Wilson loop and magnetic monopole through a non-Abelian Stokes theorem

TL;DR

This paper presents an exact reformulation of the Wilson loop in SU(N) Yang-Mills theory using a non-Abelian Stokes theorem, revealing a gauge-invariant magnetic monopole structure that offers insights into quark confinement.

Contribution

It introduces a gauge-invariant magnetic monopole derived directly from pure Yang-Mills theory and relates the Wilson loop to magnetic currents, challenging traditional views.

Findings

Wilson loop expressed via gauge-invariant magnetic currents

Magnetic charge quantization from surface independence

Wilson loop as a probe for magnetic monopoles

Abstract

We show that the Wilson loop operator for SU(N) Yang-Mills gauge connection is exactly rewritten in terms of conserved gauge-invariant magnetic and electric currents through a non-Abelian Stokes theorem of the Diakonov-Petrov type. Here the magnetic current originates from the magnetic monopole derived in the gauge-invariant way from the pure Yang--Mills theory even in the absence of the Higgs scalar field, in sharp contrast to the 't Hooft-Polyakov magnetic monopole in the Georgi-Glashow gauge-Higgs model. The resulting representation indicates that the Wilson loop operator in fundamental representations can be a probe for a single magnetic monopole irrespective of $N$ in SU(N) Yang-Mills theory, against the conventional wisdom. Moreover, we show that the quantization condition for the magnetic charge follows from the fact that the non-Abelian Stokes theorem does not depend on the surface chosen for writing the surface integral. The obtained geometrical and topological representations of the Wilson loop operator have important implications to understanding quark confinement according to the dual superconductor picture.