Pith. sign in

REVIEW 1 major objections 5 minor 217 references

Understanding Non-Split 2-Group Symmetry: (3+1)D SymTFT, Anomaly and Bordism

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read This paper computes the oriented and spin bordism groups that classify anomalies of the simplest non-split 2-group symmetry through spacetime dimension $d=5$, and shows that the anomalous 2+1D symmetry category is the fermionic fusion…

desk verdict Solid first computation of non-split 2-group bordism anomaly tables with a real soft spot: the anomalous categorical equivalence rests on an explicitly unverified twisted-supercohomology comparison. read the letter →

arxiv 2608.08766 v2 pith:OKWZNK5G submitted 2026-08-09 hep-th cond-mat.str-elmath-phmath.ATmath.MP

Zhenbang Gu , Ran Luo , Yi-Nan Wang , Yi Zhang This is my paper · ORCID
classification hep-thcond-mat.str-elmath-phmath.ATmath.MP
keywords non-split2-groupsymmetryPostnikovclass'tHooftanomalybordismclassificationSymTFTfusion2-categoryfermioniccategoricalLandauparadigm
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 works out the full anomaly structure and symmetry TFT of the simplest finite, non-split 2-group symmetry, whose $\mathbb{Z}_2$ 0-form and $\mathbb{Z}_2$ 1-form symmetries are locked together by a non-trivial Postnikov class. For physical theories in $d$ spacetime dimensions with $d\le 5$, it computes the classifying space of the 2-group and the oriented and spin bordism groups that classify 't Hooft anomalies, and it writes explicit cochain-level anomaly actions in each dimension. For $d=3$ it constructs the (3+1)D SymTFT and shows that, in the anomalous case, the symmetry category is monoidally equivalent to the fermionic fusion 2-category $2\mathrm{sVect}_{(\mathbb{Z}_4,2)}$. It then classifies the minimal topological and physical boundary conditions and organizes the resulting gapped phases into a categorical Landau paradigm table. If correct, this fixes the anomaly menu and phase diagram for this symmetry and supplies a template for non-split higher-group symmetries generally.

What carries the argument

The load-bearing machinery is the classifying space $B\mathcal{G}$ of the weak 2-group, realized as the total space of the Postnikov fibration $K(\mathbb{Z}_2,2)\to B\mathcal{G}\to K(\mathbb{Z}_2,1)$ whose k-invariant is the non-trivial Postnikov class $\beta=a^3\in H^3(B\mathbb{Z}_2;\mathbb{Z}_2)$. The paper computes the integral and mod-2 cohomology of this space with the Serre spectral sequence, then feeds the results into Atiyah–Hirzebruch and Adams spectral sequences to obtain the bordism groups. On the cochain side, the argument is carried by the cocycles $v_3=a\cup b+b\cup_1 b+b\cup_2 a^3$ and $v_4=b\cup b+b\cup_1 a^3+a^2\cup b$ built from the 2-group gauge fields, with $v_4$ generating the $d=3$ anomaly. The categorical side is carried by the classification of fusion 2-categories: the anomalous symmetry category is identified with the fermionic fusion 2-category $2\mathrm{sVect}_{(\mathbb{Z}_4,2)}$, and the SymTFT is described by Lagrangians such as $S=\frac{2\pi}{4}\int_X B\cup\delta A+\pi\int_X (B\bmod 2)\cup w_2(TX)$ with $\mathbb{Z}_4$ fields $A,B$. Minimal gapped boundaries are classified by Lagrangian algebras in the Drinfeld centers $Z_1(2\mathrm{Vect}_{\mathcal{G}})$ and $Z_1(2\mathrm{sVect}_{(\mathbb{Z}_4,2)})$.

What would settle it

Compute the full twisted spin-bordism anomaly group $\mathrm{Hom}(\widetilde{\Omega}^{\mathrm{Spin}}_5(B\mathcal{G};v),U(1))$ for the non-split extension $v=a^2$ and compare it with the paper's $\mathrm{SH}^4_v(B\mathbb{Z}_2)\cong\mathbb{Z}_2$; any additional torsion or a different group extension would break the identification $2\mathrm{Vect}^{\omega}_{\mathcal{G}}\simeq 2\mathrm{sVect}_{(\mathbb{Z}_4,2)}$. Equivalently, one could evaluate the proposed anomalous bulk action on a closed oriented 4-manifold with $w_2(TX)\neq 0$ and check that the partition function equals $(-1)^{\int_X b\cup b+b\cup_1 a^3+a^2\cup b}$.

Watch

Extended reading notes

Core claim

The paper's central claim is that the non-split 2-group $\mathcal{G}=(\mathbb{Z}_2,\mathbb{Z}_2,\mathrm{triv},1)$ has anomaly groups given by the reduced bordism groups $\widetilde{\Omega}^{\mathrm{SO}}_{d+1}(B\mathcal{G})$ and $\widetilde{\Omega}^{\mathrm{Spin}}_{d+1}(B\mathcal{G})$ for $d\le 5$, with values $\mathbb{Z}_2,0,\mathbb{Z}_2,\mathbb{Z}_2^3,\mathbb{Z}_2^2$ in the oriented cases and $\mathbb{Z}_2,\mathbb{Z}_2,\mathbb{Z}_4\oplus\mathbb{Z}_2,\mathbb{Z}_2,\mathbb{Z}_2^2$ in the spin cases. The unique $d=3$ oriented anomaly is represented by the cochain $\frac{1}{2}v_4=\frac{1}{2}(b\cup b+b\cup_1 a^3+a^2\cup b)$, where $(a,b)$ are the weak 2-group gauge fields obeying $\delta b=a^3$. When this anomaly is present, the symmetry category is not an ordinary bosonic $2\mathrm{Vect}^{\pi}_{\mathcal{G}}$ but is monoidally equivalent to the fermionic fusion 2-category $2\mathrm{sVect}_{(\mathbb{Z}_4,2)}$, whose underlying supergroup is the non-split central extension of $\mathbb{Z}_2$ by fermion parity. The paper gives explicit Lagrangian SymTFT actions for both the non-anomalous and anomalous cases, classifies their minimal Lagrangian algebras and boundary conditions, and derives the resulting gapped phases: symmetric, SSB, and SPT phases.

Load-bearing premise

The anomalous-case conclusions rest on the assumption that the three-layer twisted supercohomology theory $\mathrm{SH}^4_{a^2}(B\mathbb{Z}_2)$ captures the full twisted spin-bordism anomaly classification for the non-split extension, a comparison the paper explicitly leaves for future work.

Editorial extensions

If this is right

  • The full menu of 't Hooft anomalies of $\mathcal{G}=(\mathbb{Z}_2,\mathbb{Z}_2,\mathrm{triv},1)$ is now known in spacetime dimensions $d\le 5$, with explicit cochain actions for every bosonic anomaly.
  • In $d=3$ the unique bosonic anomaly is carried by the phase $(-1)^{\int_M b\cup b+b\cup_1 a^3+a^2\cup b}$, so any theory with this 2-group symmetry either realizes that anomaly or is the anomalous boundary of a (3+1)D invertible phase.
  • An anomalous non-split 2-group symmetry in 2+1D cannot be captured by a bosonic fusion 2-category; the anomaly forces the fermionic category $2\mathrm{sVect}_{(\mathbb{Z}_4,2)}$, so the fermionic sector is unavoidable even when the theory is nominally bosonic.
  • The SymTFT has exactly two minimal topological boundaries in the anomalous case and seven in the non-anomalous case, giving the complete set of gauging routes and the phase table for the categorical Landau paradigm.
  • No gapped phase can preserve the $\mathbb{Z}_2$ 0-form symmetry while breaking the $\mathbb{Z}_2$ 1-form symmetry, because the Postnikov class ties the 0-form anomaly to a 1-form transformation.

Reading between the lines

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

  • If the twisted supercohomology comparison flagged in Appendix B.3 holds, the same combination of Serre spectral sequence and cochain descent should extend to non-split 2-groups $\mathcal{G}=(\mathbb{Z}_2,\mathbb{Z}_N,\mathrm{triv},\beta)$, yielding anomaly actions without recomputing bordism groups by hand.
  • The equivalence $2\mathrm{Vect}^{\omega}_{\mathcal{G}}\simeq 2\mathrm{sVect}_{(\mathbb{Z}_4,2)}$ suggests that gauging the $\mathbb{Z}_2^f$ subgroup in the fermionic description should exactly reproduce the bosonic 2-group anomaly; an explicit lattice bosonization of the $\mathbb{Z}_2$ gauge-theory fractionalization example in Section 6 would provide a microscopic check.
  • A concrete lattice signature of the predicted 2-group SPT phase is the semion value of the topological spin of the dressed Wilson loop, which a stabilizer or tensor-network computation could measure directly.
  • The same bulk action with $\mathbb{Z}_4$ fields and the $w_2$ coupling may serve as a building block for SymTFTs of other fermionic 2-group symmetries, with the Postnikov class realized by a twisted cohomology generator with $v\neq 0$.
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

1 major / 5 minor

Summary. The paper analyzes the simplest finite non-split 2-group G = (Z2, Z2, triv, 1), with nontrivial Postnikov class β ∈ H³(BZ2;Z2). The anomaly classification for d ≤ 5 spacetime dimensions is obtained by computing the reduced oriented and spin bordism groups Ω̃^{SO}_{d+1}(BG) and Ω̃^{Spin}_{d+1}(BG) (Table 1), with full Serre spectral sequence, AHSS, and ASS derivations in Appendix C and explicit cochain representatives (v3, v4, and higher-degree classes) for the oriented anomalies in Section 3.2. For d = 3, the paper constructs the (3+1)D SymTFT in two cases: without the 2-group anomaly, where the Drinfeld center Z1(2Vect_G) is identified with Z1(2Vect^{πβ}_{Z2×Z2}), seven minimal Lagrangian algebras and their TQFT counterparts are classified (Tables 2 and 4), and the gapped phase structure is mapped out; with the anomaly ω (the generator of Hom(Ω̃^{SO}_4(BG),U(1)) ≅ Z2), the paper argues that the symmetry category is monoidally equivalent to the fermionic fusion 2-category 2sVect(Z4,2), proposes a Stiefel-Whitney-type Lagrangian (5.29), and classifies the two minimal topological boundary conditions (Table 3). Section 6 sketches physical realizations in Z2 gauge theory and free-fermion systems.

Significance. If the central claims hold, this is the first complete bordism classification and SymTFT/categorical-Landau analysis for a finite non-split 2-group symmetry, going beyond the split and toric 2-group cases in the literature. The paper's strengths are its explicitness and internal consistency: the spectral-sequence computations in Appendix C are laid out in unusual detail, the cochain-level anomaly actions (v3, v4, and the higher-degree representatives) are concrete and checkable, and several cross-checks are provided (e.g., SH³(BZ2) ≅ Z8 matching Ω̃^{Spin}_3(BZ2), and consistency of the twisted computation with [177, Ex. 4.13]). The conceptual highlight — a bosonic anomaly turning the 2-group symmetry category into a fermionic fusion 2-category 2sVect(Z4,2) — is interesting and potentially influential. The explicit phase diagrams (Tables 2, 3, 5, 6) and boundary condition classifications are falsifiable and of direct use for model building.

major comments (1)
  1. [§4.2 and Appendix B.3] The derivation of the load-bearing equivalence 2Vect^ω_G ≃ 2sVect(Z4,2) in Section 4.2 rests on the assertion that ϖ = 0 in SH^4_{a²}(BZ2), a fact used to identify the anomalous symmetry category and hence to drive the Drinfeld-center and boundary analyses of Sections 4.3 and 5.2. However, Appendix B.3 states that for v ≠ 0 the twisted supercohomology SH^n_v is only a three-layer approximation, that Brumfiel and Morgan did not treat v ≠ 0, and that the comparison with the full twisted spin bordism dual 'has to be checked separately.' The paper then uses SH^4_v as the anomaly-theory input for the categorical classification in Section 4.2 without performing that check, so an unmodeled differential or extension in the full twisted spin bordism dual could in principle render the class with layers (n2 = 0, n3 = a³) nontrivial, replacing 2sVect(Z4,2) by 2sVect^ϖ_{(Z4,2)} and modifying Eq. (4.46) and the Lagrangian boundary conditions of Section 5.2. I note that the concern is likely resolvable: for reduced degree 4, only the AHSS rows q = 0,1,2 contribute (the q = 3 coefficient group vanishes and the q = 4 row is the point contribution removed by reduction), so the three-layer model is complete in this degree. The authors should either add this argument, or state explicitly that the categorical claim follows from the algebraic supercohomology of [177, Examples 4.12 and 4.13], which is the input required by Theorem 4.1 and is logically independent of the bordism-theoretic identification.
minor comments (5)
  1. [Table 1] The superscript formatting in the first two rows of Table 1 (e.g., 'Z2²' in the H-row at d+1 = 5 and 'Z2³' in the Ω̃^SO row) is easy to misread; using explicit direct-sum notation such as Z2 ⊕ Z2 and Z2 ⊕ Z2 ⊕ Z2 would improve readability.
  2. [Eq. (5.11)] In Eq. (5.11), the symbol β in 'a1 ∪ a2 ∪ βa2' appears to denote the Bockstein homomorphism, which clashes with the Postnikov class β used throughout the paper; please define this symbol or rename it.
  3. [§4.2] The equality π0(2Vect^ω_G) = π0(2Vect_G) is asserted in Eq. (4.25) with only a physical intuition in its support; a brief justification (the anomaly modifies the associator, not the set of simple objects) would be helpful.
  4. [Appendix B.3] The notation SH^{n+v}(BGb), with the remark that 'n+v means degree n with twist v; it is not a sum of degrees,' is confusing; SH^n_v(BGb) is unambiguous and should be used throughout.
  5. [§3.2 and Table 1] For the spin cases, only the bordism groups in Table 1 are reported, while explicit cochain or topological actions are given only for the oriented anomalies; a sentence indicating that the spin anomaly theories are classified by Table 1 but their explicit actions are left for future work would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bordism and categorical derivations are self-contained; the acknowledged twisted-supercohomology gap is a limitation, not a circular step.

full rationale

The paper's central computations are spectral-sequence derivations of the cohomology and oriented/spin bordism groups of the classifying space BG, with explicit cochain representatives (v3, v4, v5) and independent cross-checks such as SH^3(BZ2) ~= Z8, SH^4(BZ2) = 0, and SH^3(BZ4) ~= Z8 ⊕ Z2 that match known spin bordism groups. No parameter is fitted to a subset of data and then renamed a prediction; the anomaly classes are read off from bordism computations, not declared from the desired result. The anomalous categorical equivalence 2Vect^omega_G ≃ 2sVect(Z4,2) is obtained from external theorems of Décoppet and Johnson-Freyd together with an explicit computation of the extension class (υ = a^2) and the twisted differential d^υ_2(a) = a^3 = β, so the target equivalence is not used as its own input. The only self-citation is the footnote citing the authors' earlier strict-2-group paper [64], and the paper explicitly states that it uses the weak formulation instead, so that citation is not load-bearing. Appendix B.3 honestly warns that Brumfiel and Morgan did not treat v ≠ 0 and that the comparison of the three-layer twisted supercohomology with the full twisted spin bordism dual "has to be checked separately"; this is a genuine correctness risk for the anomalous fermionic description, but it is a stated limitation rather than a circular reduction, and the bosonic H^4(BG;U(1)) ≅ Z2 anomaly classification is independent of that open comparison. Therefore the derivation chain contains no step that is equivalent to its own inputs by construction or by self-citation.

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

The central claim depends on standard spectral sequence machinery and on external classification theorems, but not on fitted parameters or new physical entities. The most delicate input is the unchecked twisted supercohomology comparison.

assumptions (6)
  • domain assumption Freed-Hopkins classification of invertible anomalies by the Anderson dual of bordism
    Used in Section 3.1 to map anomalies to Omega_tilde_{d+1}(BG).
  • standard math Serre and Adams spectral sequence differentials used to compute cohomology and bordism are correct
    Appendix C carries out these computations; small errors would change Table 1.
  • standard math Brumfiel-Morgan isomorphism SH^n is isomorphic to Hom(Omega_tilde^Spin_n, U(1)) for n=3,4
    Appendix B.1 relies on this to identify supercohomology with spin bordism duals.
  • standard math Classification theorem for fusion 2-categories by data (G,H,pi,A,psi)
    Theorem A.1 and A.2 from Deccopet et al. underlies the categorical data and center decompositions.
  • domain assumption For 2Vect_G the classifying braided category A is Vect
    Section 4.1 states this and restricts the analysis to minimal topological boundaries, excluding non-minimal boundaries with intrinsic topological order.
  • domain assumption Twisted supercohomology SH^4_v gives the correct fermionic data for the anomalous case
    Appendix B.3 explicitly says the comparison with the full twisted spin bordism dual must be checked separately.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Understanding Non-Split 2-Group Symmetry: (3+1)D SymTFT, Anomaly and Bordism." pith.science (2026). https://pith.science/paper/OKWZNK5G

@misc{pith2026260808766,
  author       = {Pith},
  title        = {Pith review of: Understanding Non-Split 2-Group Symmetry: (3+1)D SymTFT, Anomaly and Bordism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKWZNK5G}},
  note         = {Machine review of arXiv:2608.08766}
}
abstract

We present a systematic study of finite, non-split 2-group symmetries with a non-trivial Postnikov class, focusing on the simplest example with a $\mathbb{Z}_2$ 0-form symmetry and a $\mathbb{Z}_2$ 1-form symmetry, intertwined together by the non-trivial Postnikov class in $H^3(B\mathbb{Z}_2;\mathbb{Z}_2)\cong\mathbb{Z}_2$, denoted by $\mathcal{G}$. We classify the anomalies of this 2-group symmetry for physical theories in $d$-dimensional spacetime, by computing the oriented bordism groups $\Omega_{d+1}^{\rm SO}(B\mathcal{G})$ and the spin bordism groups $\Omega_{d+1}^{\rm Spin}(B\mathcal{G})$ for $d\leq 5$. For the case of $d=3$, we investigate the Symmetry TFT/TO of the 2-group symmetry $\mathcal{G}$ using the language of fusion 2-categories, as well as (3+1)D TQFT actions, for the cases without or with the 2-group anomaly classified by $\operatorname{Hom}\left(\widetilde{\Omega}_4^{\rm SO}(B\mathcal{G}),U(1)\right) \cong H^4(B\mathcal{G};U(1))\cong\mathbb{Z}_2$. We classify the minimal topological and physical boundary conditions of the Symmetry TFTs, and carry out the categorical Landau paradigm for such a non-split finite 2-group.

Figures

Figures reproduced from arXiv: 2608.08766 by the authors.

Figure 1
Figure 1. The action of the codimension-one defect labeled by a group element [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The physical meaning of the Postnikov class [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The vertical line is the single fiber Fx0 over x0. The closed loop g in the base lifts to a path gey in BG from y to Tg(y); the lifted path need not be closed. The homotopy class of Tg is the monodromy around g. For fixed Π1, Π2, and α, the Postnikov class now takes value in the β ∈ H3 α(BΠ1; Π2). (3.5) specifies the homotopy type of the middle space. When α = triv., this loop action is trivial and one recovers the … view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Gauging routes among the seven minimal topological boundary conditions of [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]
Figure 5
Figure 5. Figure 5: Hasse diagram of the connected condensable algebras of the SymTFT [PITH_FULL_IMAGE:figures/full_fig_p035_5.png]
Figure 6
Figure 6. Figure 6: Hasse diagram of the condensable algebras of the anomalous SymTFT [PITH_FULL_IMAGE:figures/full_fig_p041_6.png]
Figure 7
Figure 7. Figure 7: Illustration of the fusion 2-category 2Vectπ G. Example A.2 (2VectG). In (2+1)D system, a 2-group symmetry G = [PITH_FULL_IMAGE:figures/full_fig_p045_7.png]
Figure 8
Figure 8. Figure 8: Illustration of the fusion 2-category 2VectG. Here gi ∈ Π1 and e is the identity in Π1. 45 [PITH_FULL_IMAGE:figures/full_fig_p045_8.png]
Figure 9
Figure 9. Figure 9: A1-module cell diagrams used in the ASS. Straight and curved lines represent the actions of Sq1 and Sq2 , respectively. Sq1 and curved lines denote the action Sq2 . 74 [PITH_FULL_IMAGE:figures/full_fig_p074_9.png]
Figure 10
Figure 10. Figure 10: Contribution of He∗ (RP 2 ; Z2) to the Adams E2-page. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 [PITH_FULL_IMAGE:figures/full_fig_p076_10.png]
Figure 11
Figure 11. Figure 11: Contribution of Σ3P to the Adams E2-page. The vertical line from (3, 0) to (3, 1) represents multiplication by 2; it is the nontrivial extension producing the Z4 summand in degree 3. Reading off the limiting page gives n 1 2 3 4 5 6 ΩeSpin n (X) Z2 Z2 Z4 ⊕ Z2 Z2 Z2 ⊕ …
Figure 12
Figure 12. Figure 12: Contribution of Σ4J ′ to the Adams E2-page. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 [PITH_FULL_IMAGE:figures/full_fig_p077_12.png]
Figure 13
Figure 13. Figure 13: The combined Adams E2-page. Black classes are removed by the d2-differentials fixed by comparison with the low-degree AHSS. 77 [PITH_FULL_IMAGE:figures/full_fig_p077_13.png]
Figure 14
Figure 14. Figure 14: The Adams E3 = E∞-page in stems t − s ≤ 6. References [1] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172, [1412.5148]. [2] E. Sharpe, Notes on generalized global symmetries in QFT, Fortsch. Phys. 63 (2015) 659–682…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

217 extracted references · 13 canonical work pages

  1. [177]

    T. D. D´ ecoppet,Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier D´ ecoppet and Theo Johnson-Freyd),SIGMA20(2024) 092, [2403.03211]

    arXiv 2024

  2. [1]

    Gaiotto, A

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

    arXiv 2015

  3. [2]

    Sharpe,Notes on generalized global symmetries in QFT,Fortsch

    E. Sharpe,Notes on generalized global symmetries in QFT,Fortsch. Phys.63(2015) 659–682, [1508.04770]

    arXiv 2015

  4. [3]

    Cordova, T

    C. Cordova, T. T. Dumitrescu, K. Intriligator and S.-H. Shao,Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond, inSnowmass 2021, 5, 2022.2205.09545

    arXiv 2021

  5. [4]

    McGreevy,Generalized Symmetries in Condensed Matter,Ann

    J. McGreevy,Generalized Symmetries in Condensed Matter,Ann. Rev. Condensed Matter Phys.14(2023) 57–82, [2204.03045]

    arXiv 2023

  6. [5]

    P. R. S. Gomes,An introduction to higher-form symmetries,SciPost Phys. Lect. Notes 74(2023) 1, [2303.01817]

    arXiv 2023

  7. [6]

    Schafer-Nameki,ICTP lectures on (non-)invertible generalized symmetries,Phys

    S. Schafer-Nameki,ICTP lectures on (non-)invertible generalized symmetries,Phys. Rept.1063(2024) 1–55, [2305.18296]

    arXiv 2024

  8. [7]

    T. D. Brennan and S. Hong,Introduction to Generalized Global Symmetries in QFT and Particle Physics,2306.00912

    arXiv

Show all 217 references
  1. [8]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. W. Gould, A. Platschorre et al.,Lectures on generalized symmetries,Phys. Rept.1051(2024) 1–87, [2307.07547]. 78

    2024 arXiv

  2. [9]

    Shao,What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries,2308.00747

    S.-H. Shao,What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries,2308.00747

    arXiv

  3. [10]

    Luo, Q.-R

    R. Luo, Q.-R. Wang and Y.-N. Wang,Lecture notes on generalized symmetries and applications,Phys. Rept.1065(2024) 1–43, [2307.09215]

    2024 arXiv

  4. [11]

    Iqbal,Jena lectures on generalized global symmetries: principles and applications, 2024.2407.20815

    N. Iqbal,Jena lectures on generalized global symmetries: principles and applications, 2024.2407.20815

    2024

  5. [12]

    Costa et al.,Simons lectures on categorical symmetries,2411.09082

    D. Costa et al.,Simons lectures on categorical symmetries,2411.09082

    arXiv

  6. [13]

    Kaidi,Introduction to generalized symmetries,2603.08798

    J. Kaidi,Introduction to generalized symmetries,2603.08798

    arXiv

  7. [14]

    J. C. Baez and A. D. Lauda,Higher-Dimensional Algebra V: 2-Groups,Theory Appl. Categ.12(7, 2004) 423–491, [math/0307200]

    2004 arXiv

  8. [15]

    Baez and U

    J. Baez and U. Schreiber,Higher gauge theory: 2-connections on 2-bundles, hep-th/0412325

    arXiv

  9. [16]

    J. C. Baez and J. Huerta,An Invitation to Higher Gauge Theory,Gen. Rel. Grav.43 (2011) 2335–2392, [1003.4485]

    2011 arXiv

  10. [17]

    Gukov and A

    S. Gukov and A. Kapustin,Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories,1307.4793

    arXiv

  11. [18]

    Kapustin and R

    A. Kapustin and R. Thorngren,Higher Symmetry and Gapped Phases of Gauge Theories,Prog. Math.324(2017) 177–202, [1309.4721]

    2017 arXiv

  12. [19]

    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]

    2020 arXiv

  13. [20]

    C´ ordova, T

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

    2019 arXiv

  14. [21]

    Delcamp and A

    C. Delcamp and A. Tiwari,From gauge to higher gauge models of topological phases, JHEP10(2018) 049, [1802.10104]

    2018 arXiv

  15. [22]

    Benini, C

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

    2019 arXiv

  16. [23]

    Wan and J

    Z. Wan and J. Wang,Adjoint QCD 4, Deconfined Critical Phenomena, Symmetry-Enriched Topological Quantum Field Theory, and Higher Symmetry-Extension,Phys. Rev. D99(2019) 065013, [1812.11955]. 79

    2019 arXiv

  17. [24]

    Hsin and A

    P.-S. Hsin and A. Turzillo,Symmetry-enriched quantum spin liquids in (3 + 1)d, JHEP09(2020) 022, [1904.11550]

    2020 arXiv

  18. [25]

    Bhardwaj and S

    L. Bhardwaj and S. Sch¨ afer-Nameki,Higher-form symmetries of 6d and 5d theories, JHEP02(2021) 159, [2008.09600]

    2021 arXiv

  19. [26]

    Y.-A. Chen, T. D. Ellison and N. Tantivasadakarn,Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions,Phys. Rev. Res.3 (2021) 013056, [2008.05652]

    2021 arXiv

  20. [27]

    Iqbal and N

    N. Iqbal and N. Poovuttikul,2-group global symmetries, hydrodynamics and holography,SciPost Phys.15(2023) 063, [2010.00320]

    2023 arXiv

  21. [28]

    Cordova, T

    C. Cordova, T. T. Dumitrescu and K. Intriligator,2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories,JHEP04(2021) 252, [2009.00138]

    2021 arXiv

  22. [29]

    Del Zotto and K

    M. Del Zotto and K. Ohmori,2-Group Symmetries of 6D Little String Theories and T-Duality,Annales Henri Poincare22(2021) 2451–2474, [2009.03489]

    2021 arXiv

  23. [30]

    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]

    2021 arXiv

  24. [31]

    DeWolfe and K

    O. DeWolfe and K. Higginbotham,Generalized symmetries and 2-groups via electromagnetic duality inAdS/CF T,Phys. Rev. D103(2021) 026011, [2010.06594]

    2021 arXiv

  25. [32]

    Gukov, P.-S

    S. Gukov, P.-S. Hsin and D. Pei,Generalized global symmetries ofT[M]theories. Part I,JHEP04(2021) 232, [2010.15890]

    2021 arXiv

  26. [33]

    T. D. Brennan and C. Cordova,Axions, higher-groups, and emergent symmetry,JHEP 02(2022) 145, [2011.09600]

    2022 arXiv

  27. [34]

    Brauner,Field theories with higher-group symmetry from composite currents,JHEP 04(2021) 045, [2012.00051]

    T. Brauner,Field theories with higher-group symmetry from composite currents,JHEP 04(2021) 045, [2012.00051]

    2021 arXiv

  28. [35]

    Hidaka, M

    Y. Hidaka, M. Nitta and R. Yokokura,Higher-form symmetries and 3-group in axion electrodynamics,Phys. Lett. B808(2020) 135672, [2006.12532]

    2020 arXiv

  29. [36]

    Hidaka, M

    Y. Hidaka, M. Nitta and R. Yokokura,Global 3-group symmetry and ’t Hooft anomalies in axion electrodynamics,JHEP01(2021) 173, [2009.14368]. 80

    2021 arXiv

  30. [37]

    P.-S. Hsin, W. Ji and C.-M. Jian,Exotic invertible phases with higher-group symmetries,SciPost Phys.12(2022) 052, [2105.09454]

    2022 arXiv

  31. [38]

    Y. Lee, K. Ohmori and Y. Tachikawa,Matching higher symmetries across Intriligator-Seiberg duality,JHEP10(2021) 114, [2108.05369]

    2021 arXiv

  32. [39]

    Apruzzi, S

    F. Apruzzi, S. Schafer-Nameki, L. Bhardwaj and J. Oh,The Global Form of Flavor Symmetries and 2-Group Symmetries in 5d SCFTs,SciPost Phys.13(2022) 024, [2105.08724]

    2022 arXiv

  33. [40]

    Apruzzi, L

    F. Apruzzi, L. Bhardwaj, D. S. W. Gould and S. Schafer-Nameki,2-Group symmetries and their classification in 6d,SciPost Phys.12(2022) 098, [2110.14647]

    2022 arXiv

  34. [41]

    Bhardwaj,2-Group symmetries in class S,SciPost Phys.12(2022) 152, [2107.06816]

    L. Bhardwaj,2-Group symmetries in class S,SciPost Phys.12(2022) 152, [2107.06816]

    2022 arXiv

  35. [42]

    Del Zotto, J

    M. Del Zotto, J. J. Heckman, S. N. Meynet, R. Moscrop and H. Y. Zhang,Higher symmetries of 5D orbifold SCFTs,Phys. Rev. D106(2022) 046010, [2201.08372]

    2022 arXiv

  36. [43]

    Del Zotto, I

    M. Del Zotto, I. n. Garc ´ ıa Etxebarria and S. Schafer-Nameki,2-Group Symmetries and M-Theory,SciPost Phys.13(2022) 105, [2203.10097]

    2022 arXiv

  37. [44]

    Barkeshli, Y.-A

    M. Barkeshli, Y.-A. Chen, P.-S. Hsin and R. Kobayashi,Higher-group symmetry in finite gauge theory and stabilizer codes,SciPost Phys.16(2024) 089, [2211.11764]

    2024 arXiv

  38. [45]

    Cvetiˇ c, J

    M. Cvetiˇ c, J. J. Heckman, M. H¨ ubner and E. Torres,0-form, 1-form, and 2-group symmetries via cutting and gluing of orbifolds,Phys. Rev. D106(2022) 106003, [2203.10102]

    2022 arXiv

  39. [46]

    Cordova and S

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

    2023 arXiv

  40. [47]

    Bhardwaj and D

    L. Bhardwaj and D. S. W. Gould,Disconnected 0-form and 2-group symmetries,JHEP 07(2023) 098, [2206.01287]

    2023 arXiv

  41. [48]

    Bhardwaj, S

    L. Bhardwaj, S. Sch¨ afer-Nameki and J. Wu,Universal non-invertible symmetries, Fortschritte der Physik70(Oct., 2022) 2200143

    2022

  42. [49]

    Barkeshli, Y.-A

    M. Barkeshli, Y.-A. Chen, S.-J. Huang, R. Kobayashi, N. Tantivasadakarn and G. Zhu, Codimension-2 defects and higher symmetries in (3+1)D topological phases,SciPost Phys.14(2023) 065, [2208.07367]. 81

    2023 arXiv

  43. [50]

    Barkeshli, P.-S

    M. Barkeshli, P.-S. Hsin and R. Kobayashi,Higher-group symmetry of (3+1)d fermionicZ 2 gauge theory: Logical ccz, cs, and t gates from higher symmetry,SciPost Physics16(May, 2024) 122

    2024

  44. [51]

    Bhardwaj and S

    L. Bhardwaj and S. Sch¨ afer-Nameki,Generalized charges, part i: Invertible symmetries and higher representations,SciPost Physics16(Apr., 2024) 093

    2024

  45. [52]

    Debray,Bordism for the 2-group symmetries of the heterotic and CHL strings, Contemp

    A. Debray,Bordism for the 2-group symmetries of the heterotic and CHL strings, Contemp. Math.802(2024) 227–98, [2304.14764]

    2024 arXiv

  46. [53]

    Bartsch, M

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

    arXiv

  47. [54]

    Moradi, O

    H. Moradi, O. M. Aksoy, J. H. Bardarson and A. Tiwari,Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries, 2307.01266

    arXiv

  48. [55]

    Bartsch, M

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

    arXiv

  49. [56]

    S. D. Pace,Emergent generalized symmetries in ordered phases and applications to quantum disordering,SciPost Phys.17(2024) 080, [2308.05730]

    2024 arXiv

  50. [57]

    Cvetiˇ c, J

    M. Cvetiˇ c, J. J. Heckman, M. H¨ ubner and E. Torres,Generalized symmetries, gravity, and the swampland,Phys. Rev. D109(2024) 026012, [2307.13027]

    2024 arXiv

  51. [58]

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

    arXiv

  52. [59]

    N. Kan, O. Morikawa, Y. Nagoya and H. Wada,Higher-group structure in lattice Abelian gauge theory under instanton-sum modification,Eur. Phys. J. C83(2023) 481, [2302.13466]

    2023 arXiv

  53. [60]

    Apruzzi, F

    F. Apruzzi, F. Bedogna and N. Dondi,SymTh for non-finite symmetries,JHEP04(2,

  54. [61]

    Armas, G

    J. Armas, G. Batzios and A. Jain,Higher-group global symmetry and the bosonic M5 brane,JHEP08(2024) 003, [2402.19458]

    2024 arXiv

  55. [62]

    Nakajima, K

    T. Nakajima, K. Nakamura and T. Sakai,Note on higher-group structure in 6d self-dual gauge theory,JHEP10(2024) 093, [2406.10518]. 82

    2024 arXiv

  56. [63]

    Bartsch,On Unitary 2-Group Symmetries,2411.05067

    T. Bartsch,On Unitary 2-Group Symmetries,2411.05067

    arXiv

  57. [64]

    R. Liu, R. Luo and Y.-N. Wang,Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries,SciPost Phys.18(2025) 052, [2406.03974]

    2025 arXiv

  58. [65]

    Tian and X

    J. Tian and X. Wang,Higher Form and Higher Group Symmetries via Mirror Symmetry,JHEP10(2025) 075, [2503.09967]

    2025 arXiv

  59. [66]

    Bason, W

    D. Bason, W. Cui and L. Ruggeri,Half-Spacetime Gauging of 2-Group Symmetry in 3d,2605.06287

    arXiv

  60. [67]

    Caro-P´ erez, M

    F. Caro-P´ erez, M. P. Garcia del Moral and A. Restuccia,2-Group global symmetry in the compactified M2-brane,2607.07924

    arXiv

  61. [68]

    K. S. Brown,Cohomology of groups, vol. 87. Springer Science & Business Media, 2012, 10.1007/978-1-4684-9327-6

    2012 doi

  62. [69]

    D. N. Yetter,TQFT’s from homotopy 2 types,J. Knot Theor. Ramifications2(1993) 113–123

    1993

  63. [70]

    Kapustin and R

    A. Kapustin and R. Thorngren,Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement,Adv. Theor. Math. Phys.18(2014) 1233–1247, [1308.2926]

    2014 arXiv

  64. [71]

    Bullivant, M

    A. Bullivant, M. Cal¸ cada, Z. K´ ad´ ar, P. Martin and J. Faria Martins,Topological phases from higher gauge symmetry in 3+1 dimensions,Phys. Rev. B95(2017) 155118, [1606.06639]

    2017 arXiv

  65. [72]

    Delcamp and A

    C. Delcamp and A. Tiwari,On 2-form gauge models of topological phases,JHEP05 (2019) 064, [1901.02249]

    2019 arXiv

  66. [73]

    Carqueville and B

    N. Carqueville and B. Haake,2-Group Symmetries of 3-dimensional Defect TQFTs and Their Gauging,2506.08178

    arXiv

  67. [74]

    Huang,Finite 2-group gauge theory and its 3+1D lattice realization,JHEP03 (2026) 133, [2508.04693]

    M. Huang,Finite 2-group gauge theory and its 3+1D lattice realization,JHEP03 (2026) 133, [2508.04693]

    2026

  68. [75]

    Kapustin,Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology,1403.1467

    A. Kapustin,Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology,1403.1467

    arXiv

  69. [76]

    Kapustin, R

    A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang,Fermionic Symmetry Protected Topological Phases and Cobordisms,JHEP12(2015) 052, [1406.7329]. 83

    2015 arXiv

  70. [77]

    D. S. Freed,Anomalies and invertible field theories,Proc. Symp. Pure Math.88(2014) 25–46, [1404.7224]

    2014 arXiv

  71. [78]

    D. S. Freed and M. J. Hopkins,Reflection positivity and invertible topological phases, Geom. Topol.25(2021) 1165–1330, [1604.06527]

    2021 arXiv

  72. [79]

    Yonekura,On the cobordism classification of symmetry protected topological phases, Commun

    K. Yonekura,On the cobordism classification of symmetry protected topological phases, Commun. Math. Phys.368(2019) 1121–1173, [1803.10796]

    2019 arXiv

  73. [80]

    Garc ´ ıa-Etxebarria and M

    I. Garc ´ ıa-Etxebarria and M. Montero,Dai-Freed anomalies in particle physics,JHEP 08(2019) 003, [1808.00009]

    2019 arXiv

  74. [81]

    J. C. Baez and D. Stevenson,The classifying space of a topological 2-group,0801.3843

    arXiv

  75. [82]

    Davighi, N

    J. Davighi, N. Lohitsiri and A. Debray,Toric 2-group anomalies via cobordism,JHEP 07(2023) 019, [2302.12853]

    2023 arXiv

  76. [83]

    Wan and J

    Z. Wan and J. Wang,Higher anomalies, higher symmetries, and cobordisms I: classification of higher-symmetry-protected topological states and their boundary fermionic/bosonic anomalies via a generalized cobordism theory,Ann. Math. Sci. Appl. 4(2019) 107–311, [1812.11967]

    2019 arXiv

  77. [84]

    Z. Wan, J. Wang and Y. Zheng,Higher anomalies, higher symmetries, and cobordisms II: Lorentz symmetry extension and enriched bosonic / fermionic quantum gauge theory,Ann. Math. Sci. Appl.05(2020) 171–257, [1912.13504]

    2020 arXiv

  78. [85]

    Feng and Y.-A

    Y. Feng and Y.-A. Chen,Defining 2-group symmetry actions and anomalies on lattices, to Appear,

  79. [86]

    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

  80. [87]

    Kong, X.-G

    L. Kong, X.-G. Wen and H. Zheng,Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers,1502.01690

    arXiv

  81. [88]

    Kong and H

    L. Kong and H. Zheng,Gapless edges of 2d topological orders and enriched monoidal categories,Nucl. Phys. B927(2018) 140–165, [1705.01087]

    2018 arXiv

  82. [89]

    Kong and H

    L. Kong and H. Zheng,A mathematical theory of gapless edges of 2d topological orders. Part I,JHEP02(2020) 150, [1905.04924]. 84

    2020 arXiv

  83. [90]

    Kong and H

    L. Kong and H. Zheng,A mathematical theory of gapless edges of 2d topological orders. Part II,Nucl. Phys. B966(2021) 115384, [1912.01760]

    2021 arXiv

  84. [91]

    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

  85. [92]

    Gaiotto and J

    D. Gaiotto and J. Kulp,Orbifold groupoids,JHEP02(2021) 132, [2008.05960]

    2021 arXiv

  86. [93]

    Lichtman, R

    T. Lichtman, R. Thorngren, N. H. Lindner, A. Stern and E. Berg,Bulk anyons as edge symmetries: Boundary phase diagrams of topologically ordered states,Phys. Rev. B 104(2021) 075141, [2003.04328]

    2021 arXiv

  87. [94]

    Apruzzi, F

    F. Apruzzi, F. Bonetti, I. n. G. Etxebarria, S. S. Hosseini and S. Schafer-Nameki, Symmetry TFTs from String Theory,Commun. Math. Phys.402(12, 2023) 895–949, [2112.02092]

    2023 arXiv

  88. [95]

    Apruzzi,Higher Form Symmetries TFT in 6d,JHEP11(3, 2022) 050, [2203.10063]

    F. Apruzzi,Higher Form Symmetries TFT in 6d,JHEP11(3, 2022) 050, [2203.10063]

    2022 arXiv

  89. [96]

    Del Zotto and I

    M. Del Zotto and I. n. Garc ´ ıa Etxebarria,Global structures from the infrared,JHEP 11(2023) 058, [2204.06495]

    2023 arXiv

  90. [97]

    Moradi, S

    H. Moradi, S. F. Moosavian and A. Tiwari,Topological holography: Towards a unification of Landau and beyond-Landau physics,SciPost Phys. Core6(2023) 066, [2207.10712]

    2023 arXiv

  91. [98]

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

    arXiv

  92. [99]

    Kaidi, K

    J. Kaidi, K. Ohmori and Y. Zheng,Symmetry TFTs for Non-invertible Defects, Commun. Math. Phys.404(2023) 1021–1124, [2209.11062]

    2023 arXiv

  93. [100]

    van Beest, D

    M. van Beest, D. S. W. Gould, S. Schafer-Nameki and Y.-N. Wang,Symmetry TFTs for 3d QFTs from M-theory,JHEP02(2023) 226, [2210.03703]

    2023 arXiv

  94. [101]

    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

  95. [102]

    Bhardwaj and S

    L. Bhardwaj and S. Schafer-Nameki,Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT,SciPost Phys.19(5, 2025) 098, [2305.17159]. 85

    2025

  96. [103]

    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

  97. [104]

    Baume, J

    F. Baume, J. J. Heckman, M. H¨ ubner, E. Torres, A. P. Turner and X. Yu,SymTrees and Multi-Sector QFTs,Phys. Rev. D109(2024) 106013, [2310.12980]

    2024 arXiv

  98. [105]

    T. D. Brennan and Z. Sun,A SymTFT for continuous symmetries,JHEP12(2024) 100, [2401.06128]

    2024 arXiv

  99. [106]

    Argurio, F

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

    2024 arXiv

  100. [107]

    Bhardwaj, C

    L. Bhardwaj, C. Copetti, D. Pajer and S. Schafer-Nameki,Boundary SymTFT, SciPost Phys.19(9, 2025) 061, [2409.02166]

    2025 arXiv

  101. [108]

    Y. Choi, B. C. Rayhaun and Y. Zheng,Generalized Tube Algebras, Symmetry-Resolved Partition Functions, and Twisted Boundary States,Commun. Math. Phys.407(9,

  102. [109]

    Tian and Y.-N

    J. Tian and Y.-N. Wang,A Tale of Bulk and Branes: Symmetry TFT of 6D SCFTs from IIB/F-theory,JHEP03(10, 2025) 085, [2410.23076]

    2025 arXiv

  103. [110]

    Najjar, L

    M. Najjar, L. Santilli and Y.-N. Wang,(-1)-form symmetries from M-theory and SymTFTs,2411.19683

    arXiv

  104. [111]

    Bonetti, M

    F. Bonetti, M. Del Zotto and R. Minasian,SymTFTs and Non-Invertible Symmetries of 6d (2,0) SCFTs of TypeDfrom M-theory,2412.07842

    arXiv

  105. [112]

    Antinucci, C

    A. Antinucci, C. Copetti and S. Schafer-Nameki,SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement,SciPost Phys.18(8, 2025) 114, [2408.05585]

    2025 arXiv

  106. [113]

    Antinucci and F

    A. Antinucci and F. Benini,Anomalies and gauging of U(1) symmetries,Phys. Rev. B 111(2025) 024110, [2401.10165]

    2025 arXiv

  107. [114]

    Bonetti, M

    F. Bonetti, M. Del Zotto and R. Minasian,SymTFTs for Continuous non-Abelian Symmetries,2402.12347

  108. [115]

    Gagliano and I

    F. Gagliano and I. n. Garc ´ ıa Etxebarria,SymTFTs forU(1)symmetries from descent, 2411.15126. 86

    arXiv

  109. [116]

    Q. Jia, R. Luo, J. Tian, Y.-N. Wang and Y. Zhang,Symmetry Topological Field Theory for Flavor Symmetry,2503.04546

    arXiv

  110. [117]

    Putrov and R

    P. Putrov and R. Radhakrishnan,Braidings on topological operators, anomaly of higher-form symmetries and the SymTFT,2503.13633

    arXiv

  111. [118]

    Del Zotto, A

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

    arXiv

  112. [119]

    L.-Y. Hung, K. Ji, C. Shen, Y. Wan and Y. Zhao,A 2D-CFT Factory: Critical Lattice Models from Competing Anyon Condensation Processes in SymTO/SymTFT, 2506.05324

    arXiv

  113. [120]

    Robbins and S

    D. Robbins and S. Roy,SymTFT actions, Condensable algebras and Categorical anomaly resolutions,JHEP06(2026) 058, [2509.05408]

    2026 arXiv

  114. [121]

    Bonetti, M

    F. Bonetti, M. Del Zotto and R. Minasian,SymTFT for Continuous Symmetries: Non-linear Realizations and Spontaneous Breaking,JHEP05(2026) 048, [2509.10343]

    2026 arXiv

  115. [122]

    Jia and Y

    Q. Jia and Y. Zhang,Flat Gauging of Continuous (Non-invertible) Symmetries and Non-compact BF SymTFT for Compact Boson,2606.15732

  116. [123]

    Ji and X.-G

    W. Ji and X.-G. Wen,Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions,Phys. Rev. Res.1(2019) 033054, [1905.13279]

    2019 arXiv

  117. [124]

    Ji and X.-G

    W. Ji and X.-G. Wen,Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions,Phys. Rev. Res.2(2020) 033417, [1912.13492]

    2020 arXiv

  118. [125]

    Ji and X.-G

    W. Ji and X.-G. Wen,A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly,2106.02069

    arXiv

  119. [126]

    Chatterjee and X.-G

    A. Chatterjee and X.-G. Wen,Holographic theory for continuous phase transitions: Emergence and symmetry protection of gaplessness,Phys. Rev. B108(2023) 075105, [2205.06244]

    2023 arXiv

  120. [127]

    Chatterjee and X.-G

    A. Chatterjee and X.-G. Wen,Symmetry as a shadow of topological order and a derivation of topological holographic principle,Phys. Rev. B107(2023) 155136, [2203.03596]. 87

    2023 arXiv

  121. [128]

    Chatterjee, W

    A. Chatterjee, W. Ji and X.-G. Wen,Emergent generalized symmetry and maximal symmetry topological order,Phys. Rev. B112(2025) 115142, [2212.14432]

    2025 arXiv

  122. [129]

    R. Wen, W. Ye and A. C. Potter,Topological holography for fermions,2404.19004

    arXiv

  123. [130]

    Wen and A

    R. Wen and A. C. Potter,Classification of 1+1D gapless symmetry protected phases via topological holography,Phys. Rev. B111(2025) 115161, [2311.00050]

    2025 arXiv

  124. [131]

    Wen and A

    R. Wen and A. C. Potter,Bulk-boundary correspondence for intrinsically gapless symmetry-protected topological phases from group cohomology,Phys. Rev. B107 (2023) 245127, [2208.09001]

    2023 arXiv

  125. [132]

    Huang and M

    S.-J. Huang and M. Cheng,Topological holography, quantum criticality, and boundary states,SciPost Phys.18(2025) 213, [2310.16878]

    2025 arXiv

  126. [133]

    Wen,Topological Holography for 2+1-D Gapped and Gapless Phases with Generalized Symmetries,2503.13685

    R. Wen,Topological Holography for 2+1-D Gapped and Gapless Phases with Generalized Symmetries,2503.13685

    arXiv

  127. [134]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, D. Pajer and S. Schafer-Nameki,Categorical Landau Paradigm for Gapped Phases,Phys. Rev. Lett.133(2024) 161601, [2310.03786]

    2024 arXiv

  128. [135]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, D. Pajer and S. Sch¨ afer-Nameki,Gapped Phases with Non-Invertible Symmetries: (1+1)d,SciPost Phys.18(10, 2025) 032, [2310.03784]

    2025 arXiv

  129. [136]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, D. Pajer and S. Schafer-Nameki,The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries,SciPost Phys. 18(12, 2025) 156, [2312.17322]

    2025 arXiv

  130. [137]

    Y.-J. Hai, Z. Zhang, H. Zheng, L. Kong, J. Wu and D. Yu,Uniquely identifying topological order based on boundary-bulk duality and anyon condensation,Natl. Sci. Rev.10(2023) nwac264

    2023

  131. [138]

    Bhardwaj, D

    L. Bhardwaj, D. Pajer, S. Schafer-Nameki and A. Warman,Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries,2403.00905

  132. [139]

    Kong, Z.-H

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

  133. [140]

    Chen and Q

    J. Chen and Q. Jia,SymTFT Approach to 2D Orbifold Groupoids: ‘t Hooft Anomalies, Gauging, and Partition Functions,2411.18056. 88

    arXiv

  134. [141]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki and A. Tiwari,Illustrating the categorical Landau paradigm in lattice models,Phys. Rev. B111(2025) 054432, [2405.05302]

    2025 arXiv

  135. [142]

    Antinucci, F

    A. Antinucci, F. Benini and G. Rizi,Holographic Duals of Symmetry Broken Phases, Fortsch. Phys.72(2024) 2400172, [2408.01418]

    2024 arXiv

  136. [143]

    Bhardwaj, D

    L. Bhardwaj, D. Pajer, S. Schafer-Nameki, A. Tiwari, A. Warman and J. Wu,Gapped phases in (2+1)d with non-invertible symmetries: Part I,SciPost Phys.19(2025) 056, [2408.05266]

    2025 arXiv

  137. [144]

    Lootens, C

    L. Lootens, C. Delcamp and F. Verstraete,Entanglement and the density matrix renormalization group in the generalized Landau paradigm,Nature Phys.21(2025) 1657–1663, [2408.06334]

    2025

  138. [145]

    Bhardwaj, S

    L. Bhardwaj, S. Schafer-Nameki, A. Tiwari and A. Warman,Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part II,2502.20440

    arXiv

  139. [146]

    Chen,Essay: Generalized Landau Paradigm for Quantum Phases and Phase Transitions,Phys

    X. Chen,Essay: Generalized Landau Paradigm for Quantum Phases and Phase Transitions,Phys. Rev. Lett.135(2025) 250001, [2511.19793]

    2025

  140. [147]

    Ebisu, B

    H. Ebisu, B. Han and W. Cao,Understanding deconfined quantum critical points from crystalline categorical Landau paradigm,2606.05856

    arXiv

  141. [148]

    Ostrik,Module categories over the Drinfeld double of a finite group,Int

    V. Ostrik,Module categories over the Drinfeld double of a finite group,Int. Math. Res. Not.2003(2, 2003) 1507–1520, [math/0202130]

    2003 arXiv

  142. [149]

    Kitaev and L

    A. Kitaev and L. Kong,Models for Gapped Boundaries and Domain Walls,Commun. Math. Phys.313(2012) 351–373, [1104.5047]

    2012 arXiv

  143. [150]

    Kong,Anyon condensation and tensor categories,Nucl

    L. Kong,Anyon condensation and tensor categories,Nucl. Phys. B886(2014) 436–482, [1307.8244]

    2014 arXiv

  144. [151]

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

    2015 arXiv

  145. [152]

    I. Cong, M. Cheng and Z. Wang,Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter,Commun. Math. Phys.355(2017) 645–689, [1707.04564]

    2017 arXiv

  146. [153]

    Bhardwaj and Y

    L. Bhardwaj and Y. Tachikawa,On finite symmetries and their gauging in two dimensions,JHEP03(2018) 189, [1704.02330]. 89

    2018 arXiv

  147. [154]

    Chang, Y.-H

    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, [1802.04445]

    2019 arXiv

  148. [155]

    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

  149. [156]

    Xu and Z.-H

    R. Xu and Z.-H. Zhang,Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries,Phys. Rev. B110(2024) 155106, [2205.09656]

    2024 arXiv

  150. [157]

    C. L. Douglas and D. J. Reutter,Fusion 2-categories and a state-sum invariant for 4-manifolds,1812.11933

    arXiv

  151. [158]

    Gaiotto and T

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

    1905 arXiv

  152. [159]

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

    T. Johnson-Freyd,On the Classification of Topological Orders,Commun. Math. Phys. 393(2022) 989–1033, [2003.06663]

    2022 arXiv

  153. [160]

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

    arXiv

  154. [161]

    L. Kong, Y. Tian and S. Zhou,The center of monoidal 2-categories in 3+1D Dijkgraaf-Witten theory,Adv. Math.360(2020) 106928, [1905.04644]

    2020 arXiv

  155. [162]

    T. D. D´ ecoppet,The Morita theory of fusion 2-categories.,High. Struct.7(2023) 234–292, [2311.16827]

    2023 arXiv

  156. [163]

    T. D. D´ ecoppet,Drinfeld centers and morita equivalence classes of fusion 2-categories, Compositio Mathematica161(2025) 305–340, [2211.04917]

    2025 arXiv

  157. [164]

    Xu,On ´Etale Algebras and Bosonic Fusion 2-Categories,2411.13367

    H. Xu,On ´Etale Algebras and Bosonic Fusion 2-Categories,2411.13367

    arXiv

  158. [165]

    Bhardwaj, Y

    L. Bhardwaj, Y. Gai, S.-J. Huang, K. Inamura, S. Schafer-Nameki, A. Tiwari et al., Gapless Phases in (2+1)d with Non-Invertible Symmetries,2503.12699

    arXiv

  159. [166]

    Zhao, J.-Q

    J. Zhao, J.-Q. Lou, Z.-H. Zhang, L.-Y. Hung, L. Kong and Y. Tian,String condensations in3 + 1Dand Lagrangian algebras,Adv. Theor. Math. Phys.27(2023) 583–622, [2208.07865]

    2023 arXiv

  160. [167]

    Luo,Gapped boundaries of (3+1)-dimensional topological orders,Phys

    Z.-X. Luo,Gapped boundaries of (3+1)-dimensional topological orders,Phys. Rev. B 107(2023) 125425, [2212.09779]. 90

    2023 arXiv

  161. [168]

    W. Ji, N. Tantivasadakarn and C. Xu,Boundary states of three dimensional topological order and the deconfined quantum critical point,SciPost Phys.15(2023) 231

    2023

  162. [169]

    Ji and X

    W. Ji and X. Chen,Topological defects of 2+1d systems from line excitations in 3+1d bulk, 2024

    2024

  163. [170]

    Putrov, J

    P. Putrov, J. Wang and S.-T. Yau,Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions,Annals Phys.384(2017) 254–287, [1612.09298]

    2017 arXiv

  164. [171]

    Wang, X.-G

    J. Wang, X.-G. Wen and E. Witten,Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions,Phys. Rev. X8(2018) 031048, [1705.06728]

    2018 arXiv

  165. [172]

    Zhang, Q.-R

    Z.-F. Zhang, Q.-R. Wang and P. Ye,Non-Abelian fusion, shrinking, and quantum dimensions of Abelian gauge fluxes,Phys. Rev. B107(2023) 165117, [2208.09228]

    2023 arXiv

  166. [173]

    Zhang, Q.-R

    Z.-F. Zhang, Q.-R. Wang and P. Ye,Continuum field theory of three-dimensional topological orders with emergent fermions and braiding statistics,Phys. Rev. Res.5 (2023) 043111, [2307.09983]

    2023 arXiv

  167. [174]

    Zhang, Y

    Z.-F. Zhang, Y. Huang, Q.-R. Wang and P. Ye,Non-invertible symmetries and mixed anomalies from conserved current construction in (3+1)D twisted BF topological quantum field theories,2601.01523

  168. [175]

    Elgueta,Representation theory of 2-groups on kapranov and voevodsky’s 2-vector spaces,Advances in Mathematics213(2007) 53–92

    J. Elgueta,Representation theory of 2-groups on kapranov and voevodsky’s 2-vector spaces,Advances in Mathematics213(2007) 53–92

    2007

  169. [176]

    Huang, H

    M. Huang, H. Xu and Z.-H. Zhang,The 2-character theory for finite 2-groups, 2404.01162

    arXiv

  170. [178]

    Ambrosino, R

    F. Ambrosino, R. Luo, Y.-N. Wang and Y. Zhang,Understanding fermionic generalized symmetries,Phys. Rev. D110(2024) 105020, [2404.12301]

    2024 arXiv

  171. [179]

    Saito, Y

    S. Saito, Y. Tachikawa and Y. Zhang,Bosonic SPT and invertible phases and its relation to Steenrod’s problem,2607.14662

    arXiv

  172. [180]

    Lee and Y

    Y. Lee and Y. Tachikawa,Some comments on 6D global gauge anomalies,PTEP2021 (2021) 08B103, [2012.11622]. 91

    2021 arXiv

  173. [181]

    E. H. Brown,The serre spectral sequence theorem for continuous and ordinary cohomology,Topology and its Applications56(1994) 235–248

    1994

  174. [182]

    Datuashvili and T

    T. Datuashvili and T. Pirashvili,On (co)homology of 2-types and crossed modules, Journal of Algebra244(2001) 352–365

    2001

  175. [183]

    SPECTRAL SEQUENCES FOR LOCAL COEFFICIENTS

    J. MAY and M. G. SHULMAN, “SPECTRAL SEQUENCES FOR LOCAL COEFFICIENTS.” https://math.uchicago.edu/~may/PEOPLE/MEG/localcoeffJan26.pdf, 2009

    2009

  176. [184]

    J. W. Milnor and J. D. Stasheff,Characteristic Classes, vol. 76 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1974

    1974

  177. [185]

    W. Xi, T. Lan, L. Wang, C. Wang and W.-Q. Chen,On a class of fusion 2-category symmetry: condensation completion of braided fusion category,Adv. Theor. Math. Phys.29(2025) 151–204, [2312.15947]

    2025 arXiv

  178. [186]

    Xu,On ´Etale Algebras and Fusion 2-Categories

    H. Xu,On ´Etale Algebras and Fusion 2-Categories. PhD thesis, University of G¨ ottingen, 2026. 10.53846/goediss-12041

    2026 doi

  179. [187]

    Johnson-Freyd and M

    T. Johnson-Freyd and M. Yu,Fusion 2-categories With no Line Operators are Grouplike,Bull. Austral. Math. Soc.104(2021) 434–442, [2010.07950]

    2021 arXiv

  180. [188]

    Debray, W

    A. Debray, W. Ye and M. Yu,How to Build Anomalous (3+1)d Topological Quantum Field Theories,2510.24834

  181. [189]

    Wang and Z.-C

    Q.-R. Wang and Z.-C. Gu,Construction and classification of symmetry protected topological phases in interacting fermion systems,Phys. Rev. X10(2020) 031055, [1811.00536]

    2020 arXiv

  182. [190]

    A. P. O. Chan, P. Ye and S. Ryu,Braiding with Borromean Rings in (3+1)-Dimensional Spacetime,Phys. Rev. Lett.121(2018) 061601, [1703.01926]

    2018 arXiv

  183. [191]

    Q.-R. Wang, M. Cheng, C. Wang and Z.-C. Gu,Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in(3 + 1)-Dimensions,Phys. Rev. B99(2019) 235137, [1810.13428]

    2019 arXiv

  184. [192]

    Zhang and P

    Z.-F. Zhang and P. Ye,Compatible braidings with Hopf links, multiloop, and Borromean rings in(3 + 1)-dimensional spacetime,Phys. Rev. Res.3(2021) 023132, [2012.13761]

    2021 arXiv

  185. [193]

    D. G. Delmastro, J. Gomis, P.-S. Hsin and Z. Komargodski,Anomalies and symmetry fractionalization,SciPost Phys.15(2023) 079, [2206.15118]. 92

    2023 arXiv

  186. [194]

    Wen,Symmetry protected topological phases in non-interacting fermion systems, Phys

    X.-G. Wen,Symmetry protected topological phases in non-interacting fermion systems, Phys. Rev. B85(2012) 085103, [1111.6341]

    2012 arXiv

  187. [195]

    Y.-A. Chen, A. Kapustin, A. Turzillo and M. You,Free and interacting short-range entangled phases of fermions: Beyond the ten-fold way,Physical Review B100(2019) 195128, [1809.04958]

    2019 arXiv

  188. [196]

    Y.-A. Chen, A. Kapustin and D. Radicevic,Exact bosonization in two spatial dimensions and a new class of lattice gauge theories,Annals of Physics393(2018) 234–253, [1711.00515]

    2018 arXiv

  189. [197]

    Kong and H

    L. Kong and H. Zheng,Categories of quantum liquids I,JHEP08(2022) 070, [2011.02859]

    2022 arXiv

  190. [198]

    T. D. D´ ecoppet and M. Yu,Fiber 2-Functors and Tambara–Yamagami Fusion 2-Categories,Commun. Math. Phys.406(2025) 64, [2306.08117]

    2025 arXiv

  191. [199]

    T. D. D´ ecoppet and H. Xu,Local modules in braided monoidal 2-categories,J. Math. Phys.65(2024) 061702, [2307.02843]

    2024 arXiv

  192. [200]

    Xu,Centers of Algebras in Monoidal 2-Categories,2607.05228

    H. Xu,Centers of Algebras in Monoidal 2-Categories,2607.05228

    arXiv

  193. [201]

    Gu and X.-G

    Z.-C. Gu and X.-G. Wen,Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory,Phys. Rev. B90(2014) 115141, [1201.2648]

    2014 arXiv

  194. [202]

    Wang and Z.-C

    Q.-R. Wang and Z.-C. Gu,Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory,Phys. Rev. X8(2018) 011055, [1703.10937]

    2018 arXiv

  195. [203]

    Brumfiel and J

    G. Brumfiel and J. Morgan,The Pontrjagin Dual of 3-Dimensional Spin Bordism, 1612.02860

    arXiv

  196. [204]

    Brumfiel and J

    G. Brumfiel and J. Morgan,The Pontrjagin Dual of 4-Dimensional Spin Bordism, 1803.08147

    arXiv

  197. [205]

    M. Guo, K. Ohmori, P. Putrov, Z. Wan and J. Wang,Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms,Commun. Math. Phys.376(2020) 1073–1154, [1812.11959]

    2020 arXiv

  198. [206]

    Cartan,D´ etermination des alg` ebresh∗(π, n;zp)eth ∗(π, n;z),ppremier impair, S´ eminaire Henri Cartan7(1954–1955) 1–10

    H. Cartan,D´ etermination des alg` ebresh∗(π, n;zp)eth ∗(π, n;z),ppremier impair, S´ eminaire Henri Cartan7(1954–1955) 1–10. 93

    1954

  199. [207]

    Spectral Sequences: Preliminary Chapter 5 for Algebraic Topology

    A. Hatcher, “Spectral Sequences: Preliminary Chapter 5 for Algebraic Topology.” https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf, 2004

    2004

  200. [208]

    Cl´ ement,Integral cohomology of finite Postnikov towers

    A. Cl´ ement,Integral cohomology of finite Postnikov towers. PhD thesis, Ph. D. Thesis, University of Lausanne, Switzerland, 2002

    2002

  201. [209]

    McCleary,A User’s Guide to Spectral Sequences

    J. McCleary,A User’s Guide to Spectral Sequences. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2 ed., 2000

    2000

  202. [210]

    W. M. Singer,Steenrod squares in spectral sequences, vol. 129 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006

    2006

  203. [211]

    J. F. Adams,Stable homotopy and generalised homology. University of Chicago press, 1974

    1974

  204. [212]

    Trou´ e,The orders of the Postnikov invariants of the Thom spectrumM SO,Illinois J

    J. Trou´ e,The orders of the Postnikov invariants of the Thom spectrumM SO,Illinois J. Math.10(1966) 592–604

    1966

  205. [213]

    D. W. Anderson, E. H. Brown and F. P. Peterson,The structure of the spin cobordism ring,Annals of Mathematics86(1967) 271–298

    1967

  206. [214]

    Teichner,On the signature of four-manifolds with universal covering spin, Mathematische Annalen295(1993) 745–759

    P. Teichner,On the signature of four-manifolds with universal covering spin, Mathematische Annalen295(1993) 745–759

    1993

  207. [215]

    Beaudry and J

    A. Beaudry and J. A. Campbell,A guide for computing stable homotopy groups,arXiv preprint arXiv:1801.07530(2018)

    2018 arXiv

  208. [216]

    Barnes and C

    D. Barnes and C. Roitzheim,Foundations of stable homotopy theory, vol. 185. Cambridge University Press, 2020

    2020

  209. [217]

    Spectra and stable homotopy theory

    M. Hopkins, “Spectra and stable homotopy theory.” https://math.uchicago.edu/~amathew/256y.pdf, 2012. 94

    2012

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.