Effective Potentials and Vacuum Structure in N=1 Supersymmetric Gauge Theories

We derive the exact effective superpotential in 4d, N=1 supersymmetric SU(2) gauge theories with $N_A$ triplets and $2N_f$ doublets of matter superfields. We find the quantum vacua of these theories; the equations of motion (for $N_A=1$) can be reorganized into the singularity conditions of an elliptic curve. From the phase transition points to the Coulomb branch, we find the exact Abelian gauge couplings, $\tau$, for arbitrary bare masses and Yukawa couplings. We thus {\em derive} the result that $\tau$ is a section of an $SL(2,\Z)$ bundle over the moduli space and over the parameters space of bare masses and Yukawa couplings. For $N_c>2$, we derive the exact effective superpotential in branches of supersymmetric $SU(N_c)$ gauge theories with one supermultiplet in the adjoint representation ($N_A=1$) and zero or one flavor ($N_f=0,1$). We find the quantum vacua of these theories; the equations of motion can be reorganized into the singularity conditions of a genus $N_c-1$ hyperelliptic curve. Finally, we present the effective superpotential in the $N_A$, $N_f<N_c$ cases.