Pith. sign in

REVIEW 3 major objections 5 minor 2 cited by

On the Physics of Higher Condensation Defects

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Symmetry defects of a quantum field theory satisfy the definition of a higher fusion category, with every idempotent condensation defect splitting into a gauging and an ungauging interface.

desk verdict A serious physics-first construction of higher condensation defects with explicit 4d Lagrangian computations, but the central Karoubi-completeness claim remains conditional because the paper asserts, rather than proves, that condensation defects are the only idempotent defects. read the letter →

arxiv 2506.04346 v1 pith:546UK5Z2 submitted 2025-06-04 hep-th math-phmath.MP

Ibrahima Bah , Enoch Leung , Thomas Waddleton This is my paper
classification hep-thmath-phmath.MP MSC 81T4518M20
keywords highercondensationdefectsgaugingKaroubicompletenessfusionn-categoriesgeneralizedsymmetriestopologicalcategoricaldecoupledTQFTcoefficients
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the topological symmetry defects of a quantum field theory form a higher fusion category in the sense of the definition proposed in [44], with the hardest requirement being Karoubi completeness: every idempotent defect must factor through a pair of interfaces into an intermediate theory. For finite Abelian $p$-form symmetries, the authors show that each higher condensation defect splits as $C = \iota \otimes \rho$ into a gauging interface and an ungauging interface, with the reverse composition condensing to the identity of the gauged theory. The demonstration is carried out explicitly in a four-dimensional theory with an anomaly-free $\mathbb{Z}_N^{(1)}$ symmetry, where iterated higher gaugings generate a tree of condensation defects of every codimension, and then extended to arbitrary spacetime dimension and to anomalous symmetries. The authors explicitly disclaim proving that every QFT has this structure, and instead present it as natural structure arising in the finite Abelian case.

What carries the argument

The central objects are condensation defects: topological defects built by summing the generators of a finite Abelian $p$-form symmetry subgroup over a submanifold, with a continuum path-integral description as a Dijkgraaf-Witten-style gauge theory localized on the defect's worldvolume. The argument is carried by the splitting identity $C_G^{[n]} = \iota \otimes \rho$, where $\iota$ and $\rho$ are soft interfaces, anomaly-inflow interfaces that identify the common subgroup of symmetry generators across a boundary, realized geometrically by cylinder gauging, in which a cylinder of the ordinarily gauged theory is nucleated between the two interfaces. Idempotence of the defects, expressed by the fusion rule $C \otimes C = |\mathcal{T}_K|\, C$ with a decoupled TQFT as coefficient, is what marks them as the idempotent morphisms whose splitting witnesses Karoubi completeness.

What would settle it

Find one idempotent topological defect in a QFT with finite Abelian symmetry that is not a condensation defect built by higher gauging, for instance a direct sum or a composite of condensation defects or a defect of independent origin, and check whether it splits into interfaces; if it does not split, Karoubi completeness fails. A second probe is the Chern-Simons example with $pN = nK^2$: the paper predicts the 1-condensation defect $C_K$ stays idempotent and splittable there despite the 0-anomaly, so one can test directly whether the cylinder of gauged theory required for the splitting exists for those parameters.

Watch

Extended reading notes

Core claim

The paper's central claim is that the symmetry defects of a physical QFT, namely finite Abelian $p$-form symmetries together with the hierarchy of condensation defects built from them, naturally satisfy the definition of a separable weak higher fusion category, and its load-bearing content is Karoubi completeness. Concretely, each idempotent condensation defect factorizes as $C_G^{[n]} = \iota \otimes \rho$ into a pair of gauging and ungauging interfaces, with $\rho \otimes \iota$ condensing to the identity interface of the intermediate theory, as in equations (3.22) and (5.7)--(5.8). The authors verify this by explicit Lagrangian computation in a four-dimensional theory with an anomaly-free $\mathbb{Z}_N^{(1)}$ symmetry, constructing condensation defects of every codimension by successive higher gauging and showing each splits through a cylinder of gauged theory; they then generalize to arbitrary dimensions and arbitrary anomaly-free finite Abelian symmetries. For anomalous symmetries, they argue that a condensation defect is idempotent, and hence splittable, precisely when the relevant anomaly is trivializable, so the category structure constrains rather than breaks down in the anomalous cases.

Load-bearing premise

The argument assumes, without proof, that the condensation defects built by higher gauging are the only idempotent topological defects in the symmetry category; if other idempotents, such as direct sums or composites of condensation defects, exist, their splittings would also be needed to establish Karoubi completeness.

Editorial extensions

If this is right

  • Every global form of a theory related by gauging subgroups is encoded inside the original theory: a 1-condensation defect splits into interfaces to $T/K^{(p)}$, so ordinary gauging appears as categorical data of End$(T)$.
  • Physical higher gauging ($q$-gauging) is a special case of categorical condensation, with a $q$-gauging inducing a condensation at the level of $(q-1)$-morphisms.
  • Fusion coefficients for higher-dimensional defects are decoupled TQFTs whose effective size is a homotopy cardinality, typically a rational number rather than an integer, reconciling the fusion rules with the counting of gauge redundancies.
  • An anomalous symmetry obstructs idempotence: condensation defects built from a symmetry with a non-trivial relevant anomaly fail to be idempotent and so impose no splitting obligation, while the anomaly-free cases split as required.
  • Condensation defects are self-dual via the condensation maps, which together with finiteness and semisimplicity completes the full dualizability requirement of a fusion category.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test of the completeness premise would be to enumerate all idempotent topological defects in a lattice realization of $\mathbb{Z}_N$ gauge theory and check whether any lie outside the condensation-defect tree; the paper's conclusion depends on there being none.
  • If the splitting picture holds generally, SymTFT boundary conditions should organize into exactly the nested network described by the condensation tree, giving a concrete interval-compactification signature to look for in holographic models.
  • The decoupled-TQFT coefficients suggest that fusion data of higher symmetries should be computed by homotopy cardinality rather than integer channel counting; one could test this by computing a fusion coefficient in a gapped model via the partition function (4.16) and comparing with the categorical dimension.
  • Extending the argument to non-Abelian finite groups would require replacing Pontryagin duals with Tannaka duals $\mathrm{Rep}(G)$; the splitting conjecture then predicts that non-Abelian condensation defects factor through interfaces to theories with $\mathrm{Rep}(G)$ symmetry.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies topological defects for finite Abelian p-form symmetries and argues on physical grounds that they satisfy Johnson-Freyd's definition of a separable weak higher fusion category, with emphasis on Karoubi completeness. The authors give a concrete 4d example with a Z_N^(1) symmetry, construct a tree of nested higher condensation defects, and verify that each condensation defect C_G^[n] splits as ι ⊗ ρ into gauging and ungauging interfaces via explicit path integral computations (soft and hard interfaces, cylinder gauging). They then extend this discussion to arbitrary spacetime dimensions and anomaly-free finite Abelian symmetries, and analyze how anomalies modify idempotence and splittability. They also comment on decoupled TQFTs as fusion coefficients and on the categorical role of anomalies.

Significance. If the central claim holds, the paper would provide a physical realization of Johnson-Freyd's higher fusion category axioms, organizing condensation defects, gauging interfaces, and anomalies into a single categorical framework. The explicit Lagrangian computations in Section 3, including the splitting of C[2]_dotK in Eqs. (3.19)-(3.22) and the cylinder gauging argument, are detailed and reproducible, and the 4d tree of condensation defects is a useful concrete illustration. The anomaly analysis via the transgression map (Section 6.1) offers a testable general framework. However, the full-scope claim is not established because Karoubi completeness requires a classification of all idempotent topological defects, and the paper asserts rather than proves that condensation defects are the only ones.

major comments (3)
  1. [4.3] The sentence "As the condensation defects are the only idempotent topological defects in C, we thus see that both C and End(T) are Karoubi-complete" is the single load-bearing step for the paper's main claim, but the uniqueness assertion is not proved. Sections 2 and 3 construct splittings for the specific defects C_K^[n] (Eq. (3.22)), and Section 5 extends this to a tree of condensation defects, but no argument rules out other idempotent m-morphisms, such as direct sums of condensation defects, composites, or idempotents built from non-condensation defects. Since Karoubi completeness requires every idempotent to split for every 1 ≤ m ≤ d, the conclusion does not follow without such a classification. Section 7's own disclaimer that the paper is not a rigorous proof and that counterexamples to fusion-category descriptions exist further indicates that this gap is substantive, not merely a matter of presentation.
  2. [5] The generalization to arbitrary spacetime dimension d and arbitrary anomaly-free finite Abelian A^(p) is stated rather than demonstrated. The paper gives explicit path integral computations only in the 4d example (Sections 2-3); for general d it provides formal sums (5.3), (5.12), and says the splitting and idempotence arguments are "completely analogous" (Section 5.2) and that "the rest follows inductively" (Section 5.3). The splitting of C_G^[n] into interfaces requires constructing the relevant gauging/ungauging interfaces in all codimensions, and this is not shown. Since Karoubi completeness of C for all d is one of the paper's headline claims, this is a load-bearing gap. The claim could be saved either by supplying the general construction or by explicitly restricting the main theorem to the 4d example and leaving the general case as conjectural.
  3. [6.1 and 6.2] The treatment of anomalies inherits the same uniqueness gap. In Section 6.2, the conclusion that for a mixed anomaly the category remains Karoubi-complete because the only idempotent 1-condensation defects are the products in (6.24) again depends on the unproved assertion that these are the only idempotents. Additionally, the general claim in Section 6.1 that "the only possible q-anomalies are given by the image of τ_q" is justified only by a consistency argument (the paragraph after Eq. (6.10)) and is not established; this is a separate but related load-bearing point for the anomaly analysis.
minor comments (5)
  1. [4.3, Eq. (4.7)] In the definition of direct sums, the condition ρ_i ∘ ι_j ≅ δ_{ij} 1_{X_i} uses δ_{ij} as a morphism, but for i ≠ j the right-hand side should be the zero morphism, not 1_{X_i}; this should be corrected for clarity.
  2. [6.2] The notation U(1)_{pN} for Chern-Simons theory is not defined; please specify that it denotes the U(1) Chern-Simons theory at level pN.
  3. [Figures 3, 5, 6] The captions list arrow styles, but the figures use solid, dashed, squiggly, and tailed arrows without a complete visual legend, which makes the tree of condensation defects hard to parse.
  4. [2.1, Eq. (2.7)] The 'dressed operator' notation O_2 ⟳ 1_[2] is introduced informally; it would help to state explicitly that ⟳ denotes decorating a defect by a lower-dimensional operator on its worldvolume.
  5. [5.1, Eq. (5.3)] The sum over S, described as a minimal generating set, is not fully specified for a general finite Abelian group; the distinction between summing over all elements of K^(p) and over a chosen generating set should be clarified, since S is not unique.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Karoubi-completeness argument is a physical computation, not a definitional restatement; self-citations are corroborated and non-load-bearing.

full rationale

The paper's central claim is that symmetry defects satisfy the Johnson-Freyd higher-fusion-category definition, focusing on Karoubi completeness. The load-bearing steps are (i) idempotence of condensation defects, obtained from explicit path-integral fusion computations (e.g., Eqs. (3.14)-(3.17), with direct evaluation of (2.22)) and (ii) the splitting C_G^{[n]} = iota ⊗ rho, obtained from cylinder gauging and explicit soft-interface actions (Eqs. (3.19)-(3.22)). These are computations, not definitions: the category C is deliberately built to include gauged global forms, but the existence of gauging interfaces with the required composition properties is demonstrated by the worldvolume actions (3.5), (3.10) and (3.20), not assumed. The only relevant self-citation, [27], is used alongside the independent [45] for an explicit fusion-rule computation and is corroborated by the path integral shown in the paper; [24] appears only in the outlook. The paper explicitly disclaims a rigorous proof in Section 7 and notes known counterexamples to fusion-category descriptions, so the conclusion is not presented as forced. One logical gap is flagged: Section 4.3 asserts that 'the condensation defects are the only idempotent topological defects in C' without proof, so Karoubi completeness is conditional on that classification. That is an evidentiary gap, not a circular reduction; no idempotent is defined to be a condensation defect by fiat.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper does not fit any numerical parameters. Its central claim rests on background higher-category definitions from [44,46], on the physical assumption of a finite Abelian p-form symmetry with trivial self-anomaly, on manifold and vacuum assumptions (spin manifolds, no spontaneous symmetry breaking), and on the choice to define the category C using only theories reachable by discrete gauging. No new entities such as particles, forces, or dimensions are introduced.

assumptions (5)
  • domain assumption The definitions and recursive conditions of separable weak (multi)fusion n-categories, including categorical condensation and Karoubi completeness, as proposed by Johnson-Freyd [44] and Gaiotto-Johnson-Freyd [46], are the correct target framework.
    The paper's claim is that physical defects satisfy this specific definition; the definition itself is imported from the mathematics literature and not proved.
  • domain assumption The worldvolume theory of a condensation defect has exactly the two global symmetries listed, namely the Pontryagin dual quantum symmetry and the residual symmetry, with no additional anomalies or symmetries that would obstruct recursive gauging.
    Used throughout Sections 2.2 and 5.2 to iterate the construction; any extra symmetry or obstruction would break the tree of defects.
  • domain assumption The category C can be defined with objects restricted to theories reachable by discrete gauging of T, excluding stacks with arbitrary TQFTs.
    This choice, made in Section 4.1 with footnote 20, ensures the objects and morphisms are finite and manageable, but it limits the scope of the Karoubi completeness claim to this constructed category.
  • domain assumption All manifolds are spin and no spontaneous symmetry breaking is considered.
    These standard assumptions, stated in footnotes 5 and 6 of Section 2, avoid subtleties with spin and charge and with degenerate vacua that would modify the defect algebra.
  • standard math The theorem of Federer, pi_n Map(M^d, B^{p+1}G) is isomorphic to H^{p+1-n}(M^d; G), holds as stated.
    Invoked in Appendix A.3 to equate homotopy cardinality of the category of operators with the partition function of a finite-group p-form gauge theory.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the Physics of Higher Condensation Defects." pith.science (2026). https://pith.science/paper/546UK5Z2

@misc{pith2026250604346,
  author       = {Pith},
  title        = {Pith review of: On the Physics of Higher Condensation Defects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/546UK5Z2}},
  note         = {Machine review of arXiv:2506.04346}
}
read the original abstract

We study the structure of topological defects for finite Abelian symmetries in quantum field theories, and argue on physical grounds that they satisfy the definition of a higher fusion category proposed by Johnson-Freyd. Our primary focus is on the requirement of Karoubi completeness, i.e. the factorization conditions on higher condensation defects. We demonstrate this on a tree of such defects, constructed by successive higher gauging, explicitly using Lagrangian techniques in a concrete four-dimensional example, before turning to more general field theories. Along the way we also comment on the phenomenon where decoupled topological field theories appear as fusion coefficients. We further discuss the categorical role of anomalies, and how they may affect the properties of (higher) condensation defects.

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Non-Abelian Symmetry Operators from Hanging Branes in $AdS_5 \times S^5$

    hep-th 2025-10 conditional novelty 7.0 of 10

    In AdS5×S5, non-Abelian SO(6) symmetry operators are realized by hanging D5-brane–KK-monopole bound states, matching the Gauss-law operators from the low-energy supergravity action.

  2. Continuous symmetries and charge measurement of boundary operators in holography

    hep-th 2026-02 conditional novelty 6.0 of 10

    Continuous symmetry operators in holography are U-shaped hanging brane bound states (D5-KK in Type IIB, M5-KK in M-theory) whose worldvolume couplings reproduce the Gauss-law symmetry operators and measure charges of ...

Reference graph

Works this paper leans on

107 extracted references · 6 canonical work pages · cited by 2 Pith papers

  1. [2]

    Aharony, N

    O. Aharony, N. Seiberg and Y. Tachikawa,Reading between the lines of four-dimensional gauge theories,JHEP08(2013) 115 [1305.0318]. – 61 –

    arXiv 2013

  2. [1]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett,Generalized Global Symmetries,JHEP 02(2015) 172 [1412.5148]

    arXiv 2015

  3. [3]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg,Theta, Time Reversal, and Temperature,JHEP05(2017) 091 [1703.00501]

    arXiv 2017

  4. [4]

    Hsin, H.T

    P.-S. Hsin, H.T. Lam and N. Seiberg,Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d,SciPost Phys.6(2019) 039 [1812.04716]

    arXiv 2019

  5. [5]

    Hsieh, Y

    C.-T. Hsieh, Y. Tachikawa and K. Yonekura,Anomaly Inflow and p-Form Gauge Theories, Commun. Math. Phys.391(2022) 495 [2003.11550]

    arXiv 2022

  6. [6]

    Tachikawa,On gauging finite subgroups,SciPost Phys.8(2020) 015 [1712.09542]

    Y. Tachikawa,On gauging finite subgroups,SciPost Phys.8(2020) 015 [1712.09542]

    arXiv 2020

  7. [7]

    Córdova, T.T

    C. Córdova, T.T. Dumitrescu and K. Intriligator,Exploring 2-Group Global Symmetries, JHEP02(2019) 184 [1802.04790]

    arXiv 2019

  8. [8]

    Benini, C

    F. Benini, C. Córdova and P.-S. Hsin,On 2-Group Global Symmetries and their Anomalies, JHEP03(2019) 118 [1803.09336]

    arXiv 2019

Show all 107 references
  1. [9]

    Bhardwaj and Y

    L. Bhardwaj and Y. Tachikawa,On finite symmetries and their gauging in two dimensions, JHEP03(2018) 189 [1704.02330]

    2018 arXiv

  2. [10]

    Thorngren and Y

    R. Thorngren and Y. Wang,Fusion category symmetry. Part I. Anomaly in-flow and gapped phases,JHEP04(2024) 132 [1912.02817]

    2024 arXiv

  3. [11]

    Thorngren and Y

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

    2024 arXiv

  4. [12]

    Bhardwaj, L.E

    L. Bhardwaj, L.E. Bottini, S. Schafer-Nameki and A. Tiwari,Non-invertible higher-categorical symmetries,SciPost Phys.14(2023) 007 [2204.06564]

    2023 arXiv

  5. [13]

    Delmastro, J

    D.G. Delmastro, J. Gomis, P.-S. Hsin and Z. Komargodski,Anomalies and symmetry fractionalization,SciPost Phys.15(2023) 079 [2206.15118]

    2023 arXiv

  6. [14]

    Brennan, C

    T.D. Brennan, C. Cordova and T.T. Dumitrescu,Line Defect Quantum Numbers & Anomalies,2206.15401

    arXiv

  7. [15]

    Moore and N

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

    1989

  8. [16]

    Dijkgraaf and E

    R. Dijkgraaf and E. Witten,Topological Gauge Theories and Group Cohomology,Commun. Math. Phys.129(1990) 393

    1990

  9. [17]

    Kitaev,Anyons in an exactly solved model and beyond,Annals Phys.321(2006) 2 [cond-mat/0506438]

    A. Kitaev,Anyons in an exactly solved model and beyond,Annals Phys.321(2006) 2 [cond-mat/0506438]

    2006 arXiv

  10. [18]

    Choi, H.T

    Y. Choi, H.T. Lam and S.-H. Shao,Noninvertible Global Symmetries in the Standard Model,Phys. Rev. Lett.129(2022) 161601 [2205.05086]

    2022 arXiv

  11. [19]

    Cordova, S

    C. Cordova, S. Hong, S. Koren and K. Ohmori,Neutrino Masses from Generalized Symmetry Breaking,Phys. Rev. X14(2024) 031033 [2211.07639]

    2024 arXiv

  12. [20]

    Cordova and S

    C. Cordova and S. Koren,Higher Flavor Symmetries in the Standard Model,Annalen Phys. 535(2023) 2300031 [2212.13193]

    2023 arXiv

  13. [21]

    Y. Choi, M. Forslund, H.T. Lam and S.-H. Shao,Quantization of Axion-Gauge Couplings and Noninvertible Higher Symmetries,Phys. Rev. Lett.132(2024) 121601 [2309.03937]

    2024 arXiv

  14. [22]

    Cordova, S

    C. Cordova, S. Hong and S. Koren,Non-Invertible Peccei-Quinn Symmetry and the Massless Quark Solution to the Strong CP Problem,2402.12453

    arXiv

  15. [23]

    Apruzzi, F

    F. Apruzzi, F. Bonetti, I.n. García Etxebarria, S.S. Hosseini and S. Schafer-Nameki, Symmetry TFTs from String Theory,Commun. Math. Phys.402(2023) 895 [2112.02092]. – 62 –

    2023 arXiv

  16. [24]

    Apruzzi, I

    F. Apruzzi, I. Bah, F. Bonetti and S. Schafer-Nameki,Noninvertible Symmetries from Holography and Branes,Phys. Rev. Lett.130(2023) 121601 [2208.07373]

    2023 arXiv

  17. [25]

    Heckman, M

    J.J. Heckman, M. Hübner, E. Torres and H.Y. Zhang,The Branes Behind Generalized Symmetry Operators,Fortsch. Phys.71(2023) 2200180 [2209.03343]

    2023 arXiv

  18. [26]

    Heckman, M

    J.J. Heckman, M. Hubner, E. Torres, X. Yu and H.Y. Zhang,Top down approach to topological duality defects,Phys. Rev. D108(2023) 046015 [2212.09743]

    2023 arXiv

  19. [27]

    I. Bah, E. Leung and T. Waddleton,Non-invertible symmetries, brane dynamics, and tachyon condensation,JHEP01(2024) 117 [2306.15783]

    2024 arXiv

  20. [28]

    Apruzzi, F

    F. Apruzzi, F. Bonetti, D.S.W. Gould and S. Schafer-Nameki,Aspects of categorical symmetries from branes: SymTFTs and generalized charges,SciPost Phys.17(2024) 025 [2306.16405]

    2024 arXiv

  21. [29]

    Waddleton,U(1)R-Symmetry Topological Operators from Branes in Holography, 2408.14542

    T. Waddleton,U(1)R-Symmetry Topological Operators from Branes in Holography, 2408.14542

    arXiv

  22. [30]

    Cvetič, J.J

    M. Cvetič, J.J. Heckman, M. Hübner and C. Murdia,Metric Isometries, Holography, and Continuous Symmetry Operators,2501.17911

    arXiv

  23. [31]

    Kong,Anyon condensation and tensor categories,Nucl

    L. Kong,Anyon condensation and tensor categories,Nucl. Phys. B886(2014) 436 [1307.8244]

    2014 arXiv

  24. [32]

    Barkeshli, P

    M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang,Symmetry Fractionalization, Defects, and Gauging of Topological Phases,Phys. Rev. B100(2019) 115147 [1410.4540]

    2019 arXiv

  25. [33]

    Kong, X.-G

    L. Kong, X.-G. Wen and H. Zheng,Boundary-bulk relation in topological orders,Nucl. Phys. B922(2017) 62 [1702.00673]

    2017 arXiv

  26. [34]

    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. 2(2020) 043086 [2005.14178]

    2020 arXiv

  27. [35]

    Seiberg and S.-H

    N. Seiberg and S.-H. Shao,Majorana chain and Ising model – (non-invertible) translations, anomalies, and emanant symmetries,SciPost Phys.16(2024) 064 [2307.02534]

    2024 arXiv

  28. [36]

    S.D. Pace, A. Chatterjee and S.-H. Shao,Lattice T-duality from non-invertible symmetries in quantum spin chains,SciPost Phys.18(2025) 121 [2412.18606]

    2025 arXiv

  29. [37]

    Zhang, A

    C. Zhang, A. Vishwanath and X.-G. Wen,Hierarchy construction for non-abelian fractional quantum hall states via anyon condensation, 2025

    2025

  30. [38]

    Tantivasadakarn, A

    N. Tantivasadakarn, A. Vishwanath and R. Verresen,Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward,PRX Quantum4(2023) 020339 [2209.06202]

    2023 arXiv

  31. [39]

    Gorantla, S.-H

    P. Gorantla, S.-H. Shao and N. Tantivasadakarn,Tensor networks for non-invertible symmetries in 3+1d and beyond,2406.12978

  32. [40]

    Douglas and D.J

    C.L. Douglas and D.J. Reutter,Fusion 2-categories and a state-sum invariant for 4-manifolds,arXiv e-prints(2018) arXiv:1812.11933 [1812.11933]

    2018 arXiv

  33. [41]

    Décoppet and M

    T.D. Décoppet and M. Yu,Gauging noninvertible defects: a 2-categorical perspective,Lett. Math. Phys.113(2023) 36 [2211.08436]

    2023 arXiv

  34. [42]

    Décoppet, P

    T.D. Décoppet, P. Huston, T. Johnson-Freyd, D. Nikshych, D. Penneys, J. Plavnik et al., The Classification of Fusion 2-Categories,2411.05907. – 63 –

    arXiv

  35. [43]

    Ferrer et al.,Daggern-categories,2403.01651

    G. Ferrer et al.,Daggern-categories,2403.01651

  36. [44]

    Johnson-Freyd,On the Classification of Topological Orders,Commun

    T. Johnson-Freyd,On the Classification of Topological Orders,Commun. Math. Phys.393 (2022) 989 [2003.06663]

    2022 arXiv

  37. [45]

    Roumpedakis, S

    K. Roumpedakis, S. Seifnashri and S.-H. Shao,Higher Gauging and Non-invertible Condensation Defects,Commun. Math. Phys.401(2023) 3043 [2204.02407]

    2023 arXiv

  38. [46]

    Gaiotto and T

    D. Gaiotto and T. Johnson-Freyd,Condensations in higher categories,1905.09566

    1905 arXiv

  39. [47]

    Décoppet,Weak Fusion 2-Categories,arXiv e-prints(2021) arXiv:2103.15150 [2103.15150]

    T.D. Décoppet,Weak Fusion 2-Categories,arXiv e-prints(2021) arXiv:2103.15150 [2103.15150]

    2021 arXiv

  40. [48]

    Frohlich, J

    J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert,Defect Lines, Dualities and Generalised Orbifolds, in16th International Congress on Mathematical Physics, pp. 608–613, 2010, DOI [0909.5013]

    2010 arXiv

  41. [49]

    Carqueville and I

    N. Carqueville and I. Runkel,Orbifold completion of defect bicategories,Quantum Topol.7 (2016) 203 [1210.6363]

    2016 arXiv

  42. [50]

    Carqueville, I

    N. Carqueville, I. Runkel and G. Schaumann,Orbifolds of n-dimensional defect TQFTs, Geom. Topol.23(2019) 781 [1705.06085]

    2019 arXiv

  43. [51]

    Carqueville and L

    N. Carqueville and L. Müller,Orbifold completion of 3-categories,2307.06485

  44. [52]

    Kong, Z.-H

    L. Kong, Z.-H. Zhang, J. Zhao and H. Zheng,Higher condensation theory,2403.07813

  45. [53]

    Carqueville and T

    N. Carqueville and T. Lüders,Orbifolds, higher dagger structures, and idempotents, 2504.17764

    arXiv

  46. [54]

    Müller,On the Higher Categorical Structure of Topological Defects in Quantum Field Theories,2505.04761

    L. Müller,On the Higher Categorical Structure of Topological Defects in Quantum Field Theories,2505.04761

    arXiv

  47. [55]

    Kang and S

    M.J. Kang and S. Kang,Central extensions of higher groups: Green-Schwarz mechanism and 2-connections,2311.14666

    arXiv

  48. [56]

    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 [hep-th/9812012]

    1998 arXiv

  49. [57]

    Maldacena, G.W

    J.M. Maldacena, G.W. Moore and N. Seiberg,D-brane charges in five-brane backgrounds, JHEP10(2001) 005 [hep-th/0108152]

    2001 arXiv

  50. [58]

    Banks and N

    T. Banks and N. Seiberg,Symmetries and Strings in Field Theory and Gravity,Phys. Rev. D83(2011) 084019 [1011.5120]

    2011 arXiv

  51. [59]

    Hellerman, A

    S. Hellerman, A. Henriques, T. Pantev, E. Sharpe and M. Ando,Cluster decomposition, T-duality, and gerby CFT’s,Adv. Theor. Math. Phys.11(2007) 751 [hep-th/0606034]

    2007 arXiv

  52. [60]

    Tanizaki and M

    Y. Tanizaki and M. Ünsal,Modified instanton sum in QCD and higher-groups,JHEP03 (2020) 123 [1912.01033]

    2020 arXiv

  53. [61]

    Cherman and T

    A. Cherman and T. Jacobson,Lifetimes of near eternal false vacua,Phys. Rev. D103 (2021) 105012 [2012.10555]

    2021 arXiv

  54. [62]

    Komargodski, K

    Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri,Symmetries and strings of adjoint QCD2,JHEP03(2021) 103 [2008.07567]

    2021 arXiv

  55. [63]

    Sharpe,An introduction to decomposition, (2024), DOI [2204.09117]

    E. Sharpe,An introduction to decomposition, (2024), DOI [2204.09117]

    2024 arXiv

  56. [64]

    Kapustin and N

    A. Kapustin and N. Seiberg,Coupling a QFT to a TQFT and Duality,JHEP04(2014) 001 [1401.0740]. – 64 –

    2014 arXiv

  57. [65]

    Kaidi, K

    J. Kaidi, K. Ohmori and Y. Zheng,Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories,Phys. Rev. Lett.128(2022) 111601 [2111.01141]

    2022 arXiv

  58. [66]

    Y. Choi, C. Cordova, P.-S. Hsin, H.T. Lam and S.-H. Shao,Noninvertible duality defects in 3+1 dimensions,Phys. Rev. D105(2022) 125016 [2111.01139]

    2022 arXiv

  59. [67]

    Y. Choi, C. Cordova, P.-S. Hsin, H.T. Lam and S.-H. Shao,Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions,Commun. Math. Phys.402(2023) 489 [2204.09025]

    2023 arXiv

  60. [68]

    Copetti, M

    C. Copetti, M. Del Zotto, K. Ohmori and Y. Wang,Higher Structure of Chiral Symmetry, 2305.18282

    arXiv

  61. [69]

    Bhardwaj, T

    L. Bhardwaj, T. Décoppet, S. Schafer-Nameki and M. Yu,Fusion 3-Categories for Duality Defects,2408.13302

    arXiv

  62. [70]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik,Tensor Categories, Mathematical Surveys and Monographs, American Mathematical Society (2015)

    2015

  63. [71]

    Baez and M

    J.C. Baez and M. Shulman,Lectures on n-categories and cohomology, 2007

    2007

  64. [72]

    Cordova and K

    C. Cordova and K. Ohmori,Noninvertible Chiral Symmetry and Exponential Hierarchies, Phys. Rev. X13(2023) 011034 [2205.06243]

    2023 arXiv

  65. [73]

    Etingof, D

    P. Etingof, D. Nikshych and V. Ostrik,On fusion categories,arXiv Mathematics e-prints (2002) math/0203060 [math/0203060]

    2002 arXiv

  66. [74]

    Quinn,Lectures on axiomatic topological quantum field theory,Geometry and quantum field theory (Park City, UT, 1991)1(1995) 323

    F. Quinn,Lectures on axiomatic topological quantum field theory,Geometry and quantum field theory (Park City, UT, 1991)1(1995) 323

    1995

  67. [75]

    Baez and J

    J.C. Baez and J. Dolan,From Finite Sets to Feynman Diagrams,arXiv Mathematics e-prints(2000) math/0004133 [math/0004133]

    2000 arXiv

  68. [76]

    Baez, A.E

    J.C. Baez, A.E. Hoffnung and C.D. Walker,Higher-Dimensional Algebra VII: Groupoidification,arXiv e-prints(2009) arXiv:0908.4305 [0908.4305]

    2009 arXiv

  69. [77]

    Eilenberg and S

    S. Eilenberg and S. MacLane,Relations between homology and homotopy groups of spaces, Annals of Mathematics46(1945) 480

    1945

  70. [78]

    Eilenberg and S

    S. Eilenberg and S. MacLane,Relations between homology and homotopy groups of spaces. ii,Annals of Mathematics51(1950) 514

    1950

  71. [79]

    Eilenberg and S

    S. Eilenberg and S. MacLane,On the GroupsH(Π, n), III: Operations and Obstructions, Annals of Mathematics60(1954) 513

    1954

  72. [80]

    Horowitz,Exactly soluble diffeomorphism invariant theories,Commun

    G.T. Horowitz,Exactly soluble diffeomorphism invariant theories,Commun. Math. Phys. 125(1989) 417

    1989

  73. [81]

    Freed, G.W

    D.S. Freed, G.W. Moore and C. Teleman,Topological symmetry in quantum field theory, 2209.07471

    arXiv

  74. [82]

    Freed, M.J

    D.S. Freed, M.J. Hopkins, J. Lurie and C. Teleman,Topological Quantum Field Theories from Compact Lie Groups, inA Celebration of Raoul Bott’s Legacy in Mathematics, 5, 2009 [0905.0731]

    2009 arXiv

  75. [83]

    Breen,Monoidal categories and multiextensions, 1998

    L. Breen,Monoidal categories and multiextensions, 1998

    1998

  76. [84]

    Breen,Braidedn-categories andΣ-structures, 1998

    L. Breen,Braidedn-categories andΣ-structures, 1998

    1998

  77. [85]

    Yu,Symmetries and anomalies of (1+1)d theories: 2-groups and symmetry fractionalization,JHEP08(2021) 061 [2010.01136]

    M. Yu,Symmetries and anomalies of (1+1)d theories: 2-groups and symmetry fractionalization,JHEP08(2021) 061 [2010.01136]. – 65 –

    2021 arXiv

  78. [86]

    Bartsch, M

    T. Bartsch, M. Bullimore, A.E.V. Ferrari and J. Pearson,Non-invertible symmetries and higher representation theory I,SciPost Phys.17(2024) 015 [2208.05993]

    2024 arXiv

  79. [87]

    Bartsch, M

    T. Bartsch, M. Bullimore, A.E.V. Ferrari and J. Pearson,Non-invertible Symmetries and Higher Representation Theory II,2212.07393

    arXiv

  80. [88]

    Bartsch, M

    T. Bartsch, M. Bullimore and A. Grigoletto,Higher representations for extended operators, 2304.03789

    arXiv

  81. [89]

    Bartsch, M

    T. Bartsch, M. Bullimore and A. Grigoletto,Representation theory for categorical symmetries,2305.17165

    arXiv

  82. [90]

    Kaidi, E

    J. Kaidi, E. Nardoni, G. Zafrir and Y. Zheng,Symmetry TFTs and anomalies of non-invertible symmetries,JHEP10(2023) 053 [2301.07112]

    2023 arXiv

  83. [91]

    Choi, B.C

    Y. Choi, B.C. Rayhaun, Y. Sanghavi and S.-H. Shao,Remarks on boundaries, anomalies, and noninvertible symmetries,Phys. Rev. D108(2023) 125005 [2305.09713]

    2023 arXiv

  84. [92]

    Sun and Y

    Z. Sun and Y. Zheng,When are Duality Defects Group-Theoretical?,2307.14428

    arXiv

  85. [93]

    Cordova, P.-S

    C. Cordova, P.-S. Hsin and C. Zhang,Anomalies of non-invertible symmetries in (3+1)d, SciPost Phys.17(2024) 131 [2308.11706]

    2024 arXiv

  86. [94]

    Antinucci, F

    A. Antinucci, F. Benini, C. Copetti, G. Galati and G. Rizi,Anomalies of non-invertible self-duality symmetries: fractionalization and gauging,2308.11707

    arXiv

  87. [95]

    Choi, D.-C

    Y. Choi, D.-C. Lu and Z. Sun,Self-duality under gauging a non-invertible symmetry,JHEP 01(2024) 142 [2310.19867]

    2024 arXiv

  88. [96]

    Antinucci, C

    A. Antinucci, C. Copetti and S. Schafer-Nameki,SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement,SciPost Phys.18(2025) 114 [2408.05585]

    2025 arXiv

  89. [97]

    Del Zotto, A

    M. Del Zotto, A. Hasan and E. Riedel Gårding,SymTFT, Protected Gaplessness, and Spontaneous Breaking of Non-invertible Symmetries,2504.18501

    arXiv

  90. [98]

    García Etxebarria,Branes and Non-Invertible Symmetries,Fortsch

    I.n. García Etxebarria,Branes and Non-Invertible Symmetries,Fortsch. Phys.70(2022) 2200154 [2208.07508]

    2022 arXiv

  91. [99]

    Sen,Stable nonBPS bound states of BPS D-branes,JHEP08(1998) 010 [hep-th/9805019]

    A. Sen,Stable nonBPS bound states of BPS D-branes,JHEP08(1998) 010 [hep-th/9805019]

    1998 arXiv

  92. [100]

    Sen,Tachyon condensation on the brane anti-brane system,JHEP08(1998) 012 [hep-th/9805170]

    A. Sen,Tachyon condensation on the brane anti-brane system,JHEP08(1998) 012 [hep-th/9805170]

    1998 arXiv

  93. [101]

    Sen,NonBPS states and Branes in string theory, inAdvanced School on Supersymmetry in the Theories of Fields, Strings and Branes, pp

    A. Sen,NonBPS states and Branes in string theory, inAdvanced School on Supersymmetry in the Theories of Fields, Strings and Branes, pp. 187–234, 1, 1999 [hep-th/9904207]

    1999 arXiv

  94. [102]

    Décoppet,Compact Semisimple 2-Categories,arXiv e-prints(2021) arXiv:2111.09080 [2111.09080]

    T.D. Décoppet,Compact Semisimple 2-Categories,arXiv e-prints(2021) arXiv:2111.09080 [2111.09080]

    2021 arXiv

  95. [103]

    Johnson-Freyd and D

    T. Johnson-Freyd and D. Reutter,Minimal nondegenerate extensions,Journal of the American Mathematical Society37(2023) 81–150

    2023

  96. [104]

    Algebraically closed higher categories

    T. Johnson-Freyd, “Algebraically closed higher categories.”

  97. [105]

    Higher s-matrices and higher modular categories

    D. Reutter, “Higher s-matrices and higher modular categories.”

  98. [106]

    Private correspondence

    T. Johnson-Freyd, “Private correspondence.” Private Communication

  99. [107]

    Federer,A study of function spaces by spectral sequences,Transactions of the American Mathematical Society82(1956) 340

    H. Federer,A study of function spaces by spectral sequences,Transactions of the American Mathematical Society82(1956) 340. – 66 –

    1956

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.