  Dual representations are constructed for non-abelian lattice spin models with
U(N) and SU(N) symmetry groups, for all N and in any dimension. These models
are usually related to the effective models describing the interaction between
Polyakov loops in the strong coupled QCD. The original spin degrees of freedom
are explicitly integrated out and a dual theory appears to be a local theory
for the dual integer-valued variables. The construction is performed for the
partition function and for the most general correlation function. The latter
include the two-point function corresponding to quark-anti-quark free energy
and the N-point function related to the free energy of a baryon. We consider
both pure gauge models and models with static fermion determinant for both the
staggered and Wilson fermions with an arbitrary number of flavours. While the
Boltzmann weights of such models are complex in the presence of non-zero
chemical potential the dual Boltzmann weights appear to be strictly positive on
admissible configurations. An essential part of this work with respect to
previous studies is an extension of the dual representation to the case of 1)
an arbitrary value of the temporal coupling constant in the Wilson action and
2) an arbitrary number of flavours of static quark determinants. The
applications and extensions of the results are discussed in detail. In
particular, we outline a possible approach to Monte-Carlo simulations of the
dual theory, to the large N expansion and to the development of a tensor
renormalization group.
