Kondo and Taira in their construction of a non-Abelian version of Stokes’ theorem found that the monopole contribution to the Wilson loop depends on which representation of the gauge group the color charge belongs to. In the SU(3) gauge group for example, discussion of the fundamental representation concerns only the monopoles corresponding to the reduction from SU(3) down to U(2) symmetry, specified by the homotopy group. While for color charges in the adjoint representation we need to consider the corresponding U(1) x U(1) fundamental group. For general SU(N), the Wilson loop for gluons, which belong to the adjoint representation, depends on the full set of monopoles corresponding to the homotopy group. This is in contrast to that of quarks in the fundamental representation, which receives monopole contributions only from those corresponding to

π2[SU(N)/U(N − 1)] = π1[U(N − 1)] = π1[UN−1(1)] = ZN−1.

Article Stability of the Magnetic Monopole Condensate in three- and ...