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.