EFFECTIVE POTENTIALS AND VACUUM-STRUCTURE IN N=1 SUPERSYMMETRIC GAUGE-THEORIES

We derive the exact effective superpotential in 4d, N = 1 supersymmetr ic SU(2) gauge theories with N-A triplets and 2N(f) doublets of matter superfields. We find the quantum vacua of these theories; the equatio ns of motion (for N-A = 1) can be reorganized into the singularity con ditions of an elliptic curve. From the phase transition points to the Coulomb branch, we find the exact Abelian gauge couplings, tau, for ar bitrary bare masses and Yukawa couplings. We thus derive the result th at tau is a section of an SL(2, Z) bundle over the moduli space and ov er the parameters space of bare masses and Yukawa couplings. For N-c > 2, we derive the exact effective superpotential in branches of supers ymmetric 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 r eorganized into the singularity conditions of a genus N-c - 1 hyperell iptic curve. Finally, we present the effective superpotential in the N -A, N-f < N-c cases.