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Generalizing quantum dimensions: Symmetry-based classification of local pseudo-Hermitian systems and the corresponding domain walls

Yoshiki Fukusumi, Taishi Kawamoto

TL;DR

This work reframes CFT and TO classifications in pseudo-Hermitian contexts through SymTFTs and a generalized quantum-dimension framework defined by Verlinde line operators and fusion rings. It shows massless and massive RG flows correspond to ring homomorphisms and ideals, enabling an algebraic treatment of RG domain walls and gapped phases, illustrated via Fibonacci and Ising–Yang-Lee examples. Explicit ring-homomorphism solutions across several minimal models reveal multiple consistent RG channels and highlight anomaly/spin constraints, including potential coset decompositions and domain-wall tensor structures. The results suggest noninteger coefficients are natural in this setting, motivating generalized category-theoretic formalisms and indicating broader applicability to higher-dimensional and nonunitary QFTs.

Abstract

We study conformal field theories (CFTs) and their classifications from a modern perspective based on the abstract algebraic formalism of symmetries or conserved charges, known as symmetry topological field theories (SymTFTs). By studying the algebraic structure of the SymTFTs in detail, we found a natural generalization of the quantum dimensions associated with (pseudo-)Hermitian systems and (non)-unitary CFTs. These generalized data of SymTFTs provide classifications of massless and massive renormalization group flows, which will describe the quantum phase transitions of the corresponding pseudo-Hermitian systems. Moreover, our discussions straightforwardly enable one to relate a general class of coset constructions or level-rank dualities to domain wall problems between topological quantum field theories (or a series of corresponding quantum phase transitions related to the Higgs mechanism). Our work provides a systematic reduction and classification of algebraic data, symmetries, for pseudo-Hermitian systems based on ideas from established mathematical fields, linear algebra and ring theory.

Generalizing quantum dimensions: Symmetry-based classification of local pseudo-Hermitian systems and the corresponding domain walls

TL;DR

This work reframes CFT and TO classifications in pseudo-Hermitian contexts through SymTFTs and a generalized quantum-dimension framework defined by Verlinde line operators and fusion rings. It shows massless and massive RG flows correspond to ring homomorphisms and ideals, enabling an algebraic treatment of RG domain walls and gapped phases, illustrated via Fibonacci and Ising–Yang-Lee examples. Explicit ring-homomorphism solutions across several minimal models reveal multiple consistent RG channels and highlight anomaly/spin constraints, including potential coset decompositions and domain-wall tensor structures. The results suggest noninteger coefficients are natural in this setting, motivating generalized category-theoretic formalisms and indicating broader applicability to higher-dimensional and nonunitary QFTs.

Abstract

We study conformal field theories (CFTs) and their classifications from a modern perspective based on the abstract algebraic formalism of symmetries or conserved charges, known as symmetry topological field theories (SymTFTs). By studying the algebraic structure of the SymTFTs in detail, we found a natural generalization of the quantum dimensions associated with (pseudo-)Hermitian systems and (non)-unitary CFTs. These generalized data of SymTFTs provide classifications of massless and massive renormalization group flows, which will describe the quantum phase transitions of the corresponding pseudo-Hermitian systems. Moreover, our discussions straightforwardly enable one to relate a general class of coset constructions or level-rank dualities to domain wall problems between topological quantum field theories (or a series of corresponding quantum phase transitions related to the Higgs mechanism). Our work provides a systematic reduction and classification of algebraic data, symmetries, for pseudo-Hermitian systems based on ideas from established mathematical fields, linear algebra and ring theory.

Paper Structure

This paper contains 20 sections, 79 equations, 2 figures.

Table of Contents

  1. introduction
  2. Pseudo-Hermitian and linear dual basis
  3. Verlinde line operator and algebraic generalized quantum dimension
  4. (smeared) boundary conformal field theory and entropy
  5. Examples of our quantum dimension
  6. Possible gapped phase with Fibonacci fusion ring symmetry
  7. Massless flows and matching algebraic generalized quantum dimensions
  8. Duality between massive and massless flows
  9. Tensor decomposition and gapped domain wall problem: A benefit of introducing algebraic generalized quantum dimension
  10. Examples of ring homomorphism in unitary and nonunitary models
  11. $M(4,5)$ to $M(3,4)$
  12. $M(3,4)$ to $M(2,5)$
  13. $M(3,5)$ to $M(2,5)$
  14. $M(2,7)$ to $M(2,5)$
  15. Spin-classification for non-group-like objects: (Half) integer spin conditions
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Correspondence between massless and massive RG flows. Existence of an ideal $\mathbf{I}$ preserving $\mathbf{A}_{\text{ub}}$ implies the existence of the corresponding massive RG, by identifying $\mathbf{I}$ as the resultant module $\mathcal{H}$ in the $\mathbf{A}_{\text{ub}}$-preserving massive RG.
  • Figure 2: Tensor decomposition of UV CFT and the resultant IR theory. The red color specifies the objects and operations in the theory at the domain wall and the blue color specifies the objects in the IR theory. By tracing out the theory at the domain wall by applying the homomorphism $d_{a"_{\text{DW}}}$ realizing the algebraic generalized quantum dimensions, one can obtain the homomorphism $\rho: \mathbf{A}\rightarrow \mathbf{A'}$.