The direct link model for polymer rings as a topological field theory and the second topological moment in dense systems

Polymer rings in solution are either permanently entangled or not. Permanent topological restrictions give rise to additional entropic interactions apart from the ones arising due to mere chain flexibility or excluded volume. Conversely, entangled polymer rings systems may be formed by closing randomly entangled flexible linear chains. The dependance of linking numbers between randomly entangled rings on the chain length, more specifically the second topological moment $<n^2>$, i.e. the average squared linking number, may be determined. In this paper, an approach recently discussed in mathematical physics and called abelian BF theory, is presented which allows to express the linking constraint in its simplest form, the Gauss integral, in terms of two gauge fields. The model of Brereton and Shah for a single ring entangled with many other surrounding rings is rederived. The latter model is finally used to calculate the second topological moment, in agreement with a recent result by Ferrari, Kleinert, and Lazzizzera obtained by using ${\rm n}$-component $\phi^4$ theory in the limit ${\rm n}\to 0$.