Non-abelian higher gauge theory and categorical bundle

TL;DR

This paper introduces a new categorical bundle framework for non-abelian higher gauge theory, expanding the base space to include more complex categories and exploring its geometric and connective structures with applications to quantum dynamics.

Contribution

It proposes a novel principal categorical bundle structure with affine 2-spaces as base, extending previous models to handle non-trivial categories and non-abelian cases.

Findings

Developed the concept of affine 2-spaces as base categories.

Analyzed the geometric structure of the new categorical bundles.

Applied the framework to quantum dynamics via Bloch wave operator theory.

Abstract

A gauge theory is associated with a principal bundle endowed with a connection permitting to define horizontal lifts of paths. The horizontal lifts of surfaces cannot be defined into a principal bundle structure. An higher gauge theory is an attempt to generalize the bundle structure in order to describe horizontal lifts of surfaces. A such attempt is particularly difficult for the non-abelian case. Some structures have been proposed to realize this goal (twisted bundle, gerbes with connection, bundle gerbe, 2-bundle). Each of them uses a category in place of the total space manifold of the usual principal bundle structure. Some of them replace also the structure group by a category (more precisely a Lie crossed module viewed as a category). But the base space remains still a simple manifold (possibly viewed as a trivial category with only identity arrows). We propose a new principal categorical bundle structure, with a Lie crossed module as structure groupoid, but with a base space belonging to a bigger class of categories (which includes non-trivial categories), that we call affine 2-spaces. We study the geometric structure of the categorical bundles built on these categories (which is a more complicated structure than the 2-bundles) and the connective structures on these bundles. Finally we treat an example interesting for quantum dynamics which is associated with the Bloch wave operator theory.