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 = 1,2,3,4 supersymmetry and the internal space allows for SU(2), SU(3) or G_2 structures. In particular, we find for N = 2 supersymmetry, that the external space-time is Minkowski and the base manifold of the internal space being conformally K\"ahler for SU(2) structures, and the internal space has to be Einstein-Sasaki and no internal fluxes are allowed for SU(3) structures. Moreover, we provide a new vacuum with N = 1 supersymmetry and SU(3) structure, where all fluxes are non-zero and the first order differential equation are solved