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We show that the Wilson loop operator for SU(N) Yang-Mills gauge connection is exactly rewritten in terms of conserved gauge-invariant magnetic and electric currents through a non-Abelian Stokes theorem of the Diakonov-Petrov type. Here the magnetic current originates from the magnetic monopole derived in the gauge-invariant way from the pure Yang--Mills theory even in the absence of the Higgs scalar field, in sharp contrast to the 't Hooft-Polyakov magnetic monopole in the Georgi-Glashow gauge-Higgs model. The resulting representation indicates that the Wilson loop operator in fundamental representations can be a probe for a single magnetic monopole irrespective of $N$ in SU(N) Yang-Mills theory, against the conventional wisdom. Moreover, we show that the quantization condition for the magnetic charge follows from the fact that the non-Abelian Stokes theorem does not depend on the surface chosen for writing the surface integral. The obtained geometrical and topological representations of the Wilson loop operator have important implications to understanding quark confinement according to the dual superconductor picture.Comment: 48 pages, 8 figures; minor changes, version to be published in Physical Review
Analytic dyon solutions are obtained for SU(iV) grand unified theories broken down to SUC/V^® SUUV 2 )<8) U(l). These solutions are regular everywhere including the origin. A few comments about mass, Pontryagin-like index of dyon, and magnetic flux are made in connection with the recent report of a magnetic monopole discovery.Since the theoretical discovery of the finite-energy magnetic monopole solutions for the SU(2) gauge group, 1 " 4 the solutions of monopoles in the non-Abelian gauge theories have been much studied. 5 These monopoles with finite mass are inevitably predicted in the grand unified theories (GUT's) which unify the strong theory with the electroweak interaction. 6 Also there has been a report 7 of experimental detection of a monopole in a superconductive loop.Recently, we have obtained pointlike monopole solutions in the SU(iN0 gauge group which are regular everywhere except at the origin. 8 To obtain analytic finite-energy monopole solutions, however, we need to eliminate the remaining singularities at the origin.In this paper, we present the analytic solution in the general SU(A0 grand unification groups which are broken down to SUGVJ
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