Non-Abelian Stokes Theorem and Quark Confinement in Su (3) Yang–mills Gauge Theory

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) × U (1)) = F2, 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) × 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 by-product of these results and magnetic monopole plays the dominant role in this derivation.