Singular foliations for M-theory compactification

Abstract

We use the theory of singular foliations to study N = 1 compactifications of elevendimensional supergravity on eight-manifolds M down to AdS3 spaces, allowing for the possibility that the internal part ξ of the supersymmetry generator is chiral on some locus W which does not coincide with M. We show that the complement M \W must be a dense open subset of M and that M admits a singular foliation ¯ F endowed with a longitudinal G2 structure and defined by a closed one form !, whose geometry is determined by the supersymmetry conditions. The singular leaves are those leaves which meet W. When ! is a Morse form, the chiral locus is a finite set of points, consisting of isolated zero-dimensional leaves and of conical singularities of seven-dimensional leaves. In that case, we describe the topology of ¯ F using results from Novikov theory. We also show how this description fits in with previous formulas which were extracted by exploiting the Spin(7)± structures which exist on the complement of W.

Open Access, ⓒThe Authors.