We propose a new method to solve the Killing spinor equations of elevendimensional 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. 1 We thank O. Mac Conamhna for explaining this result to us.

We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection∇ with torsion H, the NS⊗NS three-form field strength, are Killing. We find that there are two classes of such backgrounds, the null and the timelike. The Killing spinors of the null backgrounds have stability subgroups K ⋉ R 8 in Spin(9, 1), for K = Spin(7), SU (4), Sp(2), SU (2)×SU (2) and {1}, and the Killing spinors of the timelike backgrounds have stability subgroups G 2 , SU (3), SU (2) and {1}. The former admit a single null∇-parallel vector field while the latter admit a timelike and two, three, five and nine spacelike∇-parallel vector fields, respectively. The spacetime of the null backgrounds is a Lorentzian two-parameter family of Riemannian manifolds B with skew-symmetric torsion. If the rotation of the null vector field vanishes, the holonomy of the connection with torsion of B is contained in K. The spacetime of time-like backgrounds is a principal bundle P with fibre a Lorentzian Lie group and base space a suitable Riemannian manifold with skew-symmetric torsion. The principal bundle is equipped with a connection λ which determines the non-horizontal part of the spacetime metric and of H. The curvature of λ takes values in an appropriate Lie algebra constructed from that of K. In addition dH has only horizontal components and contains the Pontrjagin class of P . We have computed in all cases the Killing spinor bilinears, expressed the fluxes in terms of the geometry and determine the field equations that are implied by the Killing spinor equations.

We investigate the Killing spinor equations of IIB supergravity for one Killing spinor. We show that there are three types of orbits of Spin(9, 1) in the space of Weyl spinors which give rise to Killing spinors with stability subgroups Spin(7) ⋉ R 8 , SU (4) ⋉ R 8 and G 2 . We solve the Killing spinor equations for the Spin(7) ⋉ R 8 and SU (4) ⋉ R 8 invariant spinors, give the fluxes in terms of the geometry and determine the conditions on the spacetime geometry imposed by supersymmetry. In both cases, the spacetime admits a null, self-parallel, Killing vector field. We also apply our formalism to examine a class of SU (4) ⋉ R 8 backgrounds which admit one and two pure spinors as Killing spinors and investigate the geometry of the spacetimes.

We find the geometry of all supersymmetric type I backgrounds by solving the gravitino and dilatino Killing spinor equations, using the spinorial geometry technique, in all cases. The solutions of the gravitino Killing spinor equation are characterized by their isotropy group in Spin(9, 1), while the solutions of the dilatino Killing spinor equation are characterized by their isotropy group in the subgroup Σ(P) of Spin(9, 1) which preserves the space of parallel spinors P. Given a solution of the gravitino Killing spinor equation with L parallel spinors, L = 1, 2, 3, 4, 5, 6, 8, the dilatino Killing spinor equation allows for solutions with N supersymmetries for any 0 < N ≤ L. Moreover for L = 16, we confirm that N = 8, 10, 12, 14, 16. We find that in most cases the Bianchi identities and the field equations of type I backgrounds imply a further reduction of the holonomy of the supercovariant connection. In addition, we show that in some cases if the holonomy group of the supercovariant connection is precisely the isotropy group of the parallel spinors, then all parallel spinors are Killing and so there are no backgrounds with N < L supersymmetries.

We adapt the spinorial geometry method to investigate supergravity backgrounds with near maximal number of supersymmetries. We then apply the formalism to show that the IIB supergravity backgrounds with 31 supersymmetries preserve an additional supersymmetry and so they are maximally supersymmetric. This rules out the existence of IIB supergravity preons.