The holonomy of the supercovariant connection and Killing spinors

TL;DR

This paper analyzes the holonomy of the supercovariant connection in M-theory, establishing conditions for Killing spinors, and explores the structure of supersymmetric backgrounds and probe brane configurations.

Contribution

It provides a detailed characterization of the holonomy reduction, Killing spinor conditions, and the algebraic structure of supersymmetric M-theory backgrounds and brane configurations.

Findings

Holonomy reduces to a subgroup of SL(32-N,R)

No topological obstruction for up to 22 Killing spinors

Supersymmetric probe configurations involve up to 31 branes

Abstract

We show that the holonomy of the supercovariant connection for M-theory backgrounds with $N$ Killing spinors reduces to a subgroup of $SL(32-N,\bR)\st (\oplus^N \bR^{32-N})$. We use this to give the necessary and sufficient conditions for a background to admit $N$ Killing spinors. We show that there is no topological obstruction for the existence of up to 22 Killing spinors in eleven-dimensional spacetime. We investigate the symmetry superalgebras of supersymmetric backgrounds and find that their structure constants are determined by an antisymmetric matrix. The Lie subalgebra of bosonic generators is related to a real form of a symplectic group. We show that there is a one-one correspondence between certain bases of the Cartan subalgebra of $sl(32, \bR)$ and supersymmetric planar probe M-brane configurations. A supersymmetric probe configuration can involve up to 31 linearly independent planar branes and preserves one supersymmetry. The space of supersymmetric planar probe M-brane configurations is preserved by an $SO(32,\bR)$ subgroup of $SL(32, \bR)$.