Holonomy and Symmetry in M-theory

TL;DR

The paper establishes that the generalized holonomy of the supercovariant derivative in 11D supergravity generically lies in ${\rm SL}(32,\mathbb{R})$, with special holonomies arising for flux, warping, and product spaces. It analyzes how holonomy controls the number of preserved supersymmetries and provides explicit background examples, illustrating when holonomy is reduced to subgroups such as ${\rm Spin}(3,2)\times{\rm Spin}(8)$ or ${\rm SL}(16,\mathbb{C})$. It argues that a consistent M-theory requires fermionic degrees of freedom to be sections of an ${\rm SL}(32,\mathbb{R})$ bundle, i.e., a local ${\rm SL}(32,\mathbb{R})$ symmetry, to accommodate phenomena like 'supersymmetry without supersymmetry' and brane-wrapping modes, and it discusses extensions to other ${D}=11$ supergravities and the need for a background-independent, symmetry-based formulation. Together, these results connect holonomy, generalized structure groups, and extended symmetry frameworks as essential ingredients in understanding M-theory vacua.

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.