Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 020, 27 pages      arXiv:2204.00798      https://doi.org/10.3842/SIGMA.2024.020

The Clifford Algebra Bundle on Loop Space

Matthias Ludewig
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Received September 01, 2023, in final form February 22, 2024; Published online March 12, 2024

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.

Key words: loop spaces; Clifford algebra; string geometry; von Neumann algebra bundles.

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