  We consider the statistical mechanics of a system of topologically linked
polymers, such as for instance a dense solution of polymer rings. If the
possible topological states of the system are distinguished using the Gauss
linking number as a topological invariant, the partition function of an
ensemble of N closed polymers coincides with the 2N point function of a field
theory containing a set of N complex replica fields and Abelian Chern-Simons
fields. Thanks to this mapping to field theories, some quantitative predictions
on the behavior of topologically entangled polymers have been obtained by
exploiting perturbative techniques. In order to go beyond perturbation theory,
a connection between polymers and anyons is established here. It is shown in
this way that the topological forces which maintain two polymers in a given
topological configuration have both attractive and repulsive components. When
these opposite components reach a sort of equilibrium, the system finds itself
in a self-dual point similar to that which, in the Landau-Ginzburg model for
superconductors, corresponds to the transition from type I to type II
superconductivity. The significance of self-duality in polymer physics is
illustrated considering the example of the so-called $4-plat$ configurations,
which are of interest in the biochemistry of DNA processes like replication,
transcription and recombination. The case of static vortex solutions of the
Euler-Lagrange equations is discussed.
