Vacuum Structures of Supersymmetric Yang-Mills Theories in $1+1$ Dimensions

TL;DR

This paper investigates the vacuum structures of 1+1 dimensional supersymmetric Yang-Mills theories with compactified space, revealing unbroken SUSY under periodic boundary conditions and instanton effects under antiperiodic conditions.

Contribution

It provides a detailed analysis of vacuum structures in SUSY Yang-Mills theories in 1+1 dimensions, including effects of different boundary conditions and instanton contributions.

Findings

Vacuum energy vanishes under periodic boundary conditions, indicating unbroken SUSY.

Nonzero gaugino condensates appear with antiperiodic boundary conditions, suggesting instanton effects.

Boundary conditions significantly influence the vacuum structure and SUSY breaking patterns.

Abstract

Vacuum structures of supersymmetric (SUSY) Yang-Mills theories in $1+1$ dimensions are studied with the spatial direction compactified. SUSY allows only periodic boundary conditions for both fermions and bosons. By using the Born-Oppenheimer approximation for the weak coupling limit, we find that the vacuum energy vanishes, and hence the SUSY is unbroken. Other boundary conditions are also studied, especially the antiperiodic boundary condition for fermions which is related to the system in finite temperatures. In that case we find for gaugino bilinears a nonvanishing vacuum condensation which indicates instanton contributions.