pith. machine review for the scientific record. sign in

arxiv: 2604.14275 · v2 · submitted 2026-04-15 · ✦ hep-th · quant-ph

Recognition: unknown

Generalized Complexity Distances and Non-Invertible Symmetries

Jonathan J. Heckman , Rebecca J. Hicks , Chitraang Murdia
Authors on Pith no claims yet

Pith reviewed 2026-05-10 12:40 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords non-invertible symmetriesquantum gatescomplexity distancesquantum field theorysymmetry categoriespost-selectiondistinguishability measures
0
0 comments X

The pith

Non-invertible symmetries in QFTs define quantum gates that include post-selection and admit generalized complexity distances.

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

The paper establishes that operations from non-invertible symmetries on quantum states function as gates in a parallel quantum computation scheme in which post-selection or projection counts as one of the allowed gates. This interpretation lets standard notions of gate complexity and geometric complexity measures carry over from ordinary unitary symmetries to the non-invertible setting. A family of distance and distinguishability measures is introduced that applies uniformly to continuous and discrete non-invertible symmetries as well as to linear combinations of unitary gates and reduces to the familiar Lie-group distances whenever the symmetry is invertible. The construction is illustrated by explicit calculations of distances between non-invertible symmetries in selected four-dimensional and two-dimensional quantum field theories. The results indicate that the simple objects of a symmetry category frequently correspond to high computational complexity.

Core claim

Non-invertible symmetries of a quantum field theory define a collection of quantum gates for a parallel quantum computation scheme that includes post-selection and projection as a gate. Gate complexity and geometric complexity measures generalize directly to this setting. A class of distance and distinguishability measures extends the standard Lie-group notion to continuous and discrete non-invertible symmetries as well as to more general linear combinations of unitary gates. Computations in four-dimensional and two-dimensional examples show that the simple objects of a symmetry category can be highly complex computationally.

What carries the argument

The mapping of non-invertible symmetry operations on states to a set of quantum gates that explicitly includes post-selection, which supplies the algebraic structure needed to define generalized complexity distances.

If this is right

  • Gate complexity can be assigned to non-invertible symmetry operations on states.
  • Distance measures apply uniformly to both invertible and non-invertible symmetries.
  • Post-selection is incorporated as a standard gate in the symmetry-based computation scheme.
  • Simple objects in symmetry categories of four- and two-dimensional QFTs can require high numbers of gates.

Where Pith is reading between the lines

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

  • The same distance construction may supply a uniform language for comparing the computational cost of different symmetry structures across quantum theories.
  • If the measures remain well-defined under renormalization-group flow, they could quantify how complexity of symmetries changes between ultraviolet and infrared descriptions.

Load-bearing premise

The algebraic structure of non-invertible symmetries still permits a consistent definition of gates and distance measures that reduce to the usual Lie-group case when the symmetry is invertible.

What would settle it

An explicit distance calculation in a model containing both invertible and non-invertible symmetries in which the proposed measure fails to recover the known Lie-group distance for the invertible subset would falsify the generalization.

Figures

Figures reproduced from arXiv: 2604.14275 by Chitraang Murdia, Jonathan J. Heckman, Rebecca J. Hicks.

Figure 1
Figure 1. Figure 1: Depiction of a set of parallel quantum computations by unitary operators [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Depiction of the arc distance vs chord distance for operators, viewed as elements [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Depiction of the (D + 1)-dimensional bulk Symmetry Theory / Symmetry Topo￾logical Field Theory Hilbert space interpretation of symmetry operators for a D-dimensional QFT. A bulk topological operator X acts on a state |Ψ⟩ of the bulk. In this depiction the radial direction serves as a time coordinate for a Euclidean boundary system [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Depiction of topological symmetry operator [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Distance between the Lm operators in the sub (2)k CFT at level k = 3. Consider next the high temperature limit β → 0. In this limit, we can use the modular S-transformation to express the character as: χi(τ ) = X j Sijχj (−1/τ ). (4.22) The sum becomes dominated by the lowest weight term at high temperature: χi(τ ) ≈ Si1χ1(−1/τ ). (4.23) Inserting this back into the trace, the character can be pulled out o… view at source ↗
Figure 6
Figure 6. Figure 6: Distance between the Lm operators in the sub (2)k CFT at level k = 4. The primary states are labeled as |r, s⟩ with 1 ≤ r ≤ p − 1 and 1 ≤ s ≤ p ′ − 1. In our conventions, the conformal weight is: hrs = (ps − p ′ r) 2 − (p − p ′ ) 2 4pp′ , (4.26) The characters are given by χ(r,s)(τ ) = 1 η(τ ) X j∈Z h q (2pp′ j+pr−p ′s) 2/4pp′ − q (2pp′ j+pr+p ′s) 2/4pp′ i (4.27) where the (r, s)-indices satisfy the restri… view at source ↗
Figure 7
Figure 7. Figure 7: Distance between the L(r,s) operators in the Ising CFT, (p, p′ ) = (4, 3). where ε = [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Distance between the Li operators in the S3 orbifold CFT with the seed theory being a free fermion CFT. Here, the line operator L1 corresponds to the trivial representation, L2 to the signed representation, and L3 corresponds to the standard representation. 5 Conclusions In this paper, we have shown that non-invertible symmetries can be interpreted as a special class of LCUs, that is, linear combinations o… view at source ↗
read the original abstract

Non-invertible symmetries of a quantum field theory (QFT) are a natural generalization of unitary symmetries, but in which the product of operators does not satisfy a group multiplication law. We show that such symmetry operations on states define a collection of quantum gates for a parallel quantum computation scheme that includes post-selection / projection as a gate. Structures such as gate complexity and more geometric complexity measures generalize to this setting. We provide a class of distance / distinguishability measures that extend the standard notion of distance for Lie groups to both continuous and discrete non-invertible symmetries, as well as more general linear combinations of unitary quantum gates. We illustrate these considerations by computing the distance between non-invertible symmetries in some 4D and 2D QFTs. We find that the simple objects of a symmetry category can be highly complex computationally.

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

0 major / 3 minor

Summary. The manuscript proposes treating non-invertible symmetry operators in QFT as quantum gates (including projections/post-selection) within a parallel computation framework. It generalizes notions of gate complexity and geometric complexity measures, introduces a class of distance/distinguishability functions that extend Lie-group distances to continuous/discrete non-invertible symmetries and linear combinations of unitary gates, illustrates the constructions via explicit distance computations in selected 4D and 2D QFT models, and concludes that simple objects of symmetry categories can be computationally complex.

Significance. If the claimed generalizations are robust, the work supplies a concrete bridge between non-invertible symmetries, quantum information, and complexity theory, offering new quantitative tools for distinguishing symmetry operators. The provision of explicit distance calculations in concrete 4D and 2D models is a positive feature that grounds the abstract constructions.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction should explicitly name the concrete 4D and 2D QFT models in which distances are computed, so that readers can immediately assess the scope of the illustrations.
  2. [Section defining distance measures] Notation for the generalized distance functions should be introduced with a clear comparison table or equation block showing the reduction to the standard Lie-group distance when the symmetry is invertible.
  3. [Section on quantum gates and post-selection] A brief discussion of potential ambiguities in the choice of post-selection gates or the precise definition of 'parallel computation scheme' would help readers reproduce the complexity assignments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. The assessment correctly captures our proposal to treat non-invertible symmetry operators as quantum gates (including projections) and the generalization of complexity and distance measures, together with the explicit computations in 4D and 2D models.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines generalized distance and complexity measures by extending standard Lie-group notions to non-invertible symmetry operators treated as quantum gates (including projections). This extension is constructed explicitly from algebraic structures of symmetry categories and reduces to the invertible case by design of the generalization, without any fitted parameters, self-referential definitions, or load-bearing self-citations that collapse the central claims. The computational complexity statement for simple objects follows directly from applying the new distance definitions to the category structure. No equations or steps reduce by construction to prior outputs within the paper; the work builds on external QFT and quantum-information concepts in a self-contained manner.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract relies on standard background from QFT symmetry categories and quantum-circuit complexity without introducing new free parameters, ad-hoc axioms, or postulated entities.

pith-pipeline@v0.9.0 · 5444 in / 1249 out tokens · 50755 ms · 2026-05-10T12:40:37.583134+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

91 extracted references · 84 canonical work pages · 4 internal anchors

  1. [1]

    Gruppentheorie und ihre Anwendung auf die Quanten mechanik der Atomspektren,

    E. P. Wigner, “Gruppentheorie und ihre Anwendung auf die Quanten mechanik der Atomspektren,”Vieweg Verlag, Braunschweig(1931) 251–254

    1931

  2. [2]

    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 arXiv 2015

  3. [3]

    What’s Done CannotBe Undone: TASILectures on Non-InvertibleSymmetries,

    S.-H. Shao, “What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries,” inTheoretical Advanced Study Institute in Elementary Particle Physics 2023: Aspects of Symmetry. 8, 2023.arXiv:2308.00747 [hep-th]

    work page arXiv 2023

  4. [4]

    Noninvertible Symmetries Act Locally by Quantum Operations,

    M. Okada and Y. Tachikawa, “Non-invertible symmetries act locally by quantum operations,”Phys. Rev. Lett.133no. 19, (2024) 191602,arXiv:2403.20062 [hep-th]

    work page arXiv 2024

  5. [5]

    Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities,

    T. Bartsch, Y. Gai, and S. Schafer-Nameki, “Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities,”arXiv:2602.07110 [quant-ph]

    work page arXiv

  6. [6]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information. Cambridge University Press, 6, 2012

    2012

  7. [7]

    Quantum computing, postselection, and probabilistic polynomial-time,

    S. Aaronson, “Quantum computing, postselection, and probabilistic polynomial-time,” Proc. Roy. Soc. Lond. A461no. 2063, (2005) 3473–3482,arXiv:quant-ph/0412187

    work page arXiv 2063

  8. [8]

    A geometric approach to quantum circuit lower bounds,

    M. A. Nielsen, “A geometric approach to quantum circuit lower bounds,”Quant. Inf. Comput.6no. 3, (2006) 213–262,arXiv:quant-ph/0502070

    work page arXiv 2006

  9. [9]

    Nielsen, M.R

    M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, “Quantum Computation as Geometry,”Science311no. 5764, (2006) 1133–1135,arXiv:quant-ph/0603161

    work page arXiv 2006

  10. [10]

    Optimal Control, Geometry, and Quantum Computing,

    M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, “Optimal Control, Geometry, and Quantum Computing,”Phys. Rev. A73no. 6, (2006) 062323, arXiv:quant-ph/0603160

    work page arXiv 2006

  11. [11]

    Dowling and Michael A

    M. R. Dowling and M. A. Nielsen, “The Geometry of Quantum Computation,”Quant. Inf. Comput.8no. 10, (2008) 0861–0899,arXiv:quant-ph/0701004

    work page arXiv 2008

  12. [12]

    Nielsen , Date-Added =

    M. Gu, A. Doherty, and M. Nielsen, “Quantum Control via Geometry: An explicit example,”arXiv:0808.3212 [quant-ph]

    work page arXiv

  13. [13]

    The Python’s Lunch: geometric obstructions to decoding Hawking radiation,

    A. R. Brown, H. Gharibyan, G. Penington, and L. Susskind, “The Python’s Lunch: geometric obstructions to decoding Hawking radiation,”JHEP08(2020) 121, arXiv:1912.00228 [hep-th]. 34

    work page arXiv 2020

  14. [14]

    Geometric Measures of Complexity for Open and Closed Quantum Systems,

    A. Acevedo and A. Falco, “Geometric Measures of Complexity for Open and Closed Quantum Systems,”arXiv:2507.18440 [quant-ph]

    work page arXiv

  15. [15]

    Gaiotto and T

    D. Gaiotto and T. Johnson-Freyd, “Condensations in Higher Categories,” arXiv:1905.09566 [math.CT]

    work page arXiv 1905

  16. [16]

    Roumpedakis, S

    K. Roumpedakis, S. Seifnashri, and S.-H. Shao, “Higher Gauging and Non-invertible Condensation Defects,”Commun. Math. Phys.401no. 3, (2023) 3043–3107, arXiv:2204.02407 [hep-th]

    work page arXiv 2023

  17. [17]

    Means and convex combinations of unitary operators,

    G. K. Pedersen and R. V. Kadison, “Means and convex combinations of unitary operators,”Mathematica Scandinavica57(1985) 249–266

    1985

  18. [18]

    Representation Theory of Solitons,

    C. Cordova, N. Holfester, and K. Ohmori, “Representation theory of solitons,”JHEP 06(2025) 001,arXiv:2408.11045 [hep-th]

    work page arXiv 2025

  19. [19]

    Interacting anyons in topological quantum liquids: The golden chain

    A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: The golden chain,” Phys. Rev. Lett.98(2007) 160409,arXiv:cond-mat/0612341

    work page Pith review arXiv 2007

  20. [20]

    Hamiltonian Simulation Using Linear Com- binations of Unitary Operations

    A. M. Childs and N. Wiebe, “Hamiltonian Simulation Using Linear Combinations of Unitary Operations,”Quant. Inf. Comput.12no. 11 & 12, (2012) 0901–0924, arXiv:1202.5822 [quant-ph]

    work page arXiv 2012

  21. [21]

    Hamiltonian Simulation by Qubitization

    G. H. Low and I. L. Chuang, “Hamiltonian Simulation by Qubitization,”Quantum3 (2019) 163,arXiv:1610.06546 [quant-ph]

    work page Pith review arXiv 2019

  22. [22]

    An algorithm for the T-count,

    D. Gosset, V. Kliuchnikov, M. Mosca, and V. Russo, “An algorithm for the T-count,” arXiv:1308.4134 [quant-ph]

    work page arXiv

  23. [23]

    Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity,

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, “Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity,”Phys. Rev. X8no. 4, (2018) 041015,arXiv:1805.03662 [quant-ph]

    work page arXiv 2018

  24. [24]

    Quantum amplitude amplification and estimation.Contemporary Mathematics, 305:53–74, 2002

    G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, “Quantum amplitude amplification and estimation,”arXiv:quant-ph/0005055

    work page arXiv

  25. [25]

    Polynomial Equivalence of Complexity Geometries,

    A. R. Brown, “Polynomial Equivalence of Complexity Geometries,”Quantum8(2024) 1391,arXiv:2205.04485 [quant-ph]

    work page arXiv 2024

  26. [26]

    Brown, D.A

    A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, “Holographic Complexity Equals Bulk Action?,”Phys. Rev. Lett.116no. 19, (2016) 191301, arXiv:1509.07876 [hep-th]

    work page arXiv 2016

  27. [27]

    Second law of quantum complexity,

    A. R. Brown and L. Susskind, “Second law of quantum complexity,”Phys. Rev. D97 no. 8, (2018) 086015,arXiv:1701.01107 [hep-th]. 35

    work page arXiv 2018

  28. [28]

    An Extension of Kakutani’s Theorem on Infinite Product Measures to the Tensor Product of Semifiniteω ∗-Algebras,

    Bures, Donald, “An Extension of Kakutani’s Theorem on Infinite Product Measures to the Tensor Product of Semifiniteω ∗-Algebras,”Transactions of the American Mathematical Society135(1969) 199

    1969

  29. [29]

    Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space,

    N. Seiberg, S. Seifnashri, and S.-H. Shao, “Non-Invertible Symmetries and LSM-Type Constraints on a Tensor Product Hilbert Space,”SciPost Phys.16(2024) 154, arXiv:2401.12281 [cond-mat.str-el]

    work page arXiv 2024

  30. [30]

    Invariants of three manifolds via link polynomials and quantum groups,

    N. Reshetikhin and V. G. Turaev, “Invariants of three manifolds via link polynomials and quantum groups,”Invent. Math.103(1991) 547–597

    1991

  31. [31]

    State sum invariants of 3 manifolds and quantum 6j symbols,

    V. G. Turaev and O. Y. Viro, “State sum invariants of 3 manifolds and quantum 6j symbols,”Topology31(1992) 865–902

    1992

  32. [32]

    Invariants of piecewise linear three manifolds,

    J. W. Barrett and B. W. Westbury, “Invariants of piecewise linear three manifolds,” Trans. Am. Math. Soc.348(1996) 3997–4022,arXiv:hep-th/9311155

    work page arXiv 1996

  33. [33]

    Witten,AdS / CFT correspondence and topological field theory,JHEP12(1998) 012 [hep-th/9812012]

    E. Witten, “AdS / CFT correspondence and topological field theory,”JHEP12 (1998) 012,arXiv:hep-th/9812012

    work page arXiv 1998

  34. [34]

    TFT construction of RCFT correlators I: Partition functions

    J. Fuchs, I. Runkel, and C. Schweigert, “TFT construction of RCFT correlators 1. Partition functions,”Nucl. Phys. B646(2002) 353–497,arXiv:hep-th/0204148

    work page Pith review arXiv 2002

  35. [35]

    Turaev-Viro invariants as an extended TQFT,

    A. K. Jr. and B. Balsam, “Turaev-Viro invariants as an extended TQFT,” arXiv:1004.1533 [math.GT]

    work page arXiv

  36. [36]

    Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory

    A. Kapustin and N. Saulina, “Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory,”arXiv:1012.0911 [hep-th]

    work page arXiv

  37. [37]

    Models for gapped boundaries and domain walls

    A. Kitaev and L. Kong, “Models for Gapped Boundaries and Domain Walls,” Commun. Math. Phys.313no. 2, (2012) 351–373,arXiv:1104.5047 [cond-mat.str-el]

    work page Pith review arXiv 2012

  38. [38]

    Bicategories for boundary conditions and for surface defects in 3-d TFT,

    J. Fuchs, C. Schweigert, and A. Valentino, “Bicategories for boundary conditions and for surface defects in 3-d TFT,”Commun. Math. Phys.321(2013) 543–575, arXiv:1203.4568 [hep-th]

    work page arXiv 2013

  39. [39]

    Relative quantum field theory

    D. S. Freed and C. Teleman, “Relative quantum field theory,”Commun. Math. Phys. 326(2014) 459–476,arXiv:1212.1692 [hep-th]

    work page Pith review arXiv 2014

  40. [40]

    Kong and X.-G

    L. Kong and X.-G. Wen, “Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions,”arXiv:1405.5858 [cond-mat.str-el]

    work page arXiv

  41. [41]

    Kong, X.-G

    L. Kong, X.-G. Wen, and H. Zheng, “Boundary-bulk relation in topological orders,” Nucl. Phys. B922(2017) 62–76,arXiv:1702.00673 [cond-mat.str-el]. 36

    work page arXiv 2017

  42. [42]

    6D Fractional Quantum Hall Effect,

    J. J. Heckman and L. Tizzano, “6D Fractional Quantum Hall Effect,”JHEP05(2018) 120,arXiv:1708.02250 [hep-th]

    work page arXiv 2018

  43. [43]

    Topological dualities in the Ising model,

    D. S. Freed and C. Teleman, “Topological dualities in the Ising model,”Geom. Topol. 26(2022) 1907–1984,arXiv:1806.00008 [math.AT]

    work page arXiv 2022

  44. [44]

    Orbifold groupoids,

    D. Gaiotto and J. Kulp, “Orbifold groupoids,”JHEP02(2021) 132, arXiv:2008.05960 [hep-th]

    work page arXiv 2021

  45. [45]

    Algebraic higher symmetry and categorical symmetry – a holographic and entanglement view of symmetry,

    L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng, “Algebraic higher symmetry and categorical symmetry – a holographic and entanglement view of symmetry,”Phys. Rev. Res.2no. 4, (2020) 043086,arXiv:2005.14178 [cond-mat.str-el]

    work page arXiv 2020

  46. [46]

    Symmetry TFTs from string theory,

    F. Apruzzi, F. Bonetti, I. Garc´ ıa Etxebarria, S. S. Hosseini, and S. Schafer-Nameki, “Symmetry TFTs from String Theory,”arXiv:2112.02092 [hep-th]

    work page arXiv

  47. [47]

    Topological symmetry in quantum field theory,

    D. S. Freed, G. W. Moore, and C. Teleman, “Topological symmetry in quantum field theory,”arXiv:2209.07471 [hep-th]

    work page arXiv

  48. [48]

    Kaidi, K

    J. Kaidi, K. Ohmori, and Y. Zheng, “Symmetry TFTs for Non-Invertible Defects,” arXiv:2209.11062 [hep-th]

    work page arXiv

  49. [49]

    The holography of non-invertible self-duality symmetries

    A. Antinucci, F. Benini, C. Copetti, G. Galati, and G. Rizi, “The holography of non-invertible self-duality symmetries,”arXiv:2210.09146 [hep-th]

    work page arXiv

  50. [50]

    Intermediate defect groups, polarization pairs, and noninvertible duality defects,

    C. Lawrie, X. Yu, and H. Y. Zhang, “Intermediate defect groups, polarization pairs, and noninvertible duality defects,”Phys. Rev. D109no. 2, (2024) 026005, arXiv:2306.11783 [hep-th]

    work page arXiv 2024

  51. [51]

    SymTrees and Multi-Sector QFTs,

    F. Baume, J. J. Heckman, M. H¨ ubner, E. Torres, A. P. Turner, and X. Yu, “SymTrees and Multi-Sector QFTs,”arXiv:2310.12980 [hep-th]

    work page arXiv

  52. [52]

    Yu, Noninvertible symmetries in 2D from type IIB string theory, Phys

    X. Yu, “Non-invertible Symmetries in 2D from Type IIB String Theory,” arXiv:2310.15339 [hep-th]

    work page arXiv

  53. [53]

    Brennan and Z

    T. D. Brennan and Z. Sun, “A SymTFT for continuous symmetries,”JHEP12(2024) 100,arXiv:2401.06128 [hep-th]

    work page arXiv 2024

  54. [54]

    Antinucci and F

    A. Antinucci and F. Benini, “Anomalies and gauging of U(1) symmetries,”Phys. Rev. B111no. 2, (2025) 024110,arXiv:2401.10165 [hep-th]

    work page arXiv 2025

  55. [55]

    On the symmetry TFT of Yang-Mills-Chern-Simons theory,

    R. Argurio, F. Benini, M. Bertolini, G. Galati, and P. Niro, “On the symmetry TFT of Yang-Mills-Chern-Simons theory,”JHEP07(2024) 130,arXiv:2404.06601 [hep-th]. 37

    work page arXiv 2024

  56. [56]

    Franco and X

    S. Franco and X. Yu, “Generalized symmetries in 2D from string theory: SymTFTs, intrinsic relativeness, and anomalies of non-invertible symmetries,”JHEP11(2024) 004,arXiv:2404.19761 [hep-th]

    work page arXiv 2024

  57. [57]

    Celestial Topology, Symmetry Theories, and Evidence for a NonSUSY D3-Brane CFT,

    J. J. Heckman and M. H¨ ubner, “Celestial Topology, Symmetry Theories, and Evidence for a NonSUSY D3-Brane CFT,”Fortsch. Phys.73no. 4, (2025) 2400270, arXiv:2406.08485 [hep-th]

    work page arXiv 2025

  58. [58]

    Gagliano and I

    F. Gagliano and I. Garc´ ıa Etxebarria, “SymTFTs forU(1) symmetries from descent,” arXiv:2411.15126 [hep-th]

    work page arXiv

  59. [59]

    Cornering relative symmetry theories,

    M. Cvetiˇ c, R. Donagi, J. J. Heckman, M. H¨ ubner, and E. Torres, “Cornering relative symmetry theories,”Phys. Rev. D111no. 8, (2025) 085026,arXiv:2408.12600 [hep-th]

    work page arXiv 2025

  60. [60]

    Boundary SymTFT,

    L. Bhardwaj, C. Copetti, D. Pajer, and S. Schafer-Nameki, “Boundary SymTFT,” SciPost Phys.19no. 2, (2025) 061,arXiv:2409.02166 [hep-th]

    work page arXiv 2025

  61. [61]

    Bonetti, M

    F. Bonetti, M. Del Zotto, and R. Minasian, “SymTFTs for Continuous non-Abelian Symmetries,”arXiv:2402.12347 [hep-th]

    work page arXiv

  62. [62]

    Apruzzi, F

    F. Apruzzi, F. Bedogna, and N. Dondi, “SymTh for non-finite symmetries,” arXiv:2402.14813 [hep-th]

    work page arXiv

  63. [63]

    Gauging in parameter space: A top-down perspective,

    X. Yu, “Gauging in parameter space: A top-down perspective,”Phys. Rev. D112 no. 2, (2025) 025020,arXiv:2411.14997 [hep-th]

    work page arXiv 2025

  64. [64]

    Symmetry Topological Field Theory for Flavor Symmetry

    Q. Jia, R. Luo, J. Tian, Y.-N. Wang, and Y. Zhang, “Symmetry Topological Field Theory for Flavor Symmetry,”arXiv:2503.04546 [hep-th]

    work page arXiv

  65. [65]

    SymTFT construction of gapless exotic-foliated dual models

    F. Apruzzi, F. Bedogna, and S. Mancani, “SymTFT construction of gapless exotic-foliated dual models,”arXiv:2504.11449 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv

  66. [66]

    Symmetry Theories, Wigner’s Function, Compactification, and Holography,

    J. J. Heckman, M. H¨ ubner, and C. Murdia, “Symmetry Theories, Wigner’s Function, Compactification, and Holography,”arXiv:2505.23887 [hep-th]

    work page arXiv

  67. [67]

    Luo, Y.-N

    R. Luo, Y.-N. Wang, and Z. Bi, “Topological Holography for Mixed-State Phases and Phase Transitions,”arXiv:2507.06218 [cond-mat.str-el]

    work page arXiv

  68. [68]

    SymTFT Entanglement and Holographic (Non)-Factorization

    E. Torres and X. Yu, “SymTFT Entanglement and Holographic (Non)-Factorization,” arXiv:2510.06319 [hep-th]

    work page arXiv

  69. [69]

    On the SymTFTs of Finite Non-Abelian Symmetries

    O. Bergman, J. J. Heckman, M. H¨ ubner, D. Migliorati, X. Yu, and H. Y. Zhang, “On the SymTFTs of Finite Non-Abelian Symmetries,”arXiv:2603.12323 [hep-th]. 38

    work page internal anchor Pith review Pith/arXiv arXiv

  70. [70]

    Heidenreich, J

    B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius, and I. Valenzuela, “Non-invertible global symmetries and completeness of the spectrum,”JHEP09 (2021) 203,arXiv:2104.07036 [hep-th]

    work page arXiv 2021

  71. [71]

    On the holographic dual of a symmetry operator at finite temperature,

    J. J. Heckman, M. H¨ ubner, and C. Murdia, “On the holographic dual of a symmetry operator at finite temperature,”Phys. Rev. D112no. 12, (2025) 126010, arXiv:2503.16427 [hep-th]

    work page arXiv 2025

  72. [72]

    Fusion Rules and Modular Transformations in 2D Conformal Field Theory,

    E. P. Verlinde, “Fusion Rules and Modular Transformations in 2D Conformal Field Theory,”Nucl. Phys. B300(1988) 360–376

    1988

  73. [73]

    CFT, BCFT, ADE and all that,

    J.-B. Zuber, “CFT, BCFT, ADE and all that,” inSchool: Bariloche 2000: Quantum Symmetries in Theoretical Physics and Mathematics. 1, 2000.arXiv:hep-th/0006151

    work page internal anchor Pith review arXiv 2000

  74. [74]

    A-D-E Classification of Conformal Field Theories,

    A. Cappelli and J.-B. Zuber, “A-D-E Classification of Conformal Field Theories,” Scholarpedia5no. 4, (2010) 10314,arXiv:0911.3242 [hep-th]

    work page arXiv 2010

  75. [75]

    Topological Defect Lines and Renormalization Group Flows in Two Dimensions

    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, “Topological Defect Lines and Renormalization Group Flows in Two Dimensions,”JHEP01(2019) 026, arXiv:1802.04445 [hep-th]

    work page Pith review arXiv 2019

  76. [76]

    Kramers-Wannier duality from conformal defects

    J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, “Kramers-Wannier duality from conformal defects,”Phys. Rev. Lett.93(2004) 070601,arXiv:cond-mat/0404051

    work page Pith review arXiv 2004

  77. [77]

    Duality and defects in rational conformal field theory

    J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, “Duality and Defects in Rational Conformal Field Theory,”Nucl. Phys. B763(2007) 354–430,arXiv:hep-th/0607247

    work page Pith review arXiv 2007

  78. [78]

    Fusion category symmetry. Part I. Anomaly in-flow and gapped phases,

    R. Thorngren and Y. Wang, “Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases,”JHEP04(2024) 132,arXiv:1912.02817 [hep-th]

    work page arXiv 2024

  79. [79]

    Generalised twisted partition functions

    V. B. Petkova and J. B. Zuber, “Generalized twisted partition functions,”Phys. Lett. B504(2001) 157–164,arXiv:hep-th/0011021

    work page Pith review arXiv 2001

  80. [80]

    Microscopic Origin of the Bekenstein-Hawking Entropy

    A. Strominger and C. Vafa, “Microscopic Origin of the Bekenstein-Hawking Entropy,” Phys. Lett. B379(1996) 99–104,arXiv:hep-th/9601029

    work page Pith review arXiv 1996

Showing first 80 references.