Topological Entanglement Entropy and Braids in Chern-Simons Theory

TL;DR

This paper investigates how topological entanglement entropy relates to braid structures in Chern-Simons theory, revealing connections between quantum entanglement, knot theory, and gauge theory through Wilson loop computations.

Contribution

It establishes a method to analyze entanglement entropy via Wilson loop expectation values in Chern-Simons theory, linking braid properties to quantum entanglement measures.

Findings

Wilson loop expectation values encode braid geometry

Entanglement entropy can be derived from braid data

Properties of auxiliary Wilson loops reflect braid topology

Abstract

We explore a web of connections between quantum entanglement and knot theory by examining how topological entanglement entropy probes the braiding data of quasi-particles in Chern-Simons theory, mainly using $SU(2)$ gauge group as our working example. The problem of determining the Renyi entropy is mapped to computing the expectation value of an auxiliary Wilson loop in $S^3$ for each braid. We study various properties of this auxiliary Wilson loop for some 2-strand and 3-strand braids, and demonstrate how they reflect some geometrical properties of the underlying braids.