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Non-Invertible Symmetries in 6d from Green-Schwarz Automorphisms

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Pith's one-line read The paper constructs non-invertible duality, triality, and S3-ality defects in the 6d (2,0) so(8) superconformal field theory and computes their fusions, claiming the first concrete 6d example of a fusion 5-category with intrinsically…

desk verdict First explicit non-invertible, non-abelian S3-ality defects in 6d with computed fusions; the fusion 5-category claim needs an associativity check but the construction itself is solid. read the letter →

arxiv 2411.09674 v1 pith:OIHWBOHC submitted 2024-11-14 hep-th cond-mat.str-elmath.CT

Fabio Apruzzi , Sakura Schafer-Nameki , Alison Warman This is my paper · ORCID
classification hep-thcond-mat.str-elmath.CT
keywords non-invertiblesymmetriesgeneralizedfusion5-categoryGreen-SchwarzautomorphismsdualitydefectstrialitySymTFT(20)superconformalfieldtheory
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 constructs new symmetries of the six-dimensional $\mathcal{N}=(2,0)$ superconformal field theory with $\mathfrak{so}(8)$ gauge algebra—symmetries that are not invertible, meaning fusing a defect with itself does not give the identity but a lower-dimensional condensation defect. The construction combines Green-Schwarz automorphisms of the BPS string charge lattice, which form the permutation group $S_3$, with gauging of a $\mathbb{Z}_2$ 2-form symmetry and stacking of symmetry-protected topological phases. From half-space gauging and independently from the symmetry topological field theory, the paper derives fusion rules for duality, triality and $S_3$-ality defects. If correct, these defects are the simple objects of a fusion 5-category whose fusions are both non-invertible and non-abelian, the first concrete example in six dimensions. The result matters because it establishes that six-dimensional superconformal theories host genuinely categorical symmetries, not merely group-like ones.

What carries the argument

The load-bearing objects are the Green-Schwarz automorphisms of the $\mathfrak{so}(8)$ charge lattice: the outer automorphism group $S_3$ acting by permutations of the three $\mathbb{Z}_2$ subgroups $S,C,V$ of the defect group, realized as invertible interfaces $D_5^{(I)}$, $T_5$, $\bar T_5$ between absolute theories with different polarizations. To these the paper attaches half-space gauging interfaces $\sigma$ (gauging the 2-form symmetry) and SPT-stacking operations $\tau$, including twisted gauging $\tau\sigma$, and computes fusions by collapsing a sandwich of two defects onto a 5-manifold. The alternative machinery is the 7d 3-form Chern-Simons theory with K the Cartan matrix of $\mathfrak{so}(8)$, where the $S_3$ action is manifest; condensation defects built by higher gauging produce twist defects, and gauging the $S_3$ symmetry in the bulk yields the same boundary fusion rules. The condensation defects $C_5^{(0)}$, $C_5^{(1)}$ and the Arf-Kervaire TQFT factor $Z(Y,M_6)$ carry the non-invertibility in the fusion algebra.

What would settle it

Search the six absolute $\mathfrak{so}(8)$ theories of eq. (2.7) for an $S_3$-invariant Lagrangian subalgebra of the defect group $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ (equivalently, an $S_3$-invariant polarization pair). If one is found, the $S_3$-ality defects would be group-theoretical and the claim of intrinsic non-invertibility would fail, even though the fusion rules (3.69)-(3.76) themselves would remain correct.

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Extended reading notes

Core claim

The paper's central claim is that the 6d $\mathcal{N}=(2,0)$ $\mathfrak{so}(8)$ SCFT admits codimension-one topological defects $D_5$, $T_5$ and $\bar T_5$—duality, triality, and $S_3$-ality defects—whose fusion products are condensation defects and TQFT coefficients rather than identities. Concretely, $D_5 \otimes D_5 = C_5^{(0)}$, $T_5 \otimes \bar T_5 = C_5^{(1)}$, and the mixed fusions obey the $S_3$ relation $D_5 \otimes T_5 = \bar T_5 \otimes D_5$, with the remaining triality fusions carrying an Arf-Kervaire TQFT factor $Z(Y,M_6)$. The same fusion table is obtained from two independent constructions: half-space (twisted) gauging of the $\mathbb{Z}_2$ 2-form symmetry and the 7d Chern-Simons SymTFT whose K-matrix is the Cartan matrix of $\mathfrak{so}(8)$. The defects are genuinely non-invertible because their squares are condensation defects, and the paper argues they are intrinsically non-invertible in that no polarization or Lagrangian subalgebra choice renders the $S_3$-ality symmetry invertible. The construction is described as the first concrete 6d example of a fusion 5-category with non-invertible, non-abelian fusions.

Load-bearing premise

The paper's strongest claim assumes that no choice of polarization or Lagrangian subalgebra makes the $S_3$-ality symmetry invertible; this assertion is stated rather than proved in detail.

Editorial extensions

If this is right

  • The 6d $\mathcal{N}=(2,0)$ $\mathfrak{so}(8)$ SCFT carries a fusion 5-category whose simple objects include the duality, triality, and $S_3$-ality defects, with fusion rules given by eqs. (3.69)-(3.76).
  • Duality defects square to condensation defects rather than the identity, so the symmetry is genuinely categorical rather than group-like.
  • Because no polarization is claimed to make the $S_3$-ality symmetry invertible, the symmetry cannot be gauged; the paper links this obstruction to the nonexistence of non-simply-laced 6d $\mathcal{N}=(2,0)$ theories of type $B_n$, $C_n$, $G_2$.
  • The construction extends to other 6d theories whose Green-Schwarz automorphisms admit polarizations to absolute theories, with duality and triality defects arising as subsectors of the $S_3$-ality structure.
  • On compactification on a two-torus the $S_3$-ality structure degenerates to the 4d $\mathcal{N}=4$ super-Yang-Mills case, where only duality or triality defects survive.

Reading between the lines

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

  • A natural next step the paper leaves open is to compute the associators, braidings, and higher-morphism data that fully specify the fusion 5-category; the object-level fusions reported here would be the zeroth-order input for that.
  • The intrinsic non-invertibility claim could be tested by searching systematically for an $S_3$-invariant Lagrangian subalgebra across all polarization choices; if one existed, the $S_3$-ality defects would become group-theoretical while the fusion rules would remain unchanged.
  • One might expect analogous $G$-ality defects in other even dimensions or in 6d theories whose Green-Schwarz automorphisms are smaller groups; the polarization-pair web in this paper provides a template for those constructions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper constructs codimension-one duality, triality, and S3-ality defects in the 6d N=(2,0) so(8) SCFT by combining Green-Schwarz automorphisms with half-space gauging and stacking of SPTs, and it computes their object-level fusion rules in eqs. (3.69)–(3.76). It also constructs the corresponding twist defects in the 7d Chern-Simons SymTFT and compares the resulting fusions with the half-space results after stacking with an Arf-Kervaire factor. The headline claim is that these defects are the first concrete example in 6d of simple objects of a fusion 5-category whose fusions are intrinsically non-invertible and non-abelian.

Significance. If the construction and fusion rules are correct, this is a substantial advance: it gives explicit, non-Lagrangian 6d duality/triality/S3-ality defects with carefully tracked normalizations, a detailed half-space gauging derivation, and a SymTFT consistency check. The half-space derivation is self-contained and the partition-function manipulations are transparent. The 6d setting and the appearance of non-abelian S3 structure are genuinely new relative to the 2d and 4d examples. The claim that the absence of an S3-invariant polarization explains the non-existence of non-simply-laced (2,0) theories is also suggestive, though it is presented as an observation rather than a proof.

major comments (3)
  1. [§3.4, eqs. (3.69)–(3.76)] The headline claim that the defects 'form a fusion 5-category' is not established by the computations presented. The paper computes pairwise fusions of the simple objects only; it does not verify the defining axioms of a fusion category at the level of 1-morphisms, in particular associativity of triple fusions such as (T5⊗T5)⊗Tbar5 versus T5⊗(T5⊗Tbar5), which would require reassociating the factors Z(Y,M≥0_6), C5^(ℓ), and (Z2)^{(I)}_{2ℓ} in a nontrivial way. The manuscript itself concedes at the end of §3.4 that a complete description of the fusion 5-category is beyond its scope. I therefore ask either that the coherence checks be supplied, or that the abstract and §1 be weakened to state that the paper computes object-level fusion rules of a candidate fusion 5-category.
  2. [§4.2.4] The assertion that 'there is no polarization or choice of Lagrangian subalgebra such that the S3-ality symmetry is invertible' is stated without proof. This assertion carries the 'intrinsically non-invertible' part of the headline claim, so it should not remain a bare statement. The missing argument is in fact short: the defect group is D=Z2×Z2, S3 permutes the three nontrivial elements freely, and the only Lagrangian subgroups of D are the three Z2 subgroups, none of which is S3-invariant. A one-line orbit argument should be included in the text.
  3. [§4.2.4, eqs. (4.73)–(4.76)] The SymTFT computation is not fully independent evidence for the half-space fusion rules: the bulk twist defects are redefined by stacking with Z(Y,M≥0_6) precisely to match the half-space results, and the paper acknowledges that the unstacked SymTFT computations do not produce this factor. The abstract's phrase 'two distinct perspectives' therefore overstates the independence. I recommend presenting the SymTFT construction as a consistency check that reproduces the fusion rules after a matching-dependent redefinition, rather than as an independent derivation of the Z(Y) terms.
minor comments (5)
  1. [Abstract] The sentence 'Applied to Z2, Z3 and S3 GS automorphisms, gives rise to...' lacks a subject; it should read 'Applying these to Z2, Z3 and S3 GS automorphisms gives rise to...'.
  2. [§2.1] There are typos: 'autopmorphisms' and 'auter automorphisms' should be 'automorphisms' and 'outer automorphisms'; similarly, 'intrisically' in §4.2.4 should be 'intrinsically'.
  3. [§3.3.2, eq. (3.57)] The notation T5(M5) = T5(M5) ⊗ τσ(M5) uses the same symbol for the non-invertible defect and for the invertible GS3 interface, which makes the definition confusing. Please use distinct symbols, e.g. T^GS_5 for the invertible interface.
  4. [§3.4, eqs. (3.74)–(3.75)] The SPT factor exp(iπ∫_{M≥0_6} qbar(C^{(I)}_3)) is introduced in the fusion summary without an explicit statement that the integral is over the limiting half-space after ϵ→0 and that it is a Z2-valued phase; this should be clarified for the reader.
  5. [§4.1.2, Figure 2 caption] The caption says that the first and second entries in S(I,J) refer to Neumann and Dirichlet boundary conditions, but this convention is not explained before equation (4.20). Please introduce the notation explicitly in the text.

Circularity Check

1 steps flagged · score 3.0 of 10

SymTFT 'derivation' of the S3-ality fusions is matched by construction to the half-space result; the half-space gauging derivation itself is self-contained.

  1. fitted input called prediction [Section 4.2.4, 'Matching with Half-space Gauging Results', around eqs. (4.73)-(4.76)]
    "To match with the half-space gauging results, we therefore define the symmetry defects on the symmetry boundary as given by the bulk twist defects stacked with Z(Y,M≥0_6): on the left-hand side of the fusion results it cancels since it is Z2-valued, but remains on the right-hand side when a non-trivial symmetry defect is present."

    The SymTFT twist-defect fusions (4.61), (4.63) and (4.64) were computed without the Z(Y,M≥0_6) factors that appear in the half-space fusions (3.72)-(3.73). The paper then redefines the boundary defects in (4.73)-(4.74) by stacking the bulk twist defects with exactly Z(Y,M≥0_6), and the resulting SymTFT fusions (4.76) reproduce (3.69)-(3.76). Thus the SymTFT derivation of the fusion rules is not independent: the defect definition was chosen after the fact so that the fusions match the half-space result. This is a fitted input presented as a second derivation; the half-space gauging computation in Section 3 and Appendix A remains self-contained and carries the central claim.

full rationale

The main construction and fusion computations in Section 3 and Appendix A are self-contained: defects are defined as compositions of GS interfaces with half-space (twisted) gauging interfaces, and the fusions (3.69)-(3.76) are obtained by explicit partition-function manipulations, including integrating out dynamical fields, using quadratic refinements, and applying Poincaré duality. This does not fit parameters or use the target result as input. The SymTFT section initially obtains different-looking fusion rules without the Z(Y,M≥0_6) factors; Section 4.2.4 then defines boundary symmetry defects by stacking with Z(Y,M≥0_6) and identifies condensation defects with specific representatives, after which the SymTFT fusions coincide with the half-space ones. Consequently the SymTFT computation is a consistency check aligned to the half-space answer rather than an independent derivation; this is a partial circularity burden on the 'two distinct perspectives' claim, but the central non-invertible and non-abelian fusion claim rests on the half-space computation. The assertion of intrinsic non-invertibility in Section 4.2.4, that no polarization or Lagrangian subalgebra makes the S3-ality symmetry invertible, is stated without proof, but it is a simple orbit argument on Z2×Z2 and is not circular. The fusion-5-category associativity and coherence axioms are not verified; this is under-support for the categorical claim, not circularity. Self-citations such as [20] and [37] are background or classification references and are not load-bearing in a way that makes the argument reduce to them.

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

The construction relies on prior classifications and mathematical structures rather than new fitted parameters. The main additional assumptions are the non-anomalous nature of S3 in the bulk SymTFT and the asserted absence of an S3-invariant polarization, both of which are used to support the strongest claims.

assumptions (5)
  • domain assumption The 6d (2,0) so(8) SCFT has defect group Z2 x Z2 and its absolute theories are specified by polarization pairs (LJ, LI).
    Invoked in Section 2.2.1, based on prior results [21,33,37], to set up the web of theories connected by gauging and GS dualities.
  • domain assumption Green-Schwarz automorphisms of so(8) form S3 and act as dualities permuting the S, C, V polarizations.
    Taken from the classification in [37] and used in Section 2.2.2 to define the interfaces D5(I), T5, Tbar5.
  • standard math A Z2-valued quadratic refinement qbar of the intersection pairing on H3(M6,Z2) exists and defines a topological action.
    Used in Section 3.3.1 to define the SPT stacking operation tau; standard mathematical structure on orientable 6-manifolds (see [33,49]).
  • ad hoc to paper The S3 symmetry in the 7d SymTFT is non-anomalous and can be gauged.
    Asserted in Section 4.2.4 ('the S3 bulk symmetry is not anomalous and can be gauged') to justify gauging S3 in the bulk; no derivation is given.
  • ad hoc to paper There is no polarization or Lagrangian subalgebra under which the S3-ality symmetry becomes invertible.
    Stated in Section 4.2.4 as 'it is easy to see'; this underpins the 'intrinsically non-invertible' claim and is not proved in the paper.

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Pith. "Pith review of Non-Invertible Symmetries in 6d from Green-Schwarz Automorphisms." pith.science (2026). https://pith.science/paper/OIHWBOHC

@misc{pith2026241109674,
  author       = {Pith},
  title        = {Pith review of: Non-Invertible Symmetries in 6d from Green-Schwarz Automorphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIHWBOHC}},
  note         = {Machine review of arXiv:2411.09674}
}
abstract

We construct non-invertible symmetries in 6d $\mathcal{N}=(2,0)$ superconformal field theories that arise from Green-Schwarz (GS) automorphisms, which form abelian or non-abelian groups. Applied to $\mathbb{Z}_2$, $\mathbb{Z}_3$ and $S_3$ GS automorphisms, gives rise to non-invertible duality, triality and $S_3$-ality defects, respectively, once combined with stacking symmetry protected topological phases (SPTs) and gauging 2-form symmetries. We derive the defects and their fusion rules from two distinct perspectives: from half-space gauging as well as from the Symmetry Topological Field Theory (SymTFT). This is the first concrete construction of symmetry defects in 6d forming a fusion 5-category whose fusions are intrinsically non-invertible and non-abelian.

Figures

Figures reproduced from arXiv: 2411.09674 by the authors.

Figure 1
Figure 1. The Green-Schwarz automorphisms of the relative 6d (2 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. The Green-Schwarz Automorphisms of the SymTFT actions [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. Action of the invertible, codimension-1, duality defects [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: figure 4 [PITH_FULL_IMAGE:figures/full_fig_p032_4.png]
Figure 4
Figure 4. Figure 4: Action of the invertible, codimension-1, triality defects [PITH_FULL_IMAGE:figures/full_fig_p033_4.png]

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Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

62 extracted references · 9 canonical work pages · cited by 2 Pith papers

  1. [21]

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

    C. Lawrie, X. Yu, and H. Y. Zhang, “Intermediate defect groups, polarization pairs, and noninvertible duality defects,” Phys. Rev. D 109 no. 2, (2024) 026005, arXiv:2306.11783 [hep-th]

    arXiv 2024

  2. [1]

    Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories,

    J. Kaidi, K. Ohmori, and Y. Zheng, “Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories,” Phys. Rev. Lett. 128 no. 11, (2022) 111601, arXiv:2111.01141 [hep-th]

    arXiv 2022

  3. [2]

    Noninvertible duality defects in 3+1 dimensions,

    Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, “Noninvertible duality defects in 3+1 dimensions,” Phys. Rev. D 105 no. 12, (2022) 125016, arXiv:2111.01139 [hep-th]

    arXiv 2022

  4. [3]

    Non-invertible higher-categorical symmetries,

    L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, “Non-invertible higher-categorical symmetries,” SciPost Phys. 14 no. 1, (2023) 007, arXiv:2204.06564 [hep-th]

    arXiv 2023

  5. [4]

    ICTP lectures on (non-)invertible generalized symmetries,

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

    arXiv 2024

  6. [5]

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

    S.-H. Shao, “What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries,” arXiv:2308.00747 [hep-th]

    arXiv

  7. [6]

    Kramers-Wannier duality from conformal defects,

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

    arXiv 2004

  8. [7]

    Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions,

    Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, “Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions,” Commun. Math. Phys. 402 no. 1, (2023) 489–542, arXiv:2204.09025 [hep-th]

    arXiv 2023

Show all 62 references
  1. [8]

    Symmetry TFTs for Non-invertible Defects,

    J. Kaidi, K. Ohmori, and Y. Zheng, “Symmetry TFTs for Non-invertible Defects,” Commun. Math. Phys. 404 no. 2, (2023) 1021–1124, arXiv:2209.11062 [hep-th]

    2023 arXiv

  2. [9]

    On triality defects in 2d CFT,

    D.-C. Lu and Z. Sun, “On triality defects in 2d CFT,” JHEP 02 (2023) 173, arXiv:2208.06077 [hep-th]

    2023 arXiv

  3. [10]

    Exploring G-ality defects in 2-dim QFTs,

    D.-C. Lu, Z. Sun, and Z. Zhang, “Exploring G-ality defects in 2-dim QFTs,” arXiv:2406.12151 [hep-th]

  4. [11]

    Orbifold groupoids,

    D. Gaiotto and J. Kulp, “Orbifold groupoids,” JHEP 02 (2021) 132, arXiv:2008.05960 [hep-th]. 59

    2021 arXiv

  5. [12]

    Symmetry TFTs from String Theory,

    F. Apruzzi, F. Bonetti, I. n. Garc ´ ıa Etxebarria, S. S. Hosseini, and S. Schafer-Nameki, “Symmetry TFTs from String Theory,” Commun. Math. Phys. 402 no. 1, (2023) 895–949, arXiv:2112.02092 [hep-th]

    2023 arXiv

  6. [13]

    Topological symmetry in quantum field theory,

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

    arXiv

  7. [14]

    The holography of non-invertible self-duality symmetries,

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

  8. [15]

    Aspects of categorical symmetries from branes: SymTFTs and generalized charges,

    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 no. 1, (2024) 025, arXiv:2306.16405 [hep-th]

    2024 arXiv

  9. [16]

    Noninvertible Symmetries from Holography and Branes,

    F. Apruzzi, I. Bah, F. Bonetti, and S. Schafer-Nameki, “Noninvertible Symmetries from Holography and Branes,” Phys. Rev. Lett. 130 no. 12, (2023) 121601, arXiv:2208.07373 [hep-th]

    2023 arXiv

  10. [17]

    Top down approach to topological duality defects,

    J. J. Heckman, M. Hubner, E. Torres, X. Yu, and H. Y. Zhang, “Top down approach to topological duality defects,” Phys. Rev. D 108 no. 4, (2023) 046015, arXiv:2212.09743 [hep-th]

    2023 arXiv

  11. [18]

    Non-invertible symmetries of class S theories,

    V. Bashmakov, M. Del Zotto, A. Hasan, and J. Kaidi, “Non-invertible symmetries of class S theories,” JHEP 05 (2023) 225, arXiv:2211.05138 [hep-th]

    2023 arXiv

  12. [19]

    “Zoology

    A. Antinucci, C. Copetti, G. Galati, and G. Rizi, ““Zoology” of non-invertible duality defects: the view from class S,” JHEP 04 (2024) 036, arXiv:2212.09549 [hep-th]

    2024 arXiv

  13. [20]

    Fusion 3-Categories for Duality Defects,

    L. Bhardwaj, T. D´ ecoppet, S. Schafer-Nameki, and M. Yu, “Fusion 3-Categories for Duality Defects,” arXiv:2408.13302 [math.CT]

    arXiv

  14. [22]

    Drinfeld centers and morita equivalence classes of fusion 2-categories,

    T. D. D´ ecoppet, “Drinfeld centers and morita equivalence classes of fusion 2-categories,” arXiv:2211.04917 [math.QA]

    arXiv

  15. [23]

    Classification of 3 + 1D bosonic topological orders (ii): The case when some pointlike excitations are fermions,

    T. Lan and X.-G. Wen, “Classification of 3 + 1D bosonic topological orders (ii): The case when some pointlike excitations are fermions,” Phys. Rev. X 9 (Apr, 2019) 021005

    2019

  16. [24]

    Classification of (3 + 1)D bosonic topological orders: The case when pointlike excitations are all bosons,

    T. Lan, L. Kong, and X.-G. Wen, “Classification of (3 + 1)D bosonic topological orders: The case when pointlike excitations are all bosons,” Phys. Rev. X 8 (Jun, 2018) 021074

    2018

  17. [25]

    Generalized Global Symmetries,

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148 [hep-th]

    2015 arXiv

  18. [26]

    On the Defect Group of a 6D SCFT,

    M. Del Zotto, J. J. Heckman, D. S. Park, and T. Rudelius, “On the Defect Group of a 6D SCFT,” Lett. Math. Phys. 106 no. 6, (2016) 765–786, arXiv:1503.04806 [hep-th]

    2016 arXiv

  19. [27]

    The anomaly field theories of six-dimensional (2,0) superconformal theories,

    S. Monnier, “The anomaly field theories of six-dimensional (2,0) superconformal theories,” Adv. Theor. Math. Phys. 22 (2018) 2035–2089, arXiv:1706.01903 [hep-th]

    2018 arXiv

  20. [28]

    6D Fractional Quantum Hall Effect,

    J. J. Heckman and L. Tizzano, “6D Fractional Quantum Hall Effect,” JHEP 05 (2018) 120, arXiv:1708.02250 [hep-th]

    2018 arXiv

  21. [29]

    Geometric Langlands From Six Dimensions,

    E. Witten, “Geometric Langlands From Six Dimensions,” arXiv:0905.2720 [hep-th]

    arXiv

  22. [30]

    Relative quantum field theory,

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

    2014 arXiv

  23. [31]

    AdS / CFT correspondence and topological field theory,

    E. Witten, “AdS / CFT correspondence and topological field theory,” JHEP 12 (1998) 012, arXiv:hep-th/9812012

    1998 arXiv

  24. [32]

    Topological field theories on manifolds with Wu structures,

    S. Monnier, “Topological field theories on manifolds with Wu structures,” Rev. Math. Phys. 29 no. 05, (2017) 1750015, arXiv:1607.01396 [math-ph]

    2017 arXiv

  25. [33]

    Generalized global symmetries of T [M ] theories. Part I,

    S. Gukov, P.-S. Hsin, and D. Pei, “Generalized global symmetries of T [M ] theories. Part I,” JHEP 04 (2021) 232, arXiv:2010.15890 [hep-th]. 60

    2021 arXiv

  26. [34]

    Five-brane effective action in M theory,

    E. Witten, “Five-brane effective action in M theory,” J. Geom. Phys. 22 (1997) 103–133, arXiv:hep-th/9610234

    1997 arXiv

  27. [35]

    2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories,

    C. Cordova, T. T. Dumitrescu, and K. Intriligator, “2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories,” JHEP 04 (2021) 252, arXiv:2009.00138 [hep-th]

    2021 arXiv

  28. [36]

    Higher-form symmetries of 6d and 5d theories,

    L. Bhardwaj and S. Sch¨ afer-Nameki, “Higher-form symmetries of 6d and 5d theories,” JHEP 02 (2021) 159, arXiv:2008.09600 [hep-th]

    2021 arXiv

  29. [37]

    Green-Schwarz Automorphisms and 6D SCFTs,

    F. Apruzzi, J. J. Heckman, and T. Rudelius, “Green-Schwarz Automorphisms and 6D SCFTs,” JHEP 02 (2018) 157, arXiv:1707.06242 [hep-th]

    2018 arXiv

  30. [38]

    2-Group symmetries and their classification in 6d,

    F. Apruzzi, L. Bhardwaj, D. S. W. Gould, and S. Schafer-Nameki, “2-Group symmetries and their classification in 6d,” SciPost Phys. 12 no. 3, (2022) 098, arXiv:2110.14647 [hep-th]

    2022 arXiv

  31. [39]

    Deformations of Superconformal Theories,

    C. Cordova, T. T. Dumitrescu, and K. Intriligator, “Deformations of Superconformal Theories,” JHEP 11 (2016) 135, arXiv:1602.01217 [hep-th]

    2016 arXiv

  32. [40]

    All AdS 7 solutions of type II supergravity,

    F. Apruzzi, M. Fazzi, D. Rosa, and A. Tomasiello, “All AdS 7 solutions of type II supergravity,” JHEP 04 (2014) 064, arXiv:1309.2949 [hep-th]

    2014 arXiv

  33. [41]

    Supersymmetric AdS7 backgrounds in half-maximal supergravity and marginal operators of (1, 0) SCFTs,

    J. Louis and S. L¨ ust, “Supersymmetric AdS7 backgrounds in half-maximal supergravity and marginal operators of (1, 0) SCFTs,” JHEP 10 (2015) 120, arXiv:1506.08040 [hep-th]

    2015 arXiv

  34. [42]

    On the Classification of 6D SCFTs and Generalized ADE Orbifolds,

    J. J. Heckman, D. R. Morrison, and C. Vafa, “On the Classification of 6D SCFTs and Generalized ADE Orbifolds,” JHEP 05 (2014) 028, arXiv:1312.5746 [hep-th]. [Erratum: JHEP 06, 017 (2015)]

    2014 arXiv

  35. [43]

    Atomic Classification of 6D SCFTs,

    J. J. Heckman, D. R. Morrison, T. Rudelius, and C. Vafa, “Atomic Classification of 6D SCFTs,” Fortsch. Phys. 63 (2015) 468–530, arXiv:1502.05405 [hep-th]

    2015 arXiv

  36. [44]

    6d Conformal Matter,

    M. Del Zotto, J. J. Heckman, A. Tomasiello, and C. Vafa, “6d Conformal Matter,” JHEP 02 (2015) 054, arXiv:1407.6359 [hep-th]

    2015 arXiv

  37. [45]

    Reading between the lines of four-dimensional gauge theories,

    O. Aharony, N. Seiberg, and Y. Tachikawa, “Reading between the lines of four-dimensional gauge theories,” JHEP 08 (2013) 115, arXiv:1305.0318 [hep-th]

    2013 arXiv

  38. [46]

    Non-invertible symmetries of N = 4 SYM and twisted compactification,

    J. Kaidi, G. Zafrir, and Y. Zheng, “Non-invertible symmetries of N = 4 SYM and twisted compactification,” JHEP 08 (2022) 053, arXiv:2205.01104 [hep-th]

    2022 arXiv

  39. [47]

    Non-invertible Gauss law and axions,

    Y. Choi, H. T. Lam, and S.-H. Shao, “Non-invertible Gauss law and axions,” JHEP 09 (2023) 067, arXiv:2212.04499 [hep-th]

    2023 arXiv

  40. [48]

    Non-Invertible Surface Defects in 2+1d QFTs from Half Spacetime Gauging,

    W. Cui, B. Haghighat, and L. Ruggeri, “Non-Invertible Surface Defects in 2+1d QFTs from Half Spacetime Gauging,” arXiv:2406.09261 [hep-th]

    arXiv

  41. [49]

    SL(2, Z) action on QFTs with Z2 symmetry and the Brown-Kervaire invariants,

    L. Bhardwaj, Y. Lee, and Y. Tachikawa, “ SL(2, Z) action on QFTs with Z2 symmetry and the Brown-Kervaire invariants,” JHEP 11 (2020) 141, arXiv:2009.10099 [hep-th]

    2020 arXiv

  42. [50]

    Exotic invertible phases with higher-group symmetries,

    P.-S. Hsin, W. Ji, and C.-M. Jian, “Exotic invertible phases with higher-group symmetries,” SciPost Phys. 12 no. 2, (2022) 052, arXiv:2105.09454 [cond-mat.str-el]

    2022 arXiv

  43. [51]

    Untersuchungen ¨ uber quadratische formen in k¨ orpern der charakteristik 2. (teil i.).,

    C. Arf, “Untersuchungen ¨ uber quadratische formen in k¨ orpern der charakteristik 2. (teil i.).,”Journal f¨ ur die reine und angewandte Mathematik 1941 no. 183, (1941) 148–167

    1941

  44. [52]

    A manifold which does not admit any differentiable structure,

    M. A. Kervaire, “A manifold which does not admit any differentiable structure,” Commentarii Mathematici Helvetici 34 (1960) 257–270

    1960

  45. [53]

    The kervaire invariant of framed manifolds and its generalization,

    W. Browder, “The kervaire invariant of framed manifolds and its generalization,” Annals of Mathematics 90 no. 1, (1969) 157–186

    1969

  46. [54]

    Generalizations of the Kervaire Invariant,

    E. H. Brown, “Generalizations of the Kervaire Invariant,” Annals of Mathematics 95 no. 2, (1972) 368–383

    1972

  47. [55]

    A Tale of Bulk and Branes: Symmetry TFT of 6D SCFTs from IIB/F-theory,

    J. Tian and Y.-N. Wang, “A Tale of Bulk and Branes: Symmetry TFT of 6D SCFTs from IIB/F-theory,” arXiv:2410.23076 [hep-th]. 61

    arXiv

  48. [56]

    Anyonic symmetries and topological defects in abelian topological phases: An application to the ADE classification,

    M. N. Khan, J. C. Y. Teo, and T. L. Hughes, “Anyonic symmetries and topological defects in abelian topological phases: An application to the ADE classification,” Phys. Rev. B 90 (Dec, 2014) 235149

    2014

  49. [57]

    Theory of Twist Liquids: Gauging an Anyonic Symmetry,

    J. C. Y. Teo, T. L. Hughes, and E. Fradkin, “Theory of Twist Liquids: Gauging an Anyonic Symmetry,” Annals Phys. 360 (2015) 349–445, arXiv:1503.06812 [cond-mat.str-el]

    2015 arXiv

  50. [58]

    Backfiring Bosonisation,

    P. Boyle Smith and Y. Zheng, “Backfiring Bosonisation,” arXiv:2403.03953 [hep-th]

    arXiv

  51. [59]

    Higher Gauging and Non-invertible Condensation Defects,

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

    2023 arXiv

  52. [60]

    Anomalies of Non-Invertible Symmetries in (3+1)d,

    C. Cordova, P.-S. Hsin, and C. Zhang, “Anomalies of Non-Invertible Symmetries in (3+1)d,” arXiv:2308.11706 [hep-th]

    arXiv

  53. [61]

    Anomalies of non-invertible self-duality symmetries: fractionalization and gauging,

    A. Antinucci, F. Benini, C. Copetti, G. Galati, and G. Rizi, “Anomalies of non-invertible self-duality symmetries: fractionalization and gauging,” arXiv:2308.11707 [hep-th]

    arXiv

  54. [62]

    When are Duality Defects Group-Theoretical?,

    Z. Sun and Y. Zheng, “When are Duality Defects Group-Theoretical?,” arXiv:2307.14428 [hep-th]. 62

    arXiv

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Reviewed August 12, 2026 · model on record in the stance chip above.