See also the Topology seminar page.
This spring we plan to go through parts of J.-L. Brylinski's book "Loop spaces, characteristic classes and geometric quantization", aiming to see how to extend the differential geometry of gerbes to 2-vector bundles of higher rank. For U(1)-gerbes, M. K. Murray's paper "Bundle gerbes" provides a more concrete approach. It is extended to non-abelian gerbes in P. Ascieri, L. Cantini and B. Jurco's paper "Nonabelian bundle gerbes, their differential geometry and gauge theory". The 2-vector bundles of N. A. Baas, B. I. Dundas and J. Rognes' paper "Two-vector bundles and forms of elliptic cohomology" generalize U(1)-gerbes in a different direction. We will to outline the relation between these notions, suggest how to define a connection in a 2-vector bundle, and explain how parallel transport with respect to such a connection may give rise to a conformal field theory.
Here are the rough seminar notes.
(Brylinski, 5.1)
(Brylinski, 5.2)
(Brylinski, 5.3)
We associate a sheaf of groupoids (= a stack) to each 2-vector bundle, show that a connective structure on a 2-vector bundle gives rise to a connective structure on the associated stack, and discuss the existence of such connective structures.
(Brylinski, 6.2)