Field Theories of Topological Random Walks

TL;DR

This paper develops a general topological field theory for linked polymers, capturing complex topological invariants beyond the Gauss linking number, and applies it to the statistical mechanics of such systems.

Contribution

It introduces a novel, highly general field theoretical framework that accounts for multiple topological invariants in polymer linkages, extending previous models.

Findings

Derived topological invariants from random walk trajectories

Constructed a field theory for linked polymers with arbitrary topological states

Applicable to systems with many polymers and complex topological interactions

Abstract

In this work we derive certain topological theories of transverse vector fields whose amplitudes reproduce topological invariants involving the interactions among the trajectories of three and four random walks. This result is applied to the construction of a field theoretical model which describes the statistical mechanics of an arbitrary number of topologically linked polymers in the context of the analytical approach of Edwards. With respect to previous attempts, our approach is very general, as it can treat a system involving an arbitrary number of polymers and the topological states are not only specified by the Gauss linking number, but also by higher order topological invariants.