Bundle gerbes

Abstract

Just as ℂ× 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 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.