Non-trivial flat connections on the 3-torus II: The exceptional groups F_4 and E_6,7,8

TL;DR

This work extends the non-trivial flat connection framework to exceptional gauge groups on the 3-torus, constructing commuting holonomies via twisted embeddings (notably $SU(3)^2$, $SU(3)^3$, and higher diagonal subgroups) to generate flat connections with distinct unbroken subgroups. It demonstrates that all non-trivial vacua in $F_4$, $E_6$, $E_7$, and $E_8$ can be accounted for by a combination of $SU(3)$-, $SU(4)$-, $SU(5)$-based twists and mixed $SU(2)$, $SU(3)$ constructions, yielding exclusively discrete or trivial unbroken subgroups in the non-trivial cases. The paper provides explicit holonomies and weight analyses to verify commuting properties and to count the resulting vacua, showing that the finite-volume Witten index $\mathrm{Tr}(-)^F$ matches the infinite-volume dual Coxeter numbers $h$ for all exceptional groups. These results support the significance of non-trivial flat connections for supersymmetric Yang–Mills theories on $T^3$ and have potential implications for heterotic string compactifications and duality studies.

Abstract

We continue the construction of non-trivial vacua for gauge theories on the 3-torus, started in hep-th/9901154. Application of constructions based on twist in SU(N) with N > 2 produce more extra vacua in theories with exceptional groups. We calculate the relevant unbroken subgroups, and their contribution to the Witten index. We show that the extra vacua we find in the exceptional groups are sufficient to solve the Witten index problem for these groups.