The Killing spinor equation with higher order potentials

Let (Mn,g)(Mn,g) be a Riemannian spin manifold. The basic equations in supergravity models of type IIa string theory with 4-form flux involve a 3-form TT, a 4-form FF, a spinorial covariant derivative ∇∇ depending on ∇g∇g, TT, FF, and a ∇∇-parallel spinor field ΨΨ. We classify and construct many explicit families of solutions to this system of spinorial field equations by means of non-integrable special geometries. The latter include αα-Sasakian structures in dimensions 5 and 7, almost Hermitian structures in dimension 6 and cocalibrated G2-structures in dimension 7. We show that there are several examples also satisfying an additional constraint for the energy-momentum tensor.