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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.
|Additional Information:||2013 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in He, Y. & McKay, J. (2013). N=2 gauge theories: Congruence subgroups, coset graphs, and modular surfaces. Journal of Mathematical Physics, 54(1), 012301 and may be found at http://dx.doi.org/10.1063/1.4772976|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||School of Engineering & Mathematical Sciences > Department of Mathematical Science|
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