Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories

Bram Vancraeynest-De Cuiper; Clement Delcamp

REVIEW 3 major objections 4 minor 3 cited by

Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The paper establishes that in (2+1)d abelian lattice gauge theories, twisted gauging preserves the charge sector $(g_1,g_2)$ and twists only the 1-form boundary condition by the character $t^2_{g_1,g_2}(\lambda)$, giving unitary…

desk verdict Strong explicit tensor-network computation of 2+1d twisted gauging, but the decorated-loop topological invariance is asserted, not shown. read the letter →

arxiv 2501.16301 v2 pith:JUQI4ZML submitted 2025-01-27 cond-mat.str-el hep-thquant-ph

Bram Vancraeynest-De Cuiper , Clement Delcamp This is my paper · ORCID

classification cond-mat.str-elhep-thquant-ph

keywords abelianlatticegaugetheorytwistedgauging1-formsymmetrycondensationdefectsdiscretetorsiontensornetworkoperatorstopologicalsectors2-group

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 seeks a precise lattice-level statement of a two-step duality for (2+1)d abelian lattice gauge theories: gauge the 1-form symmetry, then gauge the resulting 0-form symmetry in a twisted way, with the twisting labelled by a class in $H^3(G,U(1))$. Its central formula says that under this duality a topological sector $(g_1,g_2,\eta)$ is mapped to $(g_1,g_2,t^2_{g_1,g_2}(\lambda)\,\eta)$: the charge holonomies $g_1,g_2$ are untouched and only the 1-form symmetry-twisted boundary condition is multiplied by a cohomological character. The paper constructs explicit tensor-network duality operators that transmute the Hamiltonian $H$ into $H^\lambda$, and shows that once the boundary twist is promoted to a dynamical degree of freedom these operators become unitary isometries between the dual spectra. Concrete examples include the duality between toric-code-type and double-semion-type Hamiltonians for $G=\mathbb{Z}_2$, where the twist is trivial, and a $\mathbb{Z}_2^3$ example where the twist is non-trivial; the paper further argues that self-dual models acquire a 2-group symmetry. The pay-off is that a formal gauging duality becomes a concrete lattice equivalence with a computable sector map, and the symmetry structure preserved by the duality is identified.

What carries the argument

The load-bearing object is the tensor-network duality operator $D_{A,\varphi}(T^2_\triangle)$ built from a $\mathrm{Vec}^\lambda_G$-module 2-category $M(A,\varphi)$—a higher analogue of a projective representation, encoding the twisted action of $G$ on cosets $G/A$—with unit cell (3.42) and transmutation property $D_{A,\varphi}\circ H=H^\lambda\circ D_{A,\varphi}$ following from the cocycle identity (3.41). The same tensor calculus realises condensation defects $U_A^\psi(T^2_\triangle)$ as networks, and the duality operator is decorated with 't Hooft loops using the local patches (3.53). The mechanism that carries the sector mapping is the character $t^2_{g_1,g_2}(\lambda)$, defined by the slant products (3.48)-(3.50); it is what multiplies the boundary twist $\eta$ while leaving the holonomies $(g_1,g_2)$ fixed.

What would settle it

Take a small triangular-lattice torus with $G=\mathbb{Z}_2^3$ and the 3-cocycle $\lambda$ of Eq. (3.62), insert the decorated 't Hooft loop of Eq. (3.53) along one non-contractible cycle, and compute the phase (3.54) for two loops that differ by a single plaquette deformation. If the two phases differ by anything other than a factor that cancels inside the sum over $x$, the topological-invariance assumption fails and the mapping (3.55) is wrong for that datum.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is the mapping of topological sectors under twisted gauging: $(g_1,g_2,\eta)\mapsto(g_1,g_2,\eta^\lambda_{g_1,g_2})$ with $\eta^\lambda_{g_1,g_2}(-)=t^2_{g_1,g_2}(\lambda)(-)\,\eta(-)$, where $t^2_{x_1,x_2}(\lambda)(g)=t_{x_1}(\lambda)(x_2,g)/t_{x_1}(\lambda)(g,x_2)$ is the character obtained from the 3-cocycle $\lambda$ by the slant-product identities (3.48)-(3.50). This result is obtained from an explicit tensor-network realisation of the duality operator $D_{A,\varphi}(T^2_\triangle)$, including loop decorations (3.53) that allow access to every charge sector rather than only the singlet sector. On sectors compatible with the map, and with the boundary twist treated as dynamical, the restricted duality operators become unitary isometries relating the spectra of $H$ and $H^\lambda$. The paper also shows that self-duality can be promoted to an internal symmetry whose operators are organised by the fusion 2-category of 2-representations of a 2-group, a categorical generalisation of ordinary group representations, giving a lattice higher gauge theory.

Load-bearing premise

The argument rests on the claim that decorating the duality operator with 't Hooft loops is topologically invariant: the phase contributed by a decorated loop depends only on which cycle it winds around, not on its exact lattice path. The paper states this follows from the cocycle condition but does not display the deformation check; if a deformation changed the phase, the sector map (3.55) and the unitary isometries would fail.

Editorial extensions

If this is right

Charge sectors are invariant under the duality: a sector $(g_1,g_2,\eta)$ keeps its holonomies and only the 1-form twist changes, so the spectra of $H$ and $H^\lambda$ can be paired sector by sector.
Promoting the twist to a dynamical variable turns the restricted duality operators into unitary isometries, upgrading the gauging duality from a projection onto the singlet sector to a genuine lattice equivalence.
Self-dual models, such as $H+H^\lambda$ for $G=\mathbb{Z}_2$, can promote the self-duality to an internal symmetry with operators organised by the 2-representations of the 2-group $\mathbb{Z}_2^{[0]}\ltimes_\mu\mathbb{Z}_2^{[1]}$, realising a lattice higher gauge theory.
For $G=\mathbb{Z}_2$ the twist $t^2_{g_1,g_2}(\lambda)$ is trivial in every sector, while for $G=\mathbb{Z}_2^3$ it is a non-trivial character for generic sectors; formula (3.55) captures both behaviours uniformly.

Reading between the lines

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

One would expect the same character $t^2_{g_1,g_2}(\lambda)$ to control the sector map in any dimension where the ungauging/twisted-gauging sequence can be defined, since it is produced by the cocycle structure of the duality operator rather than by two-dimensional lattice details.
Because the duality operators are explicit tensor networks, the sector twist can be evaluated numerically for any finite abelian $G$ and 3-cocycle $\lambda$, giving a direct check of (3.55) beyond the analytical examples in the paper.
The promotion of self-duality to a 2-group symmetry suggests that iterating the duality could generate a tower of 2-group extensions; exploring when that tower terminates may provide a classification of self-dual abelian lattice gauge theories.

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.

Desk Editor's Note

read the letter

You should know this paper is the first to give an explicit lattice-level computation of how twisted gauging permutes topological sectors in (2+1)d abelian gauge theories, including the charge-dependent twist t^2_{g1,g2}(λ). That part is genuinely new and worth having.

The 1+1d review is clean, and the 2+1d cocycle algebra in Sec. 3.4 is explicit: the proof that D_{A,φ} intertwines H and H^λ is shown step-by-step. The tensor-network realization of condensation defects is concrete, and the worked examples (Z2 toric code/double semion and Z2^3) are useful.

The soft spot is exactly where the stress-test note lands. The decorated 't Hooft loop construction in Sec. 3.5 is load-bearing: the sector mapping (3.55) and the unitarity of the restricted isometries rest on the topological invariance of the decorated loops and on the phase evaluation (3.54). The paper says "One can check" for the former and "one finds" for the latter, but doesn't show the check. That's a real gap. The junction of two loops, which the paper acknowledges "requires some care," is exactly where an extra phase factor could change t^2_{g1,g2}(λ). I don't see an obvious error in (3.54), and the surrounding algebra (3.41)–(3.45) is credible, so the gap could presumably be filled. But as written, the central claim is conditional. The 2-group self-duality section is also argued rather than proven, though the paper is explicit about that being an argument.

This is for people working on non-invertible dualities, higher symmetries, and tensor-network lattice realizations. They will find an explicit construction and a credible conjecture, but they should treat the sector mapping as needing verification until the loop check is supplied.

I would send this to peer review. It deserves referee time, and the referees should ask for the missing topological-invariance check and a derivation of (3.54).

Referee Report

3 major / 4 minor

Summary. The paper develops a tensor-network framework for a duality operation in (2+1)d abelian lattice gauge theories: gauging the invertible 1-form Rep(G) symmetry and then gauging the resulting 0-form symmetry twisted by a 3-cocycle λ∈H^3(G,U(1)). It constructs explicit lattice realisations of condensation defects, duality operators D_A^φ, and symmetry-twisted boundary conditions, and derives the permutation of topological sectors (g1,g2,η) ↦ (g1,g2,t^2_{g1,g2}(λ)η), Eq. (3.55). From this it constructs unitary isometries relating dual Hamiltonians in compatible sectors. The paper also treats the (1+1)d analogue and explores promoting self-duality to a 2-group symmetry 2Rep(Z2[0]⋉_μ Z2[1]) in the toric-code/double-semion example.

Significance. If the asserted checks hold, the paper provides a genuinely explicit, parameter-free lattice implementation of twisted gauging in 2+1d, with concrete sector-mapping formulas and an isometry construction. The derivation of D_A^φ ∘ H = H^λ ∘ D_A^φ in Sec. 3.4 is carried out in detail through cocycle identities, and the examples (toric code, double semion, Z2^3) give concrete, checkable content. The (1+1)d section coherently re-derives the Kennedy–Tasaki duality within the same language. The main weakness is that two load-bearing steps in Sec. 3.5 — topological invariance of the decorated 't Hooft loops and the phase evaluation (3.54) — are asserted rather than proven, and the promotion to 2-group symmetry in Sec. 3.8 is only partially demonstrated. These gaps are local and appear repairable with a more detailed appendix.

major comments (3)

[Sec. 3.5, Eq. (3.53)] The topological invariance of the decorated 't Hooft loop is asserted with 'One can check that the cocycle condition d^{(3)}λ=1 ensures that the resulting duality operator is left invariant under continuous deformations of the closed loop,' but no proof is given. This invariance is load-bearing: the operator D^{t1u}_{η1→η2}[g1,g2] is defined by inserting two loops along non-contractible cycles, and without invariance under deformations and under the choice of representative of the homology class, the phrase 'projecting onto a charge sector labelled by (g1,g2)' is not well-defined, and Eq. (3.55) loses its well-defined object. A detailed proof, or at least a precise statement of the deformation equivalence used, is required.
[Sec. 3.5, Eq. (3.54)] The phase evaluation in Eq. (3.54) is presented as a single diagrammatic result. The text acknowledges that the junction of the two loop operators 'requires some care so as to be topologically invariant' and then immediately gives the final expression. The derivation should be expanded step by step, showing how the junction is resolved and why no extra phase from the ordering of the two loops appears. If an extra factor were present, the character t^2_{g1,g2}(λ) and hence the sector mapping (3.55) would be modified.
[Sec. 3.8, Eqs. (3.67)–(3.70)] The claim that any Hamiltonian built from local operators (3.67) possesses a full 2Rep(Z2[0]⋉_μ Z2[1]) symmetry is only partially substantiated. The text verifies commutation of the 0-form surface operators U_A^{φ,χ} with the local terms, but does not demonstrate the 1-form symmetry operators, the junctions between 0- and 1-form operators, or the coherence data required for a genuine 2-representation. Since the self-duality promotion is a central advertised result, this should either be derived explicitly or flagged as an argument that relies on the classification (3.66) of [Elg07] together with a precise statement of the assumptions.

minor comments (4)

[Sec. 3.5, before Eq. (3.55)] The notation 'rfsP H^1(G,T^2_△)' is a typo; the cohomology class of f should be an element of H^1(T^2_△,G).
[Sec. 3.4, Eq. (3.41)] The module associator α^φ_3 is introduced by the condition (3.41), but no explicit formula in terms of φ is given; readers need that formula to check the derivation of Eqs. (3.43)–(3.45).
[Sec. 3.8, after Eq. (3.66)] The phrase 'the monoidal associator is provided by µ' is imprecise for a 2-group; µ is a 3-cocycle that controls the associator of the monoidal category, and the statement should be made precise.
[General] There are several small typos and formatting issues ('T able 1', 'associator evaluates to the identity 1-morphisms', the author list on the first page); these should be cleaned up before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the topological-sector mapping is derived by explicit cocycle computations and does not reduce to a fit or to a self-citation.

full rationale

The paper's central claim, Eq. (3.55), is obtained from an explicit tensor-network computation rather than assumed. The duality operator D_A^φ in Eq. (3.43) is shown to transmute H into H^λ by direct application of the cocycle condition (3.41), with the intermediate algebra displayed in Eqs. (3.44)-(3.45). The sector restriction (3.51) follows from the cohomological identity (3.47)-(3.49), and the boundary-condition shift t^2_{g1,g2}(λ) in Eq. (3.55) is read off from the non-vanishing condition of the decorated-loop phase (3.54) via the orthogonality of characters. These are explicit derivations with no fitted parameters and no quantity defined in terms of the target result. Self-citations to [Del21, DT23, LDOV21, LDV22] supply tensor-network and categorical context, but the load-bearing identities are either rederived in the text (e.g., Sec. 2 explicitly verifies the duality operators announced from [LDOV21]) or proven from the stated cocycle conditions. The one notable gap is the asserted topological invariance of the decorated 't Hooft loop ('One can check that the cocycle condition dλ=1 ensures...' in Sec. 3.5); this is an omitted check that affects completeness, not an input-output equivalence, so it does not constitute circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted in this theoretical construction; the central computations are cocycle identities. The axioms are the higher-categorical classification and symmetry-structure statements imported from prior work by Delcamp and collaborators, plus the self-duality example assumption. No new physical entities are introduced.

assumptions (4)

domain assumption The symmetry structure of Hamiltonians H^λ is the fusion 2-category 2ReppGq = ModpVec_Gq, including condensation surface operators U_A^ψ.
Used throughout Sec. 3.2 to define the full set of symmetry operators; taken from [Del21, DT23].
domain assumption Indecomposable module 2-categories over 2Vec_G^λ are classified by pairs (A,φ) with λ|_A cohomologically trivial, and duality operators correspond to simple objects in 2Rep_λ(G).
Invoked in Sec. 3.4 and Sec. 3.7, especially in the derivation of Eq. (3.41); sourced from [Déc21, Del21, DT23].
domain assumption All non-trivial dualities preserving 2ReppGq are of the twisted-gauging type considered here.
Used in Sec. 3.8 to motivate self-duality and its promotion; attributed to a conjecture in [DT23] and corroboration in [DY23], not proved in this paper.
ad hoc to paper Local operators (3.67) with arbitrary coefficients generate Hamiltonians with 2ReppZ2[0] ⋉_μ Z2[1]q symmetry.
The construction in Sec. 3.8 is tailored to this paper's example; no general proof is given that all such models have the claimed symmetry.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories." pith.science (2026). https://pith.science/paper/JUQI4ZML

@misc{pith2026250116301,
  author       = {Pith},
  title        = {Pith review of: Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JUQI4ZML}},
  note         = {Machine review of arXiv:2501.16301}
}

read the original abstract

Given a two-dimensional quantum lattice model with an abelian gauge theory interpretation, we investigate a duality operation that amounts to gauging its invertible 1-form symmetry, followed by gauging the resulting 0-form symmetry in a twisted way via a choice of discrete torsion. Using tensor networks, we introduce explicit lattice realisations of the so-called condensation defects, which are obtained by gauging the 1-form symmetry along submanifolds of spacetime, and employ the same calculus to realise the duality operators. By leveraging these tensor network operators, we compute the non-trivial interplay between symmetry-twisted boundary conditions and charge sectors under the duality operation, enabling us to construct isometries relating the dual Hamiltonians. Whenever a lattice gauge theory is left invariant under the duality operation, we explore the possibility of promoting the self-duality to an internal symmetry. We argue that this results in a symmetry structure that encodes the 2-representations of a 2-group.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

Self-dual $S_3$ gauge theory in 2+1d: lattice model and topological phase transitions
cond-mat.str-el 2026-08 conditional novelty 8.0 of 10

A new sign-problem-free lattice Hamiltonian realizes the S3 quantum double with electric-magnetic duality as translation, yielding a tetracritical Ising boundary and three predicted topological transitions.
Gauging Non-Invertible Symmetries in (2+1)d Topological Orders
hep-th 2025-07 conditional novelty 7.0 of 10

A framework for gauging non-invertible symmetries in (2+1)d TQFTs, unifying 0-form and 1-form gauging via surface algebras, with constraints and toric-code examples.
Les Houches Lecture Notes on Tensor Networks
cond-mat.str-el 2025-12 unverdicted novelty 2.0 of 10

A well-organized five-lecture review of tensor networks (MPS/PEPS/MPO) covering algorithms, phase classification, string-nets, strange correlators, and dualities; it contains no new research results.

Reference graph

Works this paper leans on

78 extracted references · 11 canonical work pages · cited by 3 Pith papers

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION output.nonempty.mrnumber duplicate missing pop "" 'skip if duplicate empty 'pop " " swap * " " * write if FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'outpu...
[2]

Apruzzi, F

F. Apruzzi, F. Bonetti, I. n. Garc\' a Etxebarria, S. S. Hosseini, and S. Schafer-Nameki, Symmetry TFTs from String Theory , http://dx.doi.org/10.1007/s00220-023-04737-2 Commun. Math. Phys. 402 (2023) 895--949 , http://arxiv.org/abs/2112.02092 arXiv:2112.02092 [hep-th]

arXiv 2023
[3]

Aasen, P

D. Aasen, P. Fendley, and R. S. K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies , http://arxiv.org/abs/2008.08598 arXiv:2008.08598 [cond-mat.stat-mech]

arXiv 2008
[4]

Buerschaper, M

O. Buerschaper, M. Aguado, and G. Vidal, Explicit tensor network representation for the ground states of string-net models, Phys. Rev. B 79 (2009) 085119 https://link.aps.org/doi/10.1103/PhysRevB.79.085119

doi:10.1103/physrevb.79.085119 2009
[5]

Bartsch, M

T. Bartsch, M. Bullimore, A. E. V. Ferrari, and J. Pearson, Non-invertible symmetries and higher representation theory I , http://dx.doi.org/10.21468/SciPostPhys.17.1.015 SciPost Phys. 17 (2024) 015 , http://arxiv.org/abs/2208.05993 arXiv:2208.05993 [hep-th]

arXiv 2024
[6]

Bhardwaj, L

L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, Non-invertible higher-categorical symmetries , http://dx.doi.org/10.21468/SciPostPhys.14.1.007 SciPost Phys. 14 (2023) 007 , http://arxiv.org/abs/2204.06564 arXiv:2204.06564 [hep-th]

arXiv 2023
[7]

Bhardwaj, L

L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, Lattice Models for Phases and Transitions with Non-Invertible Symmetries , http://arxiv.org/abs/2405.05964 arXiv:2405.05964 [cond-mat.str-el]

arXiv
[8]

Benini, C

F. Benini, C. C\'ordova, and P.-S. Hsin, On 2-Group Global Symmetries and their Anomalies , http://dx.doi.org/10.1007/JHEP03(2019)118 JHEP 03 (2019) 118 , http://arxiv.org/abs/1803.09336 arXiv:1803.09336 [hep-th]

arXiv 2019

Show all 78 references

[9]

Bullivant, M

A. Bullivant, M. Cal c ada, Z. K\'ad\'ar, P. Martin, and J. a. F. Martins, Topological phases from higher gauge symmetry in 3+1 dimensions, Phys. Rev. B 95 (2017) 155118 https://link.aps.org/doi/10.1103/PhysRevB.95.155118

2017 doi
[10]

Bullivant and C

A. Bullivant and C. Delcamp, Excitations in strict 2-group higher gauge models of topological phases , http://dx.doi.org/10.1007/JHEP01(2020)107 JHEP 01 (2020) 107 , http://arxiv.org/abs/1909.07937 arXiv:1909.07937 [cond-mat.str-el]

2020 arXiv
[11]

Buican and A

M. Buican and A. Gromov, Anyonic Chains, Topological Defects, and Conformal Field Theory , http://dx.doi.org/10.1007/s00220-017-2995-6 Commun. Math. Phys. 356 (2017) 1017--1056 , http://arxiv.org/abs/1701.02800 arXiv:1701.02800 [hep-th]

2017 arXiv
[12]

J. C. Baez and J. Huerta, An Invitation to Higher Gauge Theory , http://dx.doi.org/10.1007/s10714-010-1070-9 Gen. Rel. Grav. 43 (2011) 2335--2392 , http://arxiv.org/abs/1003.4485 arXiv:1003.4485 [hep-th]

2011 arXiv
[13]

H. J. Briegel and R. Raussendorf, Persistent Entanglement in Arrays of Interacting Particles , http://dx.doi.org/10.1103/PhysRevLett.86.910 Phys. Rev. Lett. 86 (2001) 910

2001 doi
[14]

Bhardwaj, S

L. Bhardwaj, S. Schafer-Nameki, and J. Wu, Universal Non-Invertible Symmetries , http://dx.doi.org/10.1002/prop.202200143 Fortsch. Phys. 70 (2022) 2200143 , http://arxiv.org/abs/2208.05973 arXiv:2208.05973 [hep-th]

2022 arXiv
[15]

Bhardwaj and Y

L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions , http://dx.doi.org/10.1007/JHEP03(2018)189 JHEP 03 (2018) 189 , http://arxiv.org/abs/1704.02330 arXiv:1704.02330 [hep-th]

2018 arXiv
[16]

J. W. Barrett and B. W. Westbury, Invariants of piecewise linear three manifolds , http://dx.doi.org/10.1090/S0002-9947-96-01660-1 Trans. Am. Math. Soc. 348 (1996) 3997--4022 , http://arxiv.org/abs/hep-th/9311155 arXiv:hep-th/9311155

1996 arXiv
[17]

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 , http://dx.doi.org/10.1007/s00220-023-04727-4 Commun. Math. Phys. 402 (2023) 489--542 , http://arxiv.org/abs/2204.09025 arXiv:2204.09025 [hep-th]

2023 arXiv
[18]

C. Cheng, A character theory for projective representations of finite groups, Linear Algebra and its Applications 469 (2015) 230--242 https://www.sciencedirect.com/science/article/pii/S0024379514007629

2015
[19]

Chen and Q

J. Chen and Q. Jia, SymTFT Approach to 2D Orbifold Groupoids: `t Hooft Anomalies, Gauging, and Partition Functions , http://arxiv.org/abs/2411.18056 arXiv:2411.18056 [hep-th]

arXiv
[20]

Y. Choi, Y. Sanghavi, S.-H. Shao, and Y. Zheng, Non-invertible and higher-form symmetries in 2+1d lattice gauge theories , http://dx.doi.org/10.21468/SciPostPhys.18.1.008 SciPost Phys. 18 (2025) 008 , http://arxiv.org/abs/2405.13105 arXiv:2405.13105 [cond-mat.str-el]

2025 arXiv
[21]

M. B. S ahino g lu, D. Williamson, N. Bultinck, M. Mari\"en, J. Haegeman, N. Schuch, and F. Verstraete, Characterizing Topological Order with Matrix Product Operators , http://dx.doi.org/10.1007/s00023-020-00992-4 Annales Henri Poincare 22 (2021) 563--592 , http://arxiv.org/ab...

2021 arXiv
[22]

T. D. D \'e coppet, Finite semisimple module 2-categories , http://dx.doi.org/10.1007/s00029-024-00998-4 Selecta Math. 31 (2025) 5 , http://arxiv.org/abs/2107.11037 arXiv:2107.11037 [math.QA]

2025 arXiv
[23]

C. Delcamp, Tensor network approach to electromagnetic duality in (3+1)d topological gauge models , http://dx.doi.org/10.1007/JHEP08(2022)149 JHEP 08 (2022) 149 , http://arxiv.org/abs/2112.08324 arXiv:2112.08324 [cond-mat.str-el]

2022 arXiv
[24]

Diatlyk, C

O. Diatlyk, C. Luo, Y. Wang, and Q. Weller, Gauging non-invertible symmetries: topological interfaces and generalized orbifold groupoid in 2d QFT , http://dx.doi.org/10.1007/JHEP03(2024)127 JHEP 03 (2024) 127 , http://arxiv.org/abs/2311.17044 arXiv:2311.17044 [hep-th]

2024
[25]

C. L. Douglas and D. J. Reutter, Fusion 2-categories and a state-sum invariant for 4-manifolds , http://arxiv.org/abs/1812.11933 arXiv:1812.11933 [math.QA]

arXiv
[26]

Delcamp and A

C. Delcamp and A. Tiwari, From gauge to higher gauge models of topological phases , http://dx.doi.org/10.1007/JHEP10(2018)049 JHEP 10 (2018) 049 , http://arxiv.org/abs/1802.10104 arXiv:1802.10104 [cond-mat.str-el]

2018 arXiv
[27]

Delcamp and A

C. Delcamp and A. Tiwari, On 2-form gauge models of topological phases , http://dx.doi.org/10.1007/JHEP05(2019)064 JHEP 05 (2019) 064 , http://arxiv.org/abs/1901.02249 arXiv:1901.02249 [hep-th]

2019 arXiv
[28]

Delcamp and A

C. Delcamp and A. Tiwari, Higher categorical symmetries and gauging in two-dimensional spin systems , http://dx.doi.org/10.21468/SciPostPhys.16.4.110 SciPost Phys. 16 (2024) 110 , http://arxiv.org/abs/2301.01259 arXiv:2301.01259 [hep-th]

2024 arXiv
[29]

T. D. D \'e coppet and M. Yu, Fiber 2-Functors and Tambara Yamagami Fusion 2-Categories , http://dx.doi.org/10.1007/s00220-025-05249-x Commun. Math. Phys. 406 (2025) 64 , http://arxiv.org/abs/2306.08117 arXiv:2306.08117 [math.CT]

2025 arXiv
[30]

T. D. Décoppet, The Morita Theory of Fusion 2-Categories , Higher Structures 7 (2023) 234–292 http://dx.doi.org/10.21136/HS.2023.07

2023 doi
[31]

T. D. Décoppet, Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd) , Symmetry, Integrability and Geometry: Methods and Applications (2024) http://dx.doi.org/10.3842/SIGMA.2024.092

2024 doi
[32]

Etingof, S

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor categories, vol. 205, American Mathematical Soc., 2016

2016
[33]

J. Elgueta, Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces , Advances in Mathematics 213 (2007) 53--92 https://www.sciencedirect.com/science/article/pii/S0001870806003835

2007
[34]

Etingof, D

P. Etingof, D. Nikshych, and V. Ostrik, Fusion categories and homotopy theory, Quantum topology 1 (2010) 209--273 https://doi.org/10.4171/qt/6

2010 doi
[35]

Frohlich, J

J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, http://dx.doi.org/10.1142/9789814304634_0056 Defect Lines, Dualities and Generalised Orbifolds , 16th International Congress on Mathematical Physics , 2010, pp. 608--613. http://arxiv.org/abs/0909.5013 arXiv:0909.5013 [math-ph]

2010 arXiv
[36]

D. S. Freed, G. W. Moore, and C. Teleman, Topological symmetry in quantum field theory , http://arxiv.org/abs/2209.07471 arXiv:2209.07471 [hep-th]

arXiv
[37]

Feiguin, S

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, Interacting anyons in topological quantum liquids: The golden chain , http://dx.doi.org/10.1103/PhysRevLett.98.160409 Phys. Rev. Lett. 98 (2007) 160409 , http://arxiv.org/abs/cond-mat/06...

2007 arXiv
[38]

C. Gils, E. Ardonne, S. Trebst, D. A. Huse, A. W. W. Ludwig, M. Troyer, and Z. Wang, Anyonic quantum spin chains: Spin-1 generalizations and topological stability, Phys. Rev. B 87 (2013) 235120 https://link.aps.org/doi/10.1103/PhysRevB.87.235120

2013 doi
[39]

Gaiotto and T

D. Gaiotto and T. Johnson-Freyd, Condensations in higher categories , http://arxiv.org/abs/1905.09566 arXiv:1905.09566 [math.CT]

1905 arXiv
[40]

Gaiotto and J

D. Gaiotto and J. Kulp, Orbifold groupoids , http://dx.doi.org/10.1007/JHEP02(2021)132 JHEP 02 (2021) 132 , http://arxiv.org/abs/2008.05960 arXiv:2008.05960 [hep-th]

2021 arXiv
[41]

Gaiotto, A

D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries , http://dx.doi.org/10.1007/JHEP02(2015)172 JHEP 02 (2015) 172 , http://arxiv.org/abs/1412.5148 arXiv:1412.5148 [hep-th]

2015 arXiv
[42]

Greenough, Monoidal 2-structure of bimodule categories, Journal of Algebra 324 (2010) 1818--1859 https://www.sciencedirect.com/science/article/pii/S0021869310002942

J. Greenough, Monoidal 2-structure of bimodule categories, Journal of Algebra 324 (2010) 1818--1859 https://www.sciencedirect.com/science/article/pii/S0021869310002942

2010
[43]

Inamura and K

K. Inamura and K. Ohmori, Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories , http://dx.doi.org/10.21468/SciPostPhys.16.6.143 SciPost Phys. 16 (2024) 143 , http://arxiv.org/abs/2305.05774 arXiv:2305.05774 [cond-mat.str-el]

2024 arXiv
[44]

Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303 (2003) 2--30 https://www.sciencedirect.com/science/article/pii/S0003491602000180

A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303 (2003) 2--30 https://www.sciencedirect.com/science/article/pii/S0003491602000180

2003
[45]

Kaidi, K

J. Kaidi, K. Ohmori, and Y. Zheng, Symmetry TFTs for Non-invertible Defects , http://dx.doi.org/10.1007/s00220-023-04859-7 Commun. Math. Phys. 404 (2023) 1021--1124 , http://arxiv.org/abs/2209.11062 arXiv:2209.11062 [hep-th]

2023 arXiv
[46]

Kennedy and H

T. Kennedy and H. Tasaki, Hidden Z2 Z2 symmetry breaking in Haldane-gap antiferromagnets , http://dx.doi.org/10.1103/PhysRevB.45.304 Phys. Rev. B 45 (1992) 304

1992 doi
[47]

Kapustin and R

A. Kapustin and R. Thorngren, Higher Symmetry and Gapped Phases of Gauge Theories , http://dx.doi.org/10.1007/978-3-319-59939-7_5 Prog. Math. 324 (2017) 177--202 , http://arxiv.org/abs/1309.4721 arXiv:1309.4721 [hep-th]

2017 arXiv
[48]

H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. Part 1. , http://dx.doi.org/10.1103/PhysRev.60.252 Phys. Rev. 60 (1941) 252--262

1941 doi
[49]

Lootens, C

L. Lootens, C. Delcamp, G. Ortiz, and F. Verstraete, Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners , http://dx.doi.org/10.1103/PRXQuantum.4.020357 PRX Quantum 4 (2023) 020357 , http://arxiv.org/abs/2112.090...

2023 arXiv
[50]

Lootens, C

L. Lootens, C. Delcamp, and F. Verstraete, Dualities in One-Dimensional Quantum Lattice Models: Topological Sectors , http://dx.doi.org/10.1103/PRXQuantum.5.010338 PRX Quantum 5 (2024) 010338 , http://arxiv.org/abs/2211.03777 arXiv:2211.03777 [quant-ph]

2024 arXiv
[51]

Lootens, C

L. Lootens, C. Delcamp, D. Williamson, and F. Verstraete, Low-Depth Unitary Quantum Circuits for Dualities in One-Dimensional Quantum Lattice Models , http://dx.doi.org/10.1103/PhysRevLett.134.130403 Phys. Rev. Lett. 134 (2025) 130403 , http://arxiv.org/abs/2311.01439 arXiv:23...

2025 arXiv
[52]

Lootens, J

L. Lootens, J. Fuchs, J. Haegeman, C. Schweigert, and F. Verstraete, Matrix product operator symmetries and intertwiners in string-nets with domain walls, SciPost Physics 10 (2021) http://dx.doi.org/10.21468/SciPostPhys.10.3.053

2021 doi
[53]

Levin and Z.-C

M. Levin and Z.-C. Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86 (2012) 115109 https://link.aps.org/doi/10.1103/PhysRevB.86.115109

2012 doi
[54]

Y.-H. Lin, M. Okada, S. Seifnashri, and Y. Tachikawa, Asymptotic density of states in 2d CFTs with non-invertible symmetries , http://dx.doi.org/10.1007/JHEP03(2023)094 JHEP 03 (2023) 094 , http://arxiv.org/abs/2208.05495 arXiv:2208.05495 [hep-th]

2023 arXiv
[55]

L. Li, M. Oshikawa, and Y. Zheng, Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases , http://dx.doi.org/10.1103/PhysRevB.108.214429 Phys. Rev. B 108 (2023) 214429 , http://arxiv.org/abs/2301.07899 arXiv:2301.07...

2023 arXiv
[56]

L. Li, M. Oshikawa, and Y. Zheng, Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations , http://dx.doi.org/10.21468/SciPostPhys.18.5.153 SciPost Phys. 18 (2025) 153 , http://arxiv.org/abs/2307.04788 arXiv:2307.04788 [cond-mat.str-el]

2025 arXiv
[57]

L. Lin, D. G. Robbins, and E. Sharpe, Decomposition, Condensation Defects, and Fusion , http://dx.doi.org/10.1002/prop.202200130 Fortsch. Phys. 70 (2022) 2200130 , http://arxiv.org/abs/2208.05982 arXiv:2208.05982 [hep-th]

2022 arXiv
[58]

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 , http://dx.doi.org/10.1103/PhysRevB.104.075141 Phys. Rev. B 104 (2021) 075141 , http://arxiv.org/abs/2003.04328 arXiv:2003....

2021 arXiv
[59]

M. A. Levin and X.-G. Wen, String net condensation: A Physical mechanism for topological phases , http://dx.doi.org/10.1103/PhysRevB.71.045110 Phys. Rev. B 71 (2005) 045110 , http://arxiv.org/abs/cond-mat/0404617 arXiv:cond-mat/0404617

2005 arXiv
[60]

Moradi, S

H. Moradi, S. F. Moosavian, and A. Tiwari, Topological holography: Towards a unification of Landau and beyond-Landau physics , http://dx.doi.org/10.21468/SciPostPhysCore.6.4.066 SciPost Phys. Core 6 (2023) 066 , http://arxiv.org/abs/2207.10712 arXiv:2207.10712 [cond-mat.str-el]

2023 arXiv
[61]

Nikshych and B

D. Nikshych and B. Riepel, Categorical Lagrangian Grassmannians and Brauer–Picard groups of pointed fusion categories , Journal of Algebra 411 (2014) 191--214 https://www.sciencedirect.com/science/article/pii/S0021869314002373

2014
[62]

Ocneanu, Chirality for operator algebras, Subfactors (Kyuzeso, 1993) (1994) 39--63

A. Ocneanu, Chirality for operator algebras, Subfactors (Kyuzeso, 1993) (1994) 39--63

1994
[63]

Ocneanu, Operator algebras, topology and subgroups of quantum symmetry--construction of subgroups of quantum groups, Taniguchi Conference on Mathematics Nara, vol

A. Ocneanu, Operator algebras, topology and subgroups of quantum symmetry--construction of subgroups of quantum groups, Taniguchi Conference on Mathematics Nara, vol. 98, 2001, pp. 235--263

2001
[64]

M. Oshikawa, Hidden Z_2 Z_2 symmetry in quantum spin chains with arbitrary integer spin , Journal of Physics: Condensed Matter 4 (1992) 7469 https://dx.doi.org/10.1088/0953-8984/4/36/019

1992 doi
[65]

Ostrik, Module categories over the Drinfeld double of a finite group , http://arxiv.org/abs/math/0202130 arXiv:math/0202130 [math.QA]

V. Ostrik, Module categories over the Drinfeld double of a finite group , http://arxiv.org/abs/math/0202130 arXiv:math/0202130 [math.QA]

arXiv
[66]

Roumpedakis, S

K. Roumpedakis, S. Seifnashri, and S.-H. Shao, Higher Gauging and Non-invertible Condensation Defects , http://dx.doi.org/10.1007/s00220-023-04706-9 Commun. Math. Phys. 401 (2023) 3043--3107 , http://arxiv.org/abs/2204.02407 arXiv:2204.02407 [hep-th]

2023 arXiv
[67]

Schuch, I

N. Schuch, I. Cirac, and D. Pérez-García, PEPS as ground states: Degeneracy and topology , Annals of Physics 325 (2010) 2153--2192 https://www.sciencedirect.com/science/article/pii/S0003491610000990

2010
[68]

Seifnashri, Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries , http://dx.doi.org/10.21468/SciPostPhys.16.4.098 SciPost Phys

S. Seifnashri, Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries , http://dx.doi.org/10.21468/SciPostPhys.16.4.098 SciPost Phys. 16 (2024) 098 , http://arxiv.org/abs/2308.05151 arXiv:2308.05151 [cond-mat.str-el]

2024 arXiv
[69]

H. Song, A. Prem, S.-J. Huang, and M. A. Martin-Delgado, Twisted fracton models in three dimensions, Phys. Rev. B 99 (2019) 155118 https://link.aps.org/doi/10.1103/PhysRevB.99.155118

2019 doi
[70]

Scaffidi, D

T. Scaffidi, D. E. Parker, and R. Vasseur, Gapless Symmetry Protected Topological Order , http://dx.doi.org/10.1103/PhysRevX.7.041048 Phys. Rev. X 7 (2017) 041048 , http://arxiv.org/abs/1705.01557 arXiv:1705.01557 [cond-mat.str-el]

2017 arXiv
[71]

Seiberg, S

N. Seiberg, S. Seifnashri, and S.-H. Shao, Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space , http://dx.doi.org/10.21468/SciPostPhys.16.6.154 SciPost Phys. 16 (2024) 154 , http://arxiv.org/abs/2401.12281 arXiv:2401.12281 [cond-mat.str-el]

2024 arXiv
[72]

V. G. Turaev and O. Y. Viro, State sum invariants of 3 manifolds and quantum 6j symbols , http://dx.doi.org/10.1016/0040-9383(92)90015-A Topology 31 (1992) 865--902

1992 doi
[73]

Thorngren and Y

R. Thorngren and Y. Wang, Fusion category symmetry. Part I. Anomaly in-flow and gapped phases , http://dx.doi.org/10.1007/JHEP04(2024)132 JHEP 04 (2024) 132 , http://arxiv.org/abs/1912.02817 arXiv:1912.02817 [hep-th]

2024 arXiv
[74]

Tambara and S

D. Tambara and S. Yamagami, Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups , http://dx.doi.org/10.1006/jabr.1998.7558 J. Algebra 209 (1998) 692--707

1998
[75]

D. J. Williamson, N. Bultinck, and F. Verstraete, Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation , http://arxiv.org/abs/1711.07982 arXiv:1711.07982 [quant-ph]

arXiv
[76]

F. J. Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters , Journal of Mathematical Physics 12 (1971) 2259--2272 https://doi.org/10.1063/1.1665530

1971 doi
[77]

S. Willerton, The twisted Drinfeld double of a finite group via gerbes and finite groupoids , Algebraic & Geometric Topology 8 (2005) 1419--1457 https://api.semanticscholar.org/CorpusID:13997127

2005
[78]

C. Zhu, T. Lan, and X.-G. Wen, Topological nonlinear -model, higher gauge theory, and a systematic construction of 3+1 D topological orders for boson systems , Phys. Rev. B 100 (2019) 045105 https://link.aps.org/doi/10.1103/PhysRevB.100.045105

2019 doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION output.nonempty.mrnumber duplicate missing pop "" 'skip if duplicate empty 'pop " " swap * " " * write if FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'outpu...

[2] [2]

Apruzzi, F

F. Apruzzi, F. Bonetti, I. n. Garc\' a Etxebarria, S. S. Hosseini, and S. Schafer-Nameki, Symmetry TFTs from String Theory , http://dx.doi.org/10.1007/s00220-023-04737-2 Commun. Math. Phys. 402 (2023) 895--949 , http://arxiv.org/abs/2112.02092 arXiv:2112.02092 [hep-th]

arXiv 2023

[3] [3]

Aasen, P

D. Aasen, P. Fendley, and R. S. K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies , http://arxiv.org/abs/2008.08598 arXiv:2008.08598 [cond-mat.stat-mech]

arXiv 2008

[4] [4]

Buerschaper, M

O. Buerschaper, M. Aguado, and G. Vidal, Explicit tensor network representation for the ground states of string-net models, Phys. Rev. B 79 (2009) 085119 https://link.aps.org/doi/10.1103/PhysRevB.79.085119

doi:10.1103/physrevb.79.085119 2009

[5] [5]

Bartsch, M

T. Bartsch, M. Bullimore, A. E. V. Ferrari, and J. Pearson, Non-invertible symmetries and higher representation theory I , http://dx.doi.org/10.21468/SciPostPhys.17.1.015 SciPost Phys. 17 (2024) 015 , http://arxiv.org/abs/2208.05993 arXiv:2208.05993 [hep-th]

arXiv 2024

[6] [6]

Bhardwaj, L

L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, Non-invertible higher-categorical symmetries , http://dx.doi.org/10.21468/SciPostPhys.14.1.007 SciPost Phys. 14 (2023) 007 , http://arxiv.org/abs/2204.06564 arXiv:2204.06564 [hep-th]

arXiv 2023

[7] [7]

Bhardwaj, L

L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, Lattice Models for Phases and Transitions with Non-Invertible Symmetries , http://arxiv.org/abs/2405.05964 arXiv:2405.05964 [cond-mat.str-el]

arXiv

[8] [8]

Benini, C

F. Benini, C. C\'ordova, and P.-S. Hsin, On 2-Group Global Symmetries and their Anomalies , http://dx.doi.org/10.1007/JHEP03(2019)118 JHEP 03 (2019) 118 , http://arxiv.org/abs/1803.09336 arXiv:1803.09336 [hep-th]

arXiv 2019