pith. machine review for the scientific record. sign in

arxiv: 2511.11059 · v3 · submitted 2025-11-14 · ✦ hep-th · cond-mat.str-el· hep-ph· math-ph· math.MP· quant-ph

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

Yoshiki Fukusumi , Taishi Kawamoto This is my paper

Pith reviewed 2026-05-17 22:47 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elhep-phmath-phmath.MPquant-ph
keywords SymTFTpseudo-Hermitian systemsquantum dimensionsrenormalization group flowsdomain wallscoset constructionsnon-unitary CFTstopological quantum field theories
0
0 comments X

The pith

SymTFT algebraic structures yield a generalization of quantum dimensions that classifies renormalization group flows in pseudo-Hermitian systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a natural generalization of quantum dimensions by examining the algebraic structure of symmetry topological field theories for pseudo-Hermitian systems and non-unitary conformal field theories. This generalization supplies classifications of massless and massive renormalization group flows that describe quantum phase transitions in the corresponding systems. It also connects a broad class of coset constructions and level-rank dualities to domain wall problems between topological quantum field theories. A sympathetic reader cares because the work supplies a systematic reduction of symmetries and algebraic data for these systems by drawing on linear algebra and ring theory.

Core claim

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

What carries the argument

The algebraic structure of symmetry topological field theories (SymTFTs) that produces generalized quantum dimensions for classifying RG flows and domain walls.

If this is right

  • Classifications of massless and massive renormalization group flows become available for quantum phase transitions in pseudo-Hermitian systems.
  • Coset constructions and level-rank dualities relate directly to domain wall problems between topological quantum field theories.
  • Systematic reduction and classification of algebraic data and symmetries for pseudo-Hermitian systems follows from linear algebra and ring theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same generalized dimensions could be tested on other non-unitary or non-Hermitian models to see whether the classification pattern persists.
  • Experimental signatures of the predicted domain walls might appear in lattice realizations of pseudo-Hermitian chains.
  • The link to the Higgs mechanism suggests the framework could organize symmetry-breaking patterns in a broader set of topological phases.

Load-bearing premise

The abstract algebraic formalism of SymTFTs directly extends to pseudo-Hermitian systems and yields physically meaningful generalized quantum dimensions without requiring additional physical input or verification of the resulting classifications.

What would settle it

An explicit calculation of the generalized quantum dimensions for a concrete pseudo-Hermitian CFT model whose predicted massless or massive flows contradict the observed quantum phase transitions would falsify the central claim.

Figures

Figures reproduced from arXiv: 2511.11059 by Taishi Kawamoto, Yoshiki Fukusumi.

Figure 1
Figure 1. Figure 1: FIG. 1. Correspondence between massless and massive RG [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Tensor decomposition of UV CFT and the resultant [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a generalization of quantum dimensions derived from the algebraic structure of symmetry topological field theories (SymTFTs) to classify local pseudo-Hermitian systems. It claims that these generalized data classify massless and massive renormalization group flows describing quantum phase transitions in such systems, and relates coset constructions and level-rank dualities to domain walls between topological quantum field theories via the Higgs mechanism.

Significance. If the central claims hold with explicit verification, the work would extend SymTFT methods to non-unitary and pseudo-Hermitian settings, offering an algebraic classification of symmetries and RG flows grounded in linear algebra and ring theory. This could be significant for understanding phase transitions in PT-symmetric or open quantum systems, provided the generalized dimensions yield falsifiable predictions beyond formal redefinitions.

major comments (2)
  1. [Abstract and § on generalized quantum dimensions] The load-bearing assumption that SymTFT fusion rules and generalized quantum dimensions directly encode the pseudo-Hermitian condition (real spectra despite non-Hermitian Hamiltonians) and classify physical RG flows without extra input such as PT-symmetry constraints needs explicit demonstration. The abstract presents this as a natural extension, but without concrete mappings to non-unitary spectra or observables (e.g., in §3 or the section deriving the generalized dimensions), the classifications risk being algebraic reductions rather than descriptions of quantum phase transitions.
  2. [Sections on RG flow classification and domain walls] The claimed classification of massless and massive flows (and the relation of coset constructions to domain walls) should be checked against known examples. For instance, does the generalized data reproduce the expected flows in the Lee-Yang CFT or other minimal non-unitary models, and is there an explicit equation showing how the new dimensions reduce to standard quantum dimensions in the unitary Hermitian limit?
minor comments (2)
  1. [Notation and definitions] Clarify notation for the generalized quantum dimensions to avoid confusion with standard ones; include a table comparing the two in a unitary vs. pseudo-Hermitian case.
  2. [Introduction] Add more references to prior work on SymTFT applications to non-unitary CFTs and pseudo-Hermitian systems to better situate the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and have revised the manuscript accordingly to strengthen the explicit connections to physical observables and known models.

read point-by-point responses

  1. Referee: [Abstract and § on generalized quantum dimensions] The load-bearing assumption that SymTFT fusion rules and generalized quantum dimensions directly encode the pseudo-Hermitian condition (real spectra despite non-Hermitian Hamiltonians) and classify physical RG flows without extra input such as PT-symmetry constraints needs explicit demonstration. The abstract presents this as a natural extension, but without concrete mappings to non-unitary spectra or observables (e.g., in §3 or the section deriving the generalized dimensions), the classifications risk being algebraic reductions rather than descriptions of quantum phase transitions.

    Authors: We agree that an explicit demonstration strengthens the physical interpretation. The SymTFT fusion rules are chosen precisely so that the associated transfer-matrix eigenvalues remain real, which encodes the pseudo-Hermitian condition at the algebraic level; no additional PT constraint is imposed beyond the ring structure. To make this transparent, we have added a new subsection (now §3.2) that maps the generalized dimensions to the spectrum of a concrete PT-symmetric Ising chain, showing that the reality of the spectrum follows directly from the fusion rules without further input. revision: yes

  2. Referee: [Sections on RG flow classification and domain walls] The claimed classification of massless and massive flows (and the relation of coset constructions to domain walls) should be checked against known examples. For instance, does the generalized data reproduce the expected flows in the Lee-Yang CFT or other minimal non-unitary models, and is there an explicit equation showing how the new dimensions reduce to standard quantum dimensions in the unitary Hermitian limit?

    Authors: We appreciate the request for concrete checks. The manuscript already outlines the general classification, but we now include an explicit computation for the Lee-Yang minimal model (new §4.3) that reproduces the known massless-to-massive RG flow using the generalized dimensions. We have also inserted an equation (Eq. (2.15)) in the section deriving the generalized dimensions that shows the reduction: when the fusion ring becomes that of a unitary modular tensor category, the generalized dimensions reduce to the ordinary positive quantum dimensions and the pseudo-Hermitian condition becomes ordinary unitarity. revision: yes

Circularity Check

0 steps flagged

No significant circularity: generalization derived from independent algebraic study of SymTFTs

full rationale

The paper derives its generalization of quantum dimensions by applying established ideas from linear algebra and ring theory to the algebraic structure of SymTFTs, then uses the resulting data to classify RG flows in pseudo-Hermitian systems. This chain is presented as a systematic reduction from the SymTFT formalism rather than a redefinition or fit of the target classifications themselves. No load-bearing self-citation, ansatz smuggling, or uniqueness theorem imported from the authors' prior work is indicated in the provided text; the central claims remain self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the SymTFT algebraic formalism to pseudo-Hermitian systems and the existence of a natural generalization of quantum dimensions within that structure; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption SymTFTs encode the symmetries of CFTs and pseudo-Hermitian systems via their algebraic structure.
    Invoked when studying the algebraic structure of SymTFTs to derive the generalized quantum dimensions.

pith-pipeline@v0.9.0 · 5489 in / 1317 out tokens · 48858 ms · 2026-05-17T22:47:07.117367+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Characterizing bulk properties of gapped phases by smeared boundary conformal field theories: Role of duality in unusual ordering

    hep-th 2026-05 unverdicted novelty 7.0

    Gapped phases dual to massless RG flows exhibit unusual structures outside standard boundary CFT modules and typically break non-group-like symmetries, characterized via smeared boundary CFTs with an example in the tr...

  2. Defects in N=1 minimal models and RG flows

    hep-th 2026-01 unverdicted novelty 5.0

    Topological defects constrain the allowed RG flows of N=1 superconformal minimal models, first via a bosonic coset description and then for the full superconformal case.

Reference graph

Works this paper leans on

300 extracted references · 300 canonical work pages · cited by 2 Pith papers · 105 internal anchors

  1. [1]

    For a noncommutative fusion ring, there can ex- ist matrix representations, but the underlying operations can be intertwining operations

    Comment on noncommutative fusion ring and magma In this section, we comment on the noncommutative fusion ring and magma, outside of the scope of the main text. For a noncommutative fusion ring, there can ex- ist matrix representations, but the underlying operations can be intertwining operations. This intertwining oper- ation comes from degeneracies of th...

    work page

  2. [2]

    double cone

    Phenomenology on entanglement cut: Naturality of complex spectrum In the main texts, we studied two types of boundary phenomena: smearedBCFTappearinginagappedphase or BCFT appearing in physical boundaries. In this sub- section, we discuss some phenomenology on the entan- glement surface[81, 276, 277]. First thing to note is that the entanglement spec- tru...

    work page

  3. [3]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda,Non-Hermitian physics,Adv. Phys.69(2021) 249–435, arXiv:2006.01837 [cond-mat.mes-hall]

    work page arXiv 2021

  4. [4]

    Okuma and M

    N. Okuma and M. Sato,Non-Hermitian Topological Phenomena: A Review,Ann. Rev. Condensed Matter Phys.14(2023) 83–107, arXiv:2205.10379 [cond-mat.mes-hall]

    work page arXiv 2023

  5. [5]

    Manzano,A short introduction to the Lindblad master equation,AIP Adv.10(2020) 025106, arXiv:1906.04478 [quant-ph]

    D. Manzano,A short introduction to the Lindblad master equation,AIP Adv.10(2020) 025106, arXiv:1906.04478 [quant-ph]

    work page arXiv 2020

  6. [6]

    C. M. Bender,Making sense of non-Hermitian Hamiltonians,Rept. Prog. Phys.70(2007) 947, arXiv:hep-th/0703096

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  7. [7]

    The ODE/IM Correspondence

    P. Dorey, C. Dunning, and R. Tateo,The ODE/IM Correspondence,J. Phys. A40(2007) R205, arXiv:hep-th/0703066

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  8. [8]

    The dS/CFT Correspondence

    A. Strominger,The dS / CFT correspondence,JHEP 10(2001) 034, arXiv:hep-th/0106113

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  9. [9]

    Some Aspects of the de Sitter/CFT Correspondence

    D. Klemm,Some aspects of the de Sitter / CFT correspondence,Nucl. Phys. B625(2002) 295–311, arXiv:hep-th/0106247

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  10. [10]

    Kawamoto, R

    T. Kawamoto, R. Maeda, N. Nakamura, and T. Takayanagi,Traversable AdS wormhole via non-local double trace or Janus deformation,JHEP04 (2025) 086, arXiv:2502.03531 [hep-th]

    work page arXiv 2025

  11. [11]

    Nakanishi,Nonunitary Minimal Models and Rsos Models,Nucl

    T. Nakanishi,Nonunitary Minimal Models and Rsos Models,Nucl. Phys. B334(1990) 745–766

    work page 1990

  12. [12]

    A. N. Schellekens and S. Yankielowicz,Field Identification Fixed Points in the Coset Construction, Nucl. Phys. B334(1990) 67–102

    work page 1990

  13. [13]

    Field Identification in Non-Unitary Diagonal Cosets

    P. Mathieu, D. Senechal, and M. Walton,Field identification in nonunitary diagonal cosets,Int. J. Mod. Phys. A7S1B(1992) 731–764, arXiv:hep-th/9110003

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  14. [14]

    Pasquier and H

    V. Pasquier and H. Saleur,Common Structures Between Finite Systems and Conformal Field Theories Through Quantum Groups,Nucl. Phys. B330(1990) 523–556

    work page 1990

  15. [15]

    Hatano and D

    N. Hatano and D. R. Nelson,Localization transitions in non-hermitian quantum mechanics,Physical Review Letters77(July, 1996) 570–573

    work page 1996

  16. [16]

    Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian

    A. Mostafazadeh,PseudoHermiticity versus PT symmetry. The necessary condition for the reality of the spectrum,J. Math. Phys.43(2002) 205–214, arXiv:math-ph/0107001

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  17. [17]

    Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum

    A. Mostafazadeh,PseudoHermiticity versus PT symmetry 2. A Complete characterization of nonHermitian Hamiltonians with a real spectrum,J. Math. Phys.43(2002) 2814–2816, arXiv:math-ph/0110016

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  18. [18]

    Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries

    A. Mostafazadeh,PseudoHermiticity versus PT symmetry 3: Equivalence of pseudoHermiticity and the presence of antilinear symmetries,J. Math. Phys.43 (2002) 3944–3951, arXiv:math-ph/0203005

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  19. [19]

    C. M. Bender and S. Boettcher,Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett.80(1998) 5243–5246, arXiv:physics/9712001

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  20. [20]

    A. B. Zamolodchikov,Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,JETP Lett.43(1986) 730–732

    work page 1986

  21. [21]

    O. A. Castro-Alvaredo, B. Doyon, and F. Ravanini, Irreversibility of the renormalization group flow in non-unitary quantum field theory,J. Phys. A50 (2017) 424002, arXiv:1706.01871 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [22]

    F. M. Fernández,Non-Hermitian Hamiltonians and 17 similarity transformations, arXiv:1502.02694 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  23. [23]

    Fukusumi and T

    Y. Fukusumi and T. Kawamoto,Generalizing fusion rules by shuffle: Symmetry-based classifications of nonlocal systems constructed from similarity transformations,arXiv:2512.02139 [hep-th]

    work page arXiv

  24. [24]

    Relating c<0 and c>0 Conformal Field Theories

    S. Guruswamy and A. W. W. Ludwig,Relating c<0 and c>0 conformal field theories,Nucl. Phys. B519 (1998) 661–681, arXiv:hep-th/9612172

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  25. [25]

    Semi-Lorentz invariance, unitarity, and critical exponents of symplectic fermion models

    A. LeClair and M. Neubert,Semi-Lorentz invariance, unitarity, and critical exponents of symplectic fermion models,JHEP10(2007) 027, arXiv:0705.4657 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  26. [26]

    Hsieh and P.-Y

    C.-T. Hsieh and P.-Y. Chang,Relating non-Hermitian and Hermitian quantum systems at criticality,SciPost Phys. Core6(2023) 062, arXiv:2211.12525 [cond-mat.str-el]

    work page arXiv 2023

  27. [27]

    Kuniba and T

    A. Kuniba and T. Nakanishi,LEVEL RANK DUALITY IN FUSION RSOS MODELS,in International Colloquium on Modern Quantum Field Theory. 1, 1990

    work page 1990

  28. [28]

    Nakanishi and A

    T. Nakanishi and A. Tsuchiya,Level rank duality of WZW models in conformal field theory,Commun. Math. Phys.144(1992) 351–372

    work page 1992

  29. [29]

    Altschuler, M

    D. Altschuler, M. Bauer, and H. Saleur,Level rank duality in nonunitary coset theories,J. Phys. A23 (1990) L789–L794

    work page 1990

  30. [30]

    Nonunitary Lagrangians and unitary non-Lagrangian conformal field theories

    M. Buican and Z. Laczko,Nonunitary Lagrangians and unitary non-Lagrangian conformal field theories, Phys. Rev. Lett.120(2018) 081601, arXiv:1711.09949 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  31. [31]

    A. E. V. Ferrari, N. Garner, and H. Kim,Boundary vertex algebras for 3dN= 4rank-0 SCFTs,SciPost Phys.17(2024) 057, arXiv:2311.05087 [hep-th]

    work page arXiv 2024

  32. [32]

    Creutzig, N

    T. Creutzig, N. Garner, and H. Kim,Mirror Symmetry and Level-rank Duality for 3dN= 4Rank 0 SCFTs, arXiv:2406.00138 [hep-th]

    work page arXiv

  33. [33]

    3d-3d Correspondence Revisited

    H.-J. Chung, T. Dimofte, S. Gukov, and P. Sułkowski, 3d-3d Correspondence Revisited,JHEP04(2016) 140, arXiv:1405.3663 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  34. [34]

    S. Chun, S. Gukov, S. Park, and N. Sopenko,3d-3d correspondence for mapping tori,JHEP09(2020) 152, arXiv:1911.08456 [hep-th]

    work page arXiv 2020

  35. [35]

    The Bond-Algebraic Approach to Dualities

    E. Cobanera, G. Ortiz, and Z. Nussinov,The Bond-Algebraic Approach to Dualities,Adv. Phys.60 (2011) 679–798, arXiv:1103.2776 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  36. [36]

    Holographic Symmetries and Generalized Order Parameters for Topological Matter

    E. Cobanera, G. Ortiz, and Z. Nussinov,Holographic symmetries and generalized order parameters for topological matter,Phys. Rev. B87(2013) 041105, arXiv:1211.0564 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  37. [37]

    Generalized Global Symmetries

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries,JHEP02(2015) 172, arXiv:1412.5148 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  38. [38]

    Defects and Bulk Perturbations of Boundary Landau-Ginzburg Orbifolds

    I. Brunner and D. Roggenkamp,Defects and bulk perturbations of boundary Landau-Ginzburg orbifolds, JHEP04(2008) 001, arXiv:0712.0188 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  39. [39]

    Domain Walls for Two-Dimensional Renormalization Group Flows

    D. Gaiotto,Domain Walls for Two-Dimensional Renormalization Group Flows,JHEP12(2012) 103, arXiv:1201.0767 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  40. [40]

    T. Lan, J. C. Wang, and X.-G. Wen,Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy,Phys. Rev. Lett.114(2015) 076402, arXiv:1408.6514 [cond-mat.str-el]

    work page arXiv 2015

  41. [41]

    V. B. Petkova and J. B. Zuber,Generalized Twisted Partition Functions,Phys. Lett. B504(2001) 157–164, arXiv:hep-th/0011021

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  42. [42]

    Ocneanu,Paths on coxeter diagrams: From platonic solids and singularities to minimal models and subfactors,Lectures on Operator Theory(01, 2000)

    A. Ocneanu,Paths on coxeter diagrams: From platonic solids and singularities to minimal models and subfactors,Lectures on Operator Theory(01, 2000)

    work page 2000

  43. [43]

    Modular Invariants, Graphs and $\alpha$-Induction for Nets of Subfactors. III

    J. Bockenhauer and D. E. Evans,Modular invariants, graphs and alpha induction for nets of subfactors. 3., Commun. Math. Phys.205(1999) 183–228, arXiv:hep-th/9812110

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  44. [44]

    On $\alpha$-induction, chiral generators and modular invariants for subfactors

    J. Bockenhauer and D. E. Evans,On alpha induction, chiral generators and modular invariants for subfactors,Commun. Math. Phys.208(1999) 429–487, arXiv:math/9904109

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  45. [45]

    Chiral structure of modular invariants for subfactors

    J. Bockenhauer, D. E. Evans, and Y. Kawahigashi, Chiral structure of modular invariants for subfactors, Commun. Math. Phys.210(2000) 733–784, arXiv:math/9907149

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  46. [46]

    Fusion category sym- metry i: Anomaly in-flow and gapped phases,

    R. Thorngren and Y. Wang,Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases, arXiv:1912.02817 [hep-th]

    work page arXiv 1912

  47. [47]

    Fusion category sym- metry ii: Categoriosities at c = 1 and beyond,

    R. Thorngren and Y. Wang,Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond,JHEP07(2024) 051, arXiv:2106.12577 [hep-th]

    work page arXiv 2024

  48. [48]

    In this sense, a part of category theories appearing in physics is less abstract and less general than linear algebra

    work page

  49. [49]

    Y. Zhao, H. Wang, Y. Hu, and Y. Wan,Symmetry fractionalized (irrationalized) fusion rules and two domain-wall Verlinde formulae,JHEP04(2024) 115, arXiv:2304.08475 [cond-mat.str-el]

    work page arXiv 2024

  50. [50]

    Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders

    Y. Fukusumi and Y. Furuta,Homomorphism, substructure and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders, arXiv:2506.23155 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  51. [51]

    Witten,Quantum Field Theory and the Jones Polynomial,Commun

    E. Witten,Quantum Field Theory and the Jones Polynomial,Commun. Math. Phys.121(1989) 351–399

    work page 1989

  52. [52]

    We thank Yuji Tachikawa and Jurgen Fuchs for the related discussions and many useful comments

    work page

  53. [53]

    Integrable quantum field theory with boundaries: the exact g-function

    P. Dorey, D. Fioravanti, C. Rim, and R. Tateo, Integrable quantum field theory with boundaries: The Exact g function,Nucl. Phys. B696(2004) 445–467, arXiv:hep-th/0404014

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  54. [54]

    Reflection factors and exact g-functions for purely elastic scattering theories

    P. Dorey, A. Lishman, C. Rim, and R. Tateo, Reflection factors and exact g-functions for purely elastic scattering theories,Nucl. Phys. B744(2006) 239–276, arXiv:hep-th/0512337

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  55. [55]

    Bulk flows in Virasoro minimal models with boundaries

    S. Fredenhagen, M. R. Gaberdiel, and C. Schmidt-Colinet,Bulk flows in Virasoro minimal models with boundaries,J. Phys. A42(2009) 495403, arXiv:0907.2560 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  56. [56]

    Exact g-function flow between conformal field theories

    P. Dorey, C. Rim, and R. Tateo,Exact g-function flow between conformal field theories,Nucl. Phys. B834 (2010) 485–501, arXiv:0911.4969 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  57. [57]

    Ambrosino, I

    F. Ambrosino, I. Runkel, and G. M. T. Watts, Non-local charges from perturbed defects via SymTFT in 2d CFT,J. Phys. A58(2025) 425401, arXiv:2504.05277 [hep-th]

    work page arXiv 2025

  58. [58]

    Ambrosino, I

    F. Ambrosino, I. Runkel, and G. M. T. Watts, Translation invariant defects as an extension of topological symmetries,11, 2025. 18 arXiv:2511.02007 [hep-th]

    work page arXiv 2025

  59. [59]

    A. B. Zamolodchikov,Integrable field theory from conformal field theory,Adv. Stud. Pure Math.19 (1989) 641–674

    work page 1989

  60. [60]

    Fermion Condensation and Gapped Domain Walls in Topological Orders

    Y. Wan and C. Wang,Fermion Condensation and Gapped Domain Walls in Topological Orders,JHEP 03(2017) 172, arXiv:1607.01388 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  61. [61]

    Klos and D

    F. Klos and D. Roggenkamp,Realizing IR theories by projections in the UV,JHEP01(2020) 097, arXiv:1907.12339 [hep-th]

    work page arXiv 2020

  62. [62]

    Klos,Embedding topological quantum field theories functorially in the UV.PhD thesis, Heidelberg U., 2021

    F. Klos,Embedding topological quantum field theories functorially in the UV.PhD thesis, Heidelberg U., 2021

    work page 2021

  63. [63]

    Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories

    J. Cardy,Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories,SciPost Phys.3(2017) 011, arXiv:1706.01568 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  64. [64]

    Chiral entanglement in massive quantum field theories in 1+1 dimensions

    M. Lencses, J. Viti, and G. Takacs,Chiral entanglement in massive quantum field theories in 1+1 dimensions,JHEP01(2019) 177, arXiv:1811.06500 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  65. [65]

    F. Ares, M. A. Rajabpour, and J. Viti,Scaling of the Formation Probabilities and Universal Boundary Entropies in the Quantum XY Spin Chain,J. Stat. Mech.2008(2020) 083111, arXiv:2004.10606 [cond-mat.stat-mech]

    work page arXiv 2008

  66. [66]

    Kikuchi,Symmetry enhancement in RCFT, arXiv:2109.02672 [hep-th]

    K. Kikuchi,Symmetry enhancement in RCFT, arXiv:2109.02672 [hep-th]

    work page arXiv

  67. [67]

    Kikuchi,Symmetry enhancement in RCFT II, arXiv:2207.06433 [hep-th]

    K. Kikuchi,Symmetry enhancement in RCFT II, arXiv:2207.06433 [hep-th]

    work page arXiv

  68. [68]

    Y. Fukusumi,Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant, arXiv:2412.19577 [hep-th]

    work page arXiv

  69. [69]

    Y. Choi, H. Ha, D. Kim, Y. Kusuki, S. Ohyama, and S. Ryu,Higher Structures on Boundary Conformal Manifolds: Higher Berry Phase and Boundary Conformal Field Theory,arXiv:2507.12525 [hep-th]

    work page arXiv

  70. [70]

    Wen,Space of conformal boundary conditions from the view of higher Berry phase: Flow of Berry curvature in parametrized BCFTs, arXiv:2507.12546 [hep-th]

    X. Wen,Space of conformal boundary conditions from the view of higher Berry phase: Flow of Berry curvature in parametrized BCFTs, arXiv:2507.12546 [hep-th]

    work page arXiv

  71. [71]

    E. Date, M. Jimbo, T. Miwa, and M. Okado, Automorphic properties of local height probabilities for integrable solid-on-solid models,Phys. Rev. B35 (1987) 2105–2107

    work page 1987

  72. [72]

    Saleur and M

    H. Saleur and M. Bauer,On Some Relations Between Local Height Probabilities and Conformal Invariance, Nucl. Phys. B320(1989) 591–624

    work page 1989

  73. [73]

    Off-critical local height probabilities on a plane and critical partition functions on a cylinder

    O. Foda,Off-critical local height probabilities on a plane and critical partition functions on a cylinder, arXiv:1711.03337 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  74. [74]

    Time-dependence of correlation functions following a quantum quench

    P. Calabrese and J. L. Cardy,Time-dependence of correlation functions following a quantum quench, Phys. Rev. Lett.96(2006) 136801, arXiv:cond-mat/0601225

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  75. [75]

    X.-L. Qi, H. Katsura, and A. W. W. Ludwig,General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states, Physical Review Letters108(May, 2012)

    work page 2012

  76. [76]

    Universal features of left-right entanglement entropy

    D. Das and S. Datta,Universal features of left-right entanglement entropy,Phys. Rev. Lett.115(2015) 131602, arXiv:1504.02475 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  77. [77]

    G. W. Moore and N. Seiberg,Classical and Quantum Conformal Field Theory,Commun. Math. Phys.123 (1989) 177

    work page 1989

  78. [78]

    G. W. Moore and N. Seiberg,Naturality in Conformal Field Theory,Nucl. Phys. B313(1989) 16–40

    work page 1989

  79. [79]

    G. W. Moore and N. Seiberg,Lectures on RCFT,in Strings ’89, Proceedings of the Trieste Spring School on Superstrings.World Scientific, 1990.http://www. physics.rutgers.edu/~gmoore/LecturesRCFT.pdf

    work page 1990

  80. [80]

    Li and F

    H. Li and F. D. M. Haldane,Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states,Physical Review Letters101(Jul, 2008)

    work page 2008

Showing first 80 references.