A fusion structure for the spinor bundle on loop space

There is a canonical way to assign to each loop on a Riemannian manifold a Hilbert space with a conformal net that acts on it. The failure of these Hilbert spaces to constitute a locally trivial bundle over loop space is measured by the curvature of the metric. We use higher geometry (2-groups and 2-bundles) to describe this situation and outline why the Connes fusion product of Hilbert spaces should extend to a fusion structure for the spinor bundle on loop space. This is work in progress.