Gauge independence of Abelian and monopole dominances in QCD

It is believed that quantum chromodynamics (QCD) is the theory describing strong interaction. To understand confinement mechanism in QCD is one of the most important and interesting problem. The attractive scenario was proposed by 't Hooft.(1)) QCD, after abelian projection, can be regarded as an abelian theory with electric charges and magnetic monopoles. The confinement of quarks can be explained as the dual Meissner effect which is due to the condensation of these monopoles. According to this scenario, the QCD vacuum is dual to the ordinary superconductor and the monopoles play the role of the Cooper pairs. However abelian monopole condensation after abelian projection in some gauges can not explain color confinement in QCD. Abelian neutral but color non-singlet states are not confined. Gauge independence of monopole condensation is necessary for color confinement. We have shown abelian and monopole dominances numerically in maximally abelian gauge. Here the monopole dominance as well as the abelian dominance are proved generally after abelian projection even without any gauge fixing. And it is also possible to perform the numerical check of this result by using the lattice Monte Carlo simulation.