Foliated eight-manifolds for M-theory compactification

Abstract

Abstract: We characterize compact eight-manifolds M which arise as internal spaces in N=1 flux compactifications of M-theory down to AdS3 using the theory of foliations, for the case when the internal part ξ 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 G2 structure, such that the O’Neill-Gray tensors, non-adapted part of the normal connection and the torsion classes of the G2 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.