Singular foliations for M-theory compactification

We use the theory of singular foliations to study $\mathcal{N}=1$ compactifications of eleven-dimensional supergravity on eight-manifolds $M$ down to ${\mathrm{A}\mathrm{d}\mathrm{S}}_3$ spaces, allowing for the possibility that the internal part $\xi$ of the supersymmetry generator is chiral on some locus $\mathcal{W}$ which does not coincide with $M$. We show that the complement $M\setminus {\cal W}$mustbeadenseopensubsetof$M$andthat$M$admitsasingularfoliation${\bar {\cal F}}$endowedwithalongitudinal$G_2$ structure and defined by a closed one-form $\mathit{\omega }$, whose geometry is determined by the supersymmetry conditions. The singular leaves are those leaves which meet $\mathcal{W}$. When $\mathit{\omega }$ 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 $\bar{\mathcal{F}}$ using results from Novikov theory. We also show how this description fits in with previous formulas which were extracted by exploiting the $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7{)}_{\pm }$ structures which exist on the complement of $\mathcal{W}$.