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In the $SU(2)$ Yang-Mills theory on the four-dimensional Euclidean lattice,we confirm the gauge-independent "Abelian" dominance (or the restricted fielddominance) and gauge-independent magnetic-monopole dominance in the stringtension of the linear potential extracted from the Wilson loop in thefundamental representation. The dual Meissner effect is observed bydemonstrating the squeezing of the chromoelectric field flux connecting a pairof quark and antiquark. In addition, the circular magnetic-monopole current isinduced around the chromoelectric flux. The type of the dual superconductivityis also determined by fitting the result with the dual Ginzburg-Landau model.Thus the dual superconductor picture for quark confinement is supported in agauge-independent manner. These results are obtained based on a reformulationof the lattice Yang-Mills theory based on the change of variables a laCho-Duan-Ge-Faddeev-Niemi combined with a non-Abelian Stokes theorem for theWilson loop operator. We give a new procedure (called the reduction) forobtaining the color direction field which plays the central role in thisreformulation.

Gauge-independent "Abelian" and magnetic-monopole dominance, and the dual Meissner effect in lattice $SU(2)$ Yang-Mills theory