The geometry of D = 11 Killing spinors

Abstract

We propose a way to classify the local form of all bosonic supersymmetric configurations of D = 11 supergravity, using the differential forms that can be constructed as bi-linears from the Killing spinors. We show that the most general bosonic geometries either have a privileged SU(5) or a (Spin(7) × ℝ8) × ℝ structure, depending on whether the Killing vector constructed from the Killing spinor is timelike or null, respectively. In the time-like case we derive the general local form of the geometry and show that it is almost completely determined by a certain SU(5) structure on the ten-dimensional space orthogonal to the orbits of the Killing vector. We also deduce what further conditions must be imposed in order that the equations of motion are satisfied. We illustrate the formalism with some known solutions and also present some new solutions including a rotating generalisation of the resolved membrane solutions and generalisations of the recently constructed D = 11 Gödel solution. We also prove some general vanishing theorems for compactiflcations with flux. © SISSA/ISAS 2003.