A Global Geometric Approach to Parallel Transport of Strings in Gauge Theory

TL;DR

This paper develops a global geometric framework for higher parallel transport in gauge theory, deriving a 3D non-abelian Stokes' theorem and extending it to 4D volume holonomy, emphasizing gauge invariance.

Contribution

It introduces a classical principal bundle approach to higher gauge theory, providing a geometric proof of a 3D non-abelian Stokes' theorem and extending it to 4D volume holonomy.

Findings

Derived a global geometric proof of a 3D non-abelian Stokes' theorem.

Extended the theorem to 4D, establishing a formula for volume holonomy.

Ensured gauge invariance of the volume holonomy.

Abstract

This paper is motivated by recent developments of higher gauge theory. Different from its style of using higher category theory, we try to describe the concept of higher parallel transport within setting of classical principal bundle theory. From this perspective, we obtain a global geometric proof on a generalized 3-dimensional non-abelian Stokes' theorem related to parallel transport on surfaces. It can be naturally extended to four dimension when the underlying crossed module is replaced by crossed 2-module. This 4-dimensional Stokes' theorem yields a global formula for volume holonomy, and also guarantees its gauge invariance. In the process, we also find composition formulas for parallel transport on volumes which have been seen in the case of surfaces.