The four-components link invariant in the framework of topological field theories

TL;DR

This paper develops a perturbative approach to construct a second-order topological link invariant in non-Abelian Chern-Simons-Wong theory, demonstrating its ability to detect knotting in four-component links.

Contribution

It explicitly constructs a second-order on-shell action as a link invariant and introduces an Abelian model that reproduces this invariant, providing new insights into topological link detection.

Findings

The invariant depends only on closed curves.

The Abelian model reproduces the non-Abelian invariant.

The invariant detects knotting in four-component links.

Abstract

In this work, we undertake a perturbative analysis of the topological non-Abelian Chern-Simons-Wong model with the aim to explicitly construct the second-order on-shell action. The resulting action is a topological quantity depending solely on closed curves, so it correspond to an analytical expression of a link invariant. Additionally, we construct an Abelian model that reproduces the same second-order on-shell action as its non-Abelian Chern-Simons-Wong counterpart so it functions as an intermediate model, featuring Abelian fields generated by currents supported on closed paths. By geometrically analyzing each term, we demonstrate that this topological invariant effectively detects the knotting of a four-component link.