Higher Gauge Theory: 2-Connections on 2-Bundles
John Baez, Urs Schreiber
TL;DR
The paper advances a categorified gauge theory by defining principal 2-bundles with 2-connections and showing that, under the vanishing fake curvature constraint $F_A+dt(B)=0$, the local data of these 2-bundles reproduce the cocycle data of nonabelian gerbes with connection and curving. It constructs 2-holonomies as 2-functors ${ m hol}: ext{P}_2(M) o rak{G}$ from path- and surface-transport data encoded in a $ rak{g}$-valued 1-form $A$ and an $ rak{h}$-valued 2-form $B$, and connects these to the path-space holonomy formalism via Chen-type integrals. By internalizing the transition laws, the authors globalize the local 2-holonomy and show a tight link between 2-bundles with 2-connections and (twisted) nonabelian gerbes, including explicit curvature and coherence relations. The framework has potential implications for higher-dimensional gauge theories and M-theory, where nonabelian 2-form fields and membranes interact with brane stacks, and motivates future work on coherent 2-groups and more general base 2-spaces.
Abstract
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 `2-bundle' 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.