  Dual formulations of Abelian U(1) and Z(N) LGT with a static fermion
determinant are constructed at finite temperatures and non-zero chemical
potential. The dual form is valid for a broad class of lattice gauge actions,
for arbitrary number of fermion flavors and in any dimension. The distinguished
feature of the dual formulation is that the dual Boltzmann weight is strictly
positive. This allows to gain reliable results at finite density via the Monte
Carlo simulations. As a byproduct of the dual representation we outline an
exact solution for the partition function of the (1 + 1)-dimensional theory and
reveal an existence of a phase with oscillating correlations.
