  We give new descriptions of lattice SU(N) Yang-Mills theory in terms of new
lattice variables. The validity of such descriptions has already been
demonstrated in the SU(2) Yang-Mills theory by our previous works from the
viewpoint of defining and extracting topological degrees of freedom such as
gauge-invariant magnetic monopoles and vortices which play the dominant role in
quark confinement. In particular, we have found that the SU(3) lattice
Yang-Mills theory has two possible options, maximal and minimal: The existence
of the minimal option has been overlooked so far, while the maximal option
reproduces the conventional SU(3) Cho-Faddeev-Niemi-Shabanov decomposition in
the naive continuum limit. The new description gives an important framework for
understanding the mechanism of quark confinement based on the dual
superconductivity.
