  For a large class of Abelian lattice models with sign problems, including the
case of non-zero chemical potential, duality maps models with complex actions
into dual models with real actions. For extended regions of parameter space,
calculable for each model, duality resolves the sign problem for both analytic
methods and computer simulations. Explicit duality relations are given for
models for spin and gauge models based on Z(N) and U(1) symmetry groups. The
dual forms are generalizations of the Z(N) chiral clock model and the lattice
Frenkel-Kontorova model, respectively. From these equivalences, rich sets of
spatially-modulated phases are found in the strong-coupling region of the
original models.
