Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

The Classification of 3+1d Symmetry Enriched Topological Order

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Fermionic (3+1)d phases with finite symmetry are fixed by a group extension and a supercohomology class; gauging is possible exactly when the $SW^5(BG)$ anomaly vanishes.

desk verdict A serious 2-categorical classification of (3+1)d G-SETs, but Theorem 4.26's anomaly iff rests on an under-proved Lemma 4.21; worth peer review, yet that lemma needs proof or explicit scoping. read the letter →

arxiv 2509.10603 v3 pith:2LGPNEMA submitted 2025-09-12 math-ph cond-mat.str-elhep-thmath.CTmath.MPmath.QA

Thibault D. Décoppet , Matthew Yu This is my paper · ORCID
classification math-phcond-mat.str-elhep-thmath.CTmath.MPmath.QA MSC 18M2018N1081T45
keywords topologicalordersymmetryenrichedfermionicfusion2-categoriessupercohomologyequivariantizationanomalyG-crossedbraidedcategories
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

This paper claims that every (3+1)-dimensional fermionic topological order equipped with a finite symmetry group $G$ is classified by two pieces of data: a finite group $H$ that surjects onto $G$ and a class in the supercohomology group $SH^4(BH)$. It further claims that the obstruction to gauging $G$, the symmetry anomaly, is a class in $SW^5(BG)$, and that a $G$-action can be extended to a full symmetry-enriched phase exactly when that class vanishes. The argument runs through a 2-categorical version of (de-)equivariantization, which turns symmetry-enriched phases into $G$-crossed braided fusion 2-categories and reduces the classification to group theory and cohomology. A sympathetic reader would care because this replaces a difficult classification of gapped quantum phases with explicit finite-group data, and it gives a precise criterion for when a finite symmetry is anomaly-free and can be gauged.

What carries the argument

The load-bearing mechanism is 2-categorical (de-)equivariantization: an equivalence between braided fusion 2-categories containing $\mathbf{2Rep}(G)$ and $G$-crossed braided fusion 2-categories. Applied to a nondegenerate braided fusion 2-category, it reduces the classification to faithfully graded crossed extensions of strongly fusion 2-categories, whose extensions are classified homotopically by maps into a Picard space. For the fermionic anomaly, the carrying object is the fiber sequence $B\mathbf{SPic}(B) \to B\mathrm{Aut}^{\mathrm{br}}_{\mathbf{2SVect}}(B) \to B\mathrm{sWitt}$, whose final term has homotopy groups given by the super-Witt group and lower-dimensional fermionic phases; this sequence turns the question 'can $G$-defects be inserted?' into the vanishing of a class in $SW^5(BG)$.

What would settle it

Construct a nondegenerate $\mathbf{2SVect}$-enriched braided fusion 2-category that is not equivalent to the centralizer of $\mathbf{2SVect}$ inside any $Z(\mathbf{2SVect}^\varpi_G)$, or compute $SW^5(BG)$ for a small group and exhibit an anomaly class not realized by any extension $H \twoheadrightarrow G$ with the stated $SH^4(BH)$ data; either would directly contradict Proposition 4.3 or Theorem 4.26.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a classification of fermionic (3+1)d symmetry-enriched topological orders: a theory with finite symmetry $G$ is determined by a surjective homomorphism $H \twoheadrightarrow G$ together with a class in $SH^4(BH)$ (Proposition 4.3). The same framework identifies the anomaly to gauging $G$ as a class in $SW^5(BG)$, represented by a map $BG \to B\mathrm{sWitt}$, and proves that a $G$-action on a nondegenerate $\mathbf{2SVect}$-enriched braided fusion 2-category extends to a nondegenerate $\mathbf{2SVect}$-enriched $G$-crossed braided fusion 2-category if and only if this anomaly class is trivial (Theorem 4.26). This is a fermionic generalization of the bosonic symmetry-extension ansatz, and the paper shows the categorical data needed to define such theories agrees with the data of that ansatz.

Load-bearing premise

The whole classification rests on the imported result that every fermionic (3+1)d topological phase can be described by a finite group with a supercohomology action; if a phase exists outside that description, the symmetry-enriched classification and the anomaly formula would both need revision.

Editorial extensions

If this is right

  • If the classification is right, every fermionic (3+1)d $G$-SET is equivalent to gauging a finite normal subgroup $K$ of some $H$ with a Dijkgraaf-Witten action in $SH^4(BK)$, matching the fermionic symmetry-extension ansatz.
  • The anomaly to gauging a finite symmetry is not always a supercohomology cocycle: $SW^5(BG)$ contains $SH^5(BG)$ as a subgroup, and the extra layers beyond cocycles are part of the genuine gauging obstruction.
  • A $G$-action that is anomaly-free in $SW^5(BG)$ always admits a compatible faithfully graded $G$-crossed braided extension, so the symmetry can be realized by topological defects.
  • The same data classify Lagrangian algebras in $Z(3\mathrm{Vect}_G)$ and $Z(3\mathrm{SVect}_G)$, hence enumerate gapped boundaries of the corresponding (4+1)d symmetry TFTs.
  • The classification refines earlier lists by distinguishing theories that differ by the class $\varsigma \in SH^{5+\kappa}(B\mathbb{Z}/2)$, as in the pair $S$ and $T$ of nondegenerate fermionic braided fusion 2-categories.

Reading between the lines

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

  • A concrete stress test would be to compute $SW^5(BG)$ for small groups and compare the part beyond $SH^5(BG)$ with the classes realizable by some extension $H \twoheadrightarrow G$; the paper points to later work for these computations, and the first group where the two sets differ would show how often the non-cocycle anomaly layer matters physically.
  • Because the argument is an equivalence of 3-categories rather than a case-by-case check, it likely extends to anti-unitary symmetries once unitary higher fusion categories with such actions are fully developed, a direction the paper explicitly leaves open.
  • The equivariantization strategy used here should transfer one categorical level up to classify symmetry-enriched topological orders in (4+1)d, since the paper already classifies Lagrangian algebras in the relevant (4+1)d symmetry TFTs.
  • The same de-equivariantization of the symmetric center gives a route to classifying (3+1)d mixed-state topological orders, which the paper flags as a separate project; if that route works, the list of pure-state phases here would also organize the noisy/mixed-state phases.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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. This paper develops a 2-categorical (de-)equivariantization formalism to classify (3+1)d symmetry-enriched topological orders (SETs), focusing on fermionic theories with finite unitary symmetry G. The main mathematical objects are 2SVect-enriched G-crossed braided fusion 2-categories; the paper argues that these describe fermionic (3+1)d G-SETs and that their nondegenerate versions are equivalent to nondegenerate 2SVect-enriched braided fusion 2-categories equipped with a fully faithful braided 2-functor from 2Rep(G) (Propositions 4.3 and 4.4). It also states cohomological classifications of nondegenerate bosonic and fermionic braided fusion 2-categories (Theorems 3.11 and 3.20), sketches a classification of general braided fusion 2-categories (Section 3.4), and gives a homotopical description of the obstruction to gauging: a G-action extends to a 2SVect-enriched G-crossed braided extension if and only if its anomaly class in SW^5(BG) vanishes (Theorem 4.26). The paper closes with an application to Lagrangian algebras in Z(3Vect_G) and Z(3SVect_G), and it proposes a fermionic version of the Wang-Wen-Witten ansatz (Ansatz 1.8).

Significance. If the central theorems hold, this is a substantial contribution: it provides a uniform categorical framework for fermionic (3+1)d SETs, recovers and refines earlier classifications, and gives an anomaly theory valued in the 4-groupoid BsWitt that goes beyond supercohomology cocycles. The paper is explicit about the cohomological data and the relevant fiber sequences, and it identifies the precise points where the argument relies on imported results; this transparency is a strength. It contains no fitted parameters or ad hoc axioms, and several claims (e.g., the classification data in Theorems 3.20 and 4.3 and the anomaly vanishing criterion in Theorem 4.26) are concrete enough to serve as falsifiable statements for future work on explicit gauge-theory models. The main caveat is that several load-bearing steps are deferred to prior work or left as sketches, so the significance is conditional on those steps being filled in.

major comments (3)
  1. [Section 4.4, Lemma 4.21 and Theorem 4.26] The proof of Lemma 4.21 is the load-bearing step for the anomaly classification, and it is not sufficient as written. The paper cites [JF22, Cor. V.4] for the statement that every nondegenerate 2SVect-enriched braided fusion 2-category B is the centralizer of 2SVect in some nondegenerate braided fusion 2-category (see Eq. (2.17)), and then asserts that B therefore has a minimal nondegenerate extension and that Mod(B) is Morita equivalent to 3SVect. The latter two assertions are nontrivial and are neither proved nor given a reference. They are exactly what identifies the target of the composite in Definition 4.25 with BsWitt and produces the fiber sequence (4.20) used in Theorem 4.26. The lower-dimensional analogue does not make this automatic: nondegenerate SVect-enriched braided fusion 1-categories are not all Morita trivial, as the nontriviality of the super-Witt group shows. I ask the authors to prove Lemma 4.21 directly, or to state the precise external result that implies the Morita triviality of Mod(B) for every B arising from the centralizer construction.
  2. [Section 3.3, Theorem 3.20 and Remark 3.22; Section 4.2, Remark 4.5] Theorem 3.20 is presented as a classification, but the equivalence relation on the data (G, ς, τ, ϖ) is not specified; Remark 3.22 explicitly defers this to [TY25]. Without the equivalence relation, the theorem gives a complete set of invariants only up to the as-yet-unspecified identifications, so the word 'classified' is stronger than what is proved here. The same issue affects the classification claim in Remark 4.5: pairs (H, ϖ) with H a finite group surjecting onto G are not shown to be in bijection with isomorphism classes of nondegenerate 2SVect-enriched G-crossed braided fusion 2-categories until the action of Aut(G) and the autoequivalences of the categories S and T are accounted for.
  3. [Section 3.4 and Theorem 2.19] The proof of Theorem 2.19 asserts that the diagram in Eq. (2.21) is a pullback without proof; this is the compatibility condition that turns a G-graded extension into a G-crossed braided extension, and all subsequent classifications in Sections 3 and 4 rest on it. In addition, Section 3.4 is explicitly a sketch: the list of data in Eq. (3.23) is not accompanied by a statement of bijectivity or a proof that the data are complete. If the classification of all braided fusion 2-categories is claimed in the abstract, this section needs to be upgraded to a precise theorem; otherwise the claim should be softened.
minor comments (5)
  1. [Throughout] Several equations and displayed sequences contain typographical errors: in Eq. (3.14) the sequence 'SH5 B2Picbr(2SVect)' is missing arrows, and in Eq. (3.19) the map '(κ,ς)' is not displayed with a clear source and target. These should be corrected.
  2. [Throughout] The notation for 2SVect is not consistent: '2SVect', '2SV ect', and '2sVect' appear in different places; please standardize.
  3. [Various] There are several typos: 'decoherenece' in Remark 2.2, 'commmutes' after Eq. (2.7), 'nondenegerate' in Theorem 2.15, and 'topolgoical' in Section 2.5. These should be fixed by copyediting.
  4. [Section 4.4, proof of Proposition 4.19] The sentence 'It follows that pt≃ BAutsyl_{2SVect}(F)→ BAutsyl(2SVect)≃ B^2Z/2' is confusing: if F is an equivalence, the space BAutsyl_{2SVect}(F) should be equivalent to BAutsyl(2SVect), not to a point. Please clarify what is meant, since this line is part of the proof of the fiber sequence (4.20).
  5. [Section 4.4] The proof of Proposition 4.19 is quite terse in identifying the right-most pullback square in diagram (4.24) with the statement of Lemma 4.23; a sentence explaining why B4SVect^× = Mod(3SVect)^× would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the main classification and anomaly results are derived from external theorems, with same-author citations used as black boxes; the principal inherited input is the [JF22] completeness dictionary used in Lemma 4.21, a correctness risk rather than a circular reduction.

full rationale

The derivation chain is not circular at the level of the paper's own equations. The de-equivariantization equivalence (Theorem 3.2 and Proposition 4.3) is proved from the Morita equivalence 3VectG ~ 3Rep(G), and Proposition 4.3's cohomological classification follows by combining that equivalence with the external classification [JF22, Theorem 2.15 / Corollary V.4], not by assuming the G-SET conclusion. Theorem 4.26 is formally a lift criterion in the fiber sequence of Proposition 4.19; its genuine input is Lemma 4.21, which imports the [JF22] completeness dictionary (every nondegenerate 2SVect-enriched braided fusion 2-category is a centralizer of 2SVect) and then asserts a minimal nondegenerate extension and Morita equivalence to 3SVect. That lemma is not proved here and is an external dependency: if the [JF22] dictionary failed for some fermionic phase, the BsWitt target in the anomaly fiber sequence (4.20) would be wrong. This is a correctness risk, not a circular reduction by construction. Several same-author results are used as black boxes in extension and Picard-space computations, but they are parameter-free prior theorems and do not encode the G-SET or anomaly statements. No fitted parameters are promoted to predictions, and no central equation reduces to an earlier equation of this paper. The score reflects the presence of non-load-bearing self-citations and the unproven imported dictionary in Lemma 4.21, not an actual circular step.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted numerical parameters appear in this pure mathematical classification; the cohomology classes (pi, tau, upsilon, varrho) are classification data rather than fitted constants. The paper imports substantial background theorems, some proved by the same authors and some unpublished, but it does not introduce new physical degrees of freedom. The load-bearing imported assumptions are the dictionary between physical topological orders and 2SVect-enriched braided fusion 2-categories, and the extension-theoretic results used to compute Picard and anomaly spaces.

assumptions (6)
  • standard math Fusion 2-categories and their braided and sylleptic structures exist as finite semisimple higher categories with the properties reviewed in Section 2.2, following [DR18, SP11, Cra98].
    This is the background higher-category formalism on which all definitions rely.
  • standard math Deligne's theorem that symmetric fusion 1-categories split into Tannakian and super-Tannakian cases, used in Definition 1.1.
    This dichotomy underlies the bosonic versus fermionic distinction for fusion 2-categories.
  • domain assumption The prior classifications of (3+1)d topological orders and fusion 2-categories in [JF22, LKW18, LW19, DHJF+24] are correct.
    The paper uses these as input to identify physical topological orders with braided fusion 2-categories and to state the no-symmetry classification. These are not re-derived here.
  • standard math Faithfully G-graded extensions of fusion 2-categories are classified by maps into BBrPic(C), as in [Déc24, Theorem 3.11].
    This extension-theory theorem is the technical core of Theorem 2.19 and the subsequent classification of G-crossed braided extensions.
  • domain assumption The S-matrix argument from [JFR24, Theorem 2.57] determines that every nondegenerate braided fermionic strongly fusion 2-category has pi0 equal to Z/2.
    This statement, used in Section 3.3, restricts the classification data for nondegenerate fermionic braided fusion 2-categories but is only cited, not proved in this paper.
  • domain assumption The Jones-Reutter fiber sequence BC^× -> BAut(C) -> BBimod(C)^×, recorded in [BDSNY25, Theorem 5.2.24], is valid in the 2SVect-enriched setting used in Section 4.4.
    This fiber sequence underlies the anomaly obstruction calculation, and the paper notes it is unpublished work of Jones-Reutter.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Classification of 3+1d Symmetry Enriched Topological Order." pith.science (2026). https://pith.science/paper/2LGPNEMA

@misc{pith2026250910603,
  author       = {Pith},
  title        = {Pith review of: The Classification of 3+1d Symmetry Enriched Topological Order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2LGPNEMA}},
  note         = {Machine review of arXiv:2509.10603}
}
abstract

We use a 2-categorical version of (de-)equivariantization to classify (3+1)d topological orders with a finite $G$-symmetry. In particular, we argue that (3+1)d fermionic topological order with $G$-symmetry correspond to $\mathbf{2SVect}$-enriched $G$-crossed braided fusion 2-categories. We then show that the categorical data necessary to define these theories agrees with that arising from a fermionic generalization of the Wang-Wen-Witten construction of bosonic topological theories with $G$-symmetry saturating an anomaly. More generally, we also explain how 2-categorical (de-) equivariantization yields a classification of all braided fusion 2-categories.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Fermion Families and Pontryagin Class: Topological Field Theory via Colour Symmetry Extension

    hep-th 2026-05 unverdicted novelty 6.0 of 10

    From anomaly cancellation plus a stipulated minimality principle, the Standard Model is forced to N_c=N_f=3; the paper proves supporting theorems: H^d(Z_n,U(1)) cocycles split under the Z_n→Z_{n^2} extension, while A_...

Reference graph

Works this paper leans on

102 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [1]

    Categorical Anomaly Matching

    Andrea Antinucci, Christian Copetti, Yuhan Gai, and Sakura Schafer-Nameki. Categorical Anomaly Matching . 2025. arXiv:2508.00982

    arXiv 2025

  2. [2]

    Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

    Daniel Bulmash and Maissam Barkeshli. Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions . Phys. Rev. Res. , 2(4):043033, 2020. arXiv:2003.11553

    arXiv 2020

  3. [3]

    Symmetry Fractionalization, Defects, and Gauging of Topological Phases

    Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang. Symmetry Fractionalization, Defects, and Gauging of Topological Phases . Phys. Rev. B , 100(11):115147, 2019. arXiv:1410.4540

    arXiv 2019

  4. [4]

    Representation theory for categorical symmetries

    Thomas Bartsch, Mathew Bullimore, and Andrea Grigoletto. Representation theory for categorical symmetries . 2023. arXiv:2305.17165

    arXiv 2023

  5. [5]

    Bottini, Daniel Pajer, and Sakura Schafer-Nameki

    Lakshya Bhardwaj, Lea E. Bottini, Daniel Pajer, and Sakura Schafer-Nameki. Categorical L andau paradigm for gapped phases. Phys. Rev. Lett. , 133(16):161601, 2024. arXiv:2310.03786

    arXiv 2024

  6. [6]

    Bottini, Daniel Pajer, and Sakura Schafer-Nameki

    Lakshya Bhardwaj, Lea E. Bottini, Daniel Pajer, and Sakura Schafer-Nameki. The club sandwich: Gapless phases and phase transitions with non-invertible symmetries. SciPost Phys. , 18:156, 2025. arXiv:2312.17322

    arXiv 2025

  7. [7]

    Bottini, Daniel Pajer, and Sakura Sch\"afer-Nameki

    Lakshya Bhardwaj, Lea E. Bottini, Daniel Pajer, and Sakura Sch\"afer-Nameki. Gapped phases with non-invertible symmetries: (1+1)d . SciPost Phys. , 18(1):032, 2025. arXiv:2310.03784

    arXiv 2025

  8. [8]

    Relative anomalies in (2+1) D symmetry enriched topological states

    Maissam Barkeshli and Meng Cheng. Relative anomalies in (2+1) D symmetry enriched topological states. SciPost Phys. , 8:028, 2020. arXiv:1906.10691

    arXiv 2020

Show all 102 references
  1. [9]

    Classification of (2+1)D invertible fermionic topological phases with symmetry

    Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin, and Naren Manjunath. Classification of (2+1)D invertible fermionic topological phases with symmetry . Phys. Rev. B , 105(23):235143, 2022. arXiv:2109.11039

    2022 arXiv

  2. [10]

    Fusion 3-categories for duality defects

    Lakshya Bhardwaj, Thibault D\'ecoppet, Sakura Schafer-Nameki, and Matthew Yu. Fusion 3-categories for duality defects. Commun. Math. Phys. , 406:208, 2025. arXiv:2408.13302

    2025 arXiv

  3. [11]

    Comments on global symmetries, anomalies, and duality in (2+1)d

    Francesco Benini, Po-Shen Hsin, and Nathan Seiberg. Comments on global symmetries, anomalies, and duality in (2+1)d. JHEP , 04:135, 2017. arXiv:1702.07035

    2017 arXiv

  4. [12]

    Gapped phases in (2+1)d with non-invertible symmetries: P art I

    Lakshya Bhardwaj, Daniel Pajer, Sakura Schafer-Nameki, Apoorv Tiwari, Alison Warman, and Jingxiang Wu. Gapped phases in (2+1)d with non-invertible symmetries: P art I . 2024. arXiv:2408.05266

    2024 arXiv

  5. [13]

    Hasse diagrams for gapless SPT and SSB phases with non-invertible symmetries

    Lakshya Bhardwaj, Daniel Pajer, Sakura Schafer-Nameki, and Alison Warman. Hasse diagrams for gapless SPT and SSB phases with non-invertible symmetries. 2024. arXiv:2403.00905

    2024

  6. [14]

    Generalized charges, part II : Non-invertible symmetries and the symmetry TFT

    Lakshya Bhardwaj and Sakura Schafer-Nameki. Generalized charges, part II : Non-invertible symmetries and the symmetry TFT . 2023. arXiv:2305.17159

    2023

  7. [15]

    Generalized charges, part I : I nvertible symmetries and higher representations

    Lakshya Bhardwaj and Sakura Schafer-Nameki. Generalized charges, part I : I nvertible symmetries and higher representations. SciPost Phys. , 16(4):093, 2024. arXiv:2304.02660

    2024 arXiv

  8. [16]

    Gapped phases in (2+1)d with non-invertible symmetries: Part II

    Lakshya Bhardwaj, Sakura Schafer-Nameki, Apoorv Tiwari, and Alison Warman. Gapped phases in (2+1)d with non-invertible symmetries: Part II . 2025. arXiv:2502.20440

    2025 arXiv

  9. [17]

    Bais, Bernd J

    Alexander F. Bais, Bernd J. Schroers, and Joost K. Slingerland. Broken quantum symmetry and confinement phases in planar physics. Phys. Rev. Let. , 89(18):181601, 2002. arXiv:hep-th/0205117

    2002 arXiv

  10. [18]

    Bais, Bernd J

    Alexander F. Bais, Bernd J. Schroers, and Joost K. Slingerland. Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory. JHEP , 2003(05):068, 2003. arXiv:hep-th/0205114

    2003 arXiv

  11. [19]

    Dynamics of QCD _ 3 with rank-two quarks and duality

    Changha Choi, Diego Delmastro, Jaume Gomis, and Zohar Komargodski. Dynamics of QCD _ 3 with rank-two quarks and duality. JHEP , 03:078, 2020. arXiv:1810.07720

    2020 arXiv

  12. [20]

    Exceptional Chern-Simons -matter dualities

    Clay C \'o rdova, Po-Shen Hsin, and Kantaro Ohmori. Exceptional Chern-Simons -matter dualities. SciPost Phys. , 7(4):056, 2019. arXiv:1812.11705

    2019 arXiv

  13. [21]

    Global symmetries, counterterms, and duality in Chern-Simons matter theories with orthogonal gauge groups

    Clay Cordova, Po-Shen Hsin, and Nathan Seiberg. Global symmetries, counterterms, and duality in Chern-Simons matter theories with orthogonal gauge groups. SciPost Phys. , 4(4):021, 2018. arXiv:1711.10008

    2018 arXiv

  14. [22]

    Time-Reversal Symmetry, Anomalies, and Dualities in (2+1) d

    Clay C \'o rdova, Po-Shen Hsin, and Nathan Seiberg. Time-Reversal Symmetry, Anomalies, and Dualities in (2+1) d . SciPost Phys. , 5(1):006, 2018. arXiv:1712.08639

    2018 arXiv

  15. [23]

    Anomaly obstructions to symmetry preserving gapped phases

    Clay C\'ordova and Kantaro Ohmori. Anomaly obstructions to symmetry preserving gapped phases. 2019. arXiv:1910.04962

    2019 arXiv

  16. [24]

    Anomaly constraints on gapped phases with discrete chiral symmetry

    Clay C\'ordova and Kantaro Ohmori. Anomaly constraints on gapped phases with discrete chiral symmetry. Phys. Rev. D , 102(2):025011, 2020. arXiv:1912.13069

    2020 arXiv

  17. [25]

    Sjoerd E. Crans. Generalized centers of braided and sylleptic monoidal 2-categories. Advances in Mathematics , 136:183--223, 1998

    1998

  18. [26]

    (3+1)d boundary topological order of (4+1)d fermionic SPT state

    Meng Cheng, Juven Wang, and Xinping Yang. (3+1)d boundary topological order of (4+1)d fermionic SPT state . 2024. arXiv:2411.05786

    2024

  19. [27]

    D \'e coppet

    Thibault D. D \'e coppet. Rigid and separable algebras in fusion 2-categories. Advances in Mathematics , 419:108967, 2023. arXiv:2205.06453

    2023 arXiv

  20. [28]

    D \'e coppet

    Thibault D. D \'e coppet. Extension theory and fermionic strongly fusion 2-categories ( W ith an appendix joint with T heo J ohnson- F reyd). SIGMA , 20:092, 2024. arXiv:2403.03211

    2024 arXiv

  21. [29]

    D \'e coppet

    Thibault D. D \'e coppet. D rinfeld centers and M orita equivalence classes of fusion 2-categories. Compositio Mathematica , 161(2):305--340, 2025. arXiv:2211.04917

    2025 arXiv

  22. [30]

    D \'e coppet

    Thibault D. D \'e coppet. Finite semisimple module 2-categories. Selecta Mathematica , 31:5, 2025. arXiv:2107.11037

    2025 arXiv

  23. [31]

    Cat\'egories tensorielles

    Pierre Deligne. Cat\'egories tensorielles. Mosc. Math. J. , 2(2):227--248, 2002

    2002

  24. [32]

    Symmetries of abelian Chern-Simons theories and arithmetic

    Diego Delmastro and Jaume Gomis. Symmetries of abelian Chern-Simons theories and arithmetic. JHEP , 03:006, 2021. arXiv:1904.12884

    2021 arXiv

  25. [33]

    On braided fusion categories I

    Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. On braided fusion categories I . Selecta Mathematica , 16:1--119, 2010

    2010

  26. [34]

    Infrared phases of 2d QCD

    Diego Delmastro, Jaume Gomis, and Matthew Yu. Infrared phases of 2d QCD . JHEP , 02:157, 2023. arXiv:2108.02202

    2023 arXiv

  27. [35]

    D\'ecoppet, Peter Huston, Theo Johnson-Freyd, Dmitri Nikshych, David Penneys, Julia Plavnik, David Reutter, and Matthew Yu

    Thibault D. D\'ecoppet, Peter Huston, Theo Johnson-Freyd, Dmitri Nikshych, David Penneys, Julia Plavnik, David Reutter, and Matthew Yu. The classification of fusion 2-categories. 2024. arXiv:2411.05907

    2024 arXiv

  28. [36]

    The W itt group of non-degenerate braided fusion categories

    Alexei Davydov, Michael M \"u ger, Dmitri Nikshych, and Victor Ostrik. The W itt group of non-degenerate braided fusion categories. J. Reine Angew. Math. , 2013(667), 2013. arXiv:1009.2117

    2013 arXiv

  29. [37]

    Braided P icard groups and graded extensions of braided tensor categories

    Alexei Davydov and Dmitri Nikshych. Braided P icard groups and graded extensions of braided tensor categories. Selecta Mathematica , 27(4):65, 2021. arXiv:0906.0620

    2021 arXiv

  30. [38]

    On the structure of the W itt group of braided fusion categories

    Alexei Davydov, Dmitri Nikshych, and Victor Ostrik. On the structure of the W itt group of braided fusion categories. Selecta Mathematica , 19(1):237--269, 2013. arXiv:1109.5558

    2013 arXiv

  31. [39]

    Douglas and David J

    Christopher L. Douglas and David J. Reutter. Fusion 2-categories and a state-sum invariant for 4-manifolds. 2018. arXiv:1812.11933

    2018 arXiv

  32. [40]

    Topological Gauge Theories and Group Cohomology

    Robbert Dijkgraaf and Edward Witten. Topological Gauge Theories and Group Cohomology . Commun. Math. Phys. , 129:393, 1990

    1990

  33. [41]

    D\'ecoppet and Hao Xu

    Thibault D. D\'ecoppet and Hao Xu. Local modules in braided monoidal 2-categories . J. Math. Phys. , 65(6):061702, 2024. arXiv:2307.02843

    2024 arXiv

  34. [42]

    D \'e coppet and Matthew Yu

    Thibault D. D \'e coppet and Matthew Yu. Gauging noninvertible defects:\ a 2-categorical perspective. Lett. Math. Phys. , 113(2):36, 2023. arXiv:2211.08436

    2023 arXiv

  35. [43]

    D\'ecoppet and Matthew Yu

    Thibault D. D\'ecoppet and Matthew Yu. Fiber 2-functors and Tambara Yamagami fusion 2-categories. Commun. Math. Phys. , 406(3):64, 2025. arXiv:2306.08117

    2025 arXiv

  36. [44]

    Constructing anomalous 4d TQFT s

    Arun Debray, Weicheng Ye, and Matthew Yu. Constructing anomalous 4d TQFT s. In preparation

  37. [45]

    Bosonization and anomaly indicators of (2+1)-D fermionic topological orders

    Arun Debray, Weicheng Ye, and Matthew Yu. Bosonization and anomaly indicators of (2+1)-D fermionic topological orders. 2023. arXiv:2312.13341

    2023 arXiv

  38. [46]

    Ellison and Meng Cheng

    Tyler D. Ellison and Meng Cheng. Toward a Classification of Mixed-State Topological Orders in Two Dimensions . PRX Quantum , 6(1):010315, 2025. arXiv:2405.02390

    2025 arXiv

  39. [47]

    Tensor categories , volume 205

    Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. Tensor categories , volume 205. American Mathematical Soc., 2016

    2016

  40. [48]

    Fusion categories and homotopy theory

    Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. Fusion categories and homotopy theory. Quantum Topology , 1(3), 2009. arXiv:0909.3140

    2009 arXiv

  41. [49]

    Freed, Gregory W

    Daniel S. Freed, Gregory W. Moore, and Constantin Teleman. Topological symmetry in quantum field theory . 2022. arXiv:2209.07471

    2022 arXiv

  42. [50]

    Daniel S. Freed. Pions and generalized cohomology . J. Diff. Geom. , 80(1):45--77, 2008. arXiv:hep-th/0607134

    2008 arXiv

  43. [51]

    Freed and Constantin Teleman

    Daniel S. Freed and Constantin Teleman. Relative quantum field theory . Commun. Math. Phys. , 326:459--476, 2014. arXiv:1212.1692

    2014 arXiv

  44. [52]

    Condensations in higher categories

    Davide Gaiotto and Theo Johnson-Freyd. Condensations in higher categories . 2019. arXiv:1905.09566

    2019 arXiv

  45. [53]

    Phases of adjoint QCD _3 and dualities

    Jaume Gomis, Zohar Komargodski, and Nathan Seiberg. Phases of adjoint QCD _3 and dualities . SciPost Phys. , 5(1):007, 2018. arXiv:1710.03258

    2018 arXiv

  46. [54]

    Venegas-Ram \'i rez

    C \'e sar Galindo and C \'e sar F. Venegas-Ram \'i rez. Categorical fermionic actions and minimal modular extensions, 2017. arXiv:1712.07097

    2017

  47. [55]

    Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear models and a special group supercohomology theory

    Zheng-Cheng Gu and Xiao-Gang Wen. Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear models and a special group supercohomology theory . Phys. Rev. B , 90(11):115141, 2014. arXiv:1201.2648

    2014 arXiv

  48. [56]

    Jerome A. Jenquin. Spin C hern- S imons and spin TQFT s. 2006. math/0605239

    2006 arXiv

  49. [57]

    On the classification of topological orders

    Theo Johnson-Freyd. On the classification of topological orders. Commun. Math. Phys. , 393(2):989--1033, 2022. arXiv:2003.06663

    2022 arXiv

  50. [58]

    (3+1)D topological orders with only a Z _2 -charged particle

    Theo Johnson-Freyd. (3+1)D topological orders with only a Z _2 -charged particle . In Quantum Symmetries , volume 813 of Contemp. Math. , page 175–210. AMS, 2025

    2025

  51. [59]

    Theo Johnson-Freyd and David J. Reutter. S -matrices for higher fusion categories. In preparation

  52. [60]

    Theo Johnson-Freyd and David J. Reutter. Minimal non-degenerate extensions. Journ. Amer. Math. Soc. , 37:81--150, 2024. arXiv:2105.15167

    2024

  53. [61]

    Fusion 2-categories with no line operators are grouplike

    Theo Johnson-Freyd and Matthew Yu. Fusion 2-categories with no line operators are grouplike. Bull. Aust. Math. Soc. , 104(3):434--442, 2021. arXiv:2010.07950

    2021 arXiv

  54. [62]

    Topological orders in (4+1)-dimensions

    Theo Johnson-Freyd and Matthew Yu. Topological orders in (4+1)-dimensions. SciPost Phys. , 13(3):068, 2022. arXiv:2104.04534

    2022 arXiv

  55. [63]

    Extension theory for braided-enriched fusion categories

    Corey Jones, Scott Morrison, David Penneys, and Julia Plavnik. Extension theory for braided-enriched fusion categories. Int. Math. Res. Not. , 2022(20):15632--15683, 2022. arXiv:1910.03178

    2022 arXiv

  56. [64]

    Algebraic higher symmetry and categorical symmetry:\ a holographic and entanglement view of symmetry

    Liang Kong, Tian Lan, Xiao-Gang Wen, Zhi-Hao Zhang, and Hao Zheng. Algebraic higher symmetry and categorical symmetry:\ a holographic and entanglement view of symmetry. Phys. Rev. Res. , 2(4):043086, 2020. arXiv:2005.14178

    2020 arXiv

  57. [65]

    Classification of topological phases with finite internal symmetries in all dimensions

    Liang Kong, Tian Lan, Xiao-Gang Wen, Zhi-Hao Zhang, and Hao Zheng. Classification of topological phases with finite internal symmetries in all dimensions . JHEP , 09:093, 2020. arXiv:2003.08898

    2020 arXiv

  58. [66]

    Anyon condensation and tensor categories

    Liang Kong. Anyon condensation and tensor categories. Nucl. Phys. B , 886:436--482, 2014. arXiv:1307.8244

    2014 arXiv

  59. [67]

    On gapped boundaries for SPT phases beyond group cohomology

    Ryohei Kobayashi, Kantaro Ohmori, and Yuji Tachikawa. On gapped boundaries for SPT phases beyond group cohomology . JHEP , 11:131, 2019. arXiv:1905.05391

    2019 arXiv

  60. [68]

    Symmetry TFT s for non-invertible defects

    Justin Kaidi, Kantaro Ohmori, and Yunqin Zheng. Symmetry TFT s for non-invertible defects. Commun. Math. Phys. , 404(2):1021--1124, 2023. arXiv:2209.11062

    2023 arXiv

  61. [69]

    (3+1)d mixed state topological orders

    Ryohei Kobayashi, Abhinav Prem, and Matthew Yu. (3+1)d mixed state topological orders. In preparation

  62. [70]

    A symmetry breaking scenario for QCD _ 3

    Zohar Komargodski and Nathan Seiberg. A symmetry breaking scenario for QCD _ 3 . JHEP , 01:109, 2018

    2018

  63. [71]

    The center of monoidal 2-categories in 3+1 D D ijkgraaf- W itten theory

    Liang Kong, Yin Tian, and Shan Zhou. The center of monoidal 2-categories in 3+1 D D ijkgraaf- W itten theory. Advances in Mathematics , 360:106928, 2020. arXiv:1905.04644

    2020 arXiv

  64. [72]

    Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

    Liang Kong and Xiao-Gang Wen. Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions . 2014. arXiv:1405.5858

    2014 arXiv

  65. [73]

    Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers

    Liang Kong, Xiao-Gang Wen, and Hao Zheng. Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers. 2015. arXiv:1502.01690

    2015 arXiv

  66. [74]

    Boundary-bulk relation in topological orders

    Liang Kong, Xiao-Gang Wen, and Hao Zheng. Boundary-bulk relation in topological orders. Nucl. Phys. B , 922:62--76, 2017. arXiv:1702.00673

    2017 arXiv

  67. [75]

    One dimensional gapped quantum phases and enriched fusion categories

    Liang Kong, Xiao-Gang Wen, and Hao Zheng. One dimensional gapped quantum phases and enriched fusion categories . JHEP , 03:022, 2022. arXiv:2108.08835

    2022 arXiv

  68. [76]

    A mathematical theory of gapless edges of 2d topological orders.\ Part I

    Liang Kong and Hao Zheng. A mathematical theory of gapless edges of 2d topological orders.\ Part I . JHEP , 02:150, 2020. arXiv:1905.04924

    2020 arXiv

  69. [77]

    A mathematical theory of gapless edges of 2d topological orders.\ Part II

    Liang Kong and Hao Zheng. A mathematical theory of gapless edges of 2d topological orders.\ Part II . Nucl. Phys. B , 966:115384, 2021. arXiv:1912.01760

    2021 arXiv

  70. [78]

    Higher condensation theory

    Liang Kong, Zhi-Hao Zhang, Jiaheng Zhao, and Hao Zheng. Higher condensation theory . 2024. arXiv:2403.07813

    2024

  71. [79]

    Classification of (3+1)d bosonic topological orders (I) :\ the case when pointlike excitations are all bosons

    Tian Lan, Liang Kong, and Xiao-Gang Wen. Classification of (3+1)d bosonic topological orders (I) :\ the case when pointlike excitations are all bosons. Phys. Rev. X , 8(2), 2018. arXiv:1704.04221

    2018 arXiv

  72. [80]

    Classification of 3+1d bosonic topological orders (II) : The case when some pointlike excitations are fermions

    Tian Lan and Xiao-Gang Wen. Classification of 3+1d bosonic topological orders (II) : The case when some pointlike excitations are fermions. Phys. Rev. X , 9(2), 2019. arXiv:1801.08530

    2019 arXiv

  73. [81]

    Classification of fractional quantum H all states with spatial symmetries

    Naren Manjunath and Maissam Barkeshli. Classification of fractional quantum H all states with spatial symmetries . 2020. arXiv:2012.11603

    2020 arXiv

  74. [82]

    Schommer-Pries

    Christopher J. Schommer-Pries. The Classification of Two-Dimensional Extended Topological Field Theories . PhD thesis, UC Berkeley, 2011. arXiv:1112.1000

    2011 arXiv

  75. [83]

    Noisy Approach to Intrinsically Mixed-State Topological Order

    Ramanjit Sohal and Abhinav Prem. Noisy Approach to Intrinsically Mixed-State Topological Order . PRX Quantum , 6(1):010313, 2025. arXiv:2403.13879

    2025 arXiv

  76. [84]

    Gapped Boundary Phases of Topological Insulators via Weak Coupling

    Nathan Seiberg and Edward Witten. Gapped Boundary Phases of Topological Insulators via Weak Coupling . PTEP , 2016(12):12C101, 2016. arXiv:1602.04251

    2016 arXiv

  77. [85]

    A Generalized Crystalline Equivalence Principle

    Devon Stockall and Matthew Yu. A Generalized Crystalline Equivalence Principle . 2025. arXiv:2508.10978

    2025 arXiv

  78. [86]

    On gauging finite subgroups

    Yuji Tachikawa. On gauging finite subgroups . SciPost Phys. , 8(1):015, 2020. arXiv:1712.09542

    2020 arXiv

  79. [87]

    Topological quantum field theory, symmetry breaking, and finite gauge theory in 3+1D

    Ryan Thorngren. Topological quantum field theory, symmetry breaking, and finite gauge theory in 3+1D . Phys. Rev. B , 101(24):245160, 2020. arXiv:2001.11938

    2020 arXiv

  80. [88]

    Homotopy field theory in dimension 3 and crossed group-categories, 2000

    Vladimir Turaev. Homotopy field theory in dimension 3 and crossed group-categories, 2000. arXiv:math/0005291

    2000 arXiv

  81. [89]

    Mutual influence of symmetries and topological field theories

    Daniel Teixeira and Matthew Yu. Mutual influence of symmetries and topological field theories. 2025. arXiv:2507.06304

    2025 arXiv

  82. [90]

    Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders

    Xiao-Gang Wen. Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders . Phys. Rev. D , 88(4):045013, 2013. arXiv:1303.1803

    2013 arXiv

  83. [91]

    String condensation and topological holography for 2+1 D gapless SPT

    Rui Wen. String condensation and topological holography for 2+1 D gapless SPT . 2024. arXiv:2408.05801

    2024 arXiv

  84. [92]

    Topological holography for 2+1- D gapped and gapless phases with generalized symmetries

    Rui Wen. Topological holography for 2+1- D gapped and gapless phases with generalized symmetries. 2025. arXiv:2503.13685

    2025 arXiv

  85. [93]

    Towards a complete classification of symmetry-protected topological phases for interacting fermions in three dimensions and a general group supercohomology theory

    Qing-Rui Wang and Zheng-Cheng Gu. Towards a complete classification of symmetry-protected topological phases for interacting fermions in three dimensions and a general group supercohomology theory. Phys. Rev. X , 8(1):011055, 2018. arXiv:1703.10937

    2018 arXiv

  86. [94]

    The ``Parity'' Anomaly On An Unorientable Manifold

    Edward Witten. The ``Parity'' Anomaly On An Unorientable Manifold . Phys. Rev. B , 94(19):195150, 2016. arXiv:1605.02391

    2016 arXiv

  87. [95]

    Rui Wen and Andrew C. Potter. Classification of 1+1D gapless symmetry protected phases via topological holography . Phys. Rev. B , 111(11):115161, 2025. arXiv:2311.00050

    2025 arXiv

  88. [96]

    Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

    Juven Wang, Xiao-Gang Wen, and Edward Witten. Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions . Phys. Rev. X , 8(3):031048, 2018. arXiv:1705.0672

    2018

  89. [97]

    Intrinsic Mixed-State Topological Order

    Zijian Wang, Zhengzhi Wu, and Zhong Wang. Intrinsic Mixed-State Topological Order . PRX Quantum , 6(1):010314, 2025. arXiv:2307.13758

    2025 arXiv

  90. [98]

    Higher W itt groups for 2-categories I :\ centralizers, 2024

    Hao Xu. Higher W itt groups for 2-categories I :\ centralizers, 2024. arXiv:2403.07768

    2024 arXiv

  91. [99]

    On Étale algebras and bosonic fusion 2-categories, 2024

    Hao Xu. On Étale algebras and bosonic fusion 2-categories, 2024. arXiv:2411.13367

    2024 arXiv

  92. [100]

    Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase

    Xinping Yang and Meng Cheng. Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase . Phys. Rev. B , 110(4):045137, 2024. arXiv:2303.00719

    2024 arXiv

  93. [101]

    Topological mixed states:\ axiomatic approaches and phases of matter

    Tai-Hsuan Yang, Bowen Shi, and Jong Yeon Lee. Topological mixed states:\ axiomatic approaches and phases of matter. 2025. arXiv:2506.04221

    2025

  94. [102]

    String condensations in 3+1 D and L agrangian algebras

    Jiaheng Zhao, Jia-Qi Lou, Zhi-Hao Zhang, Ling-Yan Hung, Liang Kong, and Yin Tian. String condensations in 3+1 D and L agrangian algebras. Adv. Theor. Math. Phys. , 27(2):583--622, 2023. arXiv:2208.07865

    2023 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.