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