  We present a bosonization method to study generic low energy behavior of
gauge systems with finite chemical potential in 2+1 dimensions. Benefit from
the existence of gap (e.g. Gribov gap) in gauge systems at low energy, the
fermion fields can be explicitly bosonized by new gauge fields. When chemical
potential of the gauge systems is introduced, we find that topological terms
(such as Chern-Simons term in 2+1D) as constraints inevitably emerge at low
energy. The fermion sign problem at finite chemical potential and its deep
connection to the Chern-Simons theories are discussed. The Wilson's criteria of
confinement in pure gauge theories is generalized to finite chemical potential
case. The chemical potential dependence of physical quantities at strong
coupling are explicitly calculated, including the expectation value of the
Wilson loop, the confining potential and the confined/deconfined transition
temperature. The bosonization puts discussions on chiral symmetry breaking and
confined/deconfined transition on an equal footing, so it is suitable for the
study of the subtle interplay between them. We find that the chiral symmetry
breaking is a necessary (not sufficient) condition for the confinement in 2+1D,
and argue that the confined/deconfined phases are not characterized by any
local symmetries but distinguished by their non-local topologies. The low
energy modes of the strongly coupled gauge systems in a non-symmetry breaking
phase is also discussed. The results of the paper can be widely applied to real
strongly coupled gauge systems, e.g. high-temperature superconductor and
quantum chromodynamical systems in 2+1 dimensions.
