Foliated eight-manifolds for M-theory compactification

TL;DR

This paper characterizes eight-manifolds used in M-theory flux compactifications to AdS3, linking their geometry to foliations with leafwise G2 structures and analyzing the topology and noncommutative aspects of these foliations.

Contribution

It establishes a detailed geometric and topological framework connecting supersymmetric M-theory backgrounds with codimension one foliations and noncommutative tori, providing explicit formulas and criteria.

Findings

Foliations carry leafwise G2 structures determined by supergravity fields.

The C star algebra of the foliation is a noncommutative torus with dimension related to a cohomology class.

A criterion for when foliations are fibrations over the circle is provided.

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 of the supersymmetry generator is everywhere non-chiral. We prove that specifying such a supersymmetric background is equivalent with giving a codimension one foliation of M which carries a leafwise G2 structure, such that the O'Neill-Gray tensors, non-adapted part of the normal connection and 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 star algebra of the foliation is a noncommutative torus of dimension given by the irrationality rank of a certain cohomology class constructed from the four-form field strength, 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 the foliation is dense in M.