The Geometry of D=11 Killing Spinors

TL;DR

This work develops a systematic local classification of bosonic, supersymmetric backgrounds in $D=11$ supergravity by using Killing spinor bilinears to define $G$-structures. It shows that any solution preserving at least one Killing spinor locally carries either a timelike-induced $SU(5)$ structure or a null-induced $(Spin(7)\ltimes\mathbb{R}^8)\times\mathbb{R}$ structure, with the metric and four-form $F$ largely fixed by the $SU(5)$ data on the base and with an undetermined flux piece $F_{75}$ constrained by Bianchi identities and equations of motion. The authors derive comprehensive algebraic and differential conditions, connect them to intrinsic torsion, and illustrate the framework with explicit M-brane, rotating-Calabi–Yau, and Gödel-type solutions, including novel rotating and Gödel geometries. This approach lays the groundwork for a complete local classification by treating the null case and higher fractions of supersymmetry, and provides a practical recipe for constructing new supersymmetric backgrounds in M-theory.

Abstract

We propose a way to classify all supersymmetric configurations of D=11 supergravity using the G-structures defined by the Killing spinors. We show that the most general bosonic geometries admitting a Killing spinor have at least a local SU(5) or an (Spin(7)\ltimes R^8)x R structure, depending on whether the Killing vector constructed from the Killing spinor is timelike or null, respectively. In the former case we determine what kind of local SU(5) structure is present and show that almost all of the form of the geometry is determined by the structure. 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 Godel solution.