Generalized ADE Classification of Gapped Domain Walls

TL;DR

This paper connects physical rules of anyon condensation with mathematical structures called twist-free CSFA, enabling a comprehensive classification of gapped domain walls and boundaries in topological orders using generalized ADE classification.

Contribution

It demonstrates how physical rules of anyon condensation can be derived from and are equivalent to the mathematical framework of twist-free CSFA, enabling a complete classification.

Findings

Physical rules of anyon condensation are recoverable from twist-free CSFA.

The generalized ADE classification can be used to classify all gapped domain walls.

The classification aligns with previous proposals, confirming its validity.

Abstract

In this paper we would like to demonstrate how the known rules of anyon condensation motivated physically proposed by Bais \textit{et al} can be recovered by the mathematics of twist-free commutative separable Frobenius algebra (CSFA). In some simple cases, those physical rules are also sufficient conditions defining a twist-free CSFA. This allows us to make use of the generalized $ADE$ classification of CSFA's and modular invariants to classify anyon condensation, and thus characterizing all gapped domain walls and gapped boundaries of a large class of topological orders. In fact, this classification is equivalent to the classification we proposed in Ref.1.