The Infinitesimal Moduli Space of Heterotic G 2 Systems

Heterotic string compactifications on integrable G 2 structure manifolds Y with instanton bundles (V,A), (TY,θ̃) yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative D and show that it is equivalent to a heterotic G 2 system encoding the geometry of the heterotic string compactifications. This operator D acts on a bundle Q=T^*Y ⊕ End(V) ⊕ End(TY) and satisfies a nilpotency condition D²=0 , for an appropriate projection of D . Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group H¹_D(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 α' expansion.