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Nonperturbative Effective Actions of N=2 Supersymmetric Gauge Theories

We elaborate on our previous work on N=2 supersymmetric Yang-Mills theory. In particular, we show how to explicitly determine the low energy quantum effective action for $G=SU(3)$ from the underlying hyperelliptic Riemann surface, and calculate the leading instanton corrections. This is done by solving Picard-Fuchs equations and asymptotically evaluating period integrals. We find that the dynamics of the $SU(3)$ theory is governed by an Appell system of type $F_4$, and compute the exact quantum gauge coupling explicitly in terms of Appell functions.