Grothendieck's Dessins d'Enfants in a Web of Dualities. III

TL;DR

This paper connects Grothendieck's dessins d'enfants to random matrix theory and Hurwitz numbers, establishing dualities with integrable hierarchies and topological models, and introduces a correction factor for the partition function.

Contribution

It identifies the dessin partition function with the Laguerre unitary ensemble and links it to strictly monotone Hurwitz numbers, revealing new dualities and integrable structures.

Findings

Dessin partition function equals LUE partition function.

Established connection between dessin counting and monotone Hurwitz numbers.

Introduced a correction factor making the partition function a Toda tau-function.

Abstract

We identify the dessin partition function with the partition function of the Laguerre unitary ensemble (LUE). Combined with the result due to Cunden et al on the relationship between the LUE correlators and strictly monotone Hurwitz numbers introduced by Goulden et al, we then establish connection of dessin counting to strictly monotone Hurwitz numbers. We also introduce a correction factor for the dessin/LUE partition function, which plays an important role in showing that the corrected dessin/LUE partition function is a tau-function of the Toda lattice hierarchy. As an application, we use the approach of Dubrovin and Zhang for the computation of the dessin correlators. In physicists' terminology, we establish dualities among dessin counting, generalized Penner model, and $\mathbb{P}^1$-topological sigma model.