Chiral Rings and Phases of Supersymmetric Gauge Theories

TL;DR

This paper provides a geometric framework for understanding the expectation values of chiral operators in supersymmetric U(N) gauge theories, revealing continuous connections between semiclassical phases through a Riemann surface description.

Contribution

It introduces a geometric approach using meromorphic one-forms on Riemann surfaces to solve for chiral operator expectation values in supersymmetric gauge theories.

Findings

All semiclassical phases with the same number of U(1) factors are continuously connected.

The equations of motion correspond to integrality conditions on periods of a meromorphic form.

The approach simplifies understanding phase structure in supersymmetric gauge theories.

Abstract

We solve for the expectation values of chiral operators in supersymmetric U(N) gauge theories with matter in the adjoint, fundamental and anti-fundamental representations. A simple geometric picture emerges involving a description by a meromorphic one-form on a Riemann surface. The equations of motion are equivalent to a condition on the integrality of periods of this form. The solution indicates that all semiclassical phases with the same number of U(1) factors are continuously connected.