An Invitation to Higher Gauge Theory
John C. Baez, John Huerta
TL;DR
An invitation to higher gauge theory presents a structured introduction to generalizing parallel transport from points to strings via 2-connections on 2-bundles, encoded by Lie 2-groups and 2-functors on path 2-groupoids. It develops the necessary higher-categorical framework (2-categories, path 2-groupoids, crossed modules) and derives the 2-connection data $(A,B)$ constrained by $\underline{t}(B)=dA+A\wedge A$, capturing holonomies along both paths and surfaces. The paper then surveys concrete examples (shifted abelian groups, Poincaré 2-group, tangent and inner automorphism 2-groups, automorphism 2-groups, and string 2-groups) and discusses how these inform BF theory, gravity, and string theory, including potential links to gravity as a higher gauge theory via gravity 3-groups. It concludes with practical aspects such as gauge transformations, curvature notions (fake curvature and 2-curvature), and the construction of nontrivial 2-bundles, outlining a broad roadmap for applying higher gauge theory to fundamental physics and geometry.
Abstract
In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a 'tangent 2-group', which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an 'inner automorphism 2-group', which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an 'automorphism 2-group', which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a 'string 2-group'. We also touch upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra that governs 11-dimensional supergravity.