Non-trivial flat connections on the 3-torus I: G_2 and the orthogonal groups

TL;DR

This paper constructs non-trivial vacua for Yang-Mills theories on the 3-torus using twisted boundary conditions, extending previous results to exceptional groups and matching Witten index predictions.

Contribution

It introduces a new construction of vacua for Yang-Mills theories on the 3-torus based on twisted boundary conditions in SU(2) subgroups, extending to exceptional groups.

Findings

Reproduces vacua for SO(N) and G_2 theories on the 3-torus

Embeds results into F_4 and E_{6,7,8} groups

Number of vacua matches Witten index predictions

Abstract

We propose a construction of non-trivial vacua for Yang-Mills theories on the 3-torus. Although we consider theories with periodic boundary conditions, twisted boundary conditions play an essential auxiliary role in our construction. In this article we will limit ourselves to the simplest case, based on twist in SU(2) subgroups. These reproduce the recently constructed new vacua for SO(N) and G_2 theories on the 3-torus. We show how to embed the results in the other exceptional groups F_4 and E_{6,7,8} and how to compute the relevant unbroken subgroups. In a subsequent article we will generalise to SU(N > 2) subgroups. The number of vacua found this way exactly matches the number predicted by the calculation of the Witten index in the infinite volume.