Higher Gauge Theory
John C. Baez, Urs Schreiber
TL;DR
Addresses extending gauge theory to the parallel transport of 1D objects using Lie 2-groups and principal 2-bundles, with 2-connections delivering path- and surface-holonomies via a smooth 2-functor. The core development is internalization to a higher-categorical setting, yielding the path 2-groupoid $\mathcal{P}_2(M)$ and differential data $(A,B)$ constrained by $F_A+dt(B)=0$ (the fake curvature vanishes), tying to Breen and Messing's nonabelian gerbes. The work shows how such 2-connections subsume abelian and nonabelian gerbe connections and clarifies the role of the fake-curvature constraint in enabling well-defined 2-holonomies. This provides a foundational framework for higher gauge theory with potential implications for string theory and higher-dimensional parallel transport.
Abstract
Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2-groups and Lie 2-algebras. We describe a theory of 2-connections on principal 2-bundles and explain how this is related to Breen and Messing's theory of connections on nonabelian gerbes. The distinctive feature of our theory is that a 2-connection allows parallel transport along paths and surfaces in a parametrization-independent way. In terms of Breen and Messing's framework, this requires that the "fake curvature" must vanish. In this paper we summarize the main results of our theory without proofs.