The infinitesimal moduli space of heterotic $G_2$ systems

Understand

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 $\check H^1_{\check{\cal D}}(\cal Q)$.