Non-Abelian Stokes Theorem and Quark Confinement in SU(3) Yang-Mills Gauge Theory

TL;DR

This paper introduces a new SU(3) non-Abelian Stokes theorem using coherent states on the flag space, linking it to quark confinement and magnetic monopoles in Yang-Mills theory.

Contribution

It develops a novel SU(3) non-Abelian Stokes theorem and connects it to quark confinement mechanisms via topological and geometric phases.

Findings

Fundamental quark confinement persists even with partial gauge fixing.

The area law relates to the Wilczek-Zee holonomy's geometric phase.

Magnetic monopoles are central to the confinement mechanism.

Abstract

We derive a new version of SU(3) non-Abelian Stokes theorem by making use of the coherent state representation on the coset space $SU(3)/(U(1)\times U(1))=F_2$, the flag space. Then we outline a derivation of the area law of the Wilson loop in SU(3) Yang-Mills theory in the maximal Abelian gauge (The detailed exposition will be given in a forthcoming article). This derivation is performed by combining the non-Abelian Stokes theorem with the reformulation of the Yang-Mills theory as a perturbative deformation of a topological field theory recently proposed by one of the authors. Within this framework, we show that the fundamental quark is confined even if $G=SU(3)$ is broken by partial gauge fixing into $H=U(2)$ just as $G$ is broken to $H=U(1) \times U(1)$. An origin of the area law is related to the geometric phase of the Wilczek-Zee holonomy for U(2). Abelian dominance is an immediate byproduct of these results and magnetic monopole plays the dominant role in this derivation.