The homogeneity conjecture for supergravity backgrounds

TL;DR

This paper explores the construction of Lie (super)algebras from spin manifolds and demonstrates that highly supersymmetric supergravity backgrounds are necessarily locally homogeneous, advancing understanding of supergravity symmetries.

Contribution

It introduces methods to construct Lie superalgebras from geometric data and proves that sufficiently supersymmetric supergravity backgrounds must be locally homogeneous.

Findings

Construction of compact real forms of Lie algebras from spheres.

Development of the Killing superalgebra for eleven-dimensional supergravity.

Proof that highly supersymmetric backgrounds are necessarily locally homogeneous.

Abstract

These notes record three lectures given at the workshop "Higher symmetries in Physics", held at the Universidad Complutense de Madrid in November 2008. In them we explain how to construct a Lie (super)algebra associated to a spin manifold, perhaps with extra geometric data, and a notion of privileged spinors. The typical examples are supersymmetric supergravity backgrounds; although there are more classical instances of this construction. We focus on two results: the geometric constructions of compact real forms of the simple Lie algebras of type B_4, F_4 and E_8 from S^7, S^8 and S^15, respectively; and the construction of the Killing superalgebra of eleven-dimensional supergravity backgrounds. As an application of this latter construction we show that supersymmetric supergravity backgrounds with enough supersymmetry are necessarily locally homogeneous.