[Paper Review] Bundle gerbes

Michael K. Murray|arXiv (Cornell University)|Jul 25, 1994

Homotopy and Cohomology in Algebraic Topology3 references226 citations

TL;DR

This paper introduces bundle gerbes as a geometric realization of three-dimensional integral cohomology, analogous to how principal bundles realize two-dimensional cohomology. It establishes that every bundle gerbe gives rise to a gerbe via its Dixmier-Douady class, and develops the theory of bundle gerbe connections and curvature, providing a new framework for understanding higher gauge theory and characteristic classes in differential geometry.

ABSTRACT

Just as $\Cstar$ principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integral cohomology through their Dixmier-Douady class. I consider an alternative, related, geometric realisation of three dimensional cohomology called a bundle gerbe. Every bundle gerbe gives rise to a gerbe and most of the well-known examples examples of gerbes are bundle gerbes. I discuss the properties of bundle gerbes, in particular bundle gerbe connections and curvature and their associated Dixmier-Douady class.

Motivation & Objective

• To provide a new geometric realization of three-dimensional integral cohomology using bundle gerbes.
• To establish the relationship between bundle gerbes and gerbes via the Dixmier-Douady class.
• To develop the theory of bundle gerbe connections and curvature in the context of differential geometry.
• To generalize known examples of gerbes by showing they are naturally bundle gerbes.

Proposed method

• The paper constructs bundle gerbes as fibered categories over a manifold, equipped with a bundle-like structure over the fiber product.
• It defines bundle gerbe connections as connections on the bundle gerbe that are compatible with the gerbe structure.
• Curvature is defined as the curvature of the connection, which maps to the Dixmier-Douady class in de Rham cohomology.
• The theory is shown to reproduce the standard Dixmier-Douady class of a gerbe via the bundle gerbe construction.
• The framework is applied to known examples of gerbes, demonstrating that they arise naturally as bundle gerbes.

Experimental results

Research questions

• RQ1 How can three-dimensional integral cohomology be geometrically realized beyond the standard gerbe construction?
• RQ2 What is the precise relationship between bundle gerbes and gerbes in terms of their Dixmier-Douady classes?
• RQ3 How do connections and curvature on bundle gerbes relate to characteristic classes in differential cohomology?
• RQ4 Can standard examples of gerbes be naturally interpreted as bundle gerbes?

Key findings

• Every bundle gerbe gives rise to a gerbe with a well-defined Dixmier-Douady class in H^3(X, Z).
• Bundle gerbe connections and their curvature provide a differential refinement of the Dixmier-Douady class in de Rham cohomology.
• The curvature of a bundle gerbe connection is a closed 3-form that represents the image of the Dixmier-Douady class under the de Rham isomorphism.
• Most known examples of gerbes are shown to be realizable as bundle gerbes, indicating their broad applicability.