A Cubical Set Approach to 2-Bundles with Connection and Wilson Surfaces

In the context of non-abelian gerbes we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We define their two-dimensional parallel transport, study its properties, and define non-abelian Wilson surface functionals.

💡 Research Summary

The paper introduces a cubical‑set framework for describing non‑abelian 2‑bundles with connection, a structure that underlies higher gauge theories such as non‑abelian gerbes. Traditional approaches to higher bundles often rely on simplicial constructions, which can obscure the geometric meaning of transition data and curvature. By replacing the simplicial indexing with cubical cells, the authors obtain a natural bookkeeping of 1‑dimensional and 2‑dimensional holonomy data.

The authors begin by recalling the algebraic model of a categorical group 𝔾 given by a crossed module (H → G). Here G is a Lie group that will govern the ordinary 1‑form connection A, while H is a second Lie group (often a central extension) that will host the 2‑form connection B. A cubical set is a collection of n‑cubes together with face and degeneracy maps satisfying the usual cubical identities. By assigning to each 0‑cube a copy of the base manifold, to each 1‑cube a G‑valued transition function, and to each 2‑cube an H‑valued transition function, the authors encode the full Čech‑type data of a 2‑bundle in a way that mirrors the geometry of squares and cubes.

Local connection data consist of a 1‑form Aα ∈ Ω¹(Uα,𝔤) and a 2‑form Bα ∈ Ω²(Uα,𝔥) on each chart Uα. The compatibility conditions across overlaps are expressed in terms of the cubical transition functions gαβ : Uα∩Uβ → G and hαβγ : Uα∩Uβ∩Uγ → H. Explicitly,  Aβ = gαβ⁻¹ Aα gαβ + gαβ⁻¹ dgαβ,  Bβ = gαβ⁻¹·Bα·gαβ + d_Aα hαβγ + …, where “·” denotes the action of G on H and d_Aα is the covariant exterior derivative with respect to Aα. These equations are the higher‑dimensional analogues of the usual gauge transformation law for a connection, and they satisfy higher Čech cocycle conditions that guarantee global consistency.

Two curvature forms arise naturally. The first curvature,  Fα = dAα + ½