The spinorial geometry of supersymmetric backgrounds

We propose a new method to solve the Killing spinor equations of eleven-dimensional supergravity based on a description of spinors in terms of forms and on the Spin(1, 10) gauge symmetry of the supercovariant derivative. We give the canonical form of Killing spinors for backgrounds preserving two supersymmetries, N = 2, provided that one of the spinors represents the orbit of Spin(1, 10) with stability subgroup SU (5). We directly solve the Killing spinor equations of N = 1 and some N = 2, N = 3 and N = 4 backgrounds. In the N = 2 case, we investigate backgrounds with SU (5) and SU (4) invariant Killing spinors and compute the associated spacetime forms. We find that N = 2 backgrounds with SU (5) invariant Killing spinors admit a timelike Killing vector and that the space transverse to the orbits of this vector field is a Hermitian manifold with an SU (5)-structure. Furthermore, N = 2 backgrounds with SU (4) invariant Killing spinors admit two Killing vectors, one timelike and one spacelike. The space transverse to the orbits of the former is an almost Hermitian manifold with an SU (4)-structure. The spacelike Killing vector field leaves the almost complex structure invariant. We explore the canonical form of Killing spinors for backgrounds preserving more than two supersymmetries, N > 2. We investigate a class of N = 3 and N = 4 backgrounds with SU (4) invariant spinors. We find that in both cases the space transverse to a timelike vector field is a Hermitian manifold equipped with an SU (4)-structure and admits two holomorphic Killing vector fields. We also present an application to M-theory Calabi-Yau compactifications with fluxes to one-dimension.