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 $\G_2$-geometries and on the homogeneous Aloff-Wallach space. The constraint $\F \cdot Ψ= 0$ defines a non empty subfamily of solutions. We investigate the constraint $\T \cdot Ψ= 0$, too, and show that it singles out a very special choice of numerical parameters in the Killing equation, which can also be justified geometrically.

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