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

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

2012-01-17

arxiv.org/abs/1201.3633v2

Physics

High Energy Physics

High Energy Physics - Theory

42+1 pages, 5 figures; various helpful comments incorporated

Scientific paper

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.

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