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 $Spin^{c}(p,q)$ 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.