Chiral Rings, Vacua and Gaugino Condensation of Supersymmetric Gauge Theories

TL;DR

This paper derives the complete chiral ring relations for supersymmetric U(N) gauge theories with adjoint matter, linking solutions to vacua and gaugino condensation, and generalizing known formulas.

Contribution

It provides an exact characterization of the chiral ring and vacua structure in these supersymmetric gauge theories, extending previous results.

Findings

Complete chiral ring relations derived

Exact correspondence between chiral ring solutions and vacua established

Gaugino condensate relations generalized

Abstract

We find the complete chiral ring relations of the supersymmetric U(N) gauge theories with matter in adjoint representation. We demonstrate exact correspondence between the solutions of the chiral ring and the supersymmetric vacua of the gauge theory. The chiral ring determines the expectation values of chiral operators and the low energy gauge group. All the vacua have nonzero gaugino condensation. We study the chiral ring relations obeyed by the gaugino condensate. These relations are generalizations of the formula $S^N=\Lambda^{3N}$ of the pure ${\cal N} =1$ gauge theory.