Complex Lipschitz structures and bundles of complex Clifford modules

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 Spinc(p,q) structure (when p+q is odd) or a Pinc(p,q) structure (when p+q is even). When p−q≡83,4,6,7, 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.