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The Clifford Algebra Bundle on Loop Space

Matthias Ludewig

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.

The Clifford Algebra Bundle on Loop Space

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.
Paper Structure (14 sections, 23 theorems, 49 equations)

This paper contains 14 sections, 23 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Bundles of super von Neumann algebras
  3. Super factors of type I and their classification
  4. The automorphism group of super factors of type I
  5. Bundles of graded type I factors
  6. Clifford von Neumann algebras
  7. Clifford algebras and Fock spaces
  8. The restricted orthogonal group
  9. The loop space Clifford algebra bundle
  10. Definition of the bundle
  11. Transgression and the loop spin class
  12. Proof of the main theorem
  13. A twisted Clifford algebra bundle
  14. Sign discussion

Key Result

Lemma 2.2

Let $A$ be a super factor. Then the ungraded center $Z^{\mathrm{un}}(A)$ is a graded subalgebra, which is isomorphic to either $\mathbb{C}$$($trivially graded$)$ or to $\mathbb{C} \oplus \mathbb{C}$$($with the grading operator given by swapping the two summands$)$.

Theorems & Definitions (47)

  • Definition 2.1: super factor
  • Lemma 2.2
  • Definition 2.3: even/odd kind
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 37 more