# Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory

## TL;DR

This paper investigates how topological boundary conditions affect entanglement entropy in Abelian Chern-Simons theories, revealing universal sub-leading corrections and connecting edge states with bulk entanglement.

## Contribution

It introduces a continuum Hilbert space framework for entanglement in gauge theories with topological boundary conditions, providing new insights into boundary effects on entanglement entropy.

## Key findings

- Sub-leading correction to area law depends on boundary conditions
- Extended Hilbert space construction clarifies entanglement in gauge theories
- Replica path integral confirms boundary condition dependence

## Abstract

We study the entanglement entropy between (possibly distinct) topological phases across an interface using an Abelian Chern-Simons description with topological boundary conditions (TBCs) at the interface. From a microscopic point of view, these TBCs correspond to turning on particular gapping interactions between the edge modes across the interface. However, in studying entanglement in the continuum Chern-Simons description, we must confront the problem of non-factorization of the Hilbert space, which is a standard property of gauge theories. We carefully define the entanglement entropy by using an extended Hilbert space construction directly in the continuum theory. We show how a given TBC isolates a corresponding gauge invariant state in the extended Hilbert space, and hence compute the resulting entanglement entropy. We find that the sub-leading correction to the area law remains universal, but depends on the choice of topological boundary conditions. This agrees with the microscopic calculation of \cite{Cano:2014pya}. Additionally, we provide a replica path integral calculation for the entropy. In the case when the topological phases across the interface are taken to be identical, our construction gives a novel explanation of the equivalence between the left-right entanglement of (1+1)d Ishibashi states and the spatial entanglement of (2+1)d topological phases.