Holonomy and Symmetry in M-theory

TL;DR

This paper explores the role of holonomy in classifying supersymmetric solutions of 11-dimensional supergravity, emphasizing the importance of an $ ext{SL}(32, ext{R})$ symmetry in M-theory and its implications for supersymmetry and fermionic degrees of freedom.

Contribution

It demonstrates that the holonomy of solutions must be within $ ext{SL}(32, ext{R})$ and discusses the necessity of this symmetry for incorporating fermions in M-theory.

Findings

Holonomy is contained in $ ext{SL}(32, ext{R})$ for supersymmetric solutions.

Flux solutions have specific holonomy properties.

Including fermions in M-theory requires a local $ ext{SL}(32, ext{R})$ symmetry.

Abstract

Supersymmetric solutions of 11-dimensional supergravity can be classified according to the holonomy of the supercovariant derivative arising in the Killing spinor condition. It is shown that the holonomy must be contained in $\SL(32,\R)$. The holonomies of solutions with flux are discussed and examples are analysed. In extending to M-theory, account has to be taken of the phenomenon of ` supersymmetry without supersymmetry'. It is argued that including the fermionic degrees of freedom in M-theory requires a formulation with a local $\SL(32,\R)$ symmetry, analogous to the need for local Lorentz symmetry in coupling spinors to gravity.