An Invitation to Higher Gauge Theory

Higher gauge theory generalizes traditional gauge theory by extending the concept of parallel transport from point particles to higher-dimensional objects like strings. While standard gauge theory relies on Lie groups to describe the symmetries of particles, higher gauge theory utilizes Lie 2-groups to describe the symmetries of strings. This framework provides a unified language for understanding structures that appear in string theory, loop quantum gravity, and multisymplectic geometry.

The authors build the theory by first reinterpreting traditional parallel transport. A connection on a principal bundle is shown to be equivalent to a smooth functor from the path groupoid of a manifold to a Lie group. By boosting the dimension, the authors define a 2-connection as a smooth 2-functor from a path 2-groupoid to a Lie 2-group. This 2-functor assigns holonomies not only to paths but also to surfaces, providing a rigorous way to track how strings transform as they sweep out worldsheets.

A central contribution of the paper is the classification and application of various Lie 2-groups. The authors detail six primary examples, including:

• Shifted abelian groups, which relate to U(1) gerbes and string theory.
• The Poincaré 2-group, which offers a potential path toward spin foam models for 4D Minkowski spacetime.
• Tangent 2-groups, which serve as the gauge structure for 4D BF theory and topological gravity.
• Inner automorphism 2-groups, relevant for BF theory with a cosmological constant.
• Automorphism 2-groups, which are central to the theory of nonabelian gerbes.
• String 2-groups, which are essential for defining consistent string propagation on spin manifolds.

This work is significant because it bridges the gap between abstract higher category theory and concrete problems in theoretical physics. By providing a clear, step-by-step introduction to 2-connections and 2-bundles, the authors make advanced mathematical tools—such as gerbes, crossed modules, and L-infinity algebras—accessible to physicists. This framework helps clarify the geometric foundations of supergravity and string theory, offering a systematic way to handle the higher-dimensional symmetries that these theories require.