Gauge Theories and Dessins d'Enfants: Beyond the Torus

TL;DR

This paper explores the extension of dessins d'Enfants and Belyi pairs from elliptic curves to higher genus curves, revealing new connections between gauge theories, geometry, and number theory.

Contribution

It provides a partial classification of gauge theories using Belyi pairs on higher genus curves and discusses the role of Igusa and Shioda invariants.

Findings

Explicit Belyi pairs computed for higher genus cases.

Identification of invariants generalizing the elliptic j-invariant.

Enhanced understanding of gauge theories via algebraic curves.

Abstract

Dessin d'Enfants on elliptic curves are a powerful way of encoding doubly-periodic brane tilings, and thus, of four-dimensional supersymmetric gauge theories whose vacuum moduli space is toric, providing an interesting interplay between physics, geometry, combinatorics and number theory. We discuss and provide a partial classification of the situation in genera other than one by computing explicit Belyi pairs associated to the gauge theories. Important also is the role of the Igusa and Shioda invariants that generalise the elliptic $j$-invariant.