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AdS3 integrability, the Sine-Gordon model, and fractional supersymmetry

Alessandro Torrielli*

  • *Contact author: a.torrielli@surrey.ac.uk
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Phys. Rev. D 113, 065013 – Published 18 March, 2026

DOI: https://doi.org/10.1103/zbtr-ztw6

Abstract

The massless S matrix of the pure Ramond-Ramond sector of string theory on three-dimensional Anti de Sitter times a four-torus is very similar but also crucially distinct from Fendley-Intriligator’s model famous N=2 S-matrix, in turn related to Sine-Gordon model taken at a special coupling. In this article we review the reason why supersymmetry emerges but with a fractional statistics of the particles. We then use this to obtain an interpolating S matrix between massless AdS3 and Sine-Gordon model, and further find two interpolating S matrices between these and the mixed-flux relativistic S matrix, solving the Yang-Baxter equation all the way in between. They are all relativistic invariant and braiding unitary, can be written in terms of particles with fractional statistics, and embed all the AdS3 models into a parent integrable system. This also achieves the reformulation of all the AdS3 integrable scattering problems in the domain of fractional statistics.

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References (44)

  1. A. Babichenko, B. Stefański, Jr., and K. Zarembo, Integrability and the AdS3/CFT2 correspondence, J. High Energy Phys. 03 (2010) 058.
  2. A. Sfondrini, Towards integrability for AdS3/CFT2, J. Phys. A 48, 023001 (2015); R. Borsato, Integrable strings for AdS/CFT arXiv:1605.03173.
  3. F. K. Seibold and A. Sfondrini, AdS3 integrability, tensionless limits, and deformations: A review, arXiv:2408.08414.
  4. G. Arutyunov and S. Frolov, Foundations of the AdS5×S5 superstring. Part I, J. Phys. A 42, 254003 (2009); N. Beisert, C. Ahn, L. F. Alday, Z. Bajnok, J. M. Drummond, L. Freyhult, N. Gromov, R. A. Janik, V. Kazakov, T. Klose et al., Review of AdS/CFT integrability: An overview, Lett. Math. Phys. 99, 3 (2012).
  5. O. Ohlsson Sax and B. Stefański, Jr., Integrability, spin-chains, and the AdS3/CFT2 correspondence, J. High Energy Phys. 08 (2011) 029; R. Borsato, O. Ohlsson Sax, and A. Sfondrini, A dynamic su(1|1)2  S-matrix for AdS3/CFT2, 04 (2013) 113; All-loop Bethe ansatz equations for AdS3/CFT2, 04 (2013) 116; R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański, Jr., and A. Torrielli, The all-loop integrable spin-chain for strings on AdS3×S3×T4: The massive sector, 08 (2013) 043; R. Borsato, O. Ohlsson Sax, A. Sfondrini, and B. Stefański, Jr., The complete AdS3× S3× T4 worldsheet S matrix, 10 (2014) 066; M. Beccaria, F. Levkovich-Maslyuk, G. Macorini, and A. Tseytlin, Quantum corrections to spinning superstrings in AdS3×S3×M4: Determining the dressing phase, 04 (2013) 006; L. Bianchi, V. Forini, and B. Hoare, Two-dimensional S-matrices from unitarity cuts, 07 (2013) 088; O. T. Engelund, R. W. McKeown, and R. Roiban, Generalised unitarity and the worldsheet S-matrix in AdSn×Sn×M10−2n, 08 (2013) 023; R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański, Jr., and A. Torrielli, Dressing phases of AdS3/CFT2, Phys. Rev. D 88, 066004 (2013); P. Sundin and L. Wulff, The complete one-loop BMN S-matrix in AdS3×S3×T4, J. High Energy Phys. 06 (2016) 062.
  6. O. Ohlsson Sax, B. Stefański, Jr., and A. Torrielli, On the massless modes of the AdS3/CFT2 integrable systems, J. High Energy Phys. 03 (2013) 109; O. Ohlsson Sax, A. Sfondrini, and B. Stefański, Jr., Integrability and the conformal field theory of the Higgs branch, 06 (2015) 103; R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański, Jr., and A. Torrielli, On the dressing factors, Bethe equations, Yangian symmetry of strings on AdS3×S3×T4, J. Phys. A 50, 024004 (2017); M. Baggio, O. Ohlsson Sax, A. Sfondrini, B. Stefański Jr., and A. Torrielli, Protected string spectrum in AdS3/CFT2 from worldsheet integrability, J. High Energy Phys. 04 (2017) 091; O. Ohlsson Sax and B. Stefański, Jr., Closed strings and moduli in AdS3/CFT2, 05 (2018) 101.
  7. S. Majumder, O. O. Sax, B. Stefański, Jr., and A. Torrielli, Protected states in AdS3 backgrounds from integrability, J. Phys. A 54, 415401 (2021).
  8. D. Bombardelli, B. Stefański, Jr., and A. Torrielli, The low-energy limit of AdS3/CFT2 and its TBA, J. High Energy Phys. 10 (2018) 177.
  9. T. Lloyd and B. Stefański, Jr., AdS3/CFT2, finite-gap equations and massless modes, J. High Energy Phys. 04 (2014) 179; A. Pittelli, A. Torrielli, and M. Wolf, Secret symmetries of type IIB superstring theory on AdS3×S3×M4, J. Phys. A 47, 455402 (2014); M. C. Abbott and I. Aniceto, Massless Lüscher terms and the limitations of the AdS3 asymptotic Bethe ansatz, Phys. Rev. D 93, 106006 (2016); M. Baggio and A. Sfondrini, Strings on NS-NS backgrounds as integrable deformations, 98, 021902 (2018); A. Dei and A. Sfondrini, Integrable spin chain for stringy Wess-Zumino-Witten models, J. High Energy Phys. 07 (2018) 109; 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); O. Ohlsson Sax and B. Stefański, Jr., On the singularities of the RR AdS3×S3×T4 S matrix, 53, 155402 (2020); M. C. Abbott and I. Aniceto, Integrable field theories with an interacting massless sector, Phys. Rev. D 103, 086017 (2021).
  10. B. Hoare and A. A. Tseytlin, Towards the quantum S-matrix of the Pohlmeyer reduced version of AdS5×S5 superstring theory, Nucl. Phys. B851, 161 (2011); J. M. Nieto Garcia and A. Torrielli, Norms and scalar products for AdS3, J. Phys. A 53, 145401 (2020).
  11. B. Hoare, Towards a two-parameter q-deformation of AdS3×S3×M4 superstrings, Nucl. Phys. B891, 259 (2015).
  12. A. Cavaglià, N. Gromov, B. Stefański, Jr., and A. Torrielli, Quantum Spectral Curve for AdS3/CFT2: A proposal, J. High Energy Phys. 12 (2021) 048; S. Ekhammar and D. Volin, Monodromy bootstrap for su(2|2) quantum spectral curves: From Hubbard model to AdS3/CFT2, 03 (2022) 192.
  13. A. Cavaglià, S. Ekhammar, N. Gromov, and P. Ryan, Exploring the quantum spectral curve for AdS3/CFT2, J. High Energy Phys. 12 (2023) 089; S. Ekhammar, An exploration of Q-systems: From spin chains to low-dimensional AdS/CFT, Uppsala thesis, Department of Physics and Astronomy of the University of Uppsala, Sweden.
  14. S. Ekhammar, N. Gromov, and B. Stefański, Jr., Demystifying the massless sector in AdS3 quantum spectral curve, J. High Energy Phys. 10 (2025) 188.
  15. S. Frolov and A. Sfondrini, Massless S matrices for AdS3/CFT2, J. High Energy Phys. 04 (2022) 067; New dressing factors for AdS3/CFT2, 04 (2022) 162; Mirror thermodynamic Bethe Ansatz for AdS3/CFT2, 03 (2022) 138.
  16. F. K. Seibold and A. Sfondrini, Transfer matrices for AdS3/CFT2, J. High Energy Phys. 05 (2022) 089; A. Brollo, D. le Plat, A. Sfondrini, and R. Suzuki, Tensionless limit of pure–Ramond-Ramond strings and AdS3/CFT2, Phys. Rev. Lett. 131, 161604 (2023); S. Frolov, A. Pribytok, and A. Sfondrini, Ground state energy of twisted AdS3×S3×T4 superstring and the TBA, J. High Energy Phys. 09 (2023) 027; A. Brollo, D. le Plat, A. Sfondrini, and R. Suzuki, More on the tensionless limit of pure-Ramond-Ramond AdS3/CFT2, 12 (2023) 160; M. R. Gaberdiel, F. Lichtner, and B. Nairz, Anomalous dimensions in the symmetric orbifold, 05 (2025) 084; M. R. Gaberdiel, D. Kempel, and B. Nairz, AdS3×S3 magnons in the symmetric orbifold, 05 (2025) 229; M. Fabri, A. Sfondrini, and T. Skrzypek, Perturbed symmetric-product orbifold: First-order mixing and puzzles for integrability, arXiv:2504.13091.
  17. S. Frolov, D. Polvara, and A. Sfondrini, On mixed-flux worldsheet scattering in AdS3/CFT2, J. High Energy Phys. 11 (2023) 055.
  18. X. Kervyn, D. Polvara, and A. Sfondrini, Thermodynamics of integrable N=2 theories, squared, J. High Energy Phys. 09 (2025) 018.
  19. N. Baglioni, D. Polvara, A. Pone, and A. Sfondrini, On the worldsheet S matrix of the AdS3/CFT2 mixed-flux mirror model, J. High Energy Phys. 05 (2024) 237; O. Ohlsson Sax, D. Riabchenko, and B. Stefański, Jr., Worldsheet kinematics, dressing factors and odd crossing in mixed-flux AdS3 backgrounds, 09 (2024) 132; M. R. Gaberdiel, R. Gopakumar, and B. Nairz, Beyond the tensionless limit: Integrability in the symmetric orbifold, 06 (2024) 030; S. Frolov and A. Sfondrini, Comments on integrability in the symmetric orbifold, arXiv:2312.14114; B. Hoare, A. L. Retore, and F. K. Seibold, Elliptic deformations of the AdS3×S3×T4 string, J. High Energy Phys. 04 (2024) 042; S. Frolov, D. Polvara, and A. Sfondrini, Dressing factors for mixed-flux AdS3×S3×T4 superstrings, Phys. Rev. D 111, L081901 (2025); Massive dressing factors for mixed-flux AdS3/CFT2, J. High Energy Phys. 07 (2025) 171.
  20. F. K. Seibold and A. Sfondrini, Interpolating families of integrable AdS3 backgrounds, J. Phys. A 58, 235401 (2025); S. Frolov, D. Polvara, and A. Sfondrini, Dressing factors and mirror thermodynamic bethe ansatz for mixed-flux AdS3/CFT2, arXiv:2507.12191; B. Hoare and F. K. Seibold, Supersymmetry and integrability of the elliptic AdS3×S3×T4 superstring, arXiv:2507.18548.
  21. A. Torrielli, A study of integrable form factors in massless relativistic AdS3, J. Phys. A 55, 175401 (2022).
  22. M. De Leeuw, C. Paletta, A. Pribytok, A. L. Retore, and A. Torrielli, Free Fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT, J. High Energy Phys. 02 (2021) 191.
  23. M. Bocconcello, I. Masuda, F. K. Seibold, and A. Sfondrini, S-matrix for a three-parameter integrable deformation of AdS3×S3 strings, J. High Energy Phys. 11 (2020) 022.
  24. R. Borsato, S. Driezen, B. Hoare, A. L. Retore, and F. K. Seibold, Inequivalent light-cone gauge-fixings of strings on AdSn×Sn backgrounds, Phys. Rev. D 109, 106023 (2024).
  25. S. Frolov, D. Polvara, and A. Sfondrini, Exchange relations and crossing, J. Phys. A 58, 415402 (2025).
  26. A. Fontanella and A. Torrielli, Geometry of massless scattering in integrable superstring, J. High Energy Phys. 06 (2019) 116.
  27. A. Fontanella, O. Ohlsson Sax, B. Stefański, Jr., and A. Torrielli, The effectiveness of relativistic invariance in AdS3, J. High Energy Phys. 07 (2019) 105.
  28. A. Cagnazzo and K. Zarembo, B-field in AdS3/CFT2 correspondence and integrability, J. High Energy Phys. 11 (2012) 133; B. Hoare and A. A. Tseytlin, On string theory on AdS3×S3×T4 with mixed 3-form flux: Tree-level S-matrix, Nucl. Phys. B873, 682 (2013); Massive S-matrix of AdS3×S3×T4 superstring theory with mixed 3-form flux, B873, 395 (2013); A. Babichenko, A. Dekel, and O. Ohlsson Sax, Finite-gap equations for strings on AdS3×S3×T4 with mixed 3-form flux, J. High Energy Phys. 11 (2014) 122; A. Pittelli, Yangian symmetry of string theory on AdS3×S3×S3×S1 with mixed 3-form flux, Nucl. Phys. B935, 271 (2018).
  29. T. Lloyd, O. Ohlsson Sax, A. Sfondrini, and B. Stefański, Jr., The complete worldsheet S matrix of superstrings on AdS3× S3× T4 with mixed three-form flux, Nucl. Phys. B891, 570 (2015); B. Hoare, A. Stepanchuk, and A. A. Tseytlin, Giant magnon solution and dispersion relation in string theory in AdS3×S3×T4 with mixed flux, B879, 318 (2014).
  30. B. Hoare, T. J. Hollowood, and J. L. Miramontes, A relativistic relative of the magnon S-matrix, J. High Energy Phys. 11 (2011) 048; q-deformation of the AdS5×S5 superstring S-matrix and its relativistic limit, 03 (2012) 015; B. Hoare and A. A. Tseytlin, Towards the quantum S-matrix of the Pohlmeyer reduced version of AdS5×S5 superstring theory, Nucl. Phys. B851, 161 (2011); P. Sundin and L. Wulff, One- and two-loop checks for the AdS3×S3×T4 superstring with mixed flux, J. Phys. A 48, 105402 (2015); R. Hernández, J. M. Nieto, and R. Ruiz, Pulsating strings with mixed three-form flux, J. High Energy Phys. 04 (2018) 078; A. Varga, Fermion zero modes for the mixed-flux AdS3 giant magnon, 02 (2019) 135; S. Frolov, D. Polvara, and A. Sfondrini, Dressing factors for mixed-flux AdS3×S3×T4 superstrings, Phys. Rev. D 111, L081901 (2025); A. Torrielli, A study of form factors in relativistic mixed-flux AdS3, J. High Energy Phys. 03 (2024) 082; D. Bielli, V. Gautam, V. Moustakis, A. Prinsloo, and A. Torrielli, Boundary scattering in massless AdS3, 07 (2024) 266; D. Bielli, V. Moustakis, and A. Torrielli, Boundary Bethe ansatz in massless AdS3, J. Phys. A 58, 045402 (2025).
  31. A. B. Zamolodchikov and A. B. Zamolodchikov, Massless factorized scattering and sigma models with topological terms, Nucl. Phys. B379, 602 (1992); P. Fendley and H. Saleur, Massless integrable quantum field theories and massless scattering in (1+1)-dimensions, arXiv:hep-th/9310058.
  32. P. Fendley and K. A. Intriligator, Scattering and thermodynamics of fractionally charged supersymmetric solitons, Nucl. Phys. B372, 533 (1992).
  33. P. Fendley and K. A. Intriligator, Scattering and thermodynamics in integrable N=2 theories, Nucl. Phys. B380, 265 (1992).
  34. P. Fendley and K. A. Intriligator, Exact N=2 Landau-Ginzburg flows, Nucl. Phys. B413, 653 (1994).
  35. D. Bernard and A. Leclair, Quantum group symmetries and nonlocal currents in 2D QFT, Commun. Math. Phys. 142, 99 (1991).
  36. G. Mussardo, Exact matrix elements in supersymmetric theories, Nucl. Phys. B532, 529 (1998).
  37. S. Cecotti, P. Fendley, K. Intriligator, and C. Vafa, A new supersymmetric index, Nucl. Phys. B386, 405 (1992).
  38. L. Corcoran and M. de Leeuw, All regular 4×4 solutions of the Yang-Baxter equation, SciPost Phys. Core 7, 045 (2024).
  39. A. B. Zamolodchikov and A. B. Zamolodchikov, Factorized S-matrices in two-dimensions as the exact solutions of certain relativistic quantum field models, Ann. Phys. (N.Y.) 120, 253 (1979).
  40. D. Bombardelli, S-matrices and integrability, J. Phys. A 49, 323003 (2016); A. Torrielli, Integrability using the Sine-Gordon and Thirring Duality (IOP, Bristol, UK, 2024), ISBN [Amazon][WorldCat].
  41. T. R. Klassen and E. Melzer, Sine-Gordon not equal to massive Thirring, and related heresies, Int. J. Mod. Phys. A 08, 4131 (1993).
  42. P. Fendley, S. D. Mathur, C. Vafa., and N. P. Warner, Integrable deformations and scattering matrices for the N=2 supersymmetric discrete series, Phys. Lett. B 243, 257 (1990).
  43. J. M. Nieto Garcia, A. Torrielli., and L. Wyss, Boost generator in AdS3 integrable superstrings for general braiding, J. High Energy Phys. 07 (2020) 223; J. M. Nieto Garcia, A. Torrielli, and L. Wyss, Boosts superalgebras based on centrally-extended su(1|1)2, J. Geom. Phys. 164, 104172 (2021).
  44. J. Fröhlich, Statistics of fields, the Yang-Baxter equation, and the theory of knots and links, in Nonperturbative Quantum Field Theory, edited by G. ’t Hooft, A. Jaffe, G. Mack, P. K. Mitter, and R. Stora (Plenum Publishing Corporation, New York, NY, 1988).

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Phys. Rev. D 113, 065013– Published 18 March, 2026
  • Received 27 August 2025
  • Accepted 12 February 2026
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