Foliated Backgrounds for M-Theory Compactifications (I)

Using the theory of foliations, we give a geometric and topological description of compact eight-manifolds which arise as internal spaces in N = 1 flux compactifications of M-theory down to AdS(3). We prove that specifying such a supersymmetric background is equivalent with giving a codimension one foliation defined by a closed one-form and which carries a leafwise G(2) structure, whose geometry we characterize explicitly. It can be shown that the leaves of this foliation acquire singularities on the chiral locus, which is a set with empty interior when the equations of motion for the supregravity four-form field strength are imposed, but here we concentrate only on the case of an everywhere non-chiral spinor.