Magnetic monopole loops supported by a meron pair as the quark confiner
  We give a definition of gauge-invariant magnetic monopoles in Yang-Mills theory without using the Abelian projection due to 't Hooft. They automatically appear from the Wilson loop operator. This is shown by rewriting the Wilson loop operator using a non-Abelian Stokes theorem. The magnetic monopole defined in this way is a topological object of co-dimension 3, i.e., a loop in four-dimensions. We show that such magnetic loops indeed exist in four-dimensional Yang-Mills theory. In fact, we give an analytical solution representing circular magnetic monopole loops joining a pair of merons in the four-dimensional Euclidean SU(2) Yang-Mills theory. This is achieved by solving the differential equation for the adjoint color (magnetic monopole) field in the two--meron background field within the recently developed reformulation of the Yang-Mills theory. Our analytical solution corresponds to the numerical solution found by Montero and Negele on a lattice. This result strongly suggests that a meron pair is the most relevant quark confiner in the original Yang-Mills theory, as Callan, Dashen and Gross suggested long ago. 
Kei-Ichi Kondo