# Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy

## TL;DR

This paper investigates how gapped boundaries and interfaces in 2+1D topological orders influence topological entanglement entropy, using Ishibashi states and anyon condensation in Abelian and non-Abelian Chern-Simons theories.

## Contribution

It provides a detailed analysis of Ishibashi states for topological edges/interfaces and links the entanglement entropy to anyon condensation patterns in both Abelian and non-Abelian theories.

## Key findings

- Topological entanglement entropy depends on the anyon condensation pattern.
- Gapped boundaries and interfaces can be characterized by Ishibashi states.
- Conditions on edge parameters ensure the existence of conformal interfaces.

## Abstract

In this paper, we study gapped edges/interfaces in a 2+1 dimensional bosonic topological order and investigate how the topological entanglement entropy is sensitive to them. We present a detailed analysis of the Ishibashi states describing these edges/interfaces making use of the physics of anyon condensation in the context of Abelian Chern-Simons theory, which is then generalized to more non-Abelian theories whose edge RCFTs are known. Then we apply these results to computing the entanglement entropy of different topological orders. We consider cases where the system resides on a cylinder with gapped boundaries and that the entanglement cut is parallel to the boundary. We also consider cases where the entanglement cut coincides with the interface on a cylinder. In either cases, we find that the topological entanglement entropy is determined by the anyon condensation pattern that characterizes the interface/boundary. We note that conditions are imposed on some non-universal parameters in the edge theory to ensure existence of the conformal interface, analogous to requiring rational ratios of radii of compact bosons.