Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Particle production in a light-cone gauge fixed Jordanian deformation of $AdS_5\times S^5$

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read A Lax-integrable string on a Jordanian-deformed AdS5×S5 background produces particles at tree level, so its worldsheet S-matrix does not conserve particle number.

desk verdict A careful explicit computation showing cubic on-shell processes in a Jordanian-deformed sigma-model under relaxed level matching; the caveat is real and the abstract oversells it, but the paper earns its place. read the letter →

arxiv 2412.08411 v3 pith:W5BLC3FD submitted 2024-12-11 hep-th

Riccardo Borsato , Sibylle Driezen This is my paper · ORCID
classification hep-th
keywords JordaniandeformationAdS5×S5light-conegaugeworldsheetS-matrixparticleproductionintegrabilityHomogeneousYang-BaxterLaxconnection
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

In string theory, integrable sigma-models are expected to scatter particles without changing their number, a property that underpins exact S-matrix computations. This paper studies a string on a Jordanian deformation of AdS5×S5, a Homogeneous Yang-Baxter deformation that classically admits a Lax connection. After fixing light-cone gauge in an alternative way and expanding around the BMN-like pointlike vacuum, the authors find a cubic Hamiltonian. They show that the associated tree-level T-matrix elements for 1→2 and 2→1 processes are non-vanishing on-shell, meaning particles can be produced and merged. This challenges the naive identification of Lax integrability with S-matrix integrability and motivates a closer look at what 'integrability' means for such deformed string sigma-models.

What carries the argument

The central object is the cubic Hamiltonian density $H_3$ (Eq. (4.6)) that survives after light-cone gauge fixing and the perturbative decompactification expansion, together with the oscillator expansion of the transverse fields with the deformed dispersion relations (5.4). The key input is the existence of real solutions to the energy-momentum conservation conditions for the three-point vertices; in particular, the solution (5.19) deforms the $\eta\to 0$ soft-limit solutions to nonzero momenta at order $\eta^2$. After integrating out the delta functions with the resulting Jacobian (5.22), the cubic T-matrix elements (5.24) are explicitly non-vanishing at order $\eta$, showing that the cubic vertex is not kinematically forbidden. The paper verifies that no field redefinition or canonical transformation can remove these on-shell amplitudes.

What would settle it

Repeat the tree-level scattering calculation in the twisted open-string formulation obtained via the on-shell map of [22], where the Jordanian deformation is represented as an undeformed sigma-model with twisted boundary conditions; if that model's S-matrix factorizes and conserves particle number, the particle production found here would be an artifact of the light-cone gauge choice.

Watch

Extended reading notes

Core claim

The central claim is that the light-cone gauge-fixed bosonic $\sigma$-model on the Jordanian-deformed AdS5×S5 background exhibits tree-level processes in which the number of particles is not conserved. Explicitly, the T-matrix elements $T^{(x)(x)}_{(+)}(p)$ and $T^{(-)(x)}_{(x)}(p)$ of Eq. (5.24) are non-vanishing at first order in the deformation parameter $\eta$, describing the decays $(+) \to (x)+(x)$ and $(x) \to (-)+(x)$, together with their time-reversed mergers. These amplitudes satisfy energy-momentum conservation for generic real momenta when the level-matching condition is relaxed for individual factors, as is standard in factorized scattering. The result contradicts the usual axiom of S-matrix integrability that particle number is conserved, even though the undeformed limit is the well-known AdS5×S5 superstring and the deformed model admits a Lax connection.

Load-bearing premise

The conclusion relies on relaxing the level-matching condition for individual 1→2 and 2→1 factors, so that each factor can have nonzero total momentum; if level matching is enforced, no such process survives.

Editorial extensions

If this is right

  • The usual factorized S-matrix integrability condition of conserved particle number fails for this deformed sigma-model at tree level, even though the model admits a Lax connection before gauge fixing.
  • The 1→2 and 2→1 amplitudes vanish if level matching is imposed globally, but remain nonzero when it is relaxed per factor, as required for factorized scattering in the decompactified theory.
  • The gapless $(-)$ excitation is not the primary explanation for particle production, since the $(+) \to (x)+(x)$ process involves only gapped particles.
  • Field redefinitions and canonical transformations that leave the S-matrix invariant cannot eliminate the nonzero cubic T-matrix elements, so the production is not an artifact of the chosen form of $H_3$.
  • The result blocks the expectation that this Jordanian deformation's worldsheet S-matrix is obtained from the AdS5×S5 S-matrix by a Drinfel'd twist, because the scattering in the alternative light-cone gauge is not factorized.

Reading between the lines

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

  • A natural test is to compute the 2→2 unitarity relation at order $\eta^2$; if the cubic vertices contribute to intermediate states, the S-matrix could still be unitary only if additional production channels appear, which would deepen the tension with factorized integrability.
  • The level-matching subtlety suggests that the apparent production is tied to the decompactified (non-compact) scattering formalism; for closed strings on a circle, the 1→2 processes would be kinematically forbidden, and the physical statement may be that the alternative light-cone gauge does not straightforwardly admit an integrable S-matrix.
  • If these cubic processes also appear in other non-diagonal Yang-Baxter deformations that break the BMN light-cone isometries, the clean distinction between abelian TsT deformations (which twist the known S-matrix) and more general Jordanian deformations would be sharpened.
  • One could try to formulate the gauge-fixed theory in terms of the non-locally related twisted open-string variables [22]; if particle production disappears in those variables, the phenomenon would be a gauge-dependent artifact rather than a fundamental breakdown of integrability.
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 studies the bosonic sigma-model on a Jordanian deformation of AdS5 x S5. After introducing global coordinates and a pointlike classical solution that reduces to the BMN vacuum in the undeformed limit, the authors fix an alternative light-cone gauge, derive the quadratic and cubic Hamiltonians in a decompactification expansion, and quantize the quadratic theory. The central result is that certain cubic T-matrix elements, such as T^(x)(x)_(+) and T^(-)(x)_(x) in Eq. (5.24), are non-vanishing on-shell at order eta, so that 1->2 and 2->1 processes appear at tree level. The authors interpret this as particle production that clashes with the usual factorized S-matrix notion of integrability, while noting that all such processes vanish if the closed-string level-matching condition is enforced.

Significance. The paper explicitly derives, rather than assumes, a non-trivial cubic Hamiltonian from a known integrable background, with no fitted parameters and with independent numerical checks at finite eta. The ancillary Mathematica file and the detailed on-shell analysis are useful assets. If the claimed particle-production amplitudes were matrix elements between physical states, the result would challenge the common assumption that Lax integrability of a sigma-model implies factorized scattering in any light-cone gauge. However, the physical interpretation of the amplitudes is the paper's load-bearing weakness: the non-vanishing elements are computed with relaxed level matching, and the manuscript itself states that enforcing level matching eliminates all 1->2 and 2->1 processes. The central claim as stated in the abstract and Section 5.2 is therefore not established for physical closed-string states, and the paper's own caveat undercuts the headline conclusion.

major comments (3)
  1. [Abstract and Section 5.2, around Eq. (5.24)] The abstract and Section 5.2 state that the model exhibits tree-level particle production for physical particles, but the only non-vanishing amplitudes, Eq. (5.24), are obtained using the on-shell kinematics (5.19) in which the incoming particle has p1 > 0 and hence nonzero total worldsheet momentum. As the paper itself says in the last paragraph of Section 5.2, imposing the closed-string level-matching condition p1 = 0 eliminates every 1->2 and 2->1 process: the (+) -> (x)(x) decay then has no real solution, and the (x) -> (-)(x) decay reduces to a soft (-) emission whose amplitude vanishes. Consequently, Eq. (5.24) are not matrix elements between physical closed-string states of the gauge-fixed theory. The central claim should be reformulated: the calculation establishes non-vanishing cubic factorization data under a relaxed level-matching prescription, not tree-level particle production for physical particles as the abstract claims.
  2. [Section 5.2, last paragraph, and Section 6] The relaxation of level matching is justified by analogy with the standard procedure for defining 2->2 S-matrix factors in factorized scattering, where individual factors may carry nonzero total momentum. That analogy does not automatically license the interpretation of individual 1->2 and 2->1 amplitudes as physical processes, because those amplitudes do not respect the momentum constraint that defines physical states of the closed string. The paper does not provide an independent argument that nonzero-total-momentum 1->2 or 2->1 vertices appear as physical sub-processes in a consistent multi-particle S-matrix; it only states that level matching is relaxed 'in general.' Since the entire tension with S-matrix integrability depends on this step, this is a load-bearing gap that needs either a proof or an explicit, consistently caveated framing of the result as a property of factorization data rather than of physical scattering.
  3. [Table 5.1 and Section 5.2, T(-)(-)(-) entry] The table reports that T(-)(-)(-) and its reversed process 'diverge' due to collinear and IR divergences, and the text explains these divergences as a consequence of the gapless (-) dispersion. This divergent element is not needed for the main claim, but it is part of the cubic T-matrix defined by Eq. (5.12). The paper should state whether the presence of a divergent element affects the well-definedness of the other computed elements or the interpretation of the cubic vertex, or whether the divergence can be removed by an IR regulator that does not alter the finite elements such as Eq. (5.24).
minor comments (5)
  1. [Section 5.2, Eq. (5.12)] The notation T(I)(J)(K) is used both for the coefficients extracted from H3 and for the T-matrix elements after integrating delta functions; although the paper explains the distinction, the visual similarity makes equations like (5.12) and (5.23) hard to read. A different symbol or explicit subscripts would help.
  2. [Section 5.2, after Eq. (5.14)] The sentence 'the signs are uncorrelated' is ambiguous; it should be clarified that the labels + and - in the listed coefficients are independent of one another, or explicitly enumerate the distinct coefficient sets.
  3. [Table 5.1 caption] The caption labels the rows as 'creation processes,' but the table also includes decay and merger processes such as T^(x)(x)_(+), which are not creation-from-vacuum processes. Please rephrase the caption to describe all cubic processes considered.
  4. [Abstract and Section 6] The abstract's claim of 'non-trivial cubic processes for physical particles' is inconsistent with the statement in Section 6 that 'when we impose level-matching the 1 -> 2 and 2 -> 1 processes that we observed do not contribute.' The wording should be aligned to avoid giving the impression of a physical decay process that the paper itself shows disappears under the physical-state condition.
  5. [Section 4, after Eq. (4.6)] The text says that the previous coordinate redefinition (2.16) was chosen to ensure the cubic Hamiltonian vanishes in the undeformed limit, but the argument is only sketched. A brief explanation of how the coordinate shift cancels the would-be cubic terms would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the cubic T-matrix elements are derived directly from the gauge-fixed Hamiltonian, with self-citations only for conventions and the level-matching caveat explicitly disclosed.

full rationale

The central claim—non-vanishing tree-level 1->2 and 2->1 T-matrix elements—is obtained by a self-contained perturbative computation. The paper specifies the deformed background (Eqs. (2.2), (2.5)), the coordinate changes, the light-cone gauge-fixing (Sec. 3), the quadratic and cubic Hamiltonians (Eqs. (4.3), (4.6)), and the oscillator expansion (Eqs. (5.3)-(5.6)). The on-shell amplitudes in Eq. (5.24) follow from substituting this expansion into H3 and enforcing momentum/energy conservation; no parameter is fitted to data and no external 'prediction' is imported. The self-citations that occur—[37] for alternative light-cone gauge-fixings, [41] for global coordinates and the BMN limit, [44] for the background—are background/convention references; the non-zero amplitudes are not read off from them. The only assumption that shapes the headline interpretation is the relaxation of the level-matching condition, which is stated explicitly at the end of Sec. 5.2: the paper concedes that under the standard condition 'there is no single process of the type 1 -> 2 nor 2 -> 1.' This is an honest, clearly disclosed interpretive choice rather than a circular reduction: the calculation would be unchanged, and the vanishing under level matching is a documented consequence rather than a hidden premise. Thus the derivation chain is not circular; at most there is a minor self-citation in the gauge-robustness discussion (Sec. 6), which is not load-bearing for the explicit T-matrix computation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The calculation rests on the known deformed background, the standard light-cone gauge-fixing formalism, and the choice to relax level matching. No constants are fitted to data and no new entities are introduced. The deformation parameter eta is an input from the background, not a fitted constant.

assumptions (5)
  • domain assumption The Jordanian background (2.2)/(2.5), obtained from the r-matrix (2.1), is the correct type IIB supergravity background and can be studied as a bosonic sigma-model.
    Input from Refs. [24,41,43,44]; the paper does not rederive the background but relies on it.
  • domain assumption Classical integrability of the deformed sigma-model, i.e. existence of a flat Lax connection, is preserved for this Homogeneous Yang-Baxter deformation.
    Standard property cited from [2-4]; the paper uses it as background for the discussion rather than as an input to the computation.
  • domain assumption Uniform light-cone gauge fixing via the pointlike solution x+ = tau, with all other fields zero, is a valid quantization scheme, and the coordinate redefinitions (2.16) do not change the physical S-matrix.
    Load-bearing for the entire perturbative expansion; follows the standard procedure of [33,36,37].
  • domain assumption Bosonic truncation plus constant dilaton is sufficient for tree-level bosonic scattering; F3, F5, and fermionic contributions do not affect the computed processes.
    Stated in Section 6; the Fradkin-Tseytlin term is a total derivative for constant dilaton, and fermions couple to fluxes that are argued to be irrelevant at this order.
  • domain assumption The level-matching condition may be relaxed when defining the factorized S-matrix, so that individual 1-to-2 and 2-to-1 factors need not have zero total momentum.
    End of Section 5.2 and Section 6. This is the assumption that makes the observed cubic processes relevant; if strict level matching is imposed on every asymptotic state, the processes vanish.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Particle production in a light-cone gauge fixed Jordanian deformation of $AdS_5\times S^5$." pith.science (2026). https://pith.science/paper/W5BLC3FD

@misc{pith2026241208411,
  author       = {Pith},
  title        = {Pith review of: Particle production in a light-cone gauge fixed Jordanian deformation of $AdS_5\times S^5$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5BLC3FD}},
  note         = {Machine review of arXiv:2412.08411}
}
abstract

We consider a string on a Jordanian deformation of the $AdS_5\times S^5$ spacetime. This model belongs to the larger class of Homogeneous Yang-Baxter deformations, which preserve classical integrability in the sense that one can construct an explicit Lax connection. To study the scattering of bosonic worldsheet excitations, we fix light-cone gauge and expand around a pointlike classical solution that reduces to the BMN vacuum in the undeformed limit. Our analysis shows that the light-cone gauge-fixed Hamiltonian, under a perturbative field expansion, includes cubic terms that give rise to non-trivial cubic processes for physical particles. We discuss this unexpected result in relation to the property of Lax integrability of the sigma-model.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Tree-level S matrix for $\lambda$-deformed AdS3 strings

    hep-th 2026-06 unverdicted novelty 7.0 of 10

    For the λ-deformed AdS3×S3×T4 superstring, the tree-level bosonic worldsheet S matrix is purely elastic for 0≤λ<1, confirming integrability, and ill-defined at λ→1, so that limit does not capture T-dual worldsheet dynamics.

Reference graph

Works this paper leans on

65 extracted references · 20 canonical work pages · cited by 1 Pith paper

  1. [1]

    Review of AdS/CFT Integrability: An Overview

    N. Beisert et al., “Review of AdS/CFT Integrability: An Overview” , Lett. Math. Phys. 99, 3 (2012) , arxiv:1012.3982

    arXiv 2012

  2. [2]

    On integrability of the Yang-Baxter sigma-model

    C. Klimcik, “On integrability of the Yang-Baxter sigma-model” , J.Math.Phys. 50, 043508 (2009) , arxiv:0802.3518

    arXiv 2009

  3. [3]

    On classical q-deformations of integrable sigma-models

    F . Delduc, M. Magro and B. Vicedo, “On classical q-deformations of integrable sigma-models” , JHEP 1311, 192 (2013) , arxiv:1308.3581

    arXiv 2013

  4. [4]

    An integrable deformation of the AdS 5×S5 superstring action

    F . Delduc, M. Magro and B. Vicedo, “An integrable deformation of the AdS 5×S5 superstring action” , Phys.Rev.Lett. 112, 051601 (2014) , arxiv:1309.5850. 19

    arXiv 2014

  5. [5]

    Jordanian deformations of the AdS 5xS 5 superstring

    I. Kawaguchi, T . Matsumoto and K. Yoshida, “Jordanian deformations of the AdS 5xS 5 superstring” , JHEP 1404, 153 (2014) , arxiv:1401.4855

    arXiv 2014

  6. [6]

    On classical Yang-Baxter based deformations of the AdS 5 × S5 superstring

    S. J. van T ongeren, “On classical Yang-Baxter based deformations of the AdS 5 × S5 superstring” , JHEP 1506, 048 (2015) , arxiv:1504.05516

    arXiv 2015

  7. [7]

    Yang–Baxter deformations, AdS/CFT , and twist-noncommut ative gauge theory

    S. J. van T ongeren, “Yang–Baxter deformations, AdS/CFT , and twist-noncommut ative gauge theory” , Nucl. Phys. B904, 148 (2016) , arxiv:1506.01023

    arXiv 2016

  8. [8]

    Constant quasiclassical solutions of the Yang–Baxter qua ntum equation

    V . G. Drinfeld, “Constant quasiclassical solutions of the Yang–Baxter qua ntum equation” , in: “Doklady Akademii Nauk” , 531–535p

    work page

Show all 65 references
  1. [9]

    Bialgebra actions, twists, and universal deformation for mulas

    A. Giaquinto and J. J. Zhang, “Bialgebra actions, twists, and universal deformation for mulas” , J. Pure Appl. Algebra 128, 133 (1998) , hep-th/9411140

    1998 arXiv

  2. [10]

    Twist Deformations of Quantum Integrable Spin Chains

    P . Kulish, “Twist Deformations of Quantum Integrable Spin Chains” , in: “Noncommutative Spacetimes: Symmetries in Noncommutative Geometry and Field Theory” , Springer Berlin Heidelberg (2009), Berlin, Heidelberg, 167–190p, https://doi.org/10.1007/978-3-540-89793-4_9

    2009 doi

  3. [11]

    Abelian Yang–Baxter deformations and TsT transformation s

    D. Osten and S. J. van T ongeren, “Abelian Yang–Baxter deformations and TsT transformation s” , Nucl. Phys. B915, 184 (2017) , arxiv:1608.08504

    2017 arXiv

  4. [12]

    Multiparameter quantum groups and twisted quasitriangul ar Hopf algebras

    N. Reshetikhin, “Multiparameter quantum groups and twisted quasitriangul ar Hopf algebras” , Lett. Math. Phys. 20, 331 (1990)

    1990

  5. [13]

    Beauty and the twist: The Bethe ansatz for twisted N=4 SYM

    N. Beisert and R. Roiban, “Beauty and the twist: The Bethe ansatz for twisted N=4 SYM” , JHEP 0508, 039 (2005) , hep-th/0505187

    2005 arXiv

  6. [14]

    Twisted Bethe equations from a twisted S-matrix

    C. Ahn, Z. Bajnok, D. Bombardelli and R. I. Nepomechie, “Twisted Bethe equations from a twisted S-matrix” , JHEP 1102, 027 (2011) , arxiv:1010.3229

    2011 arXiv

  7. [15]

    The spectral problem for strings on twisted AdS 5 × S5

    M. de Leeuw and S. J. van T ongeren, “The spectral problem for strings on twisted AdS 5 × S5” , Nucl. Phys. B860, 339 (2012) , arxiv:1201.1451

    2012 arXiv

  8. [16]

    Quantum Spectral Curve of γ-twisted N = 4 SYM theory and fishnet CFT

    V . Kazakov ,“Quantum Spectral Curve of γ-twisted N = 4 SYM theory and fishnet CFT” , arxiv:1802.02160, in: “Ludwig Faddeev Memorial Volume” , ed.: M.-L. Ge, A. J. Niemi, K. K. Phua and L. A. T akhtajan, 293–342p

    arXiv

  9. [17]

    Do Drinfeld twists of AdS 5 × S5 survive light-cone quantization?

    S. J. van T ongeren and Y. Zimmermann, “Do Drinfeld twists of AdS 5 × S5 survive light-cone quantization?” , SciPost Phys. Core 5, 028 (2022) , arxiv:2112.10279

    2022 arXiv

  10. [18]

    Integrability in dipole-deformed N = 4 super Yang–Mills

    M. Guica, F . Levkovich-Maslyuk and K. Zarembo, “Integrability in dipole-deformed N = 4 super Yang–Mills” , J. Phys. A50, 394001 (2017) , arxiv:1706.07957

    2017 arXiv

  11. [19]

    Deformed integrable σ-models, classical R-matrices and classical exchange alge bra on Drinfel’d doubles

    B. Vicedo, “Deformed integrable σ-models, classical R-matrices and classical exchange alge bra on Drinfel’d doubles” , J. Phys. A 48, 355203 (2015) , arxiv:1504.06303

    2015 arXiv

  12. [20]

    Lax pair for strings in Lunin-Maldacena background

    S. Frolov , “Lax pair for strings in Lunin-Maldacena background” , JHEP 0505, 069 (2005) , hep-th/0503201

    2005 arXiv

  13. [21]

    On Yang–Baxter models, twist operators, and boundary cond itions

    S. J. Van T ongeren, “On Yang–Baxter models, twist operators, and boundary cond itions” , J. Phys. A 51, 305401 (2018) , arxiv:1804.05680

    2018 arXiv

  14. [22]

    Homogeneous Yang-Baxter deformations as undeformed yet twisted models

    R. Borsato, S. Driezen and J. L. Miramontes, “Homogeneous Yang-Baxter deformations as undeformed yet twisted models” , JHEP 2204, 053 (2022) , arxiv:2112.12025

    2022 arXiv

  15. [23]

    T arget space supergeometry of η and λ-deformed strings

    R. Borsato and L. Wulff, “T arget space supergeometry of η and λ-deformed strings” , JHEP 1610, 045 (2016) , arxiv:1608.03570

    2016 arXiv

  16. [24]

    All Jordanian deformations of the AdS 5 × S5 superstring

    R. Borsato and S. Driezen, “All Jordanian deformations of the AdS 5 × S5 superstring” , SciPost Phys. 14, 160 (2023) , arxiv:2212.11269

    2023 arXiv

  17. [25]

    Almost abelian twists and AdS/CFT

    S. J. van T ongeren, “Almost abelian twists and AdS/CFT” , Phys. Lett. B765, 344 (2017) , arxiv:1610.05677. 20

    2017 arXiv

  18. [26]

    Yang-Baxter σ-models, conformal twists, and noncommutative Yang-Mills theory

    T . Araujo, I. Bakhmatov , E. O. Colgáin, J. Sakamoto, M. M. Sheikh-Jabbari and K. Yoshida, “Yang-Baxter σ-models, conformal twists, and noncommutative Yang-Mills theory” , Phys. Rev. D95, 105006 (2017) , arxiv:1702.02861

    2017 arXiv

  19. [27]

    Exactly marginal operators and duality in four-dimension al N=1 supersymmetric gauge theory

    R. G. Leigh and M. J. Strassler, “Exactly marginal operators and duality in four-dimension al N=1 supersymmetric gauge theory” , Nucl. Phys. B 447, 95 (1995) , hep-th/9503121

    1995 arXiv

  20. [28]

    Deforming field theories with U (1) ×U (1) global symmetry and their gravity duals

    O. Lunin and J. M. Maldacena, “Deforming field theories with U (1) ×U (1) global symmetry and their gravity duals” , JHEP 0505, 033 (2005) , hep-th/0502086

    2005 arXiv

  21. [29]

    String theory and noncommutative geometry

    N. Seiberg and E. Witten, “String theory and noncommutative geometry” , JHEP 9909, 032 (1999) , hep-th/9908142

    1999 arXiv

  22. [30]

    Dipoles, twists and noncommutative gauge theory

    A. Bergman and O. J. Ganor, “Dipoles, twists and noncommutative gauge theory” , JHEP 0010, 018 (2000) , hep-th/0008030

    2000 arXiv

  23. [31]

    Quadratic Twist-Noncommutative Gauge Theory

    T . Meier and S. J. van T ongeren, “Quadratic Twist-Noncommutative Gauge Theory” , Phys. Rev. Lett. 131, 121603 (2023) , arxiv:2301.08757

    2023 arXiv

  24. [32]

    Gauge theory on twist-noncommutative spaces

    T . Meier and S. J. van T ongeren, “Gauge theory on twist-noncommutative spaces” , JHEP 2312, 045 (2023) , arxiv:2305.15470

    2023 arXiv

  25. [33]

    Integrable Hamiltonian for classical strings on AdS 5×S5

    G. Arutyunov and S. Frolov , “Integrable Hamiltonian for classical strings on AdS 5×S5” , JHEP 0502, 059 (2005) , hep-th/0411089

    2005 arXiv

  26. [34]

    Uniform light-cone gauge for strings in AdS 5×S5: Solving su(1|1) sector

    G. Arutyunov and S. Frolov , “Uniform light-cone gauge for strings in AdS 5×S5: Solving su(1|1) sector” , JHEP 0601, 055 (2006) , hep-th/0510208

    2006 arXiv

  27. [35]

    Finite-size Effects from Giant Magnons

    G. Arutyunov , S. Frolov and M. Zamaklar, “Finite-size Effects from Giant Magnons” , Nucl.Phys. B778, 1 (2007) , hep-th/0606126

    2007 arXiv

  28. [36]

    Foundations of the AdS 5×S5 Superstring. Part I

    G. Arutyunov and S. Frolov , “Foundations of the AdS 5×S5 Superstring. Part I” , J.Phys. A42, 254003 (2009) , arxiv:0901.4937

    2009 arXiv

  29. [37]

    Inequivalent light-cone gauge-fixings of strings on AdSn×Sn backgrounds

    R. Borsato, S. Driezen, B. Hoare, A. L. Retore and F . K. Se ibold, “Inequivalent light-cone gauge-fixings of strings on AdSn×Sn backgrounds” , Phys. Rev. D 109, 106023 (2024) , arxiv:2312.17056

    2024 arXiv

  30. [38]

    T T , ~J J , J T and ~JT deformations

    S. Frolov , “T T , ~J J , J T and ~JT deformations” , J. Phys. A 53, 025401 (2020) , arxiv:1907.12117

    2020 arXiv

  31. [39]

    Elliptic deformations of the AdS3 × S3 × T4 string

    B. Hoare, A. L. Retore and F . K. Seibold, “Elliptic deformations of the AdS3 × S3 × T4 string” , JHEP 2404, 042 (2024) , arxiv:2312.14031

    2024

  32. [40]

    Semiclassical spectrum for BMN string in Sch 5 × S5

    H. Ouyang, “Semiclassical spectrum for BMN string in Sch 5 × S5” , JHEP 1712, 126 (2017) , arxiv:1709.06844

    2017 arXiv

  33. [41]

    Semiclassical spectrum of a Jordanian deformation of AdS5×S5

    R. Borsato, S. Driezen, J. M. Nieto García and L. Wyss, “Semiclassical spectrum of a Jordanian deformation of AdS5×S5” , Phys. Rev. D 106, 066015 (2022) , arxiv:2207.14748

    2022 arXiv

  34. [42]

    Regularising spectral curves for homogeneous Yang-Baxte r strings

    S. Driezen and N. Kamath, “Regularising spectral curves for homogeneous Yang-Baxte r strings” , Phys. Lett. B 857, 138971 (2024) , arxiv:2406.09811

    2024 arXiv

  35. [43]

    Unimodular jordanian deformations of integrable superst rings

    S. J. van T ongeren, “Unimodular jordanian deformations of integrable superst rings” , SciPost Phys. 7, 011 (2019) , arxiv:1904.08892

    2019 arXiv

  36. [44]

    A Jordanian deformation of AdS space in type IIB supergravi ty

    I. Kawaguchi, T . Matsumoto and K. Yoshida, “A Jordanian deformation of AdS space in type IIB supergravi ty” , JHEP 1406, 146 (2014) , arxiv:1402.6147

    2014 arXiv

  37. [45]

    Geometry of Schrodinger Space-Times, Global Coordinates , and Harmonic Trapping

    M. Blau, J. Hartong and B. Rollier, “Geometry of Schrodinger Space-Times, Global Coordinates , and Harmonic Trapping”, JHEP 0907, 027 (2009) , arxiv:0904.3304

    2009 arXiv

  38. [46]

    Yang-Baxter deformations and string dualities

    T . Matsumoto and K. Yoshida, “Yang-Baxter deformations and string dualities” , JHEP 1503, 137 (2015) , arxiv:1412.3658

    2015 arXiv

  39. [47]

    Large spin limit of AdS 5×S5 string theory and low-energy expansion of ferromagnetic spin chains

    M. Kruczenski, A. Ryzhov and A. A. Tseytlin, “Large spin limit of AdS 5×S5 string theory and low-energy expansion of ferromagnetic spin chains” , Nucl.Phys. B692, 3 (2004) , hep-th/0403120. 21

    2004 arXiv

  40. [48]

    Semiclassical relativistic strings in S 5 and long coherent operators in N=4 SYM theory

    M. Kruczenski and A. A. Tseytlin, “Semiclassical relativistic strings in S 5 and long coherent operators in N=4 SYM theory” , JHEP 0409, 038 (2004) , hep-th/0406189

    2004 arXiv

  41. [49]

    Homogeneous Yang-Baxter deformations as non-abelian dua ls of the AdS 5 sigma-model

    B. Hoare and A. A. Tseytlin, “Homogeneous Yang-Baxter deformations as non-abelian dua ls of the AdS 5 sigma-model” , J. Phys. A49, 494001 (2016) , arxiv:1609.02550

    2016 arXiv

  42. [50]

    Integrable Deformations of T -Dual σ Models

    R. Borsato and L. Wulff, “Integrable Deformations of T -Dual σ Models” , Phys. Rev. Lett. 117, 251602 (2016) , arxiv:1609.09834

    2016 arXiv

  43. [51]

    On non-abelian T -duality and deformations of supercoset string sigma-models

    R. Borsato and L. Wulff, “On non-abelian T -duality and deformations of supercoset string sigma-models” , JHEP 1710, 024 (2017) , arxiv:1706.10169

    2017 arXiv

  44. [52]

    Quantization of the Kadomtsev–Petviashvili equation

    K. K. Kozlowski, E. Sklyanin and A. T orrielli, “Quantization of the Kadomtsev–Petviashvili equation” , Theor. Math. Phys. 192, 1162 (2017) , arxiv:1607.07685

    2017 arXiv

  45. [53]

    Absence of Particle Production and Factorization of the S M atrix in (1+1)-dimensional Models

    S. J. Parke, “Absence of Particle Production and Factorization of the S M atrix in (1+1)-dimensional Models” , Nucl. Phys. B 174, 166 (1980)

    1980

  46. [54]

    Local charges in involution and hierarchies in integrable sigma-models

    S. Lacroix, M. Magro and B. Vicedo, “Local charges in involution and hierarchies in integrable sigma-models” , JHEP 1709, 117 (2017) , arxiv:1703.01951

    2017 arXiv

  47. [55]

    Local conserved charges in principal chiral models

    J. M. Evans, M. Hassan, N. J. MacKay and A. J. Mountain, “Local conserved charges in principal chiral models” , Nucl. Phys. B 561, 385 (1999) , hep-th/9902008

    1999 arXiv

  48. [56]

    Commuting charges and symmetric spaces

    J. M. Evans and A. J. Mountain, “Commuting charges and symmetric spaces” , Phys. Lett. B 483, 290 (2000) , hep-th/0003264

    2000 arXiv

  49. [57]

    Some Properties of an Analog of the Nonlinear σ Model

    C. R. Nappi, “Some Properties of an Analog of the Nonlinear σ Model” , Phys. Rev. D21, 418 (1980)

    1980

  50. [58]

    On the massless tree-level S-matrix in 2d sigma models

    B. Hoare, N. Levine and A. A. Tseytlin, “On the massless tree-level S-matrix in 2d sigma models” , J. Phys. A 52, 144005 (2019) , arxiv:1812.02549

    2019 arXiv

  51. [59]

    The massless S-matrix of integrable σ-models

    G. Georgiou, “The massless S-matrix of integrable σ-models” , arxiv:2408.03673

    arXiv

  52. [60]

    Worldsheet spectrum in AdS(4)/CFT(3) correspondence

    K. Zarembo, “Worldsheet spectrum in AdS(4)/CFT(3) correspondence” , JHEP 0904, 135 (2009) , arxiv:0903.1747

    2009 arXiv

  53. [61]

    Classical integrability and quantum aspects of the AdS(3) x S(3) x S(3) x S(1) superstring

    P . Sundin and L. Wulff, “Classical integrability and quantum aspects of the AdS(3) x S(3) x S(3) x S(1) superstring” , JHEP 1210, 109 (2012) , arxiv:1207.5531

    2012 arXiv

  54. [62]

    The su (2|2) dynamic S-matrix

    N. Beisert, “The su (2|2) dynamic S-matrix” , Adv.Theor.Math.Phys. 12, 945 (2008) , hep-th/0511082

    2008 arXiv

  55. [63]

    The Zamolodchikov-Faddeev algebra for AdS 5×S5 superstring

    G. Arutyunov , S. Frolov and M. Zamaklar, “The Zamolodchikov-Faddeev algebra for AdS 5×S5 superstring” , JHEP 0704, 002 (2007) , hep-th/0612229

    2007 arXiv

  56. [64]

    Quantum symmetry

    M. Gerstenhaber, A. Giaquinto and S. D. Schack, “Quantum symmetry” , in: “Quantum Groups” , ed.: P . P . Kulish, Springer Berlin Heidelberg (1992), Berlin, Heidelberg, 9– 46p

    1992

  57. [65]

    Hopf structures on the Borel subalgebra of sl (2)

    O. Ogievetsky, “Hopf structures on the Borel subalgebra of sl (2)” , in: “Proceedings of the Winter School "Geometry and Physics"” , Circolo Matematico di Palermo (1994), [185]-199p, http://eudml.org/doc/221213. 22

    1994

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.