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 the sheaf of sections of 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 = g0̄+g1̄+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. © Mathematisches Seminar der Universität Hamburg and Springer 2009.