pith. sign in

arxiv: 2606.05500 · v1 · pith:OWWJZKARnew · submitted 2026-06-03 · ✦ hep-th · math.RA· math.RT

The spectrum of the bosonic ambitwistor string revisited

Jos\'e M. Figueroa-O'Farrill , Girish S. Vishwa This is my paper

Pith reviewed 2026-06-28 04:41 UTC · model grok-4.3

classification ✦ hep-th math.RAmath.RT
keywords bosonic ambitwistor stringBRST cohomologyBMS3 algebramassless spectrumPoincaré representationslittle groupunitarity
0
0 comments X

The pith

Bosonic ambitwistor string spectrum contains an extra massless vector but is non-unitary.

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

The paper recalculates the spectrum of the bosonic ambitwistor string as the BRST cohomology of the worldsheet theory, equivalently the semi-infinite cohomology of the BMS3 Lie algebra relative to its centre. All cohomology classes sit in the massless sector and reproduce the dilaton, metric and Kalb-Ramond field of the ordinary closed bosonic string together with one additional massless vector. Representation theory of the Poincaré group, obtained by viewing the cohomology as a module over the little-group stabiliser of a massless momentum, shows that the full set of states fails to be unitary.

Core claim

The BRST cohomology of the bosonic ambitwistor string resides entirely in the massless sector and induces a representation of the Poincaré group that includes the standard massless fields of the closed bosonic string together with an additional massless vector; this representation is not unitary.

What carries the argument

Semi-infinite cohomology of the BMS3 Lie algebra relative to its centre, taken in a specific module and interpreted as inducing Poincaré representations analysed via the little-group stabiliser of massless momentum.

If this is right

  • Every physical state in the spectrum is massless.
  • The spectrum contains the dilaton, graviton and Kalb-Ramond field plus one extra massless vector.
  • The extra vector cannot be interpreted as a physical Maxwell field because the representation is non-unitary.
  • The cohomology at fixed massless momentum forms a module over the Lorentz stabiliser group of that momentum.

Where Pith is reading between the lines

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

  • Changing the module in which the BMS3 cohomology is computed might remove the non-unitary vector.
  • The same representation-theoretic test could be applied to other ambitwistor or tensionless string models.
  • Non-unitarity may limit the use of this spectrum for computing gravitational scattering amplitudes.

Load-bearing premise

That the cohomology classes induce Poincaré representations whose unitarity can be diagnosed by their transformation properties under the little-group stabiliser of a massless momentum.

What would settle it

An explicit calculation that either produces massive cohomology classes or shows that the extra vector satisfies the positive-norm conditions required for unitary massless Poincaré representations.

read the original abstract

We revisit the calculation of the spectrum of the bosonic ambitwistor string, understood as the BRST cohomology or, equivalently, as the semi-infinite cohomology of the $\mathrm{BMS}_3$ Lie algebra relative to the centre with values in a particular module. We work in momentum space, which allows us to work algebraically and interpret the BRST cohomology as inducing representations of the Poincar\'e group. In agreement with the existing literature, we find that all the cohomology resides in the massless sector, but a careful representation-theoretic analysis of the spectrum reveals, in addition to the usual massless sector of the closed bosonic string (dilaton, metric and Kalb--Ramond field), also a massless vector. We devote a large part of the paper to describing the cohomology at a massless momentum $p$ as a module over the stabiliser $H$ of $p$ in the Lorentz group, a task which is made difficult due to $H$ not acting reducibly when $p\neq 0$. This allows us to conclude that the spectrum is not unitary, forbidding the interpretation of the extra massless vector as a Maxwell field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper revisits the spectrum of the bosonic ambitwistor string via BRST or equivalently semi-infinite cohomology of the BMS₃ Lie algebra relative to its centre, taken in a particular module. Working algebraically in momentum space, the cohomology is interpreted as inducing Poincaré representations. All cohomology is found to reside in the massless sector; in addition to the standard closed bosonic string massless fields (dilaton, metric, Kalb-Ramond), an extra massless vector appears. A detailed analysis treats the cohomology at massless momentum p as a module over the little-group stabiliser H (noting that H does not act reducibly for p ≠ 0), leading to the conclusion that the spectrum is non-unitary and that the extra vector cannot be interpreted as a Maxwell field.

Significance. If the module choice and the induction from cohomology to Poincaré representations are valid, the work supplies a representation-theoretic clarification of the ambitwistor spectrum that confirms its massless character while exposing an additional state and its unitarity properties. The algebraic momentum-space approach together with standard Lie-algebra cohomology tools constitutes a methodological strength, enabling parameter-free computations and direct module decompositions. The non-unitarity result, however, restricts possible physical applications of the extra vector.

major comments (2)
  1. Representation-theoretic analysis at massless momentum p: the presence of the extra massless vector and the non-unitarity verdict rest on the specific module chosen for the semi-infinite BMS₃ cohomology and on the subsequent decomposition of that cohomology as an H-module. The manuscript must supply an explicit definition of this module together with a direct comparison to the modules implicit in the prior ambitwistor literature; without this, both the extra vector and the non-unitarity conclusion risk being artifacts of the module rather than intrinsic features of the theory.
  2. The step asserting that the computed cohomology induces Poincaré representations whose restriction to the little-group stabiliser H can be used to diagnose unitarity is load-bearing for the central claim. Additional justification is required for this induction, especially given the noted fact that H does not act reducibly when p ≠ 0.
minor comments (1)
  1. The abstract and introduction would benefit from a brief parenthetical reminder of the definition of the stabiliser H to improve accessibility for readers outside the immediate representation-theory literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's thorough review and the opportunity to clarify the key aspects of our work. We address the major comments below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: Representation-theoretic analysis at massless momentum p: the presence of the extra massless vector and the non-unitarity verdict rest on the specific module chosen for the semi-infinite BMS₃ cohomology and on the subsequent decomposition of that cohomology as an H-module. The manuscript must supply an explicit definition of this module together with a direct comparison to the modules implicit in the prior ambitwistor literature; without this, both the extra vector and the non-unitarity conclusion risk being artifacts of the module rather than intrinsic features of the theory.

    Authors: The module is explicitly defined in Section 2 of the manuscript as the semi-infinite module for the BMS₃ algebra with the standard oscillator representation used in the ambitwistor string literature. We already compare to prior work by noting that our cohomology reproduces the known massless fields (dilaton, metric, Kalb-Ramond) while identifying an additional vector. However, to make the comparison more direct, we will add an explicit statement of the module and a table or paragraph contrasting it with modules in references such as the prior ambitwistor literature. This will be incorporated in the revised version. revision: partial

  2. Referee: The step asserting that the computed cohomology induces Poincaré representations whose restriction to the little-group stabiliser H can be used to diagnose unitarity is load-bearing for the central claim. Additional justification is required for this induction, especially given the noted fact that H does not act reducibly when p ≠ 0.

    Authors: The induction follows from the momentum-space computation of the BRST cohomology, which by construction carries an action of the Poincaré algebra, restricting to the little group H on the cohomology at fixed p. We handle the non-reducible action by providing an explicit decomposition of the cohomology into indecomposable modules over H in Section 4, which permits the analysis of the bilinear form to diagnose non-unitarity. We will add further explanatory text in the revised manuscript to justify this step more explicitly. revision: yes

Circularity Check

0 steps flagged

No circularity: algebraic cohomology computation in explicitly chosen module

full rationale

The derivation proceeds by direct computation of semi-infinite BMS3 cohomology relative to the centre in a specified module, performed algebraically in momentum space, followed by explicit decomposition of the resulting cohomology as an H-module (where H is the little-group stabiliser) to diagnose the Poincaré representations and unitarity. No step equates a claimed output to an input by construction, renames a fit as a prediction, or relies on a self-citation chain whose content is itself unverified; the module choice is stated as an assumption and the representation analysis is carried out independently. This is a standard self-contained algebraic derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard Lie-algebra cohomology constructions and the choice of a specific module for the BMS3 algebra; no free parameters or new postulated entities are introduced.

axioms (2)
  • standard math BRST cohomology is equivalent to the semi-infinite cohomology of the BMS3 Lie algebra relative to the centre
    Invoked to equate the two cohomology computations in the abstract.
  • domain assumption The chosen module for the cohomology values allows the states to be interpreted as inducing Poincaré representations
    Required for the momentum-space representation-theoretic analysis.

pith-pipeline@v0.9.1-grok · 5743 in / 1328 out tokens · 29058 ms · 2026-06-28T04:41:50.581287+00:00 · methodology

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

56 extracted references · 33 linked inside Pith

  1. [1]

    Ambitwistor strings and the scattering equations,

    L. Mason and D. Skinner, “Ambitwistor strings and the scattering equations,”JHEP07(2014) 048, arXiv:1311.2564 [hep-th]. [Pages 2, 3, and 6.]

    Pith/arXiv arXiv 2014

  2. [2]

    Ambitwistor strings and the scattering equations at one loop,

    T. Adamo, E. Casali, and D. Skinner, “Ambitwistor strings and the scattering equations at one loop,”JHEP04 (2014) 104,arXiv:1312.3828 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2014

  3. [3]

    Infinite Tension Limit of the Pure Spinor Superstring,

    N. Berkovits, “Infinite Tension Limit of the Pure Spinor Superstring,”JHEP03(2014) 017,arXiv:1311.4156 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2014

  4. [4]

    Perturbative gauge theory as a string theory in twistor space,

    E. Witten, “Perturbative gauge theory as a string theory in twistor space,”Commun. Math. Phys.252(2004) 189–258,arXiv:hep-th/0312171. [Page 2.]

    Pith/arXiv arXiv 2004

  5. [5]

    An Alternative string theory in twistor space for N=4 superYang-Mills,

    N. Berkovits, “An Alternative string theory in twistor space for N=4 superYang-Mills,”Phys. Rev. Lett.93(2004) 011601,arXiv:hep-th/0402045. [Page 2.]

    Pith/arXiv arXiv 2004

  6. [6]

    Scattering equations and Kawai-Lewellen-Tye orthogonality,

    F. Cachazo, S. He, and E. Y. Yuan, “Scattering equations and Kawai-Lewellen-Tye orthogonality,”Phys. Rev. D90 no. 6, (2014) 065001,arXiv:1306.6575 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2014

  7. [7]

    Scattering of Massless Particles in Arbitrary Dimensions,

    F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles in Arbitrary Dimensions,”Phys. Rev. Lett.113 no. 17, (2014) 171601,arXiv:1307.2199 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2014

  8. [8]

    Scattering in Three Dimensions from Rational Maps,

    F. Cachazo, S. He, and E. Y. Yuan, “Scattering in Three Dimensions from Rational Maps,”JHEP10(2013) 141, arXiv:1306.2962 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2013

  9. [9]

    Scattering of Massless Particles: Scalars, Gluons and Gravitons,

    F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles: Scalars, Gluons and Gravitons,”JHEP07 (2014) 033,arXiv:1309.0885 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2014

  10. [10]

    Scattering equations, supergravity integrands, and pure spinors,

    T. Adamo and E. Casali, “Scattering equations, supergravity integrands, and pure spinors,”JHEP05(2015) 120, arXiv:1502.06826 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2015

  11. [11]

    Loop Integrands for Scattering Amplitudes from the Riemann Sphere,

    Y. Geyer, L. Mason, R. Monteiro, and P. Tourkine, “Loop Integrands for Scattering Amplitudes from the Riemann Sphere,”Phys. Rev. Lett.115no. 12, (2015) 121603,arXiv:1507.00321 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2015

  12. [12]

    One-loop amplitudes on the Riemann sphere,

    Y. Geyer, L. Mason, R. Monteiro, and P. Tourkine, “One-loop amplitudes on the Riemann sphere,”JHEP03 (2016) 114,arXiv:1511.06315 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2016

  13. [13]

    Worldsheet Geometries of Ambitwistor String,

    K. Ohmori, “Worldsheet Geometries of Ambitwistor String,”JHEP06(2015) 075,arXiv:1504.02675 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2015

  14. [14]

    A Worldsheet Theory for Supergravity,

    T. Adamo, E. Casali, and D. Skinner, “A Worldsheet Theory for Supergravity,”JHEP02(2015) 116, arXiv:1409.5656 [hep-th]. [Pages 2 and 3.]

    Pith/arXiv arXiv 2015

  15. [15]

    Ambitwistor pure spinor string in a type II supergravity background,

    O. Chandia and B. C. Vallilo, “Ambitwistor pure spinor string in a type II supergravity background,”JHEP06 (2015) 206,arXiv:1505.05122 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2015

  16. [16]

    On-shell type II supergravity from the ambitwistor pure spinor string,

    O. Chandia and B. C. Vallilo, “On-shell type II supergravity from the ambitwistor pure spinor string,”Class. Quant. Grav.33no. 18, (2016) 185003,arXiv:1511.03329 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2016

  17. [17]

    Amplitudes on plane waves from ambitwistor strings,

    T. Adamo, E. Casali, L. Mason, and S. Nekovar, “Amplitudes on plane waves from ambitwistor strings,”JHEP 11(2017) 160,arXiv:1708.09249 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2017

  18. [18]

    Ambitwistor string vertex operators on curved backgrounds,

    T. Adamo, E. Casali, and S. Nekovar, “Ambitwistor string vertex operators on curved backgrounds,”JHEP01 (2019) 213,arXiv:1809.04489 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2019

  19. [19]

    Scattering equations in AdS: scalar correlators in arbitrary dimensions,

    L. Eberhardt, S. Komatsu, and S. Mizera, “Scattering equations in AdS: scalar correlators in arbitrary dimensions,”JHEP11(2020) 158,arXiv:2007.06574 [hep-th]. [Page 2.]

    arXiv 2020

  20. [20]

    Ambitwistor strings and the scattering equations on AdS3×S3,

    K. Roehrig and D. Skinner, “Ambitwistor strings and the scattering equations on AdS3×S3,”JHEP02(2022) 073, arXiv:2007.07234 [hep-th]. [Page 2.]

    arXiv 2022

  21. [21]

    Cosmological Scattering Equations,

    H. Gomez, R. L. Jusinskas, and A. Lipstein, “Cosmological Scattering Equations,”Phys. Rev. Lett.127no. 25, (2021) 251604,arXiv:2106.11903 [hep-th]. [Page 2.]

    arXiv 2021

  22. [22]

    From AdS correlators to Carrollian amplitudes with the scattering equations,

    T. Adamo, I. Surubaru, and B. Zhu, “From AdS correlators to Carrollian amplitudes with the scattering equations,”JHEP02(2026) 198,arXiv:2512.03677 [hep-th]. [Page 2.]

    arXiv 2026

  23. [23]

    The SAGEX review on scattering amplitudes Chapter 6: Ambitwistor Strings and Amplitudes from the Worldsheet,

    Y. Geyer and L. Mason, “The SAGEX review on scattering amplitudes Chapter 6: Ambitwistor Strings and Amplitudes from the Worldsheet,”J. Phys. A55no. 44, (2022) 443007,arXiv:2203.13017 [hep-th] . [Page 2.]

    arXiv 2022

  24. [24]

    Amplitudes for left-handed strings,

    W. Siegel, “Amplitudes for left-handed strings,”arXiv:1512.02569 [hep-th]. [Page 2.]

    Pith/arXiv arXiv

  25. [25]

    Doubledα′-geometry,

    O. Hohm, W. Siegel, and B. Zwiebach, “Doubledα′-geometry,”JHEP02(2014) 065,arXiv:1306.2970 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2014

  26. [26]

    A string theory which isn’t about strings,

    K. Lee, S.-J. Rey, and J. A. Rosabal, “A string theory which isn’t about strings,”JHEP11(2017) 172, arXiv:1708.05707 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2017

  27. [27]

    Factorization of Chiral String Amplitudes,

    Y.-t. Huang, W. Siegel, and E. Y. Yuan, “Factorization of Chiral String Amplitudes,”JHEP09(2016) 101, arXiv:1603.02588 [hep-th]. [Page 2.] THE SPECTRUM OF THE BOSONIC AMBITWISTOR STRING REVISITED 47

    Pith/arXiv arXiv 2016

  28. [28]

    Chiral Closed strings: Four massless states scattering amplitude,

    M. M. Leite and W. Siegel, “Chiral Closed strings: Four massless states scattering amplitude,”JHEP01(2017) 057,arXiv:1610.02052 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2017

  29. [29]

    Windings of twisted strings,

    E. Casali and P. Tourkine, “Windings of twisted strings,”Phys. Rev. D97no. 6, (2018) 061902, arXiv:1710.01241 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2018

  30. [30]

    Tensionless Strings and Galilean Conformal Algebra,

    A. Bagchi, “Tensionless Strings and Galilean Conformal Algebra,”JHEP05(2013) 141,arXiv:1303.0291 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2013

  31. [31]

    Tensionless Strings from Worldsheet Symmetries,

    A. Bagchi, S. Chakrabortty, and P. Parekh, “Tensionless Strings from Worldsheet Symmetries,”JHEP01(2016) 158,arXiv:1507.04361 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2016

  32. [32]

    Tensionless Superstrings: View from the Worldsheet,

    A. Bagchi, S. Chakrabortty, and P. Parekh, “Tensionless Superstrings: View from the Worldsheet,”JHEP10 (2016) 113,arXiv:1606.09628 [hep-th]. [Page 2.]

    Pith/arXiv arXiv 2016

  33. [33]

    A tale of three — tensionless strings and vacuum structure,

    A. Bagchi, A. Banerjee, S. Chakrabortty, S. Dutta, and P. Parekh, “A tale of three — tensionless strings and vacuum structure,”JHEP04(2020) 061,arXiv:2001.00354 [hep-th]. [Pages 2 and 6.]

    arXiv 2020

  34. [34]

    Quantum null (super)strings,

    J. Gamboa, C. Ramirez, and M. Ruiz-Altaba, “Quantum null (super)strings,”Phys. Lett. B225(1989) 335–339. [Page 2.]

    1989

  35. [35]

    On the null origin of the ambitwistor string,

    E. Casali and P. Tourkine, “On the null origin of the ambitwistor string,”JHEP11(2016) 036, arXiv:1606.05636 [hep-th]. [Pages 2 and 6.]

    Pith/arXiv arXiv 2016

  36. [36]

    Celestial operator products from the worldsheet,

    T. Adamo, W. Bu, E. Casali, and A. Sharma, “Celestial operator products from the worldsheet,”JHEP06(2022) 052,arXiv:2111.02279 [hep-th]. [Page 2.]

    arXiv 2022

  37. [37]

    All-order celestial OPE in the MHV sector,

    T. Adamo, W. Bu, E. Casali, and A. Sharma, “All-order celestial OPE in the MHV sector,”JHEP03(2023) 252, arXiv:2211.17124 [hep-th]. [Page 2.]

    arXiv 2023

  38. [38]

    The BRST quantisation of chiral BMS-like field theories,

    J. M. Figueroa-O’Farrill and G. S. Vishwa, “The BRST quantisation of chiral BMS-like field theories,”J. Math. Phys.66no. 4, (2025) 042303,arXiv:2407.12778 [hep-th]. [Pages 2, 3, 5, 6, 16, and 33.]

    arXiv 2025

  39. [39]

    Field theory actions for ambitwistor string and superstring,

    N. Berkovits and M. Lize, “Field theory actions for ambitwistor string and superstring,”JHEP09(2018) 097, arXiv:1807.07661 [hep-th]. [Pages 2, 3, 25, 27, 30, and 32.]

    Pith/arXiv arXiv 2018

  40. [40]

    Chiral strings, the sectorized description and their integrated vertex operators,

    R. Lipinski Jusinskas, “Chiral strings, the sectorized description and their integrated vertex operators,”JHEP12 (2019) 143,arXiv:1909.04069 [hep-th]. [Page 2.]

    arXiv 2019

  41. [41]

    Null spinning strings,

    J. Gamboa, C. Ramirez, and M. Ruiz-Altaba, “Null spinning strings,”Nucl. Phys. B338(1990) 143–187. [Page 2.]

    1990

  42. [42]

    Yang-Mills theory from the worldsheet,

    T. Adamo, E. Casali, and S. Nekovar, “Yang-Mills theory from the worldsheet,”Phys. Rev. D98no. 8, (2018) 086022,arXiv:1807.09171 [hep-th]. [Page 3.]

    Pith/arXiv arXiv 2018

  43. [43]

    Non-relativistic quantum strings from gauged WZW models,

    J. M. Figueroa-O’Farrill and G. S. Vishwa, “Non-relativistic quantum strings from gauged WZW models,”JHEP 08(2025) 025,arXiv:2505.03462 [hep-th]. [Page 3.]

    arXiv 2025

  44. [44]

    Worldsheet formalism for decoupling limits in string theory,

    J. Gomis and Z. Yan, “Worldsheet formalism for decoupling limits in string theory,”JHEP07(2024) 102, arXiv:2311.10565 [hep-th]. [Page 3.]

    Pith/arXiv arXiv 2024

  45. [45]

    Unification of Decoupling Limits in String and M Theory,

    C. D. A. Blair, J. Lahnsteiner, N. A. Obers, and Z. Yan, “Unification of Decoupling Limits in String and M Theory,”Phys. Rev. Lett.132no. 16, (2024) 161603,arXiv:2311.10564 [hep-th]. [Page 3.]

    arXiv 2024

  46. [46]

    Carrollian superstring in the flipped vacuum,

    B. Chen and Z. Hu, “Carrollian superstring in the flipped vacuum,”Phys. Rev. D112no. 4, (2025) 046005, arXiv:2501.11011 [hep-th]. [Page 3.]

    arXiv 2025

  47. [47]

    Quantum carrollian bosonic strings,

    J. Figueroa-O’Farrill, E. Have, and N. A. Obers, “Quantum carrollian bosonic strings,”arXiv:2509.04397 [hep-th]. [Pages 3 and 33.]

    arXiv

  48. [48]

    On unitary representations of the inhomogeneous Lorentz group,

    E. Wigner, “On unitary representations of the inhomogeneous Lorentz group,”Ann. of Math. (2)40no. 1, (1939) 149–204. [Page 4.]

    1939

  49. [49]

    G. W. Mackey,The theory of unitary group representations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955. [Page 4.]

    1976

  50. [50]

    BMS Modules in Three Dimensions,

    A. Campoleoni, H. A. Gonzalez, B. Oblak, and M. Riegler, “BMS Modules in Three Dimensions,”Int. J. Mod. Phys. A31no. 12, (2016) 1650068,arXiv:1603.03812. [Page 6.]

    Pith/arXiv arXiv 2016

  51. [51]

    D. S. Dummit and R. M. Foote,Abstract algebra. John Wiley & Sons, Inc., Hoboken, NJ, third ed., 2004. [Pages 12, 13, and 29.]

    2004

  52. [52]

    A Mathematica package for computing operator product expansions,

    K. Thielemans, “A Mathematica package for computing operator product expansions,”Int. J. Mod. Phys. C2 (1991) 787–798. [Page 16.]

    1991

  53. [53]

    A Mathematica package for computing operator product expansions,

    K. Thielemans, “A Mathematica package for computing operator product expansions,” in2nd International Workshop on Software Engineering, Artificial Intelligence and Expert Systems for High-energy and Nuclear Physics. 1992. [Page 16.]

    1992

  54. [54]

    Thielemans,An Algorithmic approach to operator product expansions, W algebras and W strings

    K. Thielemans,An Algorithmic approach to operator product expansions, W algebras and W strings. PhD thesis, Leuven U., 1994.arXiv:hep-th/9506159. [Page 16.]

    Pith/arXiv arXiv 1994

  55. [55]

    Quantum Carroll/fracton particles,

    J. Figueroa-O’Farrill, A. Pérez, and S. Prohazka, “Quantum Carroll/fracton particles,”JHEP10(2023) 041, arXiv:2307.05674 [hep-th]. [Page 25.]

    arXiv 2023

  56. [56]

    LiE, a software package for Lie group computations,

    M. A. A. van Leeuwen, “LiE, a software package for Lie group computations,”Euromath Bull.1no. 2, (1994) 83–94. [Page 44.] Maxwell Institute and School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, Scotland, United Kingdom Email address, JMF:j.m.figueroa@ed.ac.uk, ORCID: 0000-0002-930...

    1994