Systematics of M-theory spinorial geometry

TL;DR

This paper systematically classifies all supersymmetric backgrounds in eleven-dimensional supergravity by analyzing Killing spinor equations and their integrability conditions, providing a linear framework to determine fluxes and geometric restrictions.

Contribution

It introduces a comprehensive linear system approach to classify supersymmetric backgrounds in M-theory, detailing supercovariant derivatives and integrability conditions for all spinor types.

Findings

Linear systems for Killing spinor equations are established.

Fluxes are expressed in terms of geometric data.

Minimum field equations for supersymmetry are identified.

Abstract

We reduce the classification of all supersymmetric backgrounds in eleven dimensions to the evaluation of the supercovariant derivative and of an integrability condition, which contains the field equations, on six types of spinors. We determine the expression of the supercovariant derivative on all six types of spinors and give in each case the field equations that do not arise as the integrability conditions of Killing spinor equations. The Killing spinor equations of a background become a linear system for the fluxes, geometry and spacetime derivatives of the functions that determine the spinors. The solution of the linear system expresses the fluxes in terms of the geometry and specifies the restrictions on the geometry of spacetime for all supersymmetric backgrounds. We also show that the minimum number of field equations that is needed for a supersymmetric configuration to be a solution of eleven-dimensional supergravity can be found by solving a linear system. The linear systems of the Killing spinor equations and their integrability conditions are given in both a timelike and a null spinor basis. We illustrate the construction with examples.