Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection

TL;DR

This paper establishes an equivalence between abelian gerbes with connection on a manifold and certain fusion bundles over its free loop space, with applications to geometric lifting and loop group extensions.

Contribution

It introduces explicit functors for transgression and regression, linking gerbes and fusion bundles, advancing the understanding of their geometric and topological relationships.

Findings

Equivalence between abelian gerbes with connection and fusion bundles over loop space.

Explicit construction of transgression and regression functors.

Applications to lifting problems and loop group extensions.

Abstract

We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with a connection and with a "fusion" product with respect to triples of paths. The equivalence is established by explicit functors in both directions: transgression and regression. We describe applications to geometric lifting problems and loop group extensions.