Classification of supersymmetric flux vacua in M-theory

We present a comprehensive classification of supersymmetric vacua of M-theory compactification on seven-dimensional manifolds with general four-form fluxes. We analyze the cases where the resulting four-dimensional vacua have N=1,2,3,4N=1,2,3,4 supersymmetry and the internal space allows for SU(2)SU(2), SU(3)SU(3) or G2 structures. In particular, we find for N=2N=2 supersymmetry, that the external space–time is Minkowski and the base manifold of the internal space is conformally Kähler for SU(2)SU(2) structures, while for SU(3)SU(3) structures the internal space has to be Einstein–Sasaki and no internal fluxes are allowed. Moreover, we provide a new vacuum with N=1N=1 supersymmetry and SU(3)SU(3) structure, where all fluxes are non-zero and the first order differential equations are solved.