  We construct a Clifford algebra bundle formed from the tangent bundle of the
smooth loop space of a Riemannian manifold, which is a bundle of super von
Neumann algebras on the loop space. We show that this bundle is in general
non-trivial, more precisely that its triviality is obstructed by the
transgressions of the second Stiefel-Whitney class and the first (fractional)
Pontrjagin class of the manifold.
