M-theory on non-Kähler eight-manifolds

We show that M-theory admits a class of supersymmetric eight-dimensional compactification background solutions, equipped with an internal complex pure spinor, more general than the Calabi-Yau one. Building-up on this result, we obtain a a particular class of supersymmetric M-theory eight-dimensional non-geometric compactification backgrounds with external three-dimensional Minkowski space-time, proving that the global space of the non-geometric compactification is again a differentiable manifold, although with very different geometric and topological properties respect to the corresponding standard M-theory compactification background: it is a compact complex manifold admitting a Kähler covering with deck transformations acting by holomorphic homotheties with respect to the Kähler metric. We show that this class of non-geometric compactifications evade the Maldacena-Nuñez no-go theorem by means of a mechanism originally developed by Mario García-Fernández and the author for Heterotic Supergravity, and thus do not require l_P-corrections to allow for a nontrivial warp factor or four-form flux. We obtain an explicit compactification background on a complex Hopf four-fold that solves all the equations of motion of the theory, including the warp factor equation of motion. We also show that this class of non-geometric compactifications are equipped with a holomorphic principal torus fibration over a projective Kähler base as well as a codimension-one foliation with nearly-parallel G₂-leaves, making thus contact with the work of M. Babalic and C. Lazaroiu on the foliation structure of the most general M-theory supersymmetric compactifications.