[Paper Review] A note on the holonomy of connections in twisted bundles

Marco Mackaay|ArXiv.org|Jun 4, 2001

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper establishes a bijective correspondence between equivalence classes of twisted principal bundles with connection on a smooth manifold M and smooth functors from the thin fundamental categorical group $ C_2^2(M) $ to a categorical Lie group $ /mathcal{G} $. It generalizes Barrett's holonomy principle to higher gauge theory by showing that holonomy for twisted bundles is naturally encoded as functors between categorical groups, with connections reconstructed from such functors via a gauge-fixing procedure based on thin homotopy classes and path choices.

ABSTRACT

Recently twisted K-theory has received much attention due to its applications in string theory and the announced result by Freed, Hopkins and Telemann relating the twisted equivariant K-theory of a compact Lie group to its Verlinde algebra. Rather than considering gerbes as separate objects, in twisted K-theory one considers a gerbe as being part of the data for a twisted vector bundle. There is also a notion of a connection in a twisted vector bundle and Kapustin has studied some aspects of the holonomy of such connections. In this note I study the holonomy of connections in twisted principal bundles and show that it can best be defined as a functor rather than a map. Even for the case which Kapustin studied the results in this paper give a more general picture.

Motivation & Objective

• To generalize Barrett’s holonomy principle for ordinary connections to twisted principal bundles with connection.
• To identify the geometric structures on a manifold M that yield holonomy functors with values in categorical Lie groups.
• To provide a reconstruction theorem showing that such functors classify twisted principal bundles with connection up to equivalence.
• To extend the framework of higher holonomy beyond gerbes to arbitrary transitive categorical Lie groups.

Proposed method

• Define the thin fundamental categorical group $ C_2^2(M) $ using thin homotopy classes of loops and cylinders.
• Use categorical groups as target structures, with objects as thin homotopy classes and morphisms as equivalence classes of thin homotopies between cylinders.
• Construct a holonomy functor $ \mathcal{H} $ from $ C_2^2(M) $ to a categorical Lie group $ \mathcal{G} $, modeled on central extensions $ 1 \to H \to E \to G \to 1 $.
• Reconstruct local transition functions $ e_{ij} $ and connections $ A_i $ from the holonomy functor using fixed paths and homotopies in local trivializations.
• Apply gauge-fixing conventions to ensure consistency of transition functions and connection 1-forms.
• Prove that the reconstructed bundle with connection is equivalent to the original one, establishing a bijective correspondence.

Experimental results

Research questions

• RQ1 What geometric structure on a manifold M gives rise to a holonomy functor with values in a categorical Lie group?
• RQ2 How can the holonomy of a twisted principal bundle with connection be naturally formulated in terms of functors between categorical groups?
• RQ3 Can the holonomy functor fully classify twisted principal bundles with connection up to equivalence?
• RQ4 What is the relationship between the thin fundamental categorical group $ C_2^2(M) $ and the holonomy of twisted bundles?
• RQ5 How can one reconstruct the bundle and connection data from a given holonomy functor?

Key findings

• There is a bijective correspondence between equivalence classes of twisted principal $ E $-bundles with connection on $ M $ and equivalence classes of smooth functors from $ C_2^2(M) $ to the categorical group $ \mathcal{G} $ derived from a central extension $ 1 \to H \to E \to G \to 1 $.
• The transition functions $ e_{ij}(y) $ are reconstructed from the holonomy functor via representatives of $ \mathcal{H}(c_{ij}(y)) $, with gauge-fixing ensuring consistency.
• The connection 1-forms $ A_i(v) $ are derived from the derivative of the holonomy along paths, ensuring compatibility with the curvature and transition laws.
• The 2-cocycle $ h_{ijk}(y) = e_{ij}(y)e_{jk}(y)e_{ki}(y) $ lies in the kernel of $ \pi $, identifying it with $ H $, and matches the standard cocycle in higher gauge theory.
• The reconstruction procedure ensures that the resulting bundle with connection is equivalent to the original one, proving full classification.
• Two holonomy functors are equivalent if and only if they are conjugate by an element of $ G $, generalizing the standard gauge equivalence in principal bundles.