Asian Journal of Mathematics

Volume 23 (2019)

Number 5

Real spinor bundles and real Lipschitz structures

Pages: 749 – 836



C. I. Lazaroiu (Center for Geometry and Physics, Institute for Basic Science, Pohang, South Korea; and Horia Hulubei National Institute of Physics and Nuclear Engineering, Magurele, Ilfov, Romania)

C. S. Shahbazi (Institut de Physique Théorique, CEA, Saclay, France)


Let $(M, g)$ be a pseudo-Riemannian manifold of arbitrary dimension and signature. We prove that there exist mutually quasi-inverse equivalences between the groupoid of weakly faithful real pinor bundles on $(M, g)$ and the groupoid of weakly faithful real Lipschitz structures on $(M, g)$, from which follows that every bundle of weakly faithful real Clifford modules is associated to a real Lipschitz structure. The latter gives a generalization of spin structures based on certain groups which we call real Lipschitz groups. In the irreducible case, we classify real Lipschitz groups in all dimensions and signatures. Using this classification and the previous correspondence we obtain the topological obstruction to existence of a bundle of irreducible real Clifford modules over a pseudo-Riemannian manifold $(M, g)$ of arbitrary dimension and signature. As a direct application of the previous results, 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 only if the underlying eleven-manifold is orientable and spin.


spinor bundles, pseudo-Riemannian manifolds, Clifford algebras, Lipschitz structures

2010 Mathematics Subject Classification

53C10, 53C27

Received 13 November 2017

Accepted 5 July 2018

Published 30 April 2020