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Classifying fusion rules of anyons or SymTFTs: A general algebraic formula for domain wall problems and quantum phase transitions

Yoshiki Fukusumi

TL;DR

This work develops a general Verlinde-like formula to classify transformations of anyons and SymTFT objects under gapped or symmetry-preserving domain walls, by constructing ring homomorphisms between UV and IR fusion rings using the $S$-matrix and an idempotent basis. The method yields explicit coefficients $A_{eta}^{\beta'}$ for the mapping $\rho$, determines preserved sectors and an anomaly-free unbroken subalgebra, and connects massless RG flows to massive RG structures through NG-type phenomenology. It further extends to orbifolded, extended, and nonlocal models via $\,\mathbb{Z}_{N}$ extensions and bulk semionization, establishing a hierarchical sequence of homomorphisms between chiral, nonchiral, and SymTFT data. The framework provides a constructive, algebraic pathway to classify extended topological orders and their quantum phase transitions, with implications for CFT/TQFT dualities and potential future exploration of conformal interfaces and higher-dimensional analogues.

Abstract

We propose a formula for the transformation law of anyons in topologically ordered phases or topological quantum field theories (TQFTs) through a gapped or symmetry-preserving domain wall. Our formalism is based on the ring homomorphism between the $\mathbb{C}$-linear commutative fusion rings, also known as symmetry topological field theories (SymTFTs). The fundamental assumption in our formalism is the validity of the Verlinde formula, applicable to commutative fusion rings. By combining it with more specific data of the settings, our formula provides classifications of anyons compatible with developing categorical formulations. It also provides the massless renormalization group (RG) flows between conformal field theories (CFTs), or a series of measurement-induced quantum phase transitions, in the language of SymTFT, through the established correspondence between CFTs and TQFTs. Moreover, by studying the correspondence between the ideal structure in the massless RG and the module in the related massive RG, one can make the Nambu-Goldstone-type arguments for generalized symmetry. By combining our formula with orbifolding, extension, and similarity transformation, one can get a series of classifications for the corresponding extended models, or symmetry-enriched topological orders and quantum criticalities.

Classifying fusion rules of anyons or SymTFTs: A general algebraic formula for domain wall problems and quantum phase transitions

TL;DR

This work develops a general Verlinde-like formula to classify transformations of anyons and SymTFT objects under gapped or symmetry-preserving domain walls, by constructing ring homomorphisms between UV and IR fusion rings using the -matrix and an idempotent basis. The method yields explicit coefficients for the mapping , determines preserved sectors and an anomaly-free unbroken subalgebra, and connects massless RG flows to massive RG structures through NG-type phenomenology. It further extends to orbifolded, extended, and nonlocal models via extensions and bulk semionization, establishing a hierarchical sequence of homomorphisms between chiral, nonchiral, and SymTFT data. The framework provides a constructive, algebraic pathway to classify extended topological orders and their quantum phase transitions, with implications for CFT/TQFT dualities and potential future exploration of conformal interfaces and higher-dimensional analogues.

Abstract

We propose a formula for the transformation law of anyons in topologically ordered phases or topological quantum field theories (TQFTs) through a gapped or symmetry-preserving domain wall. Our formalism is based on the ring homomorphism between the -linear commutative fusion rings, also known as symmetry topological field theories (SymTFTs). The fundamental assumption in our formalism is the validity of the Verlinde formula, applicable to commutative fusion rings. By combining it with more specific data of the settings, our formula provides classifications of anyons compatible with developing categorical formulations. It also provides the massless renormalization group (RG) flows between conformal field theories (CFTs), or a series of measurement-induced quantum phase transitions, in the language of SymTFT, through the established correspondence between CFTs and TQFTs. Moreover, by studying the correspondence between the ideal structure in the massless RG and the module in the related massive RG, one can make the Nambu-Goldstone-type arguments for generalized symmetry. By combining our formula with orbifolding, extension, and similarity transformation, one can get a series of classifications for the corresponding extended models, or symmetry-enriched topological orders and quantum criticalities.
Paper Structure (8 sections, 22 equations, 1 figure)

This paper contains 8 sections, 22 equations, 1 figure.

Table of Contents

  1. introduction
  2. Classification of anyons or SymTFTs
  3. Preserved sectors and unbroken subalgebra
  4. Nambu-Goldstone type phenomenology and correspondence: Operator counting of ideal, module and unbroken symmetry
  5. Extension, orbifolding and isomorphisms
  6. Conclusion and discussion
  7. Acknowledgement
  8. Bulk semionization for Majorana fermion and related homomorphism: Structures compatible with homomorphism

Figures (1)

  • Figure 1: Picture of the transformation of anyons through a domain wall. The UV objects and structures are represented in black, and those of the IR are represented in blue. The red colour represents the homomorphism or domain wall transforming anyons. Because of the surjective property of the homomorphism, the number of types of anyon decreases under the application of the homomorphism. One can see related figures in Fukusumi:2025clrFukusumi:2025xrj, but we do not assume the coset or level-rank duality structures in the present paper.