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

Abstract

We derive a new version of the 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 that 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. Owing to dimensional reduction, the planar Wilson loop at least for 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 $C{P}^{N-1}$ model. This derivation shows that the fundamental quarks are confined by a single species of magnetic monopole. The origin of the area law is related to the geometric phase of the Wilczek-Zee holonomy for $U(N-1)$. The calculations are performed using the instanton calculus (in the dilute instanton-gas approximation) and using the large $N$ expansion (in the leading order).

77 pages, Latex, 13 eps figures, a version to appear in Prog. Theor. Phys

References in corpus (2)

• Instantons and Gribov Copies in the Maximally Abelian Gauge
• Renormalizable Abelian-projected effective gauge theory derived from Quantum Chromodynamics

Cited by in corpus (30)

• Quark confinement: dual superconductor picture based on a non-Abelian Stokes theorem and reformulations of Yang-Mills theory
• Gauge-invariant gluon mass, infrared Abelian dominance and stability of magnetic vacuum
• Magnetic condensation, Abelian dominance and instability of Savvidy vacuum
• On ghost condensation, mass generation and Abelian dominance in the Maximal Abelian Gauge
• Wilson loop and magnetic monopole through a non-Abelian Stokes theorem
• New descriptions of lattice SU(N) Yang-Mills theory towards quark confinement
• Heterotic strings on homogeneous spaces
• Non-Abelian Dual Superconductor Picture for Quark Confinement
• 4d ensembles of percolating center vortices and monopole defects: the emergence of flux tubes with N-ality and gluon confinement
• Gauge-independent "Abelian" and magnetic-monopole dominance, and the dual Meissner effect in lattice $SU(2)$ Yang-Mills theory
• Magnetic monopoles and center vortices as gauge-invariant topological defects simultaneously responsible for confinement
• Non-Abelian dual superconductivity in SU(3) Yang-Mills theory: dual Meissner effect and type of the vacuum
• Different faces of confinement
• String Theory: exact solutions, marginal deformations and hyperbolic spaces
• Berry phase in the phase space worldline representation: the axial anomaly and classical kinetic theory
• 3D Yang-Mills confining properties from a non-Abelian ensemble perspective
• Schwinger Pair Production in SL$(2,\mathbb{C})$ Topologically Non-Trivial Fields via Non-Abelian Worldline Instantons
• Magnetic Monopole Loops supported by a Meron pair as the Quark Confiner
• Non-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representation and its implication to quark confinement
• On the non-Abelian Stokes theorem for SU(2) gauge fields
• Stability of the Magnetic Monopole Condensate in three- and four-colour QCD
• Higgs-free confinement hierarchy in five colour QCD
• Abelian representation for nonabelian Wilson loops and the Non - Abelian Stokes theorem on the lattice
• Abelian Fock-Schwinger representation for the quark propagator in external gauge field
• Gauge-covariant decomposition and magnetic monopole for G(2) Yang-Mills field
• Effective Lagrangian for Non-Abelian Two-Dimensional Topological Field Theory
• How to extract the dominant part of the Wilson loop average in higher representations
• Gluonic Colour Singlets in QCD
• Topological phase states of the SU(3) QCD
• S-duality and loop operators in canonical formalism