Real pinor bundles and real Lipschitz structures

TL;DR

This paper characterizes the topological conditions for the existence of real Clifford module bundles on pseudo-Riemannian manifolds, introduces real Lipschitz structures as a generalization of spin structures, and applies these to supergravity.

Contribution

It introduces real Lipschitz structures as a new framework generalizing spin structures and classifies associated groups across all dimensions and signatures.

Findings

Topological obstructions for real Clifford module bundles identified.

Real Lipschitz structures are shown to generalize spin structures.

Application to supergravity demonstrates conditions for supersymmetry generators.

Abstract

We obtain the topological obstructions to existence of a bundle of irreducible real Clifford modules over a pseudo-Riemannian manifold $(M,g)$ of arbitrary dimension and signature and prove that bundles of Clifford modules are associated to so-called real Lipschitz structures. The latter give a generalization of spin structures based on certain groups which we call real Lipschitz groups. In the fiberwise-irreducible case, we classify the latter in all dimensions and signatures. As a simple application, we show that the supersymmetry generator of eleven-dimensional supergravity in "mostly plus" signature can be interpreted as a global section of a bundle of irreducible Clifford modules if and $\textit{only if}$ the underlying eleven-manifold is orientable and spin.