Families of solutions of the heterotic G${}_2$ system

We construct new families of solutions of the heterotic G${}_2$ system on squashed homogeneous 3-Sasakian manifolds, that is, using squashed metrics on either the 7-sphere or the Aloff-Wallach space $N_{1,1}$. We obtain AdS${}_3$ solutions for all values of the squashing parameter $s$ except for the nearly-parallel G${}_2$ value $s=1\mathrm{/}\sqrt{5}$, for which we don’t find any solutions. Along the process, we construct different G${}_2$-instanton connections on bundles over these squashed manifolds.