Skip to main content

Advertisement

Springer Nature Link
Account
Menu
Find a journal Publish with us Track your research
Search
Saved research
Cart
  1. Home
  2. Journal of High Energy Physics
  3. Article

Mirror thermodynamic Bethe ansatz for AdS3/CFT2

  • Regular Article - Theoretical Physics
  • Open access
  • Published: 22 March 2022
  • Volume 2022, article number 138 (2022)
  • Cite this article

You have full access to this open access article

Download PDF
View saved research
Journal of High Energy Physics Aims and scope Submit manuscript
Mirror thermodynamic Bethe ansatz for AdS3/CFT2
Download PDF
  • Sergey Frolov1 &
  • Alessandro Sfondrini2,3 

  • 662 Accesses

  • 37 Citations

  • 1 Altmetric

  • Explore all metrics

A preprint version of the article is available at arXiv.

Abstract

We consider superstrings on the pure-Ramond-Ramond AdS3 × S3 × T4 background. Using the recently-proposed dressing factors for the worldsheet S matrix, we formulate the string hypothesis for the mirror Bethe-Yang equations, and use it to derive the canonical mirror thermodynamic Bethe ansatz (TBA) equations of the model. For the first time, these equations account for all massive and massless modes of the model, and do not resort to any limit or special kinematics. We also discuss the simplified mirror TBA equations and the Y-system of the model.

Article PDF

Download to read the full article text

Similar content being viewed by others

Transfer matrices for AdS3/CFT2

Article Open access 16 May 2022

Dressing factors and mirror thermodynamic Bethe ansatz for mixed-flux AdS3/CFT2

Article Open access 24 February 2026

Integrable S matrix, mirror TBA and spectrum for the stringy AdS3 × S3 × S3 × S1 WZW model

Article Open access 13 February 2019

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.
  • Mirror Neuron
  • Polariton
  • Matrix Theory
  • String Theory
  • Field Theory and Polynomials
  • Crystal Field Theory
  • Quantum Field Theory and Gravitational Phenomena

References

  1. J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  2. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  3. S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  4. G. Arutyunov and S. Frolov, Foundations of the AdS5 × S5 Superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].

  5. N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  6. G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].

  7. J.A. Minahan and K. Zarembo, The Bethe ansatz for \( \mathcal{N} \) = 4 superYang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].

  8. C.-N. Yang and C.P. Yang, Thermodynamics of one-dimensional system of bosons with repulsive delta function interaction, J. Math. Phys. 10 (1969) 1115 [INSPIRE].

  9. A.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models. Scaling three state potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].

  10. P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  11. N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  12. N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].

    Google Scholar 

  13. R.A. Janik, The AdS5 × S5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].

  14. G. Arutyunov and S. Frolov, On string S-matrix, bound states and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  15. G. Arutyunov and S. Frolov, String hypothesis for the AdS5 × S5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].

    Article  ADS  Google Scholar 

  16. N. Gromov, V. Kazakov and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].

  17. D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe ansatz for planar AdS/CFT: a proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].

  18. G. Arutyunov and S. Frolov, Thermodynamic Bethe ansatz for the AdS5 × S5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].

    Article  ADS  Google Scholar 

  19. G. Arutyunov and S. Frolov, Simplified TBA equations of the AdS5 × S5 mirror model, JHEP 11 (2009) 019 [arXiv:0907.2647] [INSPIRE].

    Article  ADS  Google Scholar 

  20. G. Arutyunov, S. Frolov and R. Suzuki, Exploring the mirror TBA, JHEP 05 (2010) 031 [arXiv:0911.2224] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  21. A. Cavaglia, D. Fioravanti and R. Tateo, Extended Y-system for the AdS5/CFT4 correspondence, Nucl. Phys. B 843 (2011) 302 [arXiv:1005.3016] [INSPIRE].

    Article  ADS  Google Scholar 

  22. N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for planar \( \mathcal{N} \) = 4 Super-Yang-Mills theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].

  23. O. Ohlsson Sax and B. Stefański, Closed strings and moduli in AdS3/CFT2, JHEP 05 (2018) 101 [arXiv:1804.02023] [INSPIRE].

  24. A. Cagnazzo and K. Zarembo, B-field in AdS3/CFT2 correspondence and integrability, JHEP 11 (2012) 133 [Erratum ibid. 04 (2013) 003] [arXiv:1209.4049] [INSPIRE].

  25. 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. B 891 (2015) 570 [arXiv:1410.0866] [INSPIRE].

  26. A. Sfondrini, Towards integrability for AdS3/CFT2, J. Phys. A 48 (2015) 023001 [arXiv:1406.2971] [INSPIRE].

  27. J.M. Maldacena and H. Ooguri, Strings in AdS3 and SL(2, ℝ) WZW model. I. The spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].

  28. G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS3 at ‖ = 1, JHEP 08 (2018) 204 [arXiv:1803.04420] [INSPIRE].

    Article  ADS  Google Scholar 

  29. L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, The worldsheet dual of the symmetric product CFT, JHEP 04 (2019) 103 [arXiv:1812.01007] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  30. L. Eberhardt, A perturbative CFT dual for pure NS-NS AdS3 strings, J. Phys. A 55 (2022) 064001 [arXiv:2110.07535] [INSPIRE].

  31. M. Baggio and A. Sfondrini, Strings on NS-NS backgrounds as integrable deformations, Phys. Rev. D 98 (2018) 021902 [arXiv:1804.01998] [INSPIRE].

  32. A. Dei and A. Sfondrini, Integrable spin chain for stringy Wess-Zumino-Witten models, JHEP 07 (2018) 109 [arXiv:1806.00422] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  33. A. Sfondrini, Long strings and symmetric product orbifold from the AdS3 Bethe equations, EPL 133 (2021) 10004 [arXiv:2010.02782] [INSPIRE].

    Article  ADS  Google Scholar 

  34. L. Eberhardt and K. Ferreira, Long strings and chiral primaries in the hybrid formalism, JHEP 02 (2019) 098 [arXiv:1810.08621] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  35. L. Eberhardt and K. Ferreira, The plane-wave spectrum from the worldsheet, JHEP 10 (2018) 109 [arXiv:1805.12155] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  36. N. Berkovits, C. Vafa and E. Witten, Conformal field theory of AdS background with Ramond-Ramond flux, JHEP 03 (1999) 018 [hep-th/9902098] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  37. B. Hoare and A.A. Tseytlin, On string theory on AdS3 × S3 × T4 with mixed 3-form flux: tree-level S-matrix, Nucl. Phys. B 873 (2013) 682 [arXiv:1303.1037] [INSPIRE].

    Article  ADS  Google Scholar 

  38. A. Babichenko, A. Dekel and O. Ohlsson Sax, Finite-gap equations for strings on AdS3 × S3 × T4 with mixed 3-form flux, JHEP 11 (2014) 122 [arXiv:1405.6087] [INSPIRE].

  39. S. Frolov and A. Sfondrini, New Dressing Factors for AdS3/CFT2, arXiv:2112.08896 [INSPIRE].

  40. S. Frolov and A. Sfondrini, Dressing Factors for Mixed-flux AdS3/CFT2, in preparation.

  41. R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Towards the All-Loop Worldsheet S Matrix for AdS3 × S3 × T4, Phys. Rev. Lett. 113 (2014) 131601 [arXiv:1403.4543] [INSPIRE].

  42. M.C. Abbott and I. Aniceto, Massless Lüscher terms and the limitations of the AdS3 asymptotic Bethe ansatz, Phys. Rev. D 93 (2016) 106006 [arXiv:1512.08761] [INSPIRE].

  43. N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  44. R. Borsato, O. Ohlsson Sax and A. Sfondrini, A dynamic \( \mathfrak{su} \)(1|1)2 S-matrix for AdS3/CFT2, JHEP 04 (2013) 113 [arXiv:1211.5119] [INSPIRE].

  45. R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański and A. Torrielli, The all-loop integrable spin-chain for strings on AdS3 × S3 × T4: the massive sector, JHEP 08 (2013) 043 [arXiv:1303.5995] [INSPIRE].

  46. R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, The complete AdS3 × S3 × T4 worldsheet S matrix, JHEP 10 (2014) 066 [arXiv:1406.0453] [INSPIRE].

  47. R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski, Jr. and A. Torrielli, Dressing phases of AdS3/CFT2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].

  48. R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański, A. Torrielli and O. Ohlsson Sax, On the dressing factors, Bethe equations and Yangian symmetry of strings on AdS3 × S3 × T4, J. Phys. A 50 (2017) 024004 [arXiv:1607.00914] [INSPIRE].

  49. D. Bombardelli, B. Stefański and A. Torrielli, The low-energy limit of AdS3/CFT2 and its TBA, JHEP 10 (2018) 177 [arXiv:1807.07775] [INSPIRE].

    Article  ADS  Google Scholar 

  50. A. Fontanella, O. Ohlsson Sax, B. Stefański and A. Torrielli, The effectiveness of relativistic invariance in AdS3, JHEP 07 (2019) 105 [arXiv:1905.00757] [INSPIRE].

  51. A. Cavaglià, N. Gromov, B. Stefański, Jr., Jr. and A. Torrielli, Quantum Spectral Curve for AdS3/CFT2: a proposal, JHEP 12 (2021) 048 [arXiv:2109.05500] [INSPIRE].

  52. S. Ekhammar and D. Volin, Monodromy Bootstrap for SU(2|2) Quantum Spectral Curves: From Hubbard model to AdS3/CFT2, arXiv:2109.06164 [INSPIRE].

  53. L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Les Houches School of Physics: Astrophysical Sources of Gravitational Radiation, Les Houches France (1996), pg. 149 [hep-th/9605187] [INSPIRE].

  54. F. Woynarovich, On the S(Z) = 0 Excited States of an Anisotropic Heisenberg Chain, J. Phys. A 15 (1982) 2985 [INSPIRE].

  55. F. Woynarovich, On the eigenstates of a Heisenberg chain with complex wavenumbers not forming strings, J. Phys. C 15 (1983) 6397.

    Article  ADS  Google Scholar 

  56. O. Babelon, H.J. de Vega and C.M. Viallet, Analysis of the Bethe Ansatz Equations of the Xxz Model, Nucl. Phys. B 220 (1983) 13 [INSPIRE].

  57. A. Tsvelick and P. Wiegmann, Exact results in the theory of magnetic alloys, Adv. Phys. 32 (1983) 453.

    Article  ADS  Google Scholar 

  58. S.J. van Tongeren, Integrability of the AdS5 × S5 superstring and its deformations, J. Phys. A 47 (2014) 433001 [arXiv:1310.4854] [INSPIRE].

  59. R. Borsato, O. Ohlsson Sax and A. Sfondrini, All-loop Bethe ansatz equations for AdS3/CFT2, JHEP 04 (2013) 116 [arXiv:1212.0505] [INSPIRE].

  60. F. Seibold and A. Sfondrini, Transfer matrices of AdS3/CFT2, in preparation.

  61. G. Arutyunov and S. Frolov, The dressing factor and crossing equations, J. Phys. A 42 (2009) 425401 [arXiv:0904.4575] [INSPIRE].

  62. N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  63. S.J. van Tongeren, Introduction to the thermodynamic Bethe ansatz, J. Phys. A 49 (2016) 323005 [arXiv:1606.02951] [INSPIRE].

  64. S. Frolov and R. Suzuki, Temperature quantization from the TBA equations, Phys. Lett. B 679 (2009) 60 [arXiv:0906.0499] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  65. N. Rughoonauth, P. Sundin and L. Wulff, Near BMN dynamics of the AdS3 × S3 × S3 × S1 superstring, JHEP 07 (2012) 159 [arXiv:1204.4742] [INSPIRE].

    Article  ADS  Google Scholar 

  66. P. Sundin and L. Wulff, Worldsheet scattering in AdS3/CFT2, JHEP 07 (2013) 007 [arXiv:1302.5349] [INSPIRE].

    Article  ADS  Google Scholar 

  67. O.T. Engelund, R.W. McKeown and R. Roiban, Generalized unitarity and the worldsheet S matrix in AdSn × Sn × M10−2n, JHEP 08 (2013) 023 [arXiv:1304.4281] [INSPIRE].

    Article  ADS  Google Scholar 

  68. R. Roiban, P. Sundin, A. Tseytlin and L. Wulff, The one-loop worldsheet S-matrix for the AdSn × Sn × T10−2n superstring, JHEP 08 (2014) 160 [arXiv:1407.7883] [INSPIRE].

    Article  ADS  Google Scholar 

  69. L. Bianchi and B. Hoare, AdS3 × S3 × M4 string S-matrices from unitarity cuts, JHEP 08 (2014) 097 [arXiv:1405.7947] [INSPIRE].

    Article  ADS  Google Scholar 

  70. P. Sundin and L. Wulff, The complete one-loop BMN S-matrix in AdS3 × S3 × T4, JHEP 06 (2016) 062 [arXiv:1605.01632] [INSPIRE].

    Article  ADS  Google Scholar 

  71. S. Frolov and A. Sfondrini, Mirror Thermodynamic Bethe Ansatz for AdS3/CFT2, arXiv:2112.08898 [INSPIRE].

  72. O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Integrability and the Conformal Field Theory of the Higgs branch, JHEP 06 (2015) 103 [arXiv:1411.3676] [INSPIRE].

  73. R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, The AdS3 × S3 × S3 × S1 worldsheet S matrix, J. Phys. A 48 (2015) 415401 [arXiv:1506.00218] [INSPIRE].

  74. S. Frolov and A. Sfondrini, Massless S matrices for AdS3/CFT2, arXiv:2112.08895 [INSPIRE].

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Hamilton Mathematics Institute, Trinity College, Dublin 2, Ireland

    Sergey Frolov

  2. Institute for Advanced Study, Einstein Drive, Princeton, New Jersey, 08540, USA

    Alessandro Sfondrini

  3. Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, & Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, 35131, Padova, Italy

    Alessandro Sfondrini

Authors
  1. Sergey Frolov
    View author publications

    Search author on:PubMed Google Scholar

  2. Alessandro Sfondrini
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to Alessandro Sfondrini.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2112.08898

Correspondent fellow at Steklov Mathematical Institute, Moscow, Russian Federation. (Sergey Frolov)

IBM Einstein Fellow. (Alessandro Sfondrini)

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Frolov, S., Sfondrini, A. Mirror thermodynamic Bethe ansatz for AdS3/CFT2. J. High Energ. Phys. 2022, 138 (2022). https://doi.org/10.1007/JHEP03(2022)138

Download citation

  • Received: 19 January 2022

  • Accepted: 03 March 2022

  • Published: 22 March 2022

  • Version of record: 22 March 2022

  • DOI: https://doi.org/10.1007/JHEP03(2022)138

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Integrable Field Theories
  • AdS-CFT Correspondence

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us
  • Track your research

Footer Navigation

Discover content

  • Journals A-Z
  • Books A-Z
  • Subjects A-Z

Publish with us

  • Journal finder
  • Publish your research
  • Language editing
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our brands

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Discover

Corporate Navigation

  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support
  • Legal notice
  • Cancel contracts here

Not affiliated

Springer Nature

© 2026 Springer Nature