Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Density modes on the fuzzy sphere satisfy a genuine Lie algebra.

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

2026-08-02 22:57 UTC pith:NETTW6M2

load-bearing objection Solid algebraic core (Jacobi proof, limits, oscillator bounds), but the abstract's so(3,2) no-go overreaches what App E actually proves — fix the gap or hedge the claim. the 2 major comments →

arxiv 2602.15025 v3 pith:NETTW6M2 submitted 2026-02-16 cond-mat.str-el hep-th

3d Conformal Field Theories via Fuzzy Sphere Algebra

Luisa Eck , Zhenghan Wang This is my paper
classification cond-mat.str-el hep-th
keywords fuzzy spherelowest Landau leveldensity-mode algebraGMP algebraconformal field theoryso(3,2)Jacobi identityoscillator approximation
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.

Fuzzy-sphere models reproduce 3d CFT spectra in small systems of spinful fermions, but the reason has been unclear. This paper analyzes the algebra generated by the lowest-Landau-level density operators that build those Hamiltonians. It proves that the density modes close under commutation and obey the Jacobi identity, so they form a genuine Lie algebra. In the large-spin limit the algebra splits into two regimes: a planar limit reproducing the spin-enriched Girvin-MacDonald-Platzman algebra, and a commutative limit where low-angular-momentum modes obey a semiclassical Poisson bracket; restricting further to few spin flips above the paramagnetic state turns them into approximate harmonic oscillators with controlled error bounds. The paper also shows that a direct representation of the 3d conformal algebra so(3,2) by density modes exists only for the minimal two-electron system, and that natural extensions to larger systems do not correspond to the single-sphere thermodynamic limit. If these results hold, they give a rigorous algebraic foundation for the fuzzy-sphere approach to 3d CFTs and a precise obstruction that constrains where conformal symmetry can come from.

Core claim

The central claim is that the density modes n^A_{l,m}—the LLL-projected electron density operators that appear in fuzzy-sphere Hamiltonians—form a well-defined Lie algebra: the commutation relation (9), defined abstractly, satisfies the Jacobi identity because the fermionic bilinear representation is faithful, so the bracket is not an artifact of a particular realization. In the single-sphere thermodynamic limit, the algebra has two faces. For angular momenta l ~ √s, a local planar limit reproduces the spin-enriched GMP (Girvin–MacDonald–Platzman) algebra with noncommuting plane coordinates; for l << √s, a commutative limit gives a semiclassical algebra whose commutator reduces to a Poisson

What carries the argument

The central object is the fermionic bilinear density mode n^A_{l,m} = N_{s,l} Σ (Ŷ_{l,m})_{m1,m2} c†_{m1,σ} A_{σλ} c_{m2,λ}, where Ŷ_{l,m} are fuzzy spherical harmonics (an SO(3)-covariant basis of the matrix algebra Mat_{2s+1}). The commutation relation (9) with its 3j/6j coefficients carries the argument: it encodes both the planar GMP limit and the semiclassical Poisson bracket, and its Jacobi identity follows from the faithful fermionic realization. Also load-bearing is the so(3)-equivariant coproduct of the density-mode matrices, used to extend the s=1/2 so(3,2) representation to larger systems; it works algebraically but is structurally mismatched with the single-growing-irrep thermody

Load-bearing premise

The load-bearing premise is that the critical fuzzy-sphere Hamiltonian's low-energy sector is dominated by modes with angular momentum l much less than √s, so that the semiclassical and oscillator descriptions apply to the CFT limit; this dominance is expected from numerics but not proved in the paper.

What would settle it

Measure the weight of l ~ √s density modes in the low-energy eigenstates of the critical fuzzy-sphere Hamiltonian at the 3d Ising point as a function of s; if the weight does not tend to zero (or if the commutator (30) deviates from the delta-function by an amount that grows with s rather than vanishing), the semiclassical/oscillator identification is falsified. A direct numerical target: check the O(1/s) bound of the oscillator commutator on the few-spin-flip subspace at s=10,20,40.

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

If this is right

  • If the low-energy sector of critical fuzzy-sphere Hamiltonians is indeed dominated by l << √s modes, then the semiclassical Poisson-bracket algebra—not the planar GMP algebra—is the correct starting point for deriving the CFT limit; this validates the use of commutative-sphere intuition in that regime.
  • Within the few-spin-flip subspace, the density modes become near-independent harmonic oscillators with error bounds that vanish as s→∞ when the smallness condition kl_max = o(√s) holds; this provides a controlled analytic handle on the free-field/oscillator picture observed in numerical studies.
  • The planar limit gives a spin-enriched GMP algebra, placing fuzzy-sphere density modes within the broader quantum-Hall algebra landscape; the spinless subalgebra reduces exactly to the standard planar GMP algebra.
  • There is no so(3,2) representation generated by density modes for s>1/2, so the conformal symmetry seen in critical fuzzy-sphere spectra cannot be an exact finite-s property of the density-mode algebra; it must emerge only in a more refined scaling limit that includes the Hamiltonian and low-energy subspace.
  • The central-extension analysis exhibits Jacobi-consistent central terms of the form g(s,l,l',m) = δ_{l,l'}(-1)^m m C(s,l), leaving open whether these connect to the W-infinity central extension in the planar limit; if they do, fuzzy-sphere algebras could inherit a Virasoro-like structure.

Where Pith is reading between the lines

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

  • One testable consequence of the no-go result is that any finite-s Hamiltonian that exactly realizes so(3,2) must include operators beyond the single-mode density bilinears (e.g., higher-body terms); numerical searches for such exact conformal points would either find them or confirm the obstruction.
  • The oscillator approximation suggests a practical criterion for when the free-field description of the fuzzy-sphere critical point is valid: the product of the number of spin flips k and the angular-momentum cutoff l_max should stay well below √s. Current simulations with s~O(10) should show a crossover where oscillator behavior degrades, and that crossover could be measured directly from commutat
  • The coproduct mismatch implies that coupled or multi-sphere fuzzy-sphere constructions (e.g., tensor networks of two spheres) will not reproduce the single-sphere CFT limit with the same finite-size corrections; comparing spectra of single-sphere versus coupled-sphere systems would offer a direct check of the paper's structural claim.
  • If the low-l dominance assumption is correct, the semiclassical commutator (27) combined with the oscillator reduction provides a concrete route to analytically derive the emergent conformal generators (like Λ_z and D) directly from the density-mode algebra, which would turn the numerical observation of [Λ_z,[Λ_z,D]]=4D into a theorem within a suitable scaling limit.

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

2 major / 4 minor

Summary. The paper analyzes the Lie algebra generated by lowest-Landau-level density modes n^A_{l,m} on the fuzzy sphere. It derives the closed commutator (9), proves that the abstract bracket satisfies the Jacobi identity via a faithful fermionic bilinear realization (Sec. 4.2), and identifies commuting subalgebras. Two thermodynamic limits are studied: a planar limit with l ~ sqrt(s), where the algebra approaches a spin-enriched Girvin-MacDonald-Platzman algebra, and a commutative limit with l << sqrt(s), where it approaches a semiclassical Poisson-bracket algebra. In the low-spin-flip subspace above the paramagnetic state, the modes are shown to satisfy an approximate oscillator algebra with explicit norm bounds. The paper also constructs an explicit six-dimensional so(3,2) representation from density modes at s=1/2 and discusses an so(3)-equivariant coproduct extension to larger systems, arguing that this coproduct splits the single-sphere representation into a tensor product and therefore does not match the physical thermodynamic limit. The abstract and Sec. 6 further claim that a direct density-mode realization of so(3,2) exists only for s=1/2.

Significance. If the algebraic results stand, the paper is a useful step toward a rigorous foundation for fuzzy-sphere CFT simulations. The derivation of the density-mode commutator and the proof that it defines a genuine Lie algebra are self-contained and valuable; the two large-s limits are natural and well motivated; and the quantitative bounds in App. C go beyond the earlier heuristic oscillator observation. The explicit s=1/2 so(3,2) representation is a neat toy construction, and the discussion of the coproduct mismatch is honest. However, the headline no-go statement about so(3,2) is not fully proved: App. E proves only a unitary so(5) statement, while the so(3,2) case relevant to the paper's CFT motivation is left as a conjecture. The connection between the algebraic large-s results and actual fuzzy-sphere criticality also rests on an unproven low-angular-momentum dominance assumption. The central algebra theorems are nevertheless sound and useful.

major comments (2)
  1. [Abstract; Sec. 6.1; App. E] The assertion that an so(3,2) representation generated by density modes 'exists only in the minimal two-electron system' is not established by App. E. App. E assumes Fock-space unitarity (D_- = D_+^†) and proves only the absence of a unitary so(5) extension for s>1/2, with the so(3,2) case explicitly hedged as 'most likely also not'. For the non-unitary so(3,2) case, which is the one relevant to the CFT motivation and to the explicit s=1/2 construction of Sec. 6.1 (where D_± contain factors of i), the condition {X_+,X_+^†} ∝ I is replaced by the weaker {X_+,X_-} ∝ I, so the step forcing |k_m| = const no longer applies. The abstract and Sec. 6.1 should either be weakened to the unitary so(5) no-go, or a separate proof for non-unitary so(3,2) must be supplied. This is a load-bearing point because the paper's stated goal is to test direct density-mode realization of so(3,2).
  2. [Sec. 5.2; Sec. 5.3; Sec. 6] The physical reading of the thermodynamic-limit results depends on the statement that 'the low-energy sector of the critical fuzzy-sphere Hamiltonian is dominated by modes with angular momentum l << sqrt(s)'. This is presented in Sec. 5.2 as an expectation supported by [17,41] and numerical observations, not as a derived result. If the critical sector receives significant weight from l ~ sqrt(s) modes or from couplings between modes, the semiclassical bracket of Sec. 5.2 and the oscillator approximation of Sec. 5.3 would not describe the CFT limit, although the algebraic theorems themselves would remain true. The manuscript should label this as an explicit conjecture and, ideally, provide a quantitative diagnostic (for example, the s-dependence of spectral weight in the l ~ sqrt(s) sector of the critical Hamiltonians) so the scope of the CFT connection is clear.
minor comments (4)
  1. [Sec. 4.1, Eq. (9)] The summation condition 'l+l'+Leven' is easy to misread as a variable named Leven; please write 'l+l'+L even' with a space or introduce a phrase such as 'with l+l'+L even'.
  2. [Sec. 5.1, Eqs. (20)-(24)] The inverse angular decomposition in Eq. (24) writes planar modes in terms of sphere modes n^A_{L,M} with continuous L(Δ). This is a formal device and should be stated as such; the resulting commutator is approximate and not an exact equality at finite s.
  3. [Sec. 5.3, App. C] The norm bound in App. C is derived for fixed k and l>0; please state explicitly that the same bound is used for n^0_{l,m} and that the bound is independent of m, since the main text uses both facts.
  4. [Sec. 2.2, Eq. (2)] The R± and S± generators are nonstandard; a brief indication of how they fit into the standard so(3,2) root system, or a cross-reference to [14], would help the reader verify the signs.

Circularity Check

0 steps flagged

No significant circularity: the algebra, Jacobi proof, thermodynamic limits, and oscillator bounds are derived from independent fermionic or external inputs; the main gaps are unsupported assumptions, not circular reductions.

full rationale

The central derivation chain is self-contained. The density-mode bracket (9) is obtained from the explicit fermion-bilinear realization (11) and the u(2N) commutator structure; Jacobi is proven in Sec. 4.2 via faithfulness of the map rho, not by assuming the target algebra. The planar GMP limit (22) follows from the standard quantum-plane algebra (19), and the semiclassical limit (26) is quoted from external noncommutative-geometry results [37]; neither limit is fitted to the conclusions. The oscillator approximation (30) is derived from operator-norm bounds in Appendix C, not from the desired harmonic structure. The s=1/2 so(3,2) representation is checked directly against the defining relations (2). Self-citations ([1], [6], [44]) are motivational or future-work and are not load-bearing. Two caveats are real but are not circularity: Sec. 5.2's assertion that the low-energy sector is dominated by l << sqrt(s) is an expectation supported by [17,41], not derived here; and Appendix E proves only the unitary so(5) obstruction while explicitly hedging 'most likely also not so(3,2)' for the nonunitary case. These are proof gaps or assumptions, not cases where a result reduces to its own input by construction, and no fitted parameter is relabeled as a prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

No free parameters are fitted; the central results are structural. The paper relies on standard angular momentum and fuzzy-geometry asymptotics, the fermionic LLL realization, and the conjectural low-energy sector identification. The only place where the proof is narrower than the claim is the no-go in App E, which assumes Fock-unitarity for the so(5) version.

axioms (8)
  • domain assumption The LLL-projected fermion fields obey canonical anticommutation relations and the Fock space is built from c_{m,σ}.
    Defines the Hilbert space and the fermionic bilinear realization used throughout; all subsequent density-mode identities rest on it (Sec 3.1).
  • standard math Monopole harmonics Y^{(s)}_{s,m} are complete on S^2 and fuzzy spherical harmonics Ŷ_{l,m} form a basis of Mat_{2s+1} with the product expansion (31).
    Used to derive the mode commutator (9) and prove linear independence/faithfulness (Sec 4.1, App. B); quoted from [11,12,34,36].
  • domain assumption For l∼√s and l≪s, Clebsch-Gordan coefficients and Ŷ matrix elements are approximated by Bessel functions / semiclassical spherical harmonics (Eqs 14-17).
    This bridges the fuzzy sphere modes to the planar GMP and Poisson-bracket limits (Sec 5.1-5.2); external references [38,39] are cited, but no uniform error estimates are given.
  • domain assumption In the semiclassical limit, matrix multiplication of Ŷ tends to pointwise multiplication and commutators to the S^2 Poisson bracket with O(1/s) correction (Eq 26).
    Central to Eq (27) for small-l modes; standard in fuzzy geometry [36,37], but treated here as an asymptotic assumption rather than proved.
  • domain assumption The low-energy spectrum of critical fuzzy-sphere Hamiltonians is dominated by small-angular-momentum modes l≪√s.
    Explicitly stated as expectation in Sec 5.2 and used to motivate the oscillator approximation and the thermodynamic-limit discussion in Sec 6; if false, the CFT relevance of the derived limits is lost.
  • domain assumption Fuzzy-sphere Hamiltonians conjecturally realize 3d CFTs in the thermodynamic limit.
    Background conjecture from [7] and later works; motivates the paper but is not needed for the algebraic theorems; the paper labels it conjectural.
  • domain assumption The no-go analysis restricts to generators of the form n^A(X) with X a (2s+1)x(2s+1) orbital matrix; for the unitary so(5) version, D_-=(D_+)† with respect to the Fock inner product.
    Defines the search space in App E; the Fock-unitarity assumption applies to so(5), and the text hedges the so(3,2) conclusion, so the main-text statement that no so(3,2) extension exists is stronger than what is proved.
  • standard math Standard angular-momentum identities: 3j/6j orthogonality, triangle selection rules, Stirling approximation, Cauchy-Schwarz.
    Used in App. C for operator norm bounds and in Sec 4.2 for commuting subalgebras.

pith-pipeline@v1.3.0-alltime-deepseek · 21120 in / 18162 out tokens · 160554 ms · 2026-08-02T22:57:17.753681+00:00 · methodology

0 comments
read the original abstract

Fuzzy sphere models conjecturally realize 3d CFTs in small systems of spinful fermions, but why they work so well is still not fully understood. Their Hamiltonians are built from electron density operators projected to the lowest Landau level. We analyze the Lie algebra generated by these density modes and its large-$s$ limits. Depending on how the limit is taken, the algebra approaches either the Girvin-MacDonald-Platzman algebra in a local planar limit or a semiclassical algebra for low-angular-momentum modes in a global commutative limit. With an additional restriction to a low-excitation sector above the paramagnetic state, the density modes become approximate harmonic oscillators. We also test whether the conformal algebra $so(3,2)$ can be realized directly by density modes. Such a representation exists only in the minimal two-electron system; its natural coproduct extension does not match the physical thermodynamic limit of a single growing fuzzy sphere.

discussion (0)

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

Forward citations

Cited by 1 Pith paper

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

  1. Non-Perturbative SDiff Covariance of Fractional Quantum Hall Excitations

    cond-mat.str-el 2026-02 unverdicted novelty 7.0

    The effective Maxwell-Chern-Simons theory for FQH excitations admits a non-perturbative unitary SDiff-equivariant construction that is nevertheless non-differentiable.

Reference graph

Works this paper leans on

44 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [1]

    M. S. Zini and Z. Wang, Communications in Mathematical Physics363, 877 (2018)

    2018

  2. [2]

    Feiguin, S

    A. Feiguin, S. Trebst, A. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, Physical Review Letters98, 160409 (2007)

    2007

  3. [3]

    Wang,Topological quantum computation, 112 (American Mathematical Soc., 2010)

    Z. Wang,Topological quantum computation, 112 (American Mathematical Soc., 2010)

    2010

  4. [4]

    Koo and H

    W. Koo and H. Saleur, Nuclear Physics B426, 459 (1994)

    1994

  5. [5]

    Inamura and K

    K. Inamura and K. Ohmori, SciPost Physics16, 143 (2024)

    2024

  6. [6]

    Eck and P

    L. Eck and P. Fendley, SciPost Phys.18, 170 (2025),2408.04006

    Pith/arXiv arXiv 2025

  7. [7]

    W. Zhu, C. Han, E. Huffman, J. S. Hofmann, and Y.-C. He, Phys. Rev. X13, 021009 (2023),2210. 13482

    2023

  8. [8]

    Lao and S

    B.-X. Lao and S. Rychkov, SciPost Phys.15, 243 (2023),2307.02540

    Pith/arXiv arXiv 2023

  9. [9]

    A. M. L¨ auchli, L. Herviou, P. H. Wilhelm, and S. Rychkov, SciPost Phys.19, 076 (2025),2504.00842

    arXiv 2025

  10. [10]

    Voinea, R

    C. Voinea, R. Fan, N. Regnault, and Z. Papi´ c, Phys. Rev. X15, 031007 (2025), URLhttps://link. aps.org/doi/10.1103/bf4k-phl9

    work page doi:10.1103/bf4k-phl9 2025

  11. [11]

    Madore, Class

    J. Madore, Class. Quant. Grav.9, 69 (1992)

    1992

  12. [12]

    Grosse, C

    H. Grosse, C. Klimcik, and P. Presnajder, Int. J. Theor. Phys.35, 231 (1996),hep-th/9505175

    Pith/arXiv arXiv 1996

  13. [13]

    He (2025),2506.14904

    Y.-C. He (2025),2506.14904

    Pith/arXiv arXiv 2025

  14. [14]

    Celeghini and M

    E. Celeghini and M. Del Olmo, Annals of Physics333, 90 (2013)

    2013

  15. [15]

    P. A. Dirac, Journal of Mathematical Physics (New York)(US)4(1963)

    1963

  16. [16]

    T. T. Wu and C. N. Yang, Nucl. Phys. B107, 365 (1976)

    1976

  17. [17]

    Fardelli, A

    G. Fardelli, A. L. Fitzpatrick, and E. Katz (2024),2409.02998

    Pith/arXiv arXiv 2024

  18. [18]

    J. S. Hofmann, F. Goth, W. Zhu, Y.-C. He, and E. Huffman, SciPost Phys. Core7, 028 (2024), 2310.19880

    Pith/arXiv arXiv 2024

  19. [19]

    Taylor, C

    J. Taylor, C. Voinea, Z. Papi´ c, and R. Fan (2025),2506.22539

    Pith/arXiv arXiv 2025

  20. [20]

    A. Dey, L. Herviou, C. Mudry, and A. M. L¨ auchli (2025),2510.09755

    Pith/arXiv arXiv 2025

  21. [21]

    W. Guo, Z. Zhou, T.-C. Wei, and Y.-C. He (2025),2512.02234

    Pith/arXiv arXiv 2025

  22. [22]

    Z. Zhou, L. Hu, W. Zhu, and Y.-C. He, Phys. Rev. X14, 021044 (2024), URLhttps://link.aps. org/doi/10.1103/PhysRevX.14.021044

    doi:10.1103/physrevx.14.021044 2024

  23. [23]

    B.-B. Chen, X. Zhang, and Z. Y. Meng, Phys. Rev. B110, 125153 (2024), URLhttps://link.aps. org/doi/10.1103/PhysRevB.110.125153. 23

    work page doi:10.1103/physrevb.110.125153 2024

  24. [24]

    R. Fan, J. Dong, and A. Vishwanath (2025),2505.06342

    Pith/arXiv arXiv 2025

  25. [25]

    Elias Mir´ o and O

    J. Elias Mir´ o and O. Delouche, JHEP10, 037 (2025),2505.07655

    Pith/arXiv arXiv 2025

  26. [26]

    Arguello Cruz, I

    E. Arguello Cruz, I. R. Klebanov, G. Tarnopolsky, and Y. Xin (2025),2505.06369

    arXiv 2025

  27. [27]

    Y. Tang, C. Voinea, L. Hu, Z. Papi´ c, and W. Zhu (2025),2512.25054

    arXiv 2025

  28. [28]

    Z. Zhou, D. Gaiotto, and Y.-C. He (2025),2509.08038

    arXiv 2025

  29. [29]

    Voinea, W

    C. Voinea, W. Zhu, N. Regnault, and Z. Papi´ c, Phys. Rev. Lett.136, 076601 (2026), URLhttps: //link.aps.org/doi/10.1103/mztz-fyk3

    work page doi:10.1103/mztz-fyk3 2026

  30. [30]

    Zhou and Y.-C

    Z. Zhou and Y.-C. He, Phys. Rev. Lett.135, 026504 (2025),2410.00087

    Pith/arXiv arXiv 2025

  31. [31]

    Z. Zhou, C. Wang, and Y.-C. He (2025),2507.19580

    Pith/arXiv arXiv 2025

  32. [32]

    Fan (2024),2409.08257

    R. Fan (2024),2409.08257

    Pith/arXiv arXiv 2024

  33. [33]

    Fardelli, A

    G. Fardelli, A. L. Fitzpatrick, and E. Katz (2026),2602.04958

    arXiv 2026

  34. [34]

    S. Iso, Y. Kimura, K. Tanaka, and K. Wakatsuki, Nucl. Phys. B604, 121 (2001),hep-th/0101102

    Pith/arXiv arXiv 2001

  35. [35]

    R. K. Dora and A. C. Balram, Phys. Rev. B111, 115132 (2025), URLhttps://link.aps.org/doi/ 10.1103/PhysRevB.111.115132

    work page doi:10.1103/physrevb.111.115132 2025

  36. [36]

    C.-S. Chu, J. Madore, and H. Steinacker, JHEP08, 038 (2001),hep-th/0106205

    Pith/arXiv arXiv 2001

  37. [37]

    H. C. Steinacker,Quantum Geometry, Matrix Theory, and Gravity(Cambridge University Press, 2024)

    2024

  38. [38]

    Matsuura and F

    S. Matsuura and F. Sugino, PTEP2016, 043B01 (2016),1508.00707

    Pith/arXiv arXiv 2016

  39. [39]

    D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii,Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols(World Scientific Publishing Company, 1988), ISBN 978-981-4415-49-1, 978-9971-5-0107-5

    1988

  40. [40]

    S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. Lett.54, 581 (1985), URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.54.581

    doi:10.1103/physrevlett.54.581 1985

  41. [41]

    Zhou (2025),2503.00100

    Z. Zhou (2025),2503.00100

    arXiv 2025

  42. [42]

    A. P. Balachandran and S. Kurkcuoglu, Int. J. Mod. Phys. A19, 3395 (2004),hep-th/0310026

    Pith/arXiv arXiv 2004

  43. [43]

    Cappelli and L

    A. Cappelli and L. Maffi, JHEP05, 120 (2021),2103.04163

    Pith/arXiv arXiv 2021

  44. [44]

    Janssens and Z

    B. Janssens and Z. Wang, to appear (2026). 24

    2026