# Entanglement on multiple $S^2$ boundaries in Chern-Simons theory

## TL;DR

This paper investigates the topological entanglement arising from quasiparticle interactions within three-manifolds having multiple $S^2$ boundaries in SU(N) Chern-Simons theory, revealing representation-dependent entanglement structures.

## Contribution

It introduces a novel analysis of entanglement entropy in SU(N) Chern-Simons theory with multiple $S^2$ boundaries, emphasizing the role of SU(N) representations and entanglement structures.

## Key findings

- Entanglement entropy depends on SU(N) representations on $S^2$ boundaries.
- Distinct GHZ-like and W-like entanglement structures are observed.
- Representation multiplicities influence the entanglement structure.

## Abstract

Topological entanglement structure amongst disjoint torus boundaries of three manifolds have already been studied within the context of Chern-Simons theory. In this work, we study the topological entanglement due to interaction between the quasiparticles inside three-manifolds with one or more disjoint $S^2$ boundaries in SU($N$) Chern-Simons theory. We focus on the world-lines of quasiparticles (Wilson lines), carrying SU($N$) representations, creating four punctures on every $S^2$. We compute the entanglement entropy by partial tracing some of the boundaries. In fact, the entanglement entropy depends on the SU($N$) representations on these four-punctured $S^2$ boundaries. Further, we observe interesting features on the GHZ-like and W-like entanglement structures. Such a distinction crucially depends on the multiplicity of the irreducible representations in the tensor product of SU($N$) representations.