Effective Potentials and Vacuum Structure in N=1 Supersymmetric Gauge Theories

TL;DR

This paper derives exact effective superpotentials and analyzes vacuum structures in 4d, N=1 supersymmetric SU(N) gauge theories, revealing the geometric nature of gauge couplings and quantum vacua across various matter configurations.

Contribution

It provides new exact solutions for superpotentials and vacuum equations, connecting them to elliptic and hyperelliptic curves, and demonstrates the geometric structure of gauge couplings in these theories.

Findings

Exact effective superpotentials for SU(2) and SU(N) theories.

Quantum vacua characterized by elliptic and hyperelliptic curves.

Gauge couplings form sections of an SL(2,Z) bundle over moduli space.

Abstract

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 $N_A$, $N_f<N_c$ cases.