Virasoro constraints and topological recursion for Grothendieck's dessin counting

TL;DR

This paper establishes that the generating functions for counting certain coverings of the Riemann sphere satisfy Virasoro constraints, KP hierarchy, and topological recursion, revealing deep integrability properties.

Contribution

It demonstrates that the counting problem for Grothendieck's dessins has generating functions with integrability properties like Virasoro constraints and topological recursion.

Findings

Generating functions satisfy Virasoro constraints.

They obey the KP hierarchy.

They follow the Eynard-Orantin topological recursion.

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.