Classification of supersymmetric spacetimes in eleven dimensions

TL;DR

This paper explicitly classifies supersymmetric solutions in eleven-dimensional supergravity with Spin(7) structure, detailing the metric, four-form, and Killing spinors, and establishing conditions for solutions to satisfy all field equations.

Contribution

It provides the first explicit local classification of supersymmetric spacetimes with Spin(7) structure in eleven dimensions, including solution conditions.

Findings

Explicit form of metrics and four-forms for Spin(7) structures

Conditions linking Bianchi identity and field equations

Reduction of problem to eight-manifold with Spin(7) holonomy

Abstract

We derive, for spacetimes admitting a Spin(7) structure, the general local bosonic solution of the Killing spinor equation of eleven dimensional supergravity. The metric, four form and Killing spinors are determined explicitly, up to an arbitrary eight-manifold of Spin(7) holonomy. It is sufficient to impose the Bianchi identity and one particular component of the four form field equation to ensure that the solution of the Killing spinor equation also satisfies all the field equations, and we give these conditions explicitly.