The Geometry of D=11 Null Killing Spinors

TL;DR

This work completes the classification of bosonic $D=11$ supergravity configurations preserving at least one supersymmetry that admit a null Killing spinor. By translating supersymmetry into differential conditions on spinor bilinears, the authors reveal a privileged $(\mathrm{Spin}(7)\ltimes\mathbb{R}^8)\times\mathbb{R}$ structure and derive explicit local forms for the metric and four-form, including the conformally anti-self-dual Spin(7) base and a detailed flux decomposition. They prove both necessity and sufficiency of the constructed conditions, and demonstrate consistency with the equations of motion by presenting concrete examples: resolved membranes with waves and the basic fivebrane, as well as the zero-flux Bryant limit. The framework not only recovers known solutions within a unified formalism but also enables new generalisations of membranes with gravitational waves, providing a systematic path to generate and study 1/32-supersymmetric backgrounds in M-theory.

Abstract

We determine the necessary and sufficient conditions on the metric and the four-form for the most general bosonic supersymmetric configurations of D=11 supergravity which admit a null Killing spinor i.e. a Killing spinor which can be used to construct a null Killing vector. This class covers all supersymmetric time-dependent configurations and completes the classification of the most general supersymmetric configurations initiated in hep-th/0212008.