Foliated eight-manifolds for M-theory compactification

We characterize compact eight-manifolds M which arise as internal spaces in N = 1 flux compactifications of M-theory down to AdS(3) using the theory of foliations, for the case when the internal part xi of the supersymmetry generator is everywhere non-chiral. We prove that specifying such a supersymmetric background is equivalent with giving a codimension one foliation F of M which carries a leafwise G(2) structure, such that the O'Neill-Gray tensors, non-adapted part of the normal connection and the torsion classes of the G(2) structure are given in terms of the supergravity four-form field strength by explicit formulas which we derive. We discuss the topology of such foliations, showing that the C-* algebra C(M/F) is a noncommutative torus of dimension given by the irrationality rank of a certain cohomology class constructed from G, which must satisfy the Latour obstruction. We also give a criterion in terms of this class for when such foliations are fibrations over the circle. When the criterion is not satisfied, each leaf of F is dense in M .