Exactly solvable models and self-duality in the problem of two linked polymer rings*

In this talk the connection between a model of two linked polymers rings with fixed Gaussian linking number forming a 4-plat and the statistical mechanics of non-relativistic anyon particles is explored. 4-plats are important for biological applications. The excluded volume interactions have been switched off and only the interactions of entropic origin arising from the topological constraints are considered. An interpretation from the polymer point of view of the field equations that minimize the energy of the model in the limit in which one of the spatial dimensions of the 4-plat becomes very large is provided. It is shown that the self-dual contributions are responsible for the long-range interactions that are necessary for preserving the global topological properties of the system during the thermal fluctuations. The non self-dual part is also related to the topological constraints, and takes into account the local interactions acting on the monomers in order to prevent the breaking of the polymer lines. It turns out that the energy landscape of the two linked rings is quite complex. Assuming as a rough approximation that the monomer densities of half of the 4-plat are constant, at least two points of energy minimum are found. Classes of soliton solutions of the self-dual field equations will be discussed. One of these classes is characterized by densities of monomers that are the squared modulus of holomorphic functions. The second class is obtained under some assumptions that allow to reduce the self-dual equations to an analog of the Gouy-Chapman equation for the charge distribution of ions in a double layer capacitor. In the present case, the spatial distribution of the electric potential of the ions is replaced by the spatial distribution of the fictitious magnetic fields associated with the presence of the topological constraints. In the limit in which two of the spatial dimensions are large in comparison with the third one, exact formulas for the conformations of the monomer densities of the 4-plat will be discussed based on the elliptic, hyperbolic and trigonometric solutions of the cosh-Gordon equation which have been used for instance in the construction of classical string solutions.