The Clifford Algebra Bundle on Loop Space

TL;DR

This paper constructs a Clifford algebra bundle over the loop space of a Riemannian manifold, revealing its non-triviality is linked to topological invariants like Stiefel-Whitney and Pontrjagin classes.

Contribution

It introduces a new Clifford algebra bundle on loop space and identifies topological obstructions to its triviality.

Findings

The bundle is generally non-trivial.

Triviality is obstructed by transgressed Stiefel-Whitney and Pontrjagin classes.

Links topological invariants to algebraic bundle properties.

Abstract

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.