(M-theory-)Killing spinors on symmetric spaces

TL;DR

This paper develops an algebraic method to determine the holonomy of generalized Killing spinor derivatives on symmetric spaces, enabling the classification of supersymmetric M-theory backgrounds.

Contribution

It introduces an algebraic, algorithmic approach to compute holonomy of Killing spinor derivatives on symmetric spaces, specifically applied to classify supersymmetric M-theory backgrounds.

Findings

Derived criteria for supersymmetry preservation in symmetric M-theory backgrounds

Classified all supersymmetric symmetric M-theory backgrounds

Provided a systematic algebraic method for holonomy computation

Abstract

We show how the theory of invariant principal bundle connections for reductive homogeneous spaces can be applied to determine the holonomy of generalised Killing spinor covariant derivatives of the form $D= \nabla + \Omega$ in a purely algebraic and algorithmic way, where $\Omega : TM \rightarrow \Lambda^*(TM)$ is a left-invariant homomorphism. Specialising this to the case of symmetric M-theory backgrounds (i.e. $(M,g,F)$ with $(M,g)$ a symmetric space and $F$ an invariant closed 4-form), we derive several criteria for such a background to preserve some supersymmetry and consequently find all supersymmetric symmetric M-theory backgrounds.