Supersymmetric Gauge Theories

We derive the exact effective superpotential in 4d, N = 1 supersymmetric SU(2) gauge theories with NA triplets and 2Nf doublets of matter superfields. We find the quantum vacua of these theories; the equations of motion (for NA = 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, τ , for arbitrary bare masses and Yukawa couplings. We thus derive the result that τ 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 Nc > 2, we e-mail address: [email protected] e-mail address: [email protected] e-mail address: [email protected] Permanent address: Racah Institute of Physics, The Hebrew University, Jerusalem, 91904, Israel; e-mail address: [email protected]