On two-Dimensional Holonomy

We define the thin fundamental categorical group ${\mathcal P}_2(M,)$ of a based smooth manifold $(M,)$ as the categorical group whose objects are rank-1 homotopy classes of based loops on $M$, and whose morphisms are rank-2 homotopy classes of homotopies between based loops on $M$. Here two maps are rank-$n$ homotopic, when the rank of the differential of the homotopy between them equals $n$. Let $\C(\Gc)$ be a Lie categorical group coming from a Lie crossed module ${\Gc= (\d\colon E \to G,\tr)}$. We construct categorical holonomies, defined to be smooth morphisms ${\mathcal P}_2(M,*) \to \C(\Gc)$, by using a notion of categorical connections, being a pair $(\w,m)$, where $\w$ is a connection 1-form on $P$, a principal $G$ bundle over $M$, and $m$ is a 2-form on $P$ with values in the Lie algebra of $E$, with the pair $(\w,m)$ satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.

💡 Research Summary

The paper “On two‑Dimensional Holonomy” develops a rigorous framework for extending the classical notion of holonomy from one‑dimensional loops to two‑dimensional surfaces. The authors begin by introducing the thin fundamental categorical group 𝒫₂(M, *) associated with a based smooth manifold (M, *). In this construction, objects are rank‑1 homotopy classes of based loops—i.e., equivalence classes under “thin” homotopies whose differential has rank ≤ 1 at every point. Morphisms are rank‑2 homotopy classes of homotopies between loops, again requiring the differential of the homotopy to have rank ≤ 2 everywhere. This “thin” condition filters out deformations that change the essential differential structure, thereby preserving the fine geometric information needed for a higher‑dimensional holonomy theory.

Next, the authors recall the algebraic model of a Lie 2‑group given by a Lie crossed module 𝔊 = (∂ : E → G, ⊲). Here G is a Lie group (the “object” group) and E a Lie group (the “arrow” group), with ∂ a smooth homomorphism and ⊲ a smooth left action of G on E satisfying the usual Peiffer identities. From 𝔊 one obtains a Lie categorical group 𝒞(𝔊) whose objects are elements of G and whose morphisms are pairs (e, g) with e ∈ E, g ∈ G; composition is dictated by the crossed‑module structure.

The central geometric ingredient is a categorical connection (ω, m) on a principal G‑bundle P → M. The 1‑form ω is an ordinary connection on P, while the 2‑form m takes values in the Lie algebra 𝔢 of E. The pair must satisfy three compatibility conditions:

1. G‑equivariance of ω (the usual transformation law under the right G‑action).
2. Fake‑flatness: d m +