Vacuum structure in supersymmetric Yang-Mills theories with any gauge group

TL;DR

This paper classifies the quantum vacuum states of pure supersymmetric Yang-Mills theories on a 3-torus for various gauge groups, linking the problem to classifying flat connections and confirming the number of vacua matches previous instanton-based results.

Contribution

It provides a mathematical classification of flat connections on a 3-torus for higher orthogonal and exceptional gauge groups, determining the vacuum structure.

Findings

The moduli space of flat connections has multiple connected components for certain groups.

The total number of vacua equals the dual Coxeter number of each gauge group.

Results agree with earlier instanton-based calculations.

Abstract

We consider the pure supersymmetric Yang--Mills theories placed on a small 3-dimensional spatial torus with higher orthogonal and exceptional gauge groups. The problem of constructing the quantum vacuum states is reduced to a pure mathematical problem of classifying the flat connections on 3-torus. The latter problem is equivalent to the problem of classification of commuting triples of elements in a connected simply connected compact Lie group which is solved in this paper. In particular, we show that for higher orthogonal SO(N), N > 6, and for all exceptional groups the moduli space of flat connections involves several distinct connected components. The total number of vacuumstates is given in all cases by the dual Coxeter number of the group which agrees with the result obtained earlier with the instanton technique.