Restrictions of Heterotic $G_2$ Structures and Instanton Connections

This note revisits recent results regarding the geometry and moduli of solutions of the heterotic string on manifolds $Y$ with a $G_2$ structure. In particular, such heterotic $G_2$ systems can be rephrased in terms of a differential $\check {\cal D}$ acting on a complex $\check\Omega^*(Y , {\cal Q})$, where ${\cal Q}=T^*Y\oplus{\rm End}(TY)\oplus{\rm End}(V)$ and $\check {\cal D}$ is an appropriate projection of an exterior covariant derivative ${\cal D}$ which satisfies an instanton condition. The infinitesimal moduli are further parametrised by the first cohomology $H^1_{\check {\cal D}}(Y,{\cal Q})$. We proceed to restrict this system to manifolds $X$ with an $SU(3)$ structure corresponding to supersymmetric compactifications to four dimensional Minkowski space, often referred to as Strominger--Hull solutions. In doing so, we derive a new result: the Strominger-Hull system is equivalent to a particular holomorphic Yang-Mills covariant derivative on ${\cal Q}\vert_X=T^*X\oplus{\rm End}(TX)\oplus{\rm End}(V)$.