Seminar of Theory of Relativity and Gravitation

Physicist Figueroa-O'Farrill, Meessen and Philip showed that M-theory backgrounds with more than 24 supersymmetries (= Killing spinors) is locally homogeneous. Note that $ 24= (3/4)N$ where $N = 32$ is the rank of the spinor bundle of the Lorentzian 11-dimensional spin manifold. They conjectured that similar result is valid in other signatures and dimensions. We prove several results of such type using the natural $Spin(V)$-equivariant map $ S \otimes S \to V$ of the tensor square of the spinor $Spin(V)$-module $S$ into vectors. It allows to associate with two Killing spinors a conformal Killing vector or Killing vector. For example, we prove that a pseudo-Riemannian spin-manifold M of signature $(p,q)$ is locally homogeneous if it admits more than $(3/4)N$ ( independent) Killing spinors with the same Killing number, unless $ n = p+q \equiv 1 (mod 4)$ and $ s = p-q \equiv 3 (mod 4)$.

In these exceptional signatures we get a weaker result. In Riemannian and Lorentz cases the result is stronger : $(1/2)N$ Killing spinors are sufficient. We give also a description of (not necessarily complete) Riemannian manifolds admitting Killing spinors, which provides an inductive construction of such manifolds.