N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces

TL;DR

This paper uncovers a deep connection between four-dimensional gauge theories, congruence subgroups of the modular group, and combinatorial graphs, revealing new insights into their mathematical and physical structures.

Contribution

It establishes a novel correspondence linking gauge theories, modular subgroups, and trivalent graphs, especially highlighting genus zero cases related to Moonshine and elliptic K3 surfaces.

Findings

Enumeration and comparison of gauge theories and graphs

Identification of genus zero torsion-free congruence subgroups in Moonshine

Recasting elliptic j-invariants as dessins d'enfant for Seiberg-Witten curves

Abstract

We establish a correspondence between generalized quiver gauge theories in four dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups as exemplified by those which arise in Moonshine. We analyze in detail the case of index 24, where modular elliptic K3 surfaces emerge: here, the elliptic j-invariants can be recast as dessins d'enfant which dictate the Seiberg-Witten curves.