Heterotic string compactifications on integrable $G_2$ structure manifolds $Y$ with instanton bundles $(V,A), (TY,\tilde{ \theta})$ yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative $\cal D$ and show that it is equivalent to a heterotic $G_2$ system encoding the geometry of the heterotic string compactifications. This operator $\cal D$ acts on a bundle ${ \cal Q}=T^*Y\oplus{ \rm End}(V)\oplus{ \rm End}(TY)$ and satisfies a nilpotency condition $\check{ \cal D}^2=0$, for an appropriate projection of $\cal D$. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group $\check H^1_{ \check{ \cal D}}(\cal Q)$. We comment on the similarities and differences of our result with Atiyah's well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the $\alpha'$ expansion.
Categories: hep-th math.DG