# A new strategy to microscopic modelling of topological entanglement in polymers based on field theory

## TL;DR

This paper introduces a novel field theory approach for microscopic modeling of topological entanglement in polymers, enabling the calculation of observables related to knotted and linked polymer rings with topological constraints.

## Contribution

It develops a new method to derive field theories from topological invariants, allowing microscopic analysis of polymer entanglement and extending previous models to more complex topological constraints.

## Key findings

- Reproduces a known model of linked polymer rings
- Reduces the calculation of linking number moments to free field theory amplitudes
- Provides a framework for analyzing topological constraints in polymers

## Abstract

In this work a new strategy is proposed in order to build analytic and microscopic models of fluctuating polymer rings subjected to topological constraints. The topological invariants used to fix these constraints belong to a wide class of the so-called numerical topological invariants. For each invariant it is possible to derive a field theory that describes the statistical behavior of knotted and linked polymer rings following a straightforward algorithm. The treatment is not limited to the partition function of the system, but it allows also to express the expectation values of general observables as field theory amplitudes. Our strategy is illustrated taking as examples the Gauss linking number and a topological invariant belonging to a class of invariants due to Massey. The consistency of the new method developed here is checked by reproducing a previous field theoretical model of two linked polymer rings. After the passage to field theory, the original topological constraints imposed on the fluctuating paths of the polymers become constraints over the configurations of the topological fields that mediate the interactions of topological origin between the monomers. These constraints involve quantities like the cross-helicity which are of interest in other disciplines, like for instance in modeling the solar magnetic field. While the calculation of the vacuum expectation values of generic observables remains still challenging due to the complexity of the problem of topological entanglement in polymer systems, we succeed here to reduce the evaluation of the moments of the Gauss linking number for two linked polymer rings to the computation of the amplitudes of a free field theory.