Higher Gauge Theory: 2-Connections on 2-Bundles

Connections and curvings on gerbes are beginning to play a vital role in differential geometry and mathematical physics — first abelian gerbes, and more recently nonabelian gerbes. These concepts can be elegantly understood using the concept of ‘2bundle’ recently introduced by Bartels. A 2-bundle is a generalization of a bundle in which the fibers are categories rather than sets. Here we introduce the concept of a ‘2-connection’ on a principal 2-bundle. We describe principal 2-bundles with connection in terms of local data, and show that under certain conditions this reduces to the cocycle data for nonabelian gerbes with connection and curving subject to a certain constraint — namely, the vanishing of the ‘fake curvature’, as defined by Breen and Messing. This constraint also turns out to guarantee the existence of ‘2-holonomies’: that is, parallel transport over both curves and surfaces, fitting together to define a 2-functor from the ‘path 2-groupoid’ of the base space to the structure 2-group. We give a general theory of 2-holonomies and show how they are related to ordinary parallel transport on the path space of the base manifold.