Pith. sign in

REVIEW 1 major objections 5 minor 80 references

An order-five electromagnetic duality acts projectively on three-dimensional U(1)^2 theories, leaving a decoupled Chern-Simons phase after five steps and generating bilayer quantum Hall states at fillings 3/8+3/8 and 5/12+5/12.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 09:49 UTC pith:FWM3KUGU

load-bearing objection The bilayer hierarchy construction with the final (221) block is solid and explicitly checkable, but the order-five projective duality g5^5=U(1)_1 rests on a too-terse Appendix C derivation that may miss contact terms; the paper deserves refereeing with a request to secure that step. the 1 major comments →

arxiv 2607.20622 v1 pith:FWM3KUGU submitted 2026-07-22 hep-th cond-mat.str-el

Sp(4,Z) actions on 3d U(1)² symmetric theories: Order-five duality and bilayer quantum Hall hierarchies

Yasin F. Alam , Andreas Karch , Da-Chuan Lu , Ryan C. Spieler This is my paper
classification hep-th cond-mat.str-el
keywords Sp(4,Z) dualityU(1)^2 symmetric theoriesorder-five dualityprojective actionduality-gravity anomalybilayer quantum Hall hierarchyeven-denominator fillingsK-matrix formalism
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends the electromagnetic-duality action on three-dimensional theories with a single U(1) symmetry to theories with U(1)×U(1) global symmetry, where the duality group becomes Sp(4,Z). Its central discovery is an order-five duality transformation g5 that intrinsically mixes the two U(1) factors: applied five times it acts trivially in the bulk, but on the boundary it leaves behind a decoupled U(1)_1 Chern-Simons phase, a projective effect that cannot be removed by redefining the operation. The paper then uses these Sp(4,Z) operations to organize bilayer fractional quantum Hall hierarchies, recasting the conventional hierarchy as a sequence of SL(2,Z) steps and generalizing it to two-component systems with charge and pseudospin symmetries. A branch whose final daughter block is an interlayer-correlated (221) state yields candidate Abelian states at equal-layer fillings 3/8+3/8 and 5/12+5/12, which the paper shows are equivalent to Abelian composite-fermion descriptions through integral changes of anyon basis.

Core claim

The paper establishes that Sp(4,Z) electromagnetic duality of 4d U(1)^2 gauge theory induces a projective action on 3d boundary theories with U(1)×U(1) global symmetry, and that this structure goes beyond two independent copies of the single-U(1) construction. It identifies a genuinely two-component element g5 of order five whose bulk matrix satisfies g5^5 = I, while its boundary action satisfies g5^5 = U(1)_1, an invertible phase that cannot be removed by redefinition. This projective closure signals a mixed duality-gravity anomaly with index 2 in Z_5, consistent with an index 8 in Z_10 obtained from the lift of g5 to the genus-two mapping class group. On the quantum Hall side, the paper sh

What carries the argument

The central object is the order-five element g5 = T R2 R1 S of Sp(4,Z), a composition of a level-one Chern-Simons stacking, a rotation of the two U(1) factors, and a gauging operation that mixes them; its projective boundary closure g5^5 = U(1)_1 is derived by explicit Gaussian path integrals over auxiliary fields and carries the anomaly. On the quantum Hall side, the load-bearing machinery is the K-matrix formalism—a symmetric integral matrix encoding an Abelian Chern-Simons theory's anyons, braiding, and charges—together with the ST^p hierarchy step, where T^p stacks a level-p Chern-Simons term and S promotes the background field to a dynamical gauge field, enlarging the K-matrix by one ro

Load-bearing premise

The argument assumes that the Sp(4,Z) bulk duality transformations induce on the 3d boundary exactly the operations (2.18)–(2.20), and that the Gaussian path-integral evaluation of g5^5 in Appendix C receives no extra contact terms; if the boundary operations are defined only up to additional counterterms, the projective phase and the claimed anomaly index would shift.

What would settle it

Evaluate the fivefold application of g5 on a closed spin three-manifold such as the three-torus with a chosen spin structure, using the explicit path integral of Appendix C; if the result is the original partition function times U(1)_1 plus any extra phase or level shift, or if a nonzero contact term survives, the claimed projective closure g5^5 = U(1)_1 fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The order-five relation provides a concrete boundary signature of a mixed duality-gravity anomaly for U(1)^2 theories, with anomaly index 2 in Z_5 matching the restriction of index 8 in Z_10 from the genus-two mapping class group, thereby sharpening the anomaly classification for two-component Maxwell theories.
  • The Sp(4,Z) operations organize a web of U(1)×U(1)-symmetric gapped phases—Abelian topological orders, invertible SPT phases, and symmetry-broken phases—into a single constructive framework where symmetry-fractionalization data is tracked through the K-matrix.
  • The hierarchy construction with a final (221) daughter block yields candidate Abelian states at equal-layer fillings 3/8+3/8 and 5/12+5/12 that are equivalent to Abelian composite-fermion descriptions, providing testable topological orders for bilayer fractional quantum Hall experiments at these even-denominator fillings.
  • The spin-charge constraint—diagonal K-matrix entries obey a parity condition set by the charge vector—is satisfied by all constructed hierarchies, offering a consistency check that determines which daughter condensates are fermionic versus bosonic and constraining the allowed quantum Hall hierarchy states.
  • The construction extends to U(1)^n symmetric theories with Sp(2n,Z) duality and to non-Abelian fractional quantum Hall phases via anyon condensation on the Abelian sector, giving a framework for generating a broad web of symmetry-enriched topological phases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The projective relation g5^5 = U(1)_1 suggests that five successive gauging operations act like a discrete Z_5 symmetry in the space of theories; one could look for g5-invariant multicritical points where the five phases in an orbit meet, analogous to self-dual critical points in single-U(1) models.
  • The equivalence between the hierarchy and composite-fermion descriptions at 3/8 and 5/12 indicates that the (221) daughter block is a reorganizing principle rather than a new phase; a testable extension would be to compute non-Abelian variants where the Sp(4,Z) operations act on an Abelian sector, predicting candidate non-Abelian states at the same even-denominator fillings.
  • Because the projective phase is independent of the chosen boundary theory, it behaves as an intrinsic datum of the duality group itself; a lattice realization of bilayer quantum Hall states could look for the signature U(1)_1 edge mode after a five-step cycle.
  • The spin-charge parity constraint on diagonal K-matrix entries suggests a selection rule that limits which Abelian topological orders can be reached by hierarchy steps from a given parent, potentially ruling out certain candidate fillings in future experiments.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The manuscript studies Sp(4,Z) actions on 3d U(1)^2 symmetric theories, deriving boundary operations from 4d electromagnetic duality. Its main new result is an order-five element g5 whose bulk matrix satisfies g5^5=I but whose induced boundary operation closes only up to a decoupled U(1)_1 phase, interpreted as a projective action and related to a mixed duality-gravitational anomaly. The second half applies the same operations in K-matrix language to bilayer fractional quantum Hall hierarchies, producing candidate Abelian states at equal-layer fillings 3/8+3/8 and 5/12+5/12 from a final (221) daughter block, and demonstrates GL(N,Z) equivalence to composite-fermion descriptions via explicit matrices W4 and W6. Appendices provide the Sp(4,Z) presentation, a matrix description of the symmetry web, and the anomaly calculation.

Significance. The fractional quantum Hall applications are supported by explicit, verifiable K-matrices and integral basis changes; for example, the relation W4^T K4 W4 = K4_CF in Eq. (4.51) is directly checkable and is correct. The order-five projective relation is a genuinely new finite-order analogue of the single-U(1) (ST)^3 result. The anomaly interpretation is made plausible by two complementary arguments: the Gaussian evaluation in App. C and the restriction of the known Z10 anomaly of U(1)^2 Maxwell theory to the Z5 subgroup. The latter provides an independent anchor for the claim. The paper is constructive rather than observational, with no fitted parameters; the hierarchy states are derived from chosen K-matrices. These strengths make the result significant for the dialogue between the duality-web and fractional quantum Hall communities.

major comments (1)
  1. [Appendix C, Eqs. (C.9)-(C.13); Sec. 2.5] The derivation of g5^5=U(1)_1 is terse. The text says 'integrate over x2, x1 for the first and second equal signs,' but the first equality also eliminates x5; u is redefined after integrations; and the Gaussian steps are presented without an explicit treatment of determinant factors or contact terms. Since this is the central projective claim, please expand the derivation: specify the order of integrations, state the role of the trivial-topology assumption in footnote 1 (flat connections pure gauge, BF symmetric), and explain why all determinant factors are background-independent and absorbed into normalization. Alternatively, make the group-theoretic restriction from the Z10 anomaly the primary proof and label the path-integral evaluation as a consistency check. As written, the load-bearing step needs clarification.
minor comments (5)
  1. [Appendix C, Eq. (C.13)] Clarify that after the shift of u the expression CS(u+A+B) becomes a decoupled dynamical U(1)_1 sector and is not a background Chern-Simons term. The notation could be misread as a field-dependent phase.
  2. [Sec. 4.3, text near Eq. (4.48)] The phrase 'Z 8(r+2)' should be typeset as Z_{8(r+2)} to avoid ambiguity. For r=0 this is Z_{16}, consistent with |det K4|=16.
  3. [Abstract and Sec. 2.5] The wording 'suggesting a mixed duality-gravitational anomaly' is hedged. If the App. C derivation is intended as a proof, consider strengthening the claim; otherwise, keep the hedge consistently in the conclusion.
  4. [Footnote 1 and App. C] State explicitly that the trivial-topology assumption also underlies the Gaussian manipulations in App. C, where flat connections are pure gauge and the BF pairing is symmetric. This would preempt concerns about fractional shifts of background connections.
  5. [Eq. (2.47)] Displaying the 4x4 matrix for g5 in the main text alongside its Lagrangian action would make the g5^5=I statement immediately checkable for readers.

Circularity Check

0 steps flagged

No significant circularity: central results are explicit computations from stated definitions; the FQH fractions are candidate effective descriptions backed by verifiable K-matrix algebra.

full rationale

The paper's central new result, the order-five element g5 and its boundary projective relation g5^5 = U(1)_1, is obtained by explicit matrix multiplication and the five-fold Gaussian path integral in App. C starting from the definitions (C.7)-(C.9). The result is a computation, not a fitted parameter renamed as a prediction; nothing in the definition of g5 is chosen to force the final U(1)_1 phase. The anomaly interpretation relies on the independent classification of duality-group projective actions in [9,38] and on the external result H^2(BΓ2,Z) = Z10; these are not self-citations. The self-citations that appear ([12], [49], [75], [77]) are background analogies and are not load-bearing for the g5 calculation or the FQH hierarchies. On the FQH side, the fillings 3/8+3/8 and 5/12+5/12 are computed from explicitly written K-matrices (4.48) and (4.49) using the chosen Halperin hierarchy blocks; the paper labels these as candidate effective descriptions and explicitly states that energetic preference requires microscopic input, so they are not presented as parameter-free predictions. The equivalence to Abelian composite-fermion descriptions is demonstrated by the explicit GL(N,Z) matrices W4 and W6, which are directly verifiable. The choice of the (221) branch is post hoc relative to experiment, but that is interpretational, not circular: no equation is fitted to the experimental fractions, and no claimed derivation reduces to its own inputs. The only technical concern, namely that the App. C path-integral manipulations may be sensitive to contact-term regularization, is a correctness risk to be checked independently, not a circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The projective order-five calculation is parameter-free. The FQH section selects discrete integer parameters (m,n,p,q,r) by hand to match observed fillings; these are model choices, not continuous fits, but they do determine the final results.

free parameters (3)
  • Parent Halperin integers (m,n) for (330) = m=3, n=0
    The layer-symmetric (330) parent is chosen by hand; the central FQH results (3/8, 5/12) are obtained from this parent.
  • Final daughter block (p,q) = (2,1) = (2,1)
    The (221) interlayer-correlated bosonic block is chosen because it yields the experimental equal-layer fillings 3/8 and 5/12.
  • Intermediate daughter block (p,q) = (2,0) = (2,0)
    The (220) Jain-type blocks are standard choices for intralayer flux attachment; the number of such blocks r is varied to reach 3/8 (r=0) and 5/12 (r=1).
axioms (7)
  • domain assumption Bulk-boundary dictionary: Sp(2n,Z) electromagnetic duality of 4D Maxwell theory induces the boundary operations (2.18)-(2.20).
    Section 2.3 derives these operations following Refs. [6,39]; this is the foundational input for all boundary calculations.
  • standard math K-matrix formalism describes Abelian topological phases: even K is bosonic, odd K is spin, with anyon group Z^N/KZ^N and braiding (3.4).
    Section 3.1 reviews the standard framework; the paper relies on it for all FQH analysis.
  • domain assumption Symmetric gapped phases with U(1)xU(1) symmetry are described by symmetry-enriched topological order (SET) data.
    Section 3 assumes the SET classification of Refs. [15,16,42] without derivation.
  • standard math Anomaly classification for duality groups uses H^2(BD, Inv^3_spin) and H^2(BGamma_2,Z)=Z10 from Refs. [9,38].
    Used in Section 2.5 and App. C to interpret the projective phase as a mixed duality-gravity anomaly.
  • domain assumption Hierarchy step: tuning away from a parent FQH state creates quasiparticles whose Laughlin-like daughter state is implemented by an ST^p operation.
    Section 4.1 motivates this from the Haldane-Halperin construction; it is the physical assumption behind all generated hierarchy states.
  • domain assumption Spin-charge constraint: when coupling to a spin-c connection, the K-matrix diagonal satisfies K_ii ≡ t_i (mod 2).
    Section 4.4 applies the constraint from Refs. [2,60] to validate the constructed K-matrices.
  • domain assumption All gauge fields are considered on manifolds without homologically nontrivial cycles.
    Stated in footnote 1; this simplifies differential-form expressions and flux quantization.

pith-pipeline@v1.3.0-alltime-deepseek · 30312 in / 30713 out tokens · 217161 ms · 2026-08-01T09:49:37.936121+00:00 · methodology

0 comments
read the original abstract

The $\mathrm{SL}(2,\mathbb{Z})$ electromagnetic duality of 4d Maxwell theory induces theory-generating operations on 3d theories with $U(1)$ global symmetry. For theories with $U(1)^n$ symmetry, this structure generalizes to $\mathrm{Sp}(2n,\mathbb{Z})$. Focusing on the $U(1)^2$ case, we formulate the bulk $\mathrm{Sp}(4,\mathbb{Z})$ action and derive the corresponding boundary operations. We identify an intrinsically two-component element of order five, which generalizes the order-three $ST$ element of the single-$U(1)$ theory. Although its fifth power acts trivially in the bulk, the corresponding boundary operation closes only up to a decoupled $U(1)_1$ invertible phase, suggesting a mixed duality-gravitational anomaly. We realize the resulting theory-generating web for Abelian Chern-Simons theories within the $K$-matrix formalism and apply it to bilayer fractional quantum Hall systems. Recasting the Haldane--Halperin hierarchy construction as a sequence of $\mathrm{SL}(2,\mathbb{Z})$ operations, we generalize it to systems with charge $U(1)_c$ and pseudospin $U(1)_s$ symmetries. The resulting bilayer hierarchies contain branches terminating in an interlayer-correlated bosonic $(221)$ daughter sector, yielding candidate Abelian states at the even-denominator equal-layer fillings $3/8+3/8$ and $5/12+5/12$. Integral changes of anyon basis establish the equivalence of these states to their corresponding Abelian composite-fermion descriptions. We further discuss the spin-charge constraints that arise when the electromagnetic background field is treated as a spin-$c$ connection.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

80 extracted references · 54 linked inside Pith

  1. [1]

    E. Witten,SL(2,Z) action on three-dimensional conformal field theories with Abelian symmetry, inFrom Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan, pp. 1173–1200, 7, 2003 [hep-th/0307041]

    Pith/arXiv arXiv 2003

  2. [2]

    Seiberg, T

    N. Seiberg, T. Senthil, C. Wang and E. Witten,A Duality Web in 2+1 Dimensions and Condensed Matter Physics,Annals Phys.374(2016) 395 [1606.01989]

    Pith/arXiv arXiv 2016

  3. [3]

    Karch and D

    A. Karch and D. Tong,Particle-Vortex Duality from 3d Bosonization,Phys. Rev. X6(2016) 031043 [1606.01893]

    Pith/arXiv arXiv 2016

  4. [4]

    C. Wang, A. Nahum, M.A. Metlitski, C. Xu and T. Senthil,Deconfined quantum critical points: symmetries and dualities,Phys. Rev. X7(2017) 031051 [1703.02426]

    Pith/arXiv arXiv 2017

  5. [5]

    Gaillard and B

    M.K. Gaillard and B. Zumino,Duality rotations for interacting fields,Nucl. Phys. B193 (1981) 221

    1981

  6. [6]

    Dimofte, D

    T. Dimofte, D. Gaiotto and S. Gukov,Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys.325(2014) 367 [1108.4389]

    Pith/arXiv arXiv 2014

  7. [7]

    Kapustin and M

    A. Kapustin and M. Tikhonov,Abelian duality, walls and boundary conditions in diverse dimensions,JHEP11(2009) 006 [0904.0840]. – 34 –

    Pith/arXiv arXiv 2009

  8. [8]

    Hua and I

    L. Hua and I. Reiner,On the generators of the symplectic modular group,Transactions of the American Mathematical Society65(1949) 415

    1949

  9. [9]

    Seiberg, Y

    N. Seiberg, Y. Tachikawa and K. Yonekura,Anomalies of Duality Groups and Extended Conformal Manifolds,PTEP2018(2018) 073B04 [1803.07366]

    Pith/arXiv arXiv 2018

  10. [10]

    Korkmaz and A.I

    M. Korkmaz and A.I. Stipsicz,The second homology groups of mapping class groups of orientable surfaces,Math. Proc. Cambridge Philos. Soc.134(2003) 479

    2003

  11. [11]

    Harer,The second homology group of the mapping class group of an orientable surface, Invent

    J. Harer,The second homology group of the mapping class group of an orientable surface, Invent. Math.72(1983) 221

    1983

  12. [12]

    Karch and R.C

    A. Karch and R.C. Spieler,Critical theories connecting gapped phases withZ 2 ×Z 2 symmetry from the duality web,JHEP06(2025) 028 [2502.14032]

    Pith/arXiv arXiv 2025

  13. [13]

    Watanabe,Counting rules of nambu–goldstone modes,Annu

    H. Watanabe,Counting rules of nambu–goldstone modes,Annu. Rev. Condens. Matter Phys. 11(2020) 169 [1904.00569]

    Pith/arXiv arXiv 2020

  14. [14]

    Lu and A

    Y.-M. Lu and A. Vishwanath,Theory and classification of interacting ’integer’ topological phases in two dimensions: A Chern-Simons approach,Phys. Rev. B86(2012) 125119 [1205.3156]

    Pith/arXiv arXiv 2012

  15. [15]

    Lu and A

    Y.-M. Lu and A. Vishwanath,Classification and properties of symmetry-enriched topological phases: Chern-Simons approach with applications to Z 2 spin liquids,Phys. Rev. B93(2016) 155121 [1302.2634]

    Pith/arXiv arXiv 2016

  16. [16]

    Barkeshli, P

    M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang,Symmetry fractionalization, defects, and gauging of topological phases,Phys. Rev. B100(2019) 115147 [1410.4540]

    Pith/arXiv arXiv 2019

  17. [17]

    Wen,Topological orders and edge excitations in fractional quantum hall states,Adv

    X.-G. Wen,Topological orders and edge excitations in fractional quantum hall states,Adv. Phys.44(1995) 405 [cond-mat/9506066]

    Pith/arXiv arXiv 1995

  18. [18]

    Cheng and C.-M

    M. Cheng and C.-M. Jian,Gauging U(1) symmetry in (2+1)d topological phases,SciPost Phys.12(2022) 202 [2201.07239]

    Pith/arXiv arXiv 2022

  19. [19]

    Cheng, P.-S

    M. Cheng, P.-S. Hsin and C.-M. Jian,Gauging lie group symmetry in (2+1)d topological phases,SciPost Phys.14(2023) 100 [2205.15347]

    Pith/arXiv arXiv 2023

  20. [20]

    Burgess and B.P

    C.P. Burgess and B.P. Dolan,Particle vortex duality and the modular group: Applications to the quantum Hall effect and other 2-D systems,Phys. Rev. B63(2001) 155309 [hep-th/0010246]

    Pith/arXiv arXiv 2001

  21. [21]

    Burgess and B.P

    C.P. Burgess and B.P. Dolan,Duality and nonlinear response for quantum Hall systems, Phys. Rev. B65(2002) 155323 [cond-mat/0105621]

    Pith/arXiv arXiv 2002

  22. [22]

    Burgess and B.P

    C.P. Burgess and B.P. Dolan,Modular symmetry, the semicircle law, and quantum Hall bilayers,Phys. Rev. B76(2007) 155310 [cond-mat/0701535]

    Pith/arXiv arXiv 2007

  23. [23]

    L¨ utken and G.G

    C.A. L¨ utken and G.G. Ross,Experimental probes of emergent symmetries in the quantum Hall system,Nucl. Phys. B850(2011) 321 [1008.5257]

    Pith/arXiv arXiv 2011

  24. [24]

    Nissinen and C.A

    J. Nissinen and C.A. L¨ utken,Renormalization-group potential for quantum Hall effects, Phys. Rev. B85(2012) 155123 [1111.4902]

    Pith/arXiv arXiv 2012

  25. [25]

    Geraedts and O.I

    S.D. Geraedts and O.I. Motrunich,Exact realization of integer and fractional quantum Hall phases inU(1)×U(1)models in(2 + 1)d,Annals Phys.334(2013) 288 [1302.1436]

    Pith/arXiv arXiv 2013

  26. [26]

    Son,Is the Composite Fermion a Dirac Particle?,Phys

    D.T. Son,Is the Composite Fermion a Dirac Particle?,Phys. Rev. X5(2015) 031027 [1502.03446]. – 35 –

    Pith/arXiv arXiv 2015

  27. [27]

    Goldman and E

    H. Goldman and E. Fradkin,Loop Models, Modular Invariance, and Three Dimensional Bosonization,Phys. Rev. B97(2018) 195112 [1801.04936]

    Pith/arXiv arXiv 2018

  28. [28]

    Haldane,Fractional quantization of the hall effect: A hierarchy of incompressible quantum fluid states,Phys

    F.D.M. Haldane,Fractional quantization of the hall effect: A hierarchy of incompressible quantum fluid states,Phys. Rev. Lett.51(1983) 605

    1983

  29. [29]

    Halperin,Statistics of quasiparticles and the hierarchy of fractional quantized hall states, Phys

    B.I. Halperin,Statistics of quasiparticles and the hierarchy of fractional quantized hall states, Phys. Rev. Lett.52(1984) 1583

    1984

  30. [30]

    Halperin,Theory of the quantized hall conductance,Helv

    B.I. Halperin,Theory of the quantized hall conductance,Helv. Phys. Acta56(1983) 75

    1983

  31. [31]

    Nguyen, N.J

    R.Q. Nguyen, N.J. Zhang, N. Khurana-Batra, S. Alkidim, X. Liu, K. Watanabe et al., Bilayer Excitons in the Laughlin Fractional Quantum Hall State,2410.24208

    arXiv

  32. [32]

    Jain,Composite-fermion approach for the fractional quantum hall effect,Phys

    J.K. Jain,Composite-fermion approach for the fractional quantum hall effect,Phys. Rev. Lett.63(1989) 199

    1989

  33. [33]

    Scarola and J

    V. Scarola and J. Jain,Phase diagram of bilayer composite fermion states,Physical Review B 64(2001) 085313

    2001

  34. [34]

    Brennan and Z

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

    Pith/arXiv arXiv 2024

  35. [35]

    Antinucci and F

    A. Antinucci and F. Benini,Anomalies and gauging of U(1) symmetries,Phys. Rev. B111 (2025) 024110 [2401.10165]

    Pith/arXiv arXiv 2025

  36. [36]

    Apruzzi, F

    F. Apruzzi, F. Bedogna and N. Dondi,SymTh for non-finite symmetries,JHEP04(2026) 153 [2402.14813]

    arXiv 2026

  37. [37]

    Jia and Y

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

    arXiv

  38. [38]

    Bhardwaj, Y

    L. Bhardwaj, Y. Lee and Y. Tachikawa,SL(2,Z)action on QFTs withZ 2 symmetry and the Brown-Kervaire invariants,JHEP11(2020) 141 [2009.10099]

    Pith/arXiv arXiv 2020

  39. [39]

    Di Pietro, D

    L. Di Pietro, D. Gaiotto, E. Lauria and J. Wu,3d Abelian Gauge Theories at the Boundary, JHEP05(2019) 091 [1902.09567]

    Pith/arXiv arXiv 2019

  40. [40]

    B¨ urgisser,Elements of finite order in symplectic groups,Archiv der Mathematik39(1982) 501

    B. B¨ urgisser,Elements of finite order in symplectic groups,Archiv der Mathematik39(1982) 501

    1982

  41. [41]

    Benson, C

    D. Benson, C. Campagnolo, A. Ranicki and C. Rovi,Cohomology of symplectic groups and meyer’s signature theorem,Algebraic & Geometric Topology18(2018) 4069 [1710.04851]

    Pith/arXiv arXiv 2018

  42. [42]

    Barkeshli, P

    M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang,Symmetry Fractionalization, Defects, and Gauging of Topological Phases,Physical Review B100(2019) 115147

    2019

  43. [43]

    Wang and Z

    L. Wang and Z. Wang,In and around Abelian anyon models,J. Phys. A53(2020) 505203 [2004.12048]

    Pith/arXiv arXiv 2020

  44. [44]

    Kitaev,Anyons in an exactly solved model and beyond,Annals Phys.321(2006) 2 [cond-mat/0506438]

    A. Kitaev,Anyons in an exactly solved model and beyond,Annals Phys.321(2006) 2 [cond-mat/0506438]

    Pith/arXiv arXiv 2006

  45. [45]

    Hung and Y

    L.-Y. Hung and Y. Wan,K matrix Construction of Symmetry-Enriched Phases of Matter, Phys. Rev. B87(2013) 195103 [1302.2951]

    Pith/arXiv arXiv 2013

  46. [46]

    Turner,Dualities in 2+1 Dimensions,PoSModave2018(2019) 001 [1905.12656]

    C. Turner,Dualities in 2+1 Dimensions,PoSModave2018(2019) 001 [1905.12656]

    Pith/arXiv arXiv 2019

  47. [47]

    Moradi, S.F

    H. Moradi, S.F. Moosavian and A. Tiwari,Topological holography: Towards a unification of Landau and beyond-Landau physics,SciPost Phys. Core6(2023) 066 [2207.10712]. – 36 –

    Pith/arXiv arXiv 2023

  48. [48]

    Huang and M

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

    Pith/arXiv arXiv 2025

  49. [49]

    Spieler,Exploring two dimensionalZ 2 invariant phases with time reversal symmetry and their transitions with topological operations,JHEP10(2025) 069 [2504.20021]

    R.C. Spieler,Exploring two dimensionalZ 2 invariant phases with time reversal symmetry and their transitions with topological operations,JHEP10(2025) 069 [2504.20021]

    arXiv 2025

  50. [50]

    Blok and X.G

    B. Blok and X.G. Wen,Effective theories of the fractional quantum hall effect: Hierarchy construction,Phys. Rev. B42(1990) 8145

    1990

  51. [51]

    Read,Excitation structure of the hierarchy scheme in the fractional quantum hall effect, Phys

    N. Read,Excitation structure of the hierarchy scheme in the fractional quantum hall effect, Phys. Rev. Lett.65(1990) 1502

    1990

  52. [52]

    Wen and A

    X.G. Wen and A. Zee,A Classification of Abelian quantum Hall states and matrix formulation of topological fluids,Phys. Rev. B46(1992) 2290

    1992

  53. [53]

    Tong,Lectures on the Quantum Hall Effect, 6, 2016

    D. Tong,Lectures on the Quantum Hall Effect, 6, 2016

    2016

  54. [54]

    Kohn,Cyclotron resonance and de haas-van alphen oscillations of an interacting electron gas,Phys

    W. Kohn,Cyclotron resonance and de haas-van alphen oscillations of an interacting electron gas,Phys. Rev.123(1961) 1242

    1961

  55. [55]

    Hoyos and D.T

    C. Hoyos and D.T. Son,Hall viscosity and electromagnetic response,Phys. Rev. Lett.108 (2012) 066805 [1109.2651]

    Pith/arXiv arXiv 2012

  56. [56]

    Zhang, A

    C. Zhang, A. Vishwanath and X.-G. Wen,Hierarchy construction for non-Abelian fractional quantum Hall states via anyon condensation,Phys. Rev. B112(2025) 125116 [2406.12068]

    Pith/arXiv arXiv 2025

  57. [57]

    Balatsky and E

    A. Balatsky and E. Fradkin,Singlet quantum hall effect and chern-simons theories,Physical Review B43(1991) 10622

    1991

  58. [58]

    Milovanovi´ c and N

    M. Milovanovi´ c and N. Read,Invariant structure of the hierarchy theory of fractional quantum hall states with spin,Physical Review B56(1997) 1461

    1997

  59. [59]

    Hutasoit, A.C

    J.A. Hutasoit, A.C. Balram, S. Mukherjee, Y.-H. Wu, S.S. Mandal, A. W´ ojs et al.,The enigma of theν=2 +3 /8 fractional quantum Hall effect,Phys. Rev. B95(2017) 125302 [1605.07324]

    Pith/arXiv arXiv 2017

  60. [60]

    Seiberg and E

    N. Seiberg and E. Witten,Gapped Boundary Phases of Topological Insulators via Weak Coupling,PTEP2016(2016) 12C101 [1602.04251]

    Pith/arXiv arXiv 2016

  61. [61]

    Moore and N

    G. Moore and N. Read,Nonabelions in the fractional quantum hall effect,Nucl. Phys. B360 (1991) 362

    1991

  62. [62]

    Read and E

    N. Read and E. Rezayi,Beyond paired quantum hall states: Parafermions and incompressible states in the first excited landau level,Phys. Rev. B59(1999) 8084 [cond-mat/9809384]

    Pith/arXiv arXiv 1999

  63. [63]

    Kong,Anyon condensation and tensor categories,Nucl

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

    Pith/arXiv arXiv 2014

  64. [64]

    Bonderson and J.K

    P. Bonderson and J.K. Slingerland,Fractional quantum hall hierarchy and the second landau level,Phys. Rev. B78(2008) 125323 [0711.3204]

    Pith/arXiv arXiv 2008

  65. [65]

    Barkeshli and X.-G

    M. Barkeshli and X.-G. Wen,Bilayer quantum hall phase transitions and the orbifold non-abelian fractional quantum hall states,Phys. Rev. B84(2011) 115121 [1010.4270]

    Pith/arXiv arXiv 2011

  66. [66]

    Barkeshli and X.-G

    M. Barkeshli and X.-G. Wen,Non-abelian two-component fractional quantum hall states, Phys. Rev. B82(2010) 233301 [0906.0356]

    Pith/arXiv arXiv 2010

  67. [67]

    Senthil, D.T

    T. Senthil, D.T. Son, C. Wang and C. Xu,Duality between(2 + 1)dQuantum Critical Points,Phys. Rept.827(2019) 1 [1810.05174]. – 37 –

    Pith/arXiv arXiv 2019

  68. [68]

    Xu and Y.-Z

    C. Xu and Y.-Z. You,Self-dual Quantum Electrodynamics as Boundary State of the three dimensional Bosonic Topological Insulator,Phys. Rev. B92(2015) 220416 [1510.06032]

    Pith/arXiv arXiv 2015

  69. [69]

    Cheng and C

    M. Cheng and C. Xu,Series of (2+1)-dimensional stable self-dual interacting conformal field theories,Phys. Rev. B94(2016) 214415 [1609.02560]

    Pith/arXiv arXiv 2016

  70. [70]

    C.-M. Jian, A. Rasmussen, Y.-Z. You and C. Xu,Emergent Symmetry and Tricritical Points near the deconfined Quantum Critical Point,1708.03050

    Pith/arXiv arXiv

  71. [71]

    D.-C. Lu, C. Xu and Y.-Z. You,Self-duality protected multicriticality in deconfined quantum phase transitions,Phys. Rev. B104(2021) 205142 [2104.05147]

    Pith/arXiv arXiv 2021

  72. [72]

    Dumitrescu, P

    T.T. Dumitrescu, P. Niro and R. Thorngren,From QED 3 to Self-Dual Multicriticality in the Fradkin-Shenker Model,2602.23420

    arXiv

  73. [73]

    Behr,Eine endliche pr¨ asentation der symplektischen gruppe sp4 (z).,Mathematische Zeitschrift141(1975) 47

    H. Behr,Eine endliche pr¨ asentation der symplektischen gruppe sp4 (z).,Mathematische Zeitschrift141(1975) 47

    1975

  74. [74]

    Farb and D

    B. Farb and D. Margalit,A primer on mapping class groups, vol. 49, Princeton university press (2011)

    2011

  75. [75]

    Karch, D

    A. Karch, D. Tong and C. Turner,A Web of 2d Dualities:Z 2 Gauge Fields and Arf Invariants,SciPost Phys.7(2019) 007 [1902.05550]

    Pith/arXiv arXiv 2019

  76. [76]

    Gaiotto and J

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

    Pith/arXiv arXiv 2021

  77. [77]

    D.-C. Lu, Z. Sun and Y.-Z. You,Realizing triality andp-ality by lattice twisted gauging in (1+1)d quantum spin systems,SciPost Phys.17(2024) 136 [2405.14939]

    Pith/arXiv arXiv 2024

  78. [78]

    Huang,Fermionic quantum criticality through the lens of topological holography,Phys

    S.-J. Huang,Fermionic quantum criticality through the lens of topological holography,Phys. Rev. B111(2025) 155130 [2405.09611]

    Pith/arXiv arXiv 2025

  79. [79]

    Fidkowski and A

    L. Fidkowski and A. Kitaev,The effects of interactions on the topological classification of free fermion systems,Phys. Rev. B81(2010) 134509 [0904.2197]

    Pith/arXiv arXiv 2010

  80. [80]

    Kapustin, R

    A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang,Fermionic Symmetry Protected Topological Phases and Cobordisms,JHEP12(2015) 052 [1406.7329]. – 38 –

    Pith/arXiv arXiv 2015