Varieties of vacua in classical supersymmetric gauge theories

TL;DR

This paper provides a clear, gauge-invariant description of the classical moduli space of vacua in supersymmetric gauge theories, clarifying the structure and singularities of these spaces with or without superpotentials.

Contribution

It offers a self-contained, gauge-invariant analysis of the classical vacua moduli space, including a detailed treatment of limit points and singularities, extending previous work.

Findings

Classical moduli space is the algebraic variety of holomorphic gauge-invariant polynomials without superpotential.

With superpotential, the moduli space is defined by additional polynomial relations.

The analysis clarifies the structure and singularities of the moduli space.

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.