Virasoro constraints and topological recursion for Grothendieck's dessin counting

Abstract

We compute the number of coverings of $\mathbb{C}{P}^1\setminus \{0,1,\infty \}$ with a given monodromy type over $\infty$ and given numbers of preimages of 0 and 1. We show that the generating function for these numbers enjoys several remarkable integrability properties: it obeys the Virasoro constraints, an evolution equation, the KP (Kadomtsev-Petviashvili) hierarchy and satisfies a topological recursion in the sense of Eynard-Orantin.

22 pages, 4 figures; extended version

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