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Abelian Magnetic Monopole Dominance in Quark Confinement

We prove Abelian magnetic monopole dominance in the string tension of QCD. Abelian and monopole dominance in low energy physics of QCD has been confirmed for various quantities by recent Monte Carlo simulations of lattice gauge theory. In order to prove this dominance, we use the reformulation of continuum Yang-Mills theory in the maximal Abelian gauge as a deformation of a topological field theory of magnetic monopoles, which was proposed in the previous article by the author. This reformulation provides an efficient way for incorporating the magnetic monopole configuration as a topological non-trivial configuration in the functional integral. We derive a version of the non-Abelian Stokes theorem and use it to estimate the expectation value of the Wilson loop. This clearly exhibits the role played by the magnetic monopole as an origin of the Berry phase in the calculation of the Wilson loop in the manifestly gauge invariant manner. We show that the string tension derived from the diagonal (abelian) Wilson loop in the topological field theory (studied in the previous article) converges to that of the full non-Abelian Wilson loop in the limit of large Wilson loop. Therefore, within the above reformulation of QCD, this result (together with the previous result) completes the proof of quark confinement in QCD based on the criterion of the area law of the full non-Abelian Wilson loop.