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

Arjan Keurentjes

TL;DR

This work constructs non-trivial flat connections on the 3-torus by exploiting multi-twisted boundary conditions built from SU(2) subgroups, enabling explicit holonomy configurations beyond the trivial maximal-torus solutions. By embedding these SU(2)-based twists into $G_2$, orthogonal groups, and the exceptional groups $F_4$ and $E_{6,7,8}$, the authors determine the unbroken subgroups and identify the resulting discrete vacua or reduced gauge factors. The results reproduce and extend previous findings (notably Witten's vacua for SO(N) and G2) and demonstrate consistency with the predicted Witten index through summing $r'+1$ over vacuum components, where $r'$ is the rank of the unbroken subgroup. The paper also outlines how to generalize to SU(N) subgroups (with $N>2$) to generate additional vacua, aiming toward a complete account of the index for exceptional groups in a subsequent study. Overall, the construction provides a unified framework linking holonomy structure, diagonal subgroups, and the Witten index across a range of gauge groups including the exceptional ones.

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.

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

TL;DR

This work constructs non-trivial flat connections on the 3-torus by exploiting multi-twisted boundary conditions built from SU(2) subgroups, enabling explicit holonomy configurations beyond the trivial maximal-torus solutions. By embedding these SU(2)-based twists into , orthogonal groups, and the exceptional groups and , the authors determine the unbroken subgroups and identify the resulting discrete vacua or reduced gauge factors. The results reproduce and extend previous findings (notably Witten's vacua for SO(N) and G2) and demonstrate consistency with the predicted Witten index through summing over vacuum components, where is the rank of the unbroken subgroup. The paper also outlines how to generalize to SU(N) subgroups (with ) to generate additional vacua, aiming toward a complete account of the index for exceptional groups in a subsequent study. Overall, the construction provides a unified framework linking holonomy structure, diagonal subgroups, and the Witten index across a range of gauge groups including the exceptional ones.

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.

Paper Structure

This paper contains 29 sections, 67 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Non-trivial flat connections
  3. Holonomies and vacua
  4. Method of construction
  5. Diagonal subgroups
  6. Non-trivial vacua based on twist in $SU(2)$
  7. Twist in $SU(2)$
  8. Calculation of the unbroken subgroup
  9. Realisations of the $SU(2)$-based construction
  10. $G_2$
  11. $SO(7)$
  12. $SO(8)$
  13. $SO(N)$, $N > 8$
  14. $F_4$
  15. $E_6$
  16. ...and 14 more sections

Figures (3)

  • Figure 1: the root diagram for $G_2$
  • Figure 2: projection of the roots of $so(7)$ onto those for $g_2$
  • Figure 3: Dynkin diagrams