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Duals of lattice Abelian models with static determinant at finite density

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.