On Non-Abelian Holonomies

TL;DR

This paper presents a method for calculating the holonomy of Yang-Mills connections along triangular paths, expanding to fifth order, with potential applications in quantum gravity and lattice field theories.

Contribution

It introduces a systematic expansion method for Yang-Mills holonomies along arbitrary paths, extending previous calculations to higher order terms.

Findings

Derived a fifth-order expansion for Yang-Mills holonomies.

Provided explicit formulas for holonomy calculations in triangular loops.

Suggested applications in quantum gravity and lattice gauge theories.

Abstract

We provide a method and the results for the calculation of the holonomy of a Yang-Mills connection in an arbitrary triangular path, in an expansion (developed here to fifth order) in powers of the corresponding segments. The results might have applications in generalizing to Yang-Mills fields previous calculations of the corrections to particle dynamics induced by loop quantum gravity, as well as in the field of random lattices.