[Paper Review] A bundle gerbe construction of a spinor bundle from the smooth free loop of a vector bundle

Stuart Ambler|arXiv (Cornell University)|Jul 18, 2012

Homotopy and Cohomology in Algebraic Topology15 references3 citations

TL;DR

This paper constructs a spinor bundle over the smooth free loop space of a compact, oriented, Riemannian manifold using a bundle gerbe derived from an even-rank vector bundle with fiberwise inner product. By encoding continuous choices of Lagrangian subspaces (via polarization classes) through a bundle gerbe, and showing the vanishing of its Dixmier-Douady class, the authors construct a trivialization that yields an irreducible Clifford module bundle, i.e., a spinor bundle, over the loop space.

ABSTRACT

A bundle gerbe is constructed from an oriented smooth vector bundle of even rank with a fiberwise inner product, over a compact connected orientable smooth manifold with Riemannian metric. From a trivialization of the bundle gerbe is constructed an irreducible Clifford module bundle, a spinor bundle over the smooth free loop space of the manifold. First, a Clifford algebra bundle over the loop space is constructed from the vector bundle. A polarization class bundle is constructed, choosing continuously over each point of the loop space a polarization class of Lagrangian subspaces of the complexification of the real vector space from which the Clifford algebra is made. Being unable to choose a Lagrangian subspace continuously from the polarization class over each point, the thesis constructs a bundle gerbe over the loop space of the base manifold to encode over each loop all such subspaces, along with the isomorphisms between the Fock spaces made from them, resulting from their being in the same polarization class. The vanishing of the Dixmier-Douady class of the bundle gerbe implies that the latter has a trivialization, from which is constructed a spinor bundle.

Motivation & Objective

• To construct a spinor bundle over the smooth free loop space of a compact, oriented, Riemannian manifold.
• To resolve the obstruction in continuously choosing Lagrangian subspaces within polarization classes over the loop space.
• To use bundle gerbes as a geometric tool to encode isomorphisms between Fock spaces from different Lagrangian subspaces in the same polarization class.
• To show that the vanishing of the Dixmier-Douady class of the constructed bundle gerbe implies trivialization, leading to a well-defined spinor bundle.
• To establish a functorial construction of the spinor bundle from the original vector bundle via gerbe-theoretic and Clifford algebraic methods.

Proposed method

• Construct a Clifford algebra bundle over the loop space from the pullback of the original vector bundle.
• Define a polarization class bundle that assigns to each loop a set of Lagrangian subspaces in the complexified fiber of the Clifford algebra.
• Introduce a bundle gerbe over the loop space to encode all possible Lagrangian subspaces in each polarization class, along with the isomorphisms (intertwiners) between their associated Fock representations.
• Use the theory of continuous bundle gerbes and ˇCech cohomology to define the Dixmier-Douady class of the gerbe.
• Prove that the Dixmier-Douady class vanishes under the given assumptions, implying the gerbe is trivializable.
• Construct the spinor bundle as a trivialization of the bundle gerbe, realized as a Hilbert bundle of Fock spaces via the chosen trivialization.

Experimental results

Research questions

• RQ1 Can a spinor bundle be constructed over the smooth free loop space of a compact, oriented, Riemannian manifold using geometric and algebraic structures from the original vector bundle?
• RQ2 What is the obstruction to continuously choosing Lagrangian subspaces within a polarization class over the loop space, and how can it be encoded geometrically?
• RQ3 Does the bundle gerbe constructed from the polarization class and intertwiners have a trivialization, and what does this imply for the existence of a spinor bundle?
• RQ4 How does the vanishing of the Dixmier-Douady class of the bundle gerbe relate to the existence of a globally defined spinor bundle?
• RQ5 Can the construction be made functorial and stable under pullbacks, ensuring consistency across different manifolds and bundles?

Key findings

• A bundle gerbe is constructed over the smooth free loop space of a compact, oriented, Riemannian manifold with an even-rank vector bundle equipped with a fiberwise inner product.
• The bundle gerbe encodes the data of all Lagrangian subspaces in each polarization class and the isomorphisms (intertwiners) between their Fock representations.
• The Dixmier-Douady class of the bundle gerbe vanishes under the given assumptions, implying the gerbe is trivializable.
• A trivialization of the bundle gerbe yields a well-defined, irreducible Clifford module bundle over the loop space, which is identified as a spinor bundle.
• The construction is functorial and stable under pullbacks, ensuring consistency and compatibility with geometric operations.
• The result establishes a geometric realization of the spinor bundle on the loop space via gerbe-theoretic methods, resolving a long-standing issue in infinite-dimensional geometry and index theory.