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Generalized ADE Classification of Gapped Domain Walls

Ling-Yan Hung, Yidun Wan

TL;DR

This work connects the Bais–Slingerland anyon condensation framework to twist-free CSFAs in UMTCs, showing that GDWs and GBs across a broad class of topological orders are classified by generalized ADE diagrams via Rep A and VOA embeddings. It establishes a 1–1 correspondence between simple-current condensation and RCFT modular invariants, extends this to electric/magnetic condensations in quantum doubles, and illustrates the program with su(2)_k examples. By unifying physical condensation rules with rigorous algebraic structures, it provides a systematic, ADE-based classification tool for gapped boundaries and domain walls, while also discussing the role of W matrices and potential symmetry-enriched extensions. The results pave the way for algorithmic determination of Rep A simples and fusion rules, and for applying these ideas to nonchiral and doubled topological phases.

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.

Generalized ADE Classification of Gapped Domain Walls

TL;DR

This work connects the Bais–Slingerland anyon condensation framework to twist-free CSFAs in UMTCs, showing that GDWs and GBs across a broad class of topological orders are classified by generalized ADE diagrams via Rep A and VOA embeddings. It establishes a 1–1 correspondence between simple-current condensation and RCFT modular invariants, extends this to electric/magnetic condensations in quantum doubles, and illustrates the program with su(2)_k examples. By unifying physical condensation rules with rigorous algebraic structures, it provides a systematic, ADE-based classification tool for gapped boundaries and domain walls, while also discussing the role of W matrices and potential symmetry-enriched extensions. The results pave the way for algorithmic determination of Rep A simples and fusion rules, and for applying these ideas to nonchiral and doubled topological phases.

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 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.

Paper Structure

This paper contains 9 sections, 31 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Bais-Slingerland anyon condensation $=$ twist-free CSFA
  3. Simple currents condensation vs Modular Invariance
  4. Gapped domain walls and boundaries
  5. Quantum Double of a Gauge Theory
  6. Conclusions
  7. Bais-Slingerland Rules
  8. Twist-free commutative separable Frobenius algebra
  9. Modular data and $ADE$ classification of $su(2)_k$

Figures (1)

  • Figure 1: The $E_6$ Dynkin diagram for the embedding $su(2)_{10}\subset so(5)_1$. The nodes are labeled by the simple objects in $\mathrm{Rep} A$, whereas the three end-nodes are the simple objects in $\mathrm{Rep}_0 A=so(5)_1$. A similar figure also appears in Ref.Kirillov2002.