Transgression to loop spaces and its inverse, II: Gerbes and fusion bundles with connection

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.

Keywords

connections on gerbes, diffeological spaces, gerbes, surface holonomy, loop space, orientability of loop spaces, spin structures, transgression

2010 Mathematics Subject Classification

Primary 53C08. Secondary 53C27, 55P35.

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Published 28 January 2016