We summarize the foliation approach to ${ \cal N}=1$ compactifications of eleven-dimensional supergravity on eight-manifolds $M$ down to $\mathrm{ AdS}_3$ spaces for the case when the internal part $\xi$ of the supersymmetry generator is chiral on some proper subset ${ \cal W}$ of $M$. In this case, a topological no-go theorem implies that the complement $M\setminus { \cal W}$ must be a dense open subset, while $M$ admits a singular foliation ${ \bar { \cal F}}$ (in the sense of Haefliger) which is defined by a closed one-form $\boldsymbol{ \omega}$ and is endowed with a longitudinal $G_2$ structure. The geometry of this foliation is determined by the supersymmetry conditions. We also describe the topology of ${ \bar { \cal F}}$ in the case when $\boldsymbol{ \omega}$ is a Morse form.
Categories: hep-th math.DG