Effective potentials and vacuum structure in N = 1 supersymmetric gauge theories Original Research Article

We derive the exact effective superpotential in 4d, N = 1 supersymmetric SU(2) gauge theories with NA triplets and 2Nƒ 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 derive the exact effective superpotential in branches of supersymmetric SU(Nc) gauge theories with one supermultiplet in the adjoint representation (NA = 1) and zero or one flavor (Nƒ = 0,1). We find the quantum vacua of these theories; the equations of motion can be reorganized into the singularity conditions of a genus Nc − 1 hyperelliptic curve. Finally, we present the effective superpotential in the NaA, Nƒ < Nc cases.