Singular foliations for M-theory compactification

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