The Infinitesimal Moduli Space of Heterotic G$_{2}$ Systems
Résumé
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 ${\mathcal{D}}$ and show that it is equivalent to a heterotic G$_{2}$ system encoding the geometry of the heterotic string compactifications. This operator ${\mathcal{D}}$ acts on a bundle ${\mathcal{Q}=T^*Y \oplus {\rm End}(V) \oplus {\rm End}(TY)}$ and satisfies a nilpotency condition ${\check{{\mathcal{D}}}^2=0}$ , for an appropriate projection of ${\mathcal D}$ . Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group ${\check H^1_{\check{{\mathcal{D}}}}(\mathcal{Q})}$ . 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.