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

We derive a new version of non-Abelian Stokes theorem for the Wilson loop in the SU(N) case by making use of the coherent state representation on the coset space SU(N)/U(1)^{N-1}=F_{N-1}, the flag space. We consider the SU(N) Yang-Mills theory in the maximal Abelian gauge in which SU(N) is broken down to U(1)^{N-1}. First, we show that the Abelian dominance in the string tension follows from this theorem and the Abelian-projected effective gauge theory which was derived by one of the authors. Next (but independently), combining the non-Abelian Stokes theorem with a novel reformulation of the Yang-Mills theory recently proposed by one of the authors, we proceed to derive the area law of the Wilson loop in four-dimensional SU(N) Yang-Mills theory in the maximal Abelian gauge. Due to dimensional reduction, the planar Wilson loop in the fundamental representation in four-dimensional SU(N) Yang-Mills theory can be estimated by the diagonal (Abelian) Wilson loop defined in the two-dimensional CP^{N-1} model. This derivation shows that the fundamental quarks are confined by a single species of magnetic monopole. The non-Abelian Stokes theorem extracts the gauge-invariant magnetic monopole contribution and gives the SU(N) generalization of the 't Hooft-Polyakov tensor for the magnetic monopole. An origin of the area law is related to the geometric phase of the Wilczek-Zee holonomy for U(N-1). The calculations are performed by the instanton calculus (in the dilute instanton-gas approximation) or by the large N expansion (in the leading order)