Complex Lipschitz structures and bundles of complex Clifford modules

TL;DR

This paper establishes a correspondence between bundles of complex Clifford modules and Lipschitz structures on pseudo-Riemannian manifolds, providing new insights into their geometric and topological classifications.

Contribution

It constructs an equivalence between complex Clifford module bundles and Lipschitz structures, and characterizes when such bundles exist via Spin^c and Pin^c structures.

Findings

Equivalence between complex Clifford modules and Lipschitz structures

Characterization of bundles of irreducible complex Clifford modules via Spin^c and Pin^c structures

Comparison with real Clifford module classifications in specific signatures

Abstract

Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We construct mutually quasi-inverse equivalences between the groupoid of bundles of weakly-faithful complex Clifford modules on $(M,g)$ and the groupoid of reduced complex Lipschitz structures on $(M,g)$. As an application, we show that $(M,g)$ admits a bundle of irreducible complex Clifford modules if and only if it admits either a $Spin^{c}(p,q)$ structure (when $p+q$ is odd) or a $Pin^{c}(p,q)$ structure (when $p+q$ is even). When $p-q\equiv_8 3,4,6, 7$, we compare with the classification of bundles of irreducible real Clifford modules which we obtained in previous work. The results obtained in this note form a counterpart of the classification of bundles of faithful complex Clifford modules which was previously given by T. Friedrich and A. Trautman.