Superization of homogeneous spin manifolds and geometry of homogeneous supermanifolds

Abstract

Let M ₀=G ₀/H be a (pseudo)-Riemannian homogeneous spin manifold, with reductive decomposition $\mathfrak {g}_{0}=\mathfrak {h}+\mathfrak {m}$ and let S(M ₀) be the spin bundle defined by the spin representation $\tilde{ \operatorname {Ad}}:H\rightarrow \mathrm {GL}_{\mathbb {R}}(S)$ of the stabilizer H. This article studies the superizations of M ₀, 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 ^*(M ₀)). Here G is the Lie supergroup associated with a certain extension of the Lie algebra of symmetry $\mathfrak {g}_{0}$ to an algebra of supersymmetry $\mathfrak {g}=\mathfrak {g}_{\overline {0}}+\mathfrak {g}_{\overline {1}}=\mathfrak {g}_{0}+S$ via the Kostant-Koszul construction. Each algebra of supersymmetry naturally determines a flat connection $\nabla^{\mathcal {S}}$ in the spin bundle S(M ₀). Killing vectors together with generalized Killing spinors (i.e. $\nabla^{\mathcal {S}}$ -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 $\nabla^{\mathcal {S}}$ defines a superconnection on M, via the super-version of a theorem of Wang.