We determine the nilpotent BRST and anti-BRST transformations for the Cho-Faddeev-Niemi variables for the SU(2) Yang-Mills theory based on the new interpretation given in the previous paper of the Cho-Faddeev-Niemi decomposition. This gives a firm ground for performing the BRST quantization of the Yang-Mills theory written in terms of the Cho-Faddeev-Niemi variables. We propose also a modified version of the new Maximal Abelian gauge which could play an important role in the reduction to the original Yang-Mills theory.

We give a new way of looking at the Cho-Faddeev-Niemi (CFN) decomposition of the Yang-Mills theory to answer how the enlarged local gauge symmetry respected by the CFN variables is restricted to obtain another Yang-Mills theory with the same local and global gauge symmetries as the original Yang-Mills theory. This may shed new light on the fundamental issue of the discrepancy between two theories for independent degrees of freedom and the role of the Maximal Abelian gauge in Yang-Mills theory. As a byproduct, this consideration gives new insight into the meaning of the gauge invariance and the observables, e.g., a gauge-invariant mass term and vacuum condensates of mass dimension two. We point out the implications for the Skyrme-Faddeev model.

In this paper we begin on a new lattice formulation of the non-linear change of variables called the Cho-Faddeev-Niemi decomposition in SU(2) Yang-Mills theory. This is a compact lattice formulation improving the non-compact lattice formulation proposed in our previous paper. Based on this formulation, we propose a new gauge-invariant definition of the magnetic monopole current which guarantees the magnetic charge quantization and reproduces the conventional magnetic-current density obtained in the Abelian projection based on the DeGrand-Toussaint method. Finally, we demonstrate the magnetic monopole dominance in the string tension in SU(2) Yang-Mills theory on a lattice. Our formulation enables one to reproduce in the gauge-invariant way remarkable results obtained so far only in the Maximally Abelian gauge.

We present the first implementation of the Cho-Faddeev-Niemi decomposition of the SU(2) Yang-Mills field on a lattice. Our construction retains the color symmetry (global SU(2) gauge invariance) even after a new type of Maximally Abelian gauge, as explicitly demonstrated by numerical simulations. Moreover, we propose a gauge-invariant definition of the magnetic monopole current using this formulation and compare the new definition with the conventional one by DeGrand and Toussaint to exhibit its validity.

We propose a new version of SU (N ) Yang-Mills theory reformulated in terms of new field variables which are obtained by a nonlinear change of variables from the original Yang-Mills gauge field. The reformulated Yang-Mills theory enables us to study the low-energy dynamics by explicitly extracting the topological degrees of freedom such as magnetic monopoles and vortices to clarify the mechanism for quark confinement. The dual superconductivity in Yang-Mills theory is understood in a gauge-invariant manner, as demonstrated recently by a non-Abelian Stokes theorem for the Wilson loop operator, although the basic idea of this reformulation is based on the Cho-Faddeev-Niemi decomposition of the gauge potential.