Dessins d'Enfants in $\mathcal{N}=2$ Generalised Quiver Theories

TL;DR

This paper explores the role of Grothendieck's dessins d'enfants as ribbon graphs in certain $ ext{SU}(2)$ gauge theories with $ ext{N}=2$ supersymmetry, revealing their emergence at specific Coulomb branch points and connecting to broader mathematical structures.

Contribution

It identifies the precise conditions under which dessins d'enfants appear in $ ext{N}=2$ supersymmetric gauge theories, linking them to ribbon graphs at isolated Coulomb branch points.

Findings

Dessins d'enfants correspond to ribbon graphs in these theories.

They appear at specific isolated points in the Coulomb branch.

Connections to modular group and trivalent dessins are established.

Abstract

We study Grothendieck's dessins d'enfants in the context of the $\mathcal{N}=2$ supersymmetric gauge theories in $\left(3+1\right)$ dimensions with product $SU\left(2\right)$ gauge groups which have recently been considered by Gaiotto et al. We identify the precise context in which dessins arise in these theories: they are the so-called ribbon graphs of such theories at certain isolated points in the Coulomb branch of the moduli space. With this point in mind, we highlight connections to other work on trivalent dessins, gauge theories, and the modular group.