$N=2$ Super Yang-Mills and Subgroups of $SL(2,Z)$

Abstract

We discuss $SL(2,Z)$ subgroups appropriate for the study of $N=2$ Super Yang-Mills with $N_f=2n$ flavors. Hyperelliptic curves describing such theories should have coefficients that are modular forms of these subgroups. In particular, uniqueness arguments are sufficient to construct the $SU(3)$ curve, up to two numerical constants, which can be fixed by making some assumptions about strong coupling behavior. We also discuss the situation for higher groups. We also include a derivation of the closed form $β$-function for the $SU(2)$ and $SU(3)$ theories without matter, and the massless theories with $N_f=n$.