summary

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 necessary conditions for the Chekhov-Eynard-Orantin topological recursion.

Mathematics Subject Classification

14N10, 14H57, 05E05

Keywords/Phrases

Hurwitz numbers, dessins d'enfants, spectral curves, enumerative geometry

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Affiliation

Kramer, Reinier
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Postbus 94248, 1090 GE Amsterdam, The Netherlands
Lewański, Danilo
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Shadrin, Sergey
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Postbus 94248, 1090 GE Amsterdam, The Netherlands

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