Supersymmetric M-theory compactifications with fluxes on seven-manifolds and G-structures

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We consider Minkowski compactifications of M-theory on generic seven-dimensional manifolds. After analyzing the conditions on the four-form flux, we establish a set of relations between the components of the intrinsic torsion of the internal manifold and the components of the four-form flux needed for preserving supersymmetry. The existence of two nowhere vanishing vectors on any seven-manifold with G₂ structure plays a crucial role in our analysis, leading to the possibility of four-dimensional compactifications with N=1 and N=2 supersymmetry.