The Killing Spinor Equation with Higher Order Potentials

Let (M, 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 T , a 4-form F , a spinorial covariant derivative ∇ depending on ∇, T , F , and a ∇-parallel spinor field Ψ. 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 α-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 energymomentum tensor.