A classification of compactifying solutions for d=11 supergravity

Supergravity in eleven dimensions is known to have classical solutions of the type (anti-de Sitter space-time)x(7-dimensional Einstein space). We give a list of all homogeneous 7-manifolds which admit an Einstein metric. Known solutions are reviewed, with some emphasis on the SU(3)xSU(2)xU(1) compactifications. Their topology is discussed in detail. The list includes three new solutions, with symmetry groups SU(3)xSU(2), SO(5) and SO(5)xU(1). The first solution has no supersymmetry, while the second and third yield respectively N=1 and N=2 supersymmetry in four dimensions. The last two solutions may be extended to solutions with nonzero internal photon curl, breaking all supersymmetry. The existence of a spin structure on homogeneous manifolds G/H is discussed and related to topological properties of G/H. As an illustration, we treat the coset spaces SU(m+1)xSU(n+1)/SU(m)xSU(n)xU(1), which include the spaces with SU(3)xSU(2)xU(1) symmetry. (orig.)