Pith. sign in

REVIEW 3 major objections 5 minor 123 references

Entanglement Spectrum Resolved by Loop Symmetries

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read For any state symmetric under non-invertible loop symmetries, the entanglement spectrum decomposes into a topological block structure determined by the manifold's fundamental groupoid.

desk verdict The central block-decomposition theorem is sound and genuinely new, but the advertised Li–Haldane 'full entanglement spectrum' proof only fixes degeneracies, not eigenvalues. read the letter →

arxiv 2510.18350 v1 pith:UMLSPXPZ submitted 2025-10-21 quant-ph cond-mat.stat-mechcond-mat.str-elhep-thmath-phmath.MP

Haruki Yagi , Zongping Gong This is my paper
classification quant-phcond-mat.stat-mechcond-mat.str-elhep-thmath-phmath.MP MSC 81P4281T4520C15
keywords entanglementspectrumloopsymmetryhigher-formnon-invertibleSeifert–vanKampentheoremfundamentalgroupoidtopologicalentropyquantumdoublemodel
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

Quantum states invariant under the group-representation loop symmetry generated by Wilson loops — a higher-form, generally non-invertible symmetry — are shown to have a bipartite entanglement structure that is completely fixed by topology. The paper proves that the matrix encoding the entanglement spectrum decomposes into blocks labeled by holonomies on the boundary between the two subsystems, with each block a tensor product of a topological part (determined by homomorphisms from the fundamental groupoid into the symmetry group) and a geometric part counting gauge degrees. An explicit algorithm is given to compute this block structure for any regular bipartition of any manifold, and the paper works out the spectra for tori, Klein bottles, lens spaces, and higher-dimensional tori. Imposing gauge invariance turns the same machinery into an exact description of entanglement in topological gauge theories, reproducing the topological entanglement entropy and confirming the conjectured correspondence between the full entanglement spectrum and a rational conformal field theory in two dimensions.

What carries the argument

The load-bearing objects are the fundamental groupoid π_1(M, A) of the manifold with base points on the bipartition boundary, and the contravariant functor Hom(·, G) that maps groupoids to sets of group homomorphisms. The Seifert–van Kampen theorem converts the gluing X ∪_∂ Y into a pushout of groupoids; Hom(·, G) then converts this pushout into a pullback of sets, whose elements are exactly the holonomy assignments compatible across the boundary. The sizes of the fibers |r_X^{-1}(φ)| and |r_Y^{-1}(φ)| set the dimensions of the topological blocks, while the gauge transformations on vertices away from the base points produce the geometric multiplicity. In the gauge-invariant refinement, the p

What would settle it

Take G = S_3 and a 4×4 lattice discretizing a torus bipartitioned into two cylinders. Construct an explicit flat (loop-symmetric) state by gauge-transforming a representative holonomy configuration, compute the bipartite matrix W numerically, and compare its singular value degeneracies with the predicted block dimensions (|C_c| times powers of |G| for each conjugacy class c). Any block of the wrong size, or a missing/extra block, would refute the claim.

Watch

Extended reading notes

Core claim

The central result is a block decomposition theorem for the matrix W whose singular values give the entanglement spectrum. For a Rep(G) loop-symmetric state on a bipartite manifold M = X ∪ Y with common boundary ∂, W decomposes as a direct sum over compatible boundary holonomies φ of |G|^{|V∂|-|A|} copies of a topological block of dimension |r_X^{-1}(φ)||G|^{|V_X|} × |r_Y^{-1}(φ)||G|^{|V_Y|}. Here r_X and r_Y are the restriction maps from the subsystem fundamental groupoid homomorphisms to the boundary, and |A| is the number of base points, one per boundary component. The derivation shows loop-symmetric states are exactly flat connections, parameterized by gauge transformations modulo non-Ab

Load-bearing premise

The block-counting formula assumes the bipartition is regular — the boundary between subsystems is a manifold and the lattice is a good discretization whose fundamental groupoid is homotopy-equivalent to the continuum manifold; if the cut is irregular or too coarse, the formula may break.

Editorial extensions

If this is right

  • The full entanglement spectrum, not just entropy, of any loop-symmetric state on a regular bipartition is exactly determined by this block structure.
  • Gauge-invariant loop-symmetric states in any dimension have reduced density matrices whose degeneracies are governed by the representation-theoretic data of the group (conjugacy classes and centralizer representations), and their topological entanglement entropy is the base-point count minus the logarithm of the corresponding quantum dimension.
  • In two dimensions, the block multiplicities coincide with the boundary Hilbert-space dimensions of the rational conformal field theory, proving the full-spectrum correspondence for the quantum double model.
  • Non-orientable manifolds are covered, with the Frobenius–Schur indicator determining which topological sectors contribute.
  • The algorithm reduces entanglement spectrum computation to counting group homomorphisms and stabilizer orbits, and applies to arbitrary dimensions.

Reading between the lines

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

  • The same groupoid-pullback strategy may extend to higher-form symmetries via higher homotopy groupoids, potentially yielding analogous block structures for higher-form gauge theories.
  • Because the block structure is topology-only (parameter-free), it offers a benchmark for numerical tensor-network simulations of topological order, independent of microscopic details.
  • The rigid sector structure suggests that in systems with higher-form symmetries, the entanglement spectrum has protected degeneracies that may survive at finite energy density, possibly constraining thermalization.
  • The framework could be adapted to compute symmetry-resolved entanglement for non-invertible symmetries in experimental platforms that realize loop-symmetric states.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a general algebraic-topological framework for the bipartite entanglement structure of states with a Rep(G) loop symmetry, i.e., states whose contractible Wilson-loop holonomies are trivial. The central result, Eqs. (11)–(12), is an exact block decomposition of the bipartite matrix W into a topological part, obtained from the fiber product of Hom(π1(X,A),G) and Hom(π1(Y,A),G) over Hom(π1(∂,A),G), tensored with a geometric gauge-transform part of size |G|^{|V∂|−|A|}. This is illustrated for tori, genus-γ surfaces, Klein bottles, general non-orientable surfaces, Heegaard splittings of lens spaces, and higher-dimensional tori. After imposing gauge invariance, the paper claims to reproduce the topological entanglement entropy and to verify the Li–Haldane conjecture for the full entanglement spectrum of the Kitaev quantum double model. The algebraic-topology core is carefully derived, with supporting material in the appendices, and the dimension-counting checks against known ground-state degeneracies are convincing. The main problem is that the strongest advertised corollary, the exact Li–Haldane correspondence for the full entanglement spectrum, is not established: the computation fixes only block multiplicities, not the eigenvalue spectrum.

Significance. If the central block-decomposition theorem, Eq. (11), is taken as the paper's main contribution, this is a significant and largely rigorous result. It provides an exact, algorithmic description of how non-invertible loop symmetries constrain reduced density matrices on arbitrary bipartite manifolds, including non-orientable manifolds and higher dimensions. The gauge-invariant refinement correctly recovers known topological entanglement entropies and ground-state degeneracies, and the appearance of the Drinfel'd-double modular S-matrix in Eqs. (51)–(53) is a valuable structural insight. The appendices are unusually careful: Appendix B gives a torsor proof of the geometric-block structure, and Appendix E explains the non-Abelian cohomology dictionary. However, the paper overstates its result in the abstract and in §V.C: the Li–Haldane correspondence for the full entanglement spectrum is not proved. What is proved is a correspondence at the level of block degeneracies and multiplicities, which is an important but strictly weaker statement.

major comments (3)
  1. [Abstract and §V.C] The abstract claims that for the Kitaev quantum double model 'the Li–Haldane conjecture concerning the full entanglement spectrum holds exactly.' This is not supported by the proof. In §V.C, Eqs. (51) and (53) determine only the multiplicities x_{α_φ} and y_{α_φ}, i.e., the sizes of the blocks C^{x_{α_φ}×y_{α_φ}} in Eq. (44). The actual Schmidt spectrum is the singular-value spectrum of the unconstrained matrices Ψ_{α_φ} in Eq. (G2). Those matrices are arbitrary inputs in the general ground-state decomposition; the proof never fixes them. For a generic superposition of two topological sectors with weights a and b, the Schmidt eigenvalues vary continuously with |a|² and |b|². Thus the paper establishes a degeneracy-level, not eigenvalue-level, correspondence.
  2. [Eqs. (47)–(49) and (G2)] The state-dependence of the full spectrum is made explicit by the paper itself. Eq. (49) gives the topological-entanglement-entropy correction as a function of |ψ_{α_φ,i}|², the singular values of Ψ_{α_φ}. Equation (G2) leaves Ψ_{α_φ} completely free subject only to normalization. Therefore the set of possible reduced-density-matrix spectra for gauge-invariant loop-symmetric states is a continuum. A fixed RCFT modular datum predicts a fixed spectrum (or at least fixed degeneracy structure), not a continuum of eigenvalue magnitudes. The proof can at most claim that the degeneracy pattern of the entanglement spectrum matches the anyon-content/RCFT fusion data.
  3. [§V.C, Eq. (51)] The identification of Eq. (51) with the generalized Verlinde formula is a statement about dimensions of spaces of intertwiners (anyon Hilbert-space dimensions on a genus-γ surface), not about the list of Schmidt eigenvalues. Even in the minimally entangled case, the reduced density matrix is proportional to the identity on its support (flat spectrum), so the 'full spectrum' contains no energy-scale information beyond the overall degeneracy. This is far weaker than the usual Li–Haldane correspondence, which concerns a nontrivial tower of low-lying entanglement levels. The manuscript should be revised to state clearly that only the degeneracy/multiplicity part of the Li–Haldane correspondence is derived.
minor comments (5)
  1. [Eq. (12)] The notation C^{m×n} is used for the vector space of m×n matrices, but the reader may initially confuse it with a dimension. Consider writing Mat_{m×n}(C) or adding an explicit sentence.
  2. [Fig. 3 caption] The caption refers to 'the blue square' in a way that is hard to parse. The figure would be clearer if the block-support region were labeled explicitly and the direct-sum decomposition stated in the caption.
  3. [Eq. (18), general orientable case] The direct sum over c∈G^{×n} is not restricted to the image Im until after the formula is introduced. It would help to state immediately that terms with R_{γ_X,n}(c)=0 or R_{γ_Y,n}(c)=0 are absent, as is done later.
  4. [Appendix D.8] The symbol 'holn' appears in the text without definition. Clarify that it denotes the holonomy along the n-th circle direction.
  5. [Throughout] There are minor typographical issues, e.g., 'degneracy' in the summary section and inconsistent use of 'γ' vs 'Γ' for the TEE. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the block structure follows from the defining loop symmetry via external algebraic-topology theorems; the Li–Haldane full-spectrum claim overreaches but is not a circular reduction.

full rationale

The derivation chain is self-contained and non-circular. Loop-symmetric states are defined by Eq. (3); Appendix A proves their equivalence to flat configurations using Schur orthogonality, after which Appendix B establishes that configurations with fixed topological holonomy form a torsor under gauge transformations on V\A. The main block decomposition (Eqs. 11–12) is obtained by applying the Seifert–van Kampen theorem and the contravariant Hom(·,G) functor to the bipartition diagram (Eqs. 8–10), reproducing the standard pullback of Hom-sets. These are external results in algebraic topology and group representation theory, not assumptions of the conclusion. The gauge-invariant refinement (Eqs. 39–44) uses Schur's lemma and orbit–stabilizer counting, with the total degrees of freedom verified independently against |Hom(π1(M),G)| counts (Eqs. 20, 29, 32, I9). No fitted parameter is relabeled as a prediction, and no load-bearing self-citation appears; the cited S-matrix, Verlinde, and Drinfel'd-double data (Refs. 84–91, 65) are external benchmarks. The main legitimate concern is that §V.C's claim of verifying the Li–Haldane correspondence for the full entanglement spectrum overreaches: Eqs. (51)/(53) determine only block dimensions x_αφ, y_αφ, while the actual Schmidt eigenvalues depend on the arbitrary matrices Ψ_αφ in Eq. (G2). That is a scope/correctness gap, however, not circularity, because the spectrum was not used as an input to derive the block structure.

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

The derivation is essentially self-contained given standard algebraic topology and finite-group representation theory; no parameters are fitted and no new physical entities are introduced. The main domain inputs are the loop-symmetry/flatness condition and the regularity of the bipartition.

assumptions (6)
  • standard math Seifert–van Kampen theorem for fundamental groupoids with multiple base points
    Used in Sec. III to turn the manifold bipartition diagram (8) into the pushout diagram (9) of fundamental groupoids.
  • standard math Nerve theorem / good discretization: a smooth manifold can be replaced by a homotopy-equivalent simplicial complex with contractible intersections
    Appendix B1; needed so the lattice discretization preserves π1 and non-Abelian cohomology data.
  • standard math Non-Abelian H^1(M;G) ≅ Hom(π1(M),G)/G and Hom(π1(−,A),G) sends pushouts to pullbacks
    Appendix E and Sec. III Eq. (10); central to identifying topological degrees of freedom.
  • standard math Finite-group character orthogonality, regular-representation trick, Peter–Weyl, Frobenius–Schur indicators, and unitarity of the modular S-matrix
    Used throughout Secs. IV–V and Appendices D, H, I to evaluate block sizes and degeneracies.
  • domain assumption The physical Hilbert space is a tensor product of G-spins on edges, and loop symmetry Eq. (3) imposes flatness on every contractible Wilson loop
    This defines the class of states analyzed; equivalence with flatness is proved in Appendix A, but the starting point is a model assumption rather than a derived Hamiltonian result.
  • domain assumption Bipartition is regular: X and Y are complementary submanifolds with common boundary ∂, with at least one base point per boundary component
    Stated in Sec. II; the Seifert–van Kampen pullback diagram (10) and the counting |G|^{|V∂|−|A|} depend on it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Entanglement Spectrum Resolved by Loop Symmetries." pith.science (2026). https://pith.science/paper/UMLSPXPZ

@misc{pith2026251018350,
  author       = {Pith},
  title        = {Pith review of: Entanglement Spectrum Resolved by Loop Symmetries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMLSPXPZ}},
  note         = {Machine review of arXiv:2510.18350}
}
read the original abstract

A rigorous analysis is presented for the entanglement spectrum of quantum many-body states possessing a higher-form group-representation symmetry generated by topological Wilson loops, which is generally non-invertible. A general framework based on elementary algebraic topology and category theory is developed to determine the block structure of reduced density matrices for arbitrary bipartite manifolds on which the states are defined. Within this framework, we scrutinize the impact of topology on the entanglement structure for low-dimensional manifolds, including especially the torus, the Klein bottle, and lens spaces. By further incorporating gauge invariance, we refine our framework to determine the entanglement structure for topological gauge theories in arbitrary dimensions. In particular, in two dimensions, it is shown for the Kitaev quantum double model that not only the topological entanglement entropy can be reproduced, but also the Li-Haldane conjecture concerning the full entanglement spectrum holds exactly.

Figures

Figures reproduced from arXiv: 2510.18350 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Procedure for defining a lattice and its biparti [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Torus has essentially two topologically different [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Nonzero elements in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Constructing the Klein bottle from a square. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Heegaard splitting of lens space is achieved by gluing [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

123 extracted references · 23 canonical work pages

  1. [1]

    Entanglement in many-body systems

    Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys., 80: 0 517--576, May 2008. doi:10.1103/RevModPhys.80.517. URL https://link.aps.org/doi/10.1103/RevModPhys.80.517

    doi:10.1103/revmodphys.80.517 2008

  2. [2]

    Quantum entanglement in condensed matter systems

    Nicolas Laflorencie. Quantum entanglement in condensed matter systems. Physics Reports, 646: 0 1--59, 2016. ISSN 0370-1573. doi:https://doi.org/10.1016/j.physrep.2016.06.008. URL https://www.sciencedirect.com/science/article/pii/S0370157316301582. Quantum entanglement in condensed matter systems

    doi:10.1016/j.physrep.2016.06.008 2016

  3. [3]

    D. Harlow. Jerusalem lectures on black holes and quantum information. Rev. Mod. Phys., 88: 0 015002, Feb 2016. doi:10.1103/RevModPhys.88.015002. URL https://link.aps.org/doi/10.1103/RevModPhys.88.015002

    doi:10.1103/revmodphys.88.015002 2016

  4. [4]

    Entanglement entropy: Holography and renormalization group

    Tatsuma Nishioka. Entanglement entropy: Holography and renormalization group. Reviews of Modern Physics, 90 0 (3): 0 035007, September 2018. ISSN 0034-6861, 1539-0756. doi:10.1103/RevModPhys.90.035007

    doi:10.1103/revmodphys.90.035007 2018

  5. [5]

    Aps medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory

    Edward Witten. Aps medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory. Rev. Mod. Phys., 90: 0 045003, Oct 2018. doi:10.1103/RevModPhys.90.045003. URL https://link.aps.org/doi/10.1103/RevModPhys.90.045003

    doi:10.1103/revmodphys.90.045003 2018

  6. [6]

    Lectures on entanglement in quantum field theory, March 2023

    Horacio Casini and Marina Huerta. Lectures on entanglement in quantum field theory, March 2023

    2023

  7. [7]

    Holographic Derivation of Entanglement Entropy from AdS / CFT

    Shinsei Ryu and Tadashi Takayanagi. Holographic Derivation of Entanglement Entropy from AdS / CFT . Physical Review Letters, 96 0 (18): 0 181602, May 2006. ISSN 0031-9007, 1079-7114. doi:10.1103/PhysRevLett.96.181602

    doi:10.1103/physrevlett.96.181602 2006

  8. [8]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys., 82: 0 277--306, Feb 2010. doi:10.1103/RevModPhys.82.277. URL https://link.aps.org/doi/10.1103/RevModPhys.82.277

    doi:10.1103/revmodphys.82.277 2010

Show all 123 references
  1. [9]

    Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn

    Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. Colloquium: Many-body localization, thermalization, and entanglement. Rev. Mod. Phys., 91: 0 021001, May 2019. doi:10.1103/RevModPhys.91.021001. URL https://link.aps.org/doi/10.1103/RevModPhys.91.021001

    2019 doi

  2. [10]

    Holzhey, F

    C. Holzhey, F. Larsen, and F. Wilczek. Geometric and Renormalized Entropy in Conformal Field Theory . Nuclear Physics B, 424 0 (3): 0 443--467, August 1994. ISSN 05503213. doi:10.1016/0550-3213(94)90402-2

    1994 doi

  3. [11]

    Vidal, J

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev. Entanglement in quantum critical phenomena. Phys. Rev. Lett., 90: 0 227902, Jun 2003. doi:10.1103/PhysRevLett.90.227902. URL https://link.aps.org/doi/10.1103/PhysRevLett.90.227902

    2003 doi

  4. [12]

    Entanglement entropy and quantum field theory

    Pasquale Calabrese and John Cardy. Entanglement entropy and quantum field theory. Journal of Statistical Mechanics: Theory and Experiment, 2004 0 (06): 0 P06002, June 2004. ISSN 1742-5468. doi:10.1088/1742-5468/2004/06/P06002

    2004 doi

  5. [13]

    X. G. WEN. Topological orders in rigid states. International Journal of Modern Physics B, 04 0 (02): 0 239--271, 1990. doi:10.1142/S0217979290000139. URL https://doi.org/10.1142/S0217979290000139

    1990 doi

  6. [14]

    Kim, Michael Levin, Ting-Chun Lin, Daniel Ranard, and Bowen Shi

    Isaac H. Kim, Michael Levin, Ting-Chun Lin, Daniel Ranard, and Bowen Shi. Universal lower bound on topological entanglement entropy. Phys. Rev. Lett., 131: 0 166601, Oct 2023. doi:10.1103/PhysRevLett.131.166601. URL https://link.aps.org/doi/10.1103/PhysRevLett.131.166601

    2023 doi

  7. [15]

    Topological Entanglement Entropy

    Alexei Kitaev and John Preskill. Topological Entanglement Entropy . Physical Review Letters, 96 0 (11): 0 110404, March 2006. doi:10.1103/PhysRevLett.96.110404

    2006 doi

  8. [16]

    Detecting Topological Order in a Ground State Wave Function

    Michael Levin and Xiao-Gang Wen. Detecting Topological Order in a Ground State Wave Function . Physical Review Letters, 96 0 (11): 0 110405, March 2006. doi:10.1103/PhysRevLett.96.110405

    2006 doi

  9. [17]

    Colloquium: Zoo of quantum-topological phases of matter

    Xiao-Gang Wen. Colloquium: Zoo of quantum-topological phases of matter. Rev. Mod. Phys., 89: 0 041004, Dec 2017. doi:10.1103/RevModPhys.89.041004. URL https://link.aps.org/doi/10.1103/RevModPhys.89.041004

    2017 doi

  10. [18]

    Hui Li and F. D. M. Haldane. Entanglement Spectrum as a Generalization of Entanglement Entropy : Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States . Physical Review Letters, 101 0 (1): 0 010504, July 2008. ISSN 0031-9007, 1079-7114. doi:1...

    2008 doi

  11. [19]

    Turner, Erez Berg, and Masaki Oshikawa

    Frank Pollmann, Ari M. Turner, Erez Berg, and Masaki Oshikawa. Entanglement spectrum of a topological phase in one dimension. Physical Review B, 81 0 (6): 0 064439, February 2010. doi:10.1103/PhysRevB.81.064439

    2010 doi

  12. [20]

    Geraedts, Rahul Nandkishore, and Nicolas Regnault

    Scott D. Geraedts, Rahul Nandkishore, and Nicolas Regnault. Many-body localization and thermalization: Insights from the entanglement spectrum. Physical Review B, 93 0 (17): 0 174202, May 2016. ISSN 2469-9950, 2469-9969. doi:10.1103/PhysRevB.93.174202

    2016 doi

  13. [21]

    Eugene P. Wigner. Symmetry and conservation laws. Proceedings of the National Academy of Sciences, 51 0 (5): 0 956--965, 1964. doi:10.1073/pnas.51.5.956. URL https://www.pnas.org/doi/abs/10.1073/pnas.51.5.956

    1964 doi

  14. [22]

    David J. Gross. The role of symmetry in fundamental physics. Proceedings of the National Academy of Sciences, 93 0 (25): 0 14256--14259, 1996. doi:10.1073/pnas.93.25.14256. URL https://www.pnas.org/doi/abs/10.1073/pnas.93.25.14256

    1996 doi

  15. [23]

    The Quantum Theory of Fields : Volume 1: Foundations , volume 1

    Steven Weinberg. The Quantum Theory of Fields : Volume 1: Foundations , volume 1. Cambridge University Press, Cambridge, 1995. ISBN 978-0-521-67053-1. doi:10.1017/CBO9781139644167

    1995 doi

  16. [24]

    The Quantum Theory of Fields : Volume 2: Modern Applications , volume 2

    Steven Weinberg. The Quantum Theory of Fields : Volume 2: Modern Applications , volume 2. Cambridge University Press, Cambridge, 1996. ISBN 978-0-521-55002-4. doi:10.1017/CBO9781139644174

    1996 doi

  17. [25]

    Group Theory and its Application to Physical Problems

    Morton Hamermesh. Group Theory and its Application to Physical Problems. Addison-Wesley, Reading, MA, 1964

    1964

  18. [26]

    Generalized global symmetries

    Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett. Generalized global symmetries. Journal of High Energy Physics, 2015 0 (2): 0 172, February 2015. ISSN 1029-8479. doi:10.1007/JHEP02(2015)172

    2015 doi

  19. [27]

    Bottini, Ludovic Fraser-Taliente , Liam Gladden, Dewi S

    Lakshya Bhardwaj, Lea E. Bottini, Ludovic Fraser-Taliente , Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. Lectures on Generalized Symmetries , September 2023 a

    2023

  20. [28]

    ICTP Lectures on ( Non- ) Invertible Generalized Symmetries , June 2023

    Sakura Schafer-Nameki . ICTP Lectures on ( Non- ) Invertible Generalized Symmetries , June 2023

    2023

  21. [29]

    What's Done Cannot Be Undone : TASI Lectures on Non-Invertible Symmetries , March 2024

    Shu-Heng Shao. What's Done Cannot Be Undone : TASI Lectures on Non-Invertible Symmetries , March 2024

    2024

  22. [30]

    Generalized Symmetries in Condensed Matter

    John McGreevy. Generalized Symmetries in Condensed Matter . Annual Review of Condensed Matter Physics, 14 0 (1): 0 57--82, March 2023. ISSN 1947-5454, 1947-5462. doi:10.1146/annurev-conmatphys-040721-021029

    2023 doi

  23. [31]

    Coupling a QFT to a TQFT and duality

    Anton Kapustin and Nathan Seiberg. Coupling a QFT to a TQFT and duality. Journal of High Energy Physics, 2014 0 (4): 0 1, April 2014. ISSN 1029-8479. doi:10.1007/JHEP04(2014)001

    2014 doi

  24. [32]

    Higher SPT 's and a generalization of anomaly in-flow, November 2015

    Ryan Thorngren and Curt von Keyserlingk. Higher SPT 's and a generalization of anomaly in-flow, November 2015

    2015

  25. [33]

    Unifying Constructions of Non-Invertible Symmetries

    Lakshya Bhardwaj, Sakura Schafer-Nameki , and Apoorv Tiwari. Unifying Constructions of Non-Invertible Symmetries . SciPost Physics, 15 0 (3): 0 122, September 2023 b . ISSN 2542-4653. doi:10.21468/SciPostPhys.15.3.122

    2023 doi

  26. [34]

    Physics of Symmetry Protected Topological phases involving Higher Symmetries and their Applications

    Chao-Ming Jian, Xiao-Chuan Wu, Yichen Xu, and Cenke Xu. Physics of Symmetry Protected Topological phases involving Higher Symmetries and their Applications . Physical Review B, 103 0 (6): 0 064426, February 2021. ISSN 2469-9950, 2469-9969. doi:10.1103/PhysRevB.103.064426

    2021 doi

  27. [35]

    Fracton phases via exotic higher-form symmetry-breaking

    Marvin Qi, Leo Radzihovsky, and Michael Hermele. Fracton phases via exotic higher-form symmetry-breaking. Annals of Physics, 424: 0 168360, January 2021. ISSN 00034916. doi:10.1016/j.aop.2020.168360

    2021

  28. [36]

    On lattice models of gapped phases with fusion category symmetries

    Kansei Inamura. On lattice models of gapped phases with fusion category symmetries. Journal of High Energy Physics, 2022 0 (03): 0 036, March 2022. doi:10.1007/JHEP03(2022)036. URL https://doi.org/10.1007/JHEP03(2022)036

    2022 doi

  29. [37]

    Pace and Xiao-Gang Wen

    Salvatore D. Pace and Xiao-Gang Wen. Exact emergent higher-form symmetries in bosonic lattice models. Physical Review B, 108 0 (19): 0 195147, November 2023. ISSN 2469-9950, 2469-9969. doi:10.1103/PhysRevB.108.195147

    2023 doi

  30. [38]

    Anomalies of (1+1) -dimensional categorical symmetries

    Carolyn Zhang and Clay C\'ordova. Anomalies of (1+1) -dimensional categorical symmetries. Phys. Rev. B, 110: 0 035155, Jul 2024. doi:10.1103/PhysRevB.110.035155. URL https://link.aps.org/doi/10.1103/PhysRevB.110.035155

    2024 doi

  31. [39]

    Cluster state as a non-invertible symmetry protected topological phase, September 2024

    Sahand Seifnashri and Shu-Heng Shao. Cluster state as a non-invertible symmetry protected topological phase, September 2024

    2024

  32. [40]

    Non-invertible and higher-form symmetries in 2+1d lattice gauge theories

    Yichul Choi, Yaman Sanghavi, Shu-Heng Shao, and Yunqin Zheng. Non-invertible and higher-form symmetries in 2+1d lattice gauge theories . SciPost Phys., 18: 0 008, 2025. doi:10.21468/SciPostPhys.18.1.008. URL https://scipost.org/10.21468/SciPostPhys.18.1.008

    2025 doi

  33. [41]

    Topological defects on the lattice: I

    David Aasen, Roger S K Mong, and Paul Fendley. Topological defects on the lattice: I . The Ising model. Journal of Physics A: Mathematical and Theoretical, 49 0 (35): 0 354001, August 2016. ISSN 1751-8121. doi:10.1088/1751-8113/49/35/354001

    2016 doi

  34. [42]

    Higher-form symmetries and spontaneous symmetry breaking, February 2018

    Ethan Lake. Higher-form symmetries and spontaneous symmetry breaking, February 2018

    2018

  35. [43]

    Group theory: and its application to the quantum mechanics of atomic spectra

    Eugene Wigner. Group theory: and its application to the quantum mechanics of atomic spectra. Academic Press, New York, 1959

    1959

  36. [44]

    Freeman J. Dyson. The Threefold Way . Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics . Journal of Mathematical Physics, 3 0 (6): 0 1199--1215, November 1962 a . ISSN 0022-2488. doi:10.1063/1.1703863

    1962 doi

  37. [45]

    Symmetry-resolved entanglement in many-body systems

    Moshe Goldstein and Eran Sela. Symmetry-resolved entanglement in many-body systems. Phys. Rev. Lett., 120: 0 200602, May 2018. doi:10.1103/PhysRevLett.120.200602. URL https://link.aps.org/doi/10.1103/PhysRevLett.120.200602

    2018 doi

  38. [46]

    Symmetry resolved entanglement in free fermionic systems

    Riccarda Bonsignori, Paola Ruggiero, and Pasquale Calabrese. Symmetry resolved entanglement in free fermionic systems. J. Phys. A: Math. Theor., 52 0 (47): 0 475302, oct 2019. doi:10.1088/1751-8121/ab4b77. URL https://doi.org/10.1088/1751-8121/ab4b77

    2019 doi

  39. [47]

    Symmetry-resolved entanglement detection using partial transpose moments

    Antoine Neven, Jose Carrasco, Vittorio Vitale, Christian Kokail, Andreas Elben, Marcello Dalmonte, Pasquale Calabrese, Peter Zoller, Beno\^it Vermersch, Richard Kueng, and Barbara Kraus. Symmetry-resolved entanglement detection using partial transpose moments. npj Quantum Info...

    2021 doi

  40. [48]

    Threefold Way for Typical Entanglement

    Haruki Yagi, Ken Mochizuki, and Zongping Gong. Threefold Way for Typical Entanglement . Physical Review Letters, 134 0 (15): 0 150401, April 2025. doi:10.1103/PhysRevLett.134.150401

    2025 doi

  41. [49]

    Anyonic entanglement and topological entanglement entropy

    Parsa Bonderson, Christina Knapp, and Kaushal Patel. Anyonic entanglement and topological entanglement entropy. Annals of Physics, 385: 0 399--468, October 2017. ISSN 00034916. doi:10.1016/j.aop.2017.07.018

    2017 doi

  42. [50]

    Saura-Bastida , A

    P. Saura-Bastida , A. Das, G. Sierra, and J. Molina-Vilaplana . Categorical-symmetry resolved entanglement in conformal field theory. Physical Review D, 109 0 (10): 0 105026, May 2024. doi:10.1103/PhysRevD.109.105026

    2024 doi

  43. [51]

    Hamiltonian formulation of Wilson 's lattice gauge theories

    John Kogut and Leonard Susskind. Hamiltonian formulation of Wilson 's lattice gauge theories. Physical Review D, 11 0 (2): 0 395--408, January 1975. ISSN 0556-2821. doi:10.1103/PhysRevD.11.395

    1975 doi

  44. [52]

    This is a higher-form symmetry what is called (d-2)-form symmetry in the field theory community under the convention of considering spacetime as d-dimensional

    Note1. This is a higher-form symmetry what is called (d-2)-form symmetry in the field theory community under the convention of considering spacetime as d-dimensional. However, we determined that the emphasis should be on the fact that the symmetry operator is defined on a clos...

  45. [53]

    This setup is reminiscent of lattice gauge theory

    Note2. This setup is reminiscent of lattice gauge theory. However, we do not impose gauge invariance. In contrast, we require that the Wilson loops are topological. The theory can be interpreted as a topological gauge theory (in the Kogut-Susskind formalism) without gauge invariance

  46. [54]

    Our framework is directly applicable to open manifolds as well, but this paper focuses on the case of closed manifolds

    Note3. Our framework is directly applicable to open manifolds as well, but this paper focuses on the case of closed manifolds. Therefore, it should not cause confusion to interpret as the common boundary of X and Y obtained as a partition of M . However, when a manifold follow...

  47. [55]

    The proof of this fact is also provided in the Supplemental Materials

    Note4. The proof of this fact is also provided in the Supplemental Materials

  48. [56]

    Entanglement Properties of Gauge Theories from Higher-Form Symmetries

    Wen-Tao Xu, Tibor Rakovszky, Michael Knap, and Frank Pollmann. Entanglement Properties of Gauge Theories from Higher-Form Symmetries . Physical Review X, 15 0 (1): 0 011001, January 2025. ISSN 2160-3308. doi:10.1103/PhysRevX.15.011001

    2025 doi

  49. [57]

    Violation of eigenstate thermalization hypothesis in quantum field theories with higher-form symmetry

    Osamu Fukushima and Ryusuke Hamazaki. Violation of eigenstate thermalization hypothesis in quantum field theories with higher-form symmetry. Phys. Rev. Lett., 131: 0 131602, Sep 2023. doi:10.1103/PhysRevLett.131.131602. URL https://link.aps.org/doi/10.1103/PhysRevLett.131.131602

    2023 doi

  50. [58]

    We do not demonstrate the case of S^1 as it does not allow discretization with plaquettes, implying no symmetry is actually imposed

    Note5. We do not demonstrate the case of S^1 as it does not allow discretization with plaquettes, implying no symmetry is actually imposed. Interestingly, even this no-symmetry result can be reproduced in our framework, as we demonstrate in Appendix app:S1

  51. [59]

    Bridson and Henry Wilton

    Martin R. Bridson and Henry Wilton. The triviality problem for profinite completions. Invent. Math., 202: 0 839, Nov 2015. doi:10.1007/s00222-015-0578-8. URL https://doi.org/10.1007/s00222-015-0578-8

    2015 doi

  52. [60]

    A Guide to the Classification Theorem for Compact Surfaces

    Jean Gallier and Dianna Xu. A Guide to the Classification Theorem for Compact Surfaces . Geometry and Computing . Springer, Berlin, Heidelberg, 2013. ISBN 978-3-642-34363-6 978-3-642-34364-3. doi:10.1007/978-3-642-34364-3

    2013 doi

  53. [61]

    Francis and Jeffrey R

    George K. Francis and Jeffrey R. Weeks. Conway's ZIP Proof . The American Mathematical Monthly, 106 0 (5): 0 393--399, May 1999. ISSN 0002-9890. doi:10.1080/00029890.1999.12005061

    1999

  54. [62]

    http://www.w3.org/1998/math/mathml

    Yuji Tachikawa and Kazuya Yonekura. Derivation of the time-reversal anomaly for ( <mml:math xmlns:mml="http://www.w3.org/1998/math/mathml" display="inline"><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> )-dimensional topological phases. ...

    1998 doi

  55. [63]

    On the dimensions of (2+1)d abelian bosonic topological systems on nonorientable manifolds

    Ippo Orii. On the dimensions of (2+1)d abelian bosonic topological systems on nonorientable manifolds. Progress of Theoretical and Experimental Physics, 2025 0 (5), April 2025 a . ISSN 2050-3911. doi:10.1093/ptep/ptaf056. URL http://dx.doi.org/10.1093/ptep/ptaf056

    2025 doi

  56. [64]

    Generalization of anomaly formula for time reversal symmetry in (2+1)d abelian bosonic tqfts, 2025 b

    Ippo Orii. Generalization of anomaly formula for time reversal symmetry in (2+1)d abelian bosonic tqfts, 2025 b . URL https://arxiv.org/abs/2508.04990

    2025

  57. [65]

    Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter : Path Integrals , Non-orientable Manifolds , and Anomalies

    Maissam Barkeshli, Parsa Bonderson, Meng Cheng, Chao-Ming Jian, and Kevin Walker. Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter : Path Integrals , Non-orientable Manifolds , and Anomalies . Communications in Mathematical Physics, 374 0 (2): 0 1021...

    2020 doi

  58. [66]

    Problems in low dimensional manifold theory

    Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part, volume 2, pages 273--312, 1978

    1976

  59. [67]

    Problems around 3–manifolds

    Joachim Hyam Rubinstein. Problems around 3–manifolds. In Heegaard splittings of 3-manifolds (Haifa 2005), page 285–298. Mathematical Sciences Publishers, December 2007. doi:10.2140/gtm.2007.12.285. URL http://dx.doi.org/10.2140/gtm.2007.12.285

    2005 doi

  60. [68]

    Three dimensional manifolds, kleinian groups and hyperbolic geometry

    William P Thurston. Three dimensional manifolds, kleinian groups and hyperbolic geometry. 1982

    1982

  61. [69]

    The entropy formula for the ricci flow and its geometric applications, 2002

    Grisha Perelman. The entropy formula for the ricci flow and its geometric applications, 2002. URL https://arxiv.org/abs/math/0211159

    2002 arXiv

  62. [70]

    Ricci flow with surgery on three-manifolds, 2003 a

    Grisha Perelman. Ricci flow with surgery on three-manifolds, 2003 a . URL https://arxiv.org/abs/math/0303109

    2003 arXiv

  63. [71]

    Finite extinction time for the solutions to the ricci flow on certain three-manifolds, 2003 b

    Grisha Perelman. Finite extinction time for the solutions to the ricci flow on certain three-manifolds, 2003 b . URL https://arxiv.org/abs/math/0307245

    2003 arXiv

  64. [72]

    Forstudier til en topologisk teori for de algebraiske fladers sammenhaeng

    Poul Heegaard. Forstudier til en topologisk teori for de algebraiske fladers sammenhaeng. Bojesen, 1898

  65. [73]

    Algorithmic and computer methods for three-manifolds, volume 425

    Anatolij Timofeevi c Fomenko and Serge Vladimirovich Matveev. Algorithmic and computer methods for three-manifolds, volume 425. Springer Science & Business Media, 2013

    2013

  66. [74]

    Lectures on the topology of 3-manifolds: an introduction to the Casson invariant

    Nikolai Saveliev. Lectures on the topology of 3-manifolds: an introduction to the Casson invariant. Walter de Gruyter, 2011

    2011

  67. [75]

    Zur dreidimensionalen Topologie

    Kurt Reidemeister. Zur dreidimensionalen Topologie . Abhandlungen aus dem Mathematischen Seminar der Universit \"a t Hamburg , 9 0 (1): 0 189--194, December 1933. ISSN 1865-8784. doi:10.1007/BF02940644. URL https://doi.org/10.1007/BF02940644

    1933 doi

  68. [76]

    James W. Singer. Three-dimensional manifolds and their heegaard diagrams. Transactions of the American Mathematical Society, 35: 0 88--111, 1933. URL https://api.semanticscholar.org/CorpusID:53542340

    1933

  69. [77]

    3-manifold groups, 2013

    Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton. 3-manifold groups, 2013. URL https://arxiv.org/abs/1205.0202

    2013 arXiv

  70. [78]

    Yu Kitaev

    A. Yu Kitaev. Fault-tolerant quantum computation by anyons . https://arxiv.org/abs/quant-ph/9707021v1, July 1997

    1997 arXiv

  71. [79]

    Leigh, and Sean Nowling

    Shiying Dong, Eduardo Fradkin, Robert G. Leigh, and Sean Nowling. Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids . Journal of High Energy Physics, 2008 0 (05): 0 016--016, May 2008. ISSN 1029-8479. doi:10.1088/1126-6708/2008/05/016

    2008 doi

  72. [80]

    Quasiparticle statistics and braiding from ground-state entanglement

    Yi Zhang, Tarun Grover, Ari Turner, Masaki Oshikawa, and Ashvin Vishwanath. Quasiparticle statistics and braiding from ground-state entanglement. Physical Review B, 85 0 (23): 0 235151, June 2012. doi:10.1103/PhysRevB.85.235151

    2012 doi

  73. [81]

    Entanglement entropy of gapped phases and topological order in three dimensions

    Tarun Grover. Entanglement entropy of gapped phases and topological order in three dimensions. Physical Review B, 84 0 (19), 2011. doi:10.1103/PhysRevB.84.195120

    2011 doi

  74. [82]

    John C. Baez. Spin Networks in Gauge Theory . Advances in Mathematics, 117 0 (2): 0 253--272, February 1996. ISSN 00018708. doi:10.1006/aima.1996.0012

    1996

  75. [83]

    Note6. Their paper does not mention the explicit block structure of the density matrix, and since the multiplicity of the amplitudes does not appear, the amplitudes differ by a quantum dimensional factor from our convention, but they are consistent

  76. [84]

    Classical and quantum conformal field theory

    Gregory Moore and Nathan Seiberg. Classical and quantum conformal field theory. Communications in Mathematical Physics, 123 0 (2): 0 177--254, June 1989. ISSN 1432-0916. doi:10.1007/BF01238857

    1989 doi

  77. [85]

    Lectures on RCFT

    Gregory Moore and Nathan Seiberg. Lectures on RCFT . In H. C. Lee, editor, Physics, Geometry and Topology , pages 263--361. Springer US, Boston, MA, 1990. ISBN 978-1-4615-3802-8. doi:10.1007/978-1-4615-3802-8_8

    1990 doi

  78. [86]

    Quantum field theory and the Jones polynomial

    Edward Witten. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics, 121 0 (3): 0 351--399, September 1989. ISSN 1432-0916. doi:10.1007/BF01217730

    1989 doi

  79. [87]

    TFT construction of RCFT correlators I : Partition functions

    J \"u rgen Fuchs, Ingo Runkel, and Christoph Schweigert. TFT construction of RCFT correlators I : Partition functions. Nuclear Physics B, 646 0 (3): 0 353--497, December 2002. ISSN 05503213. doi:10.1016/S0550-3213(02)00744-7

    2002 doi

  80. [88]

    TFT construction of RCFT correlators II : Unoriented world sheets

    J \"u rgen Fuchs, Ingo Runkel, and Christoph Schweigert. TFT construction of RCFT correlators II : Unoriented world sheets. Nuclear Physics B, 678 0 (3): 0 511--637, February 2004. ISSN 05503213. doi:10.1016/j.nuclphysb.2003.11.026

    2004 doi

  81. [89]

    Entanglement spectrum of topological insulators and superconductors

    Lukasz Fidkowski. Entanglement spectrum of topological insulators and superconductors. Phys. Rev. Lett., 104: 0 130502, Apr 2010. doi:10.1103/PhysRevLett.104.130502. URL https://link.aps.org/doi/10.1103/PhysRevLett.104.130502

    2010 doi

  82. [90]

    Finite group modular data

    Antoine Coste, Terry Gannon, and Philippe Ruelle. Finite group modular data. Nuclear Physics B, 581 0 (3): 0 679--717, August 2000. ISSN 05503213. doi:10.1016/S0550-3213(00)00285-6

    doi

  83. [91]

    Dijkgraaf, V

    R. Dijkgraaf, V. Pasquier, and P. Roche. Quasi hope algebras, group cohomology and orbifold models. Nuclear Physics B - Proceedings Supplements, 18 0 (2): 0 60--72, 1991. ISSN 0920-5632. doi:https://doi.org/10.1016/0920-5632(91)90123-V. URL https://www.sciencedirect.com/scienc...

    1991

  84. [92]

    Gruppentheorie und ihre anwendung auf die quantenmechanik der atomspektren

    Eugen Wigner. Gruppentheorie und ihre anwendung auf die quantenmechanik der atomspektren. 1931

    1931

  85. [93]

    The threefold way

    Freeman J Dyson. The threefold way. algebraic structure of symmetry groups and ensembles in quantum mechanics. Journal of Mathematical Physics, 3 0 (6): 0 1199--1215, 1962 b

    1962

  86. [94]

    Asaeda and U

    M. Asaeda and U. Haagerup. Exotic Subfactors of Finite Depth with Jones Indices and. Communications in Mathematical Physics, 202 0 (1): 0 1--63, April 1999. ISSN 1432-0916. doi:10.1007/s002200050574

    1999 doi

  87. [95]

    Ibieta-Jimenez , M

    J.P. Ibieta-Jimenez , M. Petrucci, L.N. Queiroz Xavier, and P. Teotonio-Sobrinho . Topological entanglement entropy in d-dimensions for Abelian higher gauge theories. Journal of High Energy Physics, 2020 0 (3): 0 167, March 2020. ISSN 1029-8479. doi:10.1007/JHEP03(2020)167

    2020 doi

  88. [96]

    Jacob Lurie. Kerodon. https://kerodon.net/, 2018

    2018

  89. [97]

    Nonabelian algebraic topology

    Ronald Brown. Nonabelian algebraic topology. arXiv preprint math/0407275, 2004

    2004 arXiv

  90. [98]

    Tannakian categories

    Pierre Deligne and James S Milne. Tannakian categories. In Hodge cycles, motives, and Shimura varieties, pages 101--228. Springer, 1982

    1982

  91. [99]

    Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions, October 2024

    Bowei Liu, Junjia Zhang, Shuhei Ohyama, Yuya Kusuki, and Shinsei Ryu. Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions, October 2024

    2024

  92. [100]

    Lessa, Meng Cheng, and Chong Wang

    Leonardo A. Lessa, Meng Cheng, and Chong Wang. Mixed- State Quantum Anomaly and Multipartite Entanglement . Physical Review X, 15 0 (1): 0 011069, March 2025. doi:10.1103/PhysRevX.15.011069

    2025 doi

  93. [101]

    Amanda Gatto Lamas, Jacopo Gliozzi, and Taylor L. Hughes. Higher-form entanglement asymmetry and topological order, October 2025

    2025

  94. [102]

    Entanglement sum rule from higher-form symmetries, 2025

    Pei-Yao Liu. Entanglement sum rule from higher-form symmetries, 2025. URL https://arxiv.org/abs/2510.17317

    2025

  95. [103]

    Differential forms in algebraic topology, volume 82

    Raoul Bott and Loring W Tu. Differential forms in algebraic topology, volume 82. Springer Science & Business Media, 1982

    1982

  96. [104]

    Algebraic Topology

    Allen Hatcher. Algebraic Topology . Cambridge University Press, 2002. ISBN 978-0-521-79540-1

    2002

  97. [105]

    Topology and Groupoids

    Ronald Brown. Topology and Groupoids. www. groupoids. org, 2006

    2006

  98. [106]

    Non- Abelian Cohomology and Van Kampen 's Theorem

    Paul Olum. Non- Abelian Cohomology and Van Kampen 's Theorem . Annals of Mathematics, 68 0 (3): 0 658--668, 1958. ISSN 0003-486X. doi:10.2307/1970160

    1958 doi

  99. [107]

    Nikolai V. Ivanov. Non-abelian cohomology and seifert-van kampen theorem, 2023. URL https://arxiv.org/abs/2310.18886

    2023 arXiv

  100. [108]

    \'E quations diff \'e rentielles \`a points singuliers r \'e guliers , volume 163

    Pierre Deligne. \'E quations diff \'e rentielles \`a points singuliers r \'e guliers , volume 163. Springer, 2006

    2006

  101. [109]

    Foundations of differential geometry, volume 2

    Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differential geometry, volume 2. John Wiley & Sons, 1996

    1996

  102. [110]

    Fibre Bundles , volume 20 of Graduate Texts in Mathematics

    Dale Husemoller. Fibre Bundles , volume 20 of Graduate Texts in Mathematics . Springer, New York, NY, 1994. doi:10.1007/978-1-4757-2261-1. URL http://link.springer.com/10.1007/978-1-4757-2261-1

    1994 doi

  103. [111]

    Higgs bundles and local systems

    Carlos T Simpson. Higgs bundles and local systems. Publications Math \'e matiques de l'IH \'E S , 75: 0 5--95, 1992

    1992

  104. [112]

    H. A. Kramers and G. H. Wannier. Statistics of the Two-Dimensional Ferromagnet . Part I . Physical Review, 60 0 (3): 0 252--262, August 1941 a . ISSN 0031-899X. doi:10.1103/PhysRev.60.252

    1941 doi

  105. [113]

    H. A. Kramers and G. H. Wannier. Statistics of the Two-Dimensional Ferromagnet . Part II . Physical Review, 60 0 (3): 0 263--276, August 1941 b . ISSN 0031-899X. doi:10.1103/PhysRev.60.263

    1941 doi

  106. [114]

    Franz J. Wegner. Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters . Journal of Mathematical Physics, 12 0 (10): 0 2259--2272, October 1971. ISSN 0022-2488, 1089-7658. doi:10.1063/1.1665530

    1971 doi

  107. [115]

    John B. Kogut. An introduction to lattice gauge theory and spin systems. Reviews of Modern Physics, 51 0 (4): 0 659--713, October 1979. ISSN 0034-6861. doi:10.1103/RevModPhys.51.659

    1979 doi

  108. [116]

    Bellissard

    J. Bellissard. A remark about the duality for non- Abelian lattice fields. Journal of Mathematical Physics, 20 0 (7): 0 1490--1493, July 1979. ISSN 0022-2488, 1089-7658. doi:10.1063/1.524206

    1979 doi

  109. [117]

    Drouffe, C

    J.M. Drouffe, C. Itzykson, and J.B. Zuber. Lattice models with a solvable symmetry group. Nuclear Physics B, 147 0 (1-2): 0 132--134, January 1979. ISSN 05503213. doi:10.1016/0550-3213(79)90418-8

    1979 doi

  110. [118]

    Duality for non-abelian lattice fields

    Peter Orland. Duality for non-abelian lattice fields. Nuclear Physics B, 163: 0 275--288, January 1980. ISSN 05503213. doi:10.1016/0550-3213(80)90403-4

    1980 doi

  111. [119]

    Hierarchy of Topological Order From Finite-Depth Unitaries , Measurement , and Feedforward

    Nathanan Tantivasadakarn, Ashvin Vishwanath, and Ruben Verresen. Hierarchy of Topological Order From Finite-Depth Unitaries , Measurement , and Feedforward . PRX Quantum, 4 0 (2): 0 020339, June 2023. doi:10.1103/PRXQuantum.4.020339

    2023 doi

  112. [120]

    UNIPOTENT REPRESENTATIONS OF A FINITE CHEVALLEY GROUP OF TYPE E8

    GEORGE LUSZTIG. UNIPOTENT REPRESENTATIONS OF A FINITE CHEVALLEY GROUP OF TYPE E8 . The Quarterly Journal of Mathematics, 30 0 (3): 0 315--338, September 1979. ISSN 0033-5606. doi:10.1093/qmath/30.3.315

    1979 doi

  113. [121]

    Tensor Categories

    Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. Tensor Categories . American Mathematical Soc., July 2015. ISBN 978-1-4704-2024-6

    2015

  114. [122]

    Steven H. Simon. Topological Quantum . Oxford University Press, 2023. ISBN 978-0-19-888672-3

    2023

  115. [123]

    Simon, Julien Vidal, and Jean-No \"e l Fuchs

    Anna Ritz-Zwilling , Beno \^i t Dou c ot, Steven H. Simon, Julien Vidal, and Jean-No \"e l Fuchs. Partition function of the Kitaev quantum double model, September 2025

    2025

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.