Varieties of vacua in classical supersymmetric gauge theories

TL;DR

This work provides a unified, rigorous framework for the classical moduli space of vacua in four-dimensional supersymmetric gauge theories by exploiting invariance under the complexified gauge group $G^{\rm c}$. It proves that the classical vacua form the holomorphic quotient ${\cal M}_0={\cal F}//G^{\rm c}$, and that in the absence of a superpotential this space is the algebraic variety generated by holomorphic gauge-invariant polynomials; when a superpotential is present, additional polynomial relations arise from stationary conditions. Through explicit examples (SUSY QED, a $U(1)\times U(1)$ model, and SUSY QCD with $N_F=N$), the paper illustrates the orbit structure, the appearance of singularities, and the role of limit points and enhanced symmetry. The analysis strengthens the link to geometric invariant theory and provides a self-contained description of the vacuum manifold, including its fine geometric features relevant for quantum considerations. Overall, the results offer a robust, algebraic-geometric classification of classical vacua and a foundation for exploring quantum moduli spaces in SUSY gauge theories.

Abstract

We give a simple description of the classical moduli space of vacua for supersymmetric gauge theories with or without a superpotential. The key ingredient in our analysis is the observation that the lagrangian is invariant under the action of the complexified gauge group $\Gc$. From this point of view the usual $D$-flatness conditions are an artifact of Wess--Zumino gauge. By using a gauge that preserves $\Gc$ invariance we show that every constant matter field configuration that extremizes the superpotential is $\Gc$ gauge-equivalent (in a sense that we make precise) to a unique classical vacuum. This result is used to prove that in the absence of a superpotential the classical moduli space is the algebraic variety described by the set of all holomorphic gauge-invariant polynomials. When a superpotential is present, we show that the classical moduli space is a variety defined by imposing additional relations on the holomorphic polynomials. Many of these points are already contained in the existing literature. The main contribution of the present work is that we give a careful and self-contained treatment of limit points and singularities.