Gauge theory on Aloff-Wallach spaces

TL;DR

This paper classifies invariant G_2-instantons on Aloff-Wallach spaces with specific structures, revealing how these instantons distinguish different G_2-structures and exhibit interesting phenomena like merging and non-minimal energy configurations.

Contribution

It provides a classification of invariant G_2-instantons on Aloff-Wallach spaces for certain gauge groups, highlighting their role in differentiating G_2-structures and uncovering novel instanton behaviors.

Findings

Existence of G_2-instantons distinguishes different G_2-structures.

Certain instantons exist on one structure but not another.

Examples of instantons merging and not minimizing energy.

Abstract

For gauge groups $U(1)$ and $SO(3)$ we classify invariant $G_2$-instantons for homogeneous coclosed $G_2$-structures on Aloff-Wallach spaces $X_{k,l}$. As a consequence, we give examples where $G_2$-instantons can be used to distinguish between different strictly nearly parallel $G_2$-structures on the same Aloff-Wallach space. In addition to this, we find that while certain $G_2$-instantons exist for the strictly nearly parallel $G_2$-structure on $X_{1,1}$, no such $G_2$-instantons exist for the tri-Sasakian one. As a further consequence of the classification, we produce examples of some other interesting phenomena, such as: irreducible $G_2$-instantons that, as the structure varies, merge into the same reducible and obstructed one; and $G_2$-instantons on nearly parallel $G_2$-manifolds that are not locally energy minimizing.