Restrictions of heterotic G <inf>2</inf> structures and instanton connections

This note revisits recent results regarding the geometry and moduli of solutions of the heterotic string onmanifolds Y with aG2 structure. In particular, such heterotic G 2 systems can be rephrased in terms of a differential Ď acting on a complex Ω ∗ (Y, Q), where Q = T∗Y ⊕End (TY) ⊕ End (V), and Ď is an appropriate projection of an exterior covariant derivativeD which satisfies an instanton condition. The infinitesimal moduli are further parametrized by the first cohomologyH 1Ď (Y, Q).We proceed to restrict this system to manifolds X with an SU(3) structure corresponding to supersymmetric compactifications to four-dimensionalMinkowski space, often referred to as Strominger–Hull solutions. In doing so, we derive a new result: the Strominger-Hull systemis equivalent to a particular holomorphic Yang–Mills covariant derivative on Q|X = T∗X ⊕End (TX) ⊕ End(V).