Abstract
We use the theory of singular foliations to study \( \mathcal{N}=1 \) compactifications of eleven-dimensional 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 \( \mathcal{W} \) which does not coincide with M . We show that the complement M \ \( \mathcal{W} \) must be a dense open subset of M and that M admits a singular foliation \( \overline{\mathrm{\mathcal{F}}} \) endowed with a longitudinal G 2 structure and defined by a closed one-form ω, whose geometry is determined by the supersymmetry conditions. The singular leaves are those leaves which meet \( \mathcal{W} \). When ω 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 \( \overline{\mathrm{\mathcal{F}}} \) 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 \( \mathcal{W} \).
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Babalic, E.M., Lazaroiu, C.I. Singular foliations for M-theory compactification. J. High Energ. Phys. 2015, 116 (2015). https://doi.org/10.1007/JHEP03(2015)116
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DOI: https://doi.org/10.1007/JHEP03(2015)116