Superization of Homogeneous Spin Manifolds and Geometry of Homogeneous Supermanifolds

Let M0 = G0/H be a (pseudo)-Riemannian homogeneous spin manifold, with reductive decomposition g0 = h + m and let S(M0) be the spin bundle defined by the spin representation Ãd : H → GLR(S) of the stabilizer H . This article studies the superizations of M0, i.e. its extensions to a homogeneous supermanifold M = G/H whose sheaf of superfunctions is isomorphic to Λ(S(M0)). Here G is the Lie supergroup associated with a certain extension of the Lie algebra of symmetry g0 to an algebra of supersymmetry g = g0 + g1 = g0 + S via the Kostant-Koszul construction. Each algebra of supersymmetry naturally determines a flat connection ∇S in the spin bundle S(M0). Killing vectors together with generalized Killing spinors (i.e. ∇S-parallel spinors) are interpreted as the values of appropriate geometric symmetries of M , namely even and odd Killing fields. An explicit formula for the Killing representation of the algebra of supersymmetry is obtained, generalizing some results of Koszul. The generalized spin connection ∇S defines a superconnection on M , via the super-version of a theorem of Wang.