Killing Spinors in Supergravity with 4-fluxes

We study the spinorial Killing equation of supergravity involving a torsion 3-form T as well as a flux 4-form F. In dimension seven, we construct explicit families of compact solutions out of 3-Sasakian geometries, nearly parallel G2-geometries and on the homogeneous Aloff-Wallach space. The constraint F · Ψ = 0 defines a non empty subfamily of solutions. We investigate the constraint T ·Ψ = 0, too, and show that it singles out a very special choice of numerical parameters in the Killing equation, which can also be justified geometrically.