The geometry of extended null supersymmetry in M-theory

TL;DR

This paper develops a comprehensive formalism to analyze extended null supersymmetry in eleven-dimensional supergravity, providing explicit conditions for additional Killing spinors and classifying solutions with specific G2 structures.

Contribution

It derives necessary and sufficient algebraic and differential conditions for extended null supersymmetry in M-theory and classifies solutions with certain G2 structures.

Findings

Solutions with G2 structure are either products of 4D Minkowski space and G2 holonomy manifolds.

Classified supersymmetric spacetimes with rac{G_2 times \u211d^7}{ imes} rac{\u211d^2}{ ext{structure groups}}.

Provided a systematic method for analyzing all extended null supersymmetric solutions in eleven dimensions.

Abstract

For supersymmetric spacetimes in eleven dimensions admitting a null Killing spinor, a set of explicit necessary and sufficient conditions for the existence of any number of arbitrary additional Killing spinors is derived. The necessary and sufficient conditions are comprised of algebraic relationships, linear in the spinorial components, between the spinorial components and their first derivatives, and the components of the spin connection and four-form. The integrability conditions for the Killing spinor equation are also analysed in detail, to determine which components of the field equations are implied by arbitrary additional supersymmetries and the four-form Bianchi identity. This provides a complete formalism for the systematic and exhaustive investigation of all spacetimes with extended null supersymmetry in eleven dimensions. The formalism is employed to show that the general bosonic solution of eleven dimensional supergravity admitting a $G_2$ structure defined by four Killing spinors is either locally the direct product of $\mathbb{R}^{1,3}$ with a seven-manifold of $G_2$ holonomy, or locally the Freund-Rubin direct product of $AdS_4$ with a seven-manifold of weak $G_2$ holonomy. In addition, all supersymmetric spacetimes admitting a $(G_2\ltimes\mathbb{R}^7)\times\mathbb{R}^2$ structure are classified.