Killing spinors are Killing vector fields in Riemannian Supergeometry

Abstract

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is formulated as G-structure on M, where G is a supergroup with even part G_0\cong Spin(k,l); (k,l) the signature of (M_0,g_0). Killing vector fields on (M,g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field X_s on M. Our main result is that X_s is a Killing vector field on (M,g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field X_s.