Effective Superpotentials, Geometry and Integrable Systems

Abstract

We consider the effective superpotentials of N; = 1 SU(N_c) and U(N_c) supersymmetric gauge theories that are obtained from the N = 2 theory by adding a tree-level superpotential. We show that several of the techniques for computing the effective superpotential are implicitly regularized by 2N_c massive chiral multiplets in the fundamental representation, i.e. the gauge theory is embedded in the finite theory with nontrivial UV fixed point. In the maximally confining phase we obtain explicit general formulae for the effective superpotential, which reduce to previously known results in particular cases. In order to study N = 1 and N = 2 theories with fundamentals, we explicitly factorize the Seiberg-Witten curve for 0 ≤ N_f < 2N _c and use the results to rederive the N = 1 superpotential. N = 2 gauge theories have an underlying integrable structure, and we obtain results on a new Lax matrix for N_f = N_c .