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.
- Pub Date:
- March 2010
- Mathematics - Differential Geometry;
- Mathematical Physics;
- Mathematics - Algebraic Topology;
- 80 pages. v2: more corollaries of the main result, exposition improved, terminology changed slightly. v3: proof of the main theorem restructured. v4: some minor corrections included