Gauge theory on Aloff–Wallach spaces

Abstract

For gauge groups $U\left(1\right)$ and $SO\left(3\right)$ 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 $3$–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.