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

TL;DR

This work analyzes the vacuum structure of pure $N=1$ supersymmetric Yang–Mills theories for arbitrary gauge groups by reducing to flat connections on a three-torus, i.e., commuting triples in the gauge group. It develops a Lie-theoretic framework using Dynkin diagrams, conjugacy classes, and Heisenberg pairs to classify non-Cartan flat connections, revealing extra vacua for higher orthogonal and exceptional groups. The main result is that the total number of quantum vacua equals the dual Coxeter number $h^\lor$ for all groups, with explicit moduli-space descriptions for exceptional groups. This resolves historical mismatches and yields a detailed description of the vacuum spectrum in these theories.

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.

Figures (8)

• Figure 1: Dynkin diagram for $Spin(7)$ with its Dynkin labels.
• Figure 2: Coroot lattices for $G = G_2$ and its subgroup $\tilde{G} = [SU(2)]^2/{\bf Z}_2$. Blank circles mark out the nodes of $Q^\lor( G)$ not belonging to $Q^\lor(\tilde{G})$.
• Figure 3: Dynkin diagrams for $Spin(2r+1)$.
• Figure 4: Dynkin diagram for $F_4$.
• Figure 5: Dynkin diagram for $E_6$.