Topologically Linked Polymers are Anyon Systems

Abstract

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.