- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Phase diagram of the D1-D5 CFT and localized black holes

Ofer Aharony*, Ronny Frumkin†, and Jonathan Mehl‡

  • *Contact author: ofer.aharony@weizmann.ac.il
  • †Contact author: ronny.frumkin@weizmann.ac.il
  • ‡Contact author: jonathan.mehl@weizmann.ac.il
PDF

Phys. Rev. D 113, 126024 – Published 22 June, 2026

DOI: https://doi.org/10.1103/7cjq-cwc1

Abstract

In this paper we analyze the phases that dominate the microcanonical ensemble at various energies in the D1-D5 CFT, which is dual to type II string theory on AdS3×S3×T4. We focus on black hole solutions, and on the dependence of the phase structure on the ratio of the size of the torus to the Anti-de Sitter (AdS) scale; as small localized black holes (with horizon topology S8) grow, they can start to fill the S3 or the T4 or both, and we analyze the general aspects of the transitions between the various phases of uniform and nonuniform black holes, incorporating known solutions and discussing the properties of additional unknown solutions. Some features of the transitions between these phases are similar to higher dimensional AdS spaces, while other features are different. We provide evidence that when the torus is much larger than the AdS radius, there is a large range of energies where the typical states are a novel phase, described by a lattice (in the T4 directions) of black holes with horizon topology S5×S3. In this phase the entropy is linear in the energy, with a coefficient that is of order the AdS radius.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (69)

  1. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  2. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998).
  3. E. Witten, Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  4. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Large N field theories, string theory and gravity, Phys. Rep. 323, 183 (2000).
  5. T. Banks, M. R. Douglas, G. T. Horowitz, and E. J. Martinec, AdS dynamics from conformal field theory, arXiv:hep-th/9808016.
  6. A. W. Peet and S. F. Ross, Microcanonical phases of string theory on AdS(m)×Sn, J. High Energy Phys. 12 (1998) 020.
  7. V. E. Hubeny and M. Rangamani, Unstable horizons, J. High Energy Phys. 05 (2002) 027.
  8. Ó. J. C. Dias, J. E. Santos, and B. Way, Lumpy AdS5×S5 black holes and black belts, J. High Energy Phys. 04 (2015) 060.
  9. Ó. J. C. Dias, J. E. Santos, and B. Way, Localised AdS5×S5 black holes, Phys. Rev. Lett. 117, 151101 (2016).
  10. S. W. Hawking and D. N. Page, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys. 87, 577 (1983).
  11. E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2, 505 (1998).
  12. B. Sundborg, The Hagedorn transition, deconfinement and N=4 SYM theory, Nucl. Phys. B573, 349 (2000).
  13. O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, and M. Van Raamsdonk, The Hagedorn—deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys. 8, 603 (2004).
  14. O. Aharony and E. Y. Urbach, Type II string theory on AdS3×S3×T4 and symmetric orbifolds, Phys. Rev. D 110, 046028 (2024).
  15. M. Banados, C. Teitelboim, and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69, 1849 (1992).
  16. M. Banados, M. Henneaux, C. Teitelboim, and J. Zanelli, Geometry of the (2+1) black hole, Phys. Rev. D 48, 1506 (1993).
  17. O. J. C. Dias and J. E. Santos, Localised AdS3×S3×T4 black holes, Phys. Rev. Lett. 136, 031501 (2026).
  18. T. Hartman, C. A. Keller, and B. Stoica, Universal spectrum of 2d conformal field theory in the large c limit, J. High Energy Phys. 09 (2014) 118.
  19. I. Bah and P. Heidmann, Non-BPS bubbling geometries in AdS3, J. High Energy Phys. 02 (2023) 133.
  20. I. Bena, R. Dulac, P. Heidmann, and Z. Wei, Localized black holes in AdS3: What happens below c12 stays below c12, J. High Energy Phys. 10 (2025) 165.
  21. J. D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity, Commun. Math. Phys. 104, 207 (1986).
  22. L. Landau and E. Lifshitz, Statistical Physics: Volume 5 (Butterworth-Heinemannl, London, 2013).
  23. J. Polchinski, String Theory. Vol. 1: An Introduction to the Bosonic String, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2007), 10.1017/CBO9780511816079.
  24. B. Zwiebach, A First Course in String Theory (Cambridge University Press, Cambridge, England, 2006).
  25. J. L. F. Barbon and E. Rabinovici, Closed string tachyons and the Hagedorn transition in AdS space, J. High Energy Phys. 03 (2002) 057.
  26. A. Giveon, D. Kutasov, E. Rabinovici, and A. Sever, Phases of quantum gravity in AdS3 and linear dilaton backgrounds, Nucl. Phys. B719, 3 (2005).
  27. F.-L. Lin, T. Matsuo, and D. Tomino, Hagedorn strings and correspondence principle in AdS3, J. High Energy Phys. 09 (2007) 042.
  28. M. Berkooz, Z. Komargodski, and D. Reichmann, Thermal AdS3, BTZ and competing winding modes condensation, J. High Energy Phys. 12 (2007) 020.
  29. T. G. Mertens, Hagedorn string thermodynamics in curved spacetimes and near black hole horizons, Ph.D. thesis, Gent University, 2015, arXiv:1506.07798.
  30. T. Harmark and M. Wilhelm, Solving the Hagedorn temperature of AdS5/CFT4 via the quantum spectral curve: Chemical potentials and deformations, J. High Energy Phys. 07 (2022) 136.
  31. S. K. Ashok and J. Troost, String scale thermal anti-de Sitter spaces, J. High Energy Phys. 05 (2021) 024.
  32. E. Y. Urbach, String stars in anti de Sitter space, J. High Energy Phys. 04 (2022) 072.
  33. T. Harmark, Hagedorn temperature from the thermal scalar in AdS and pp-wave backgrounds, J. High Energy Phys. 06 (2024) 140.
  34. G. T. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D 55, 6189 (1997).
  35. L. Susskind, Some speculations about black hole entropy in string theory, arXiv:hep-th/9309145.
  36. B. Kol, Topology change in general relativity, and the black hole black string transition, J. High Energy Phys. 10 (2005) 049.
  37. T. Wiseman, From black strings to black holes, Classical Quantum Gravity 20, 1177 (2003).
  38. E. Sorkin, B. Kol, and T. Piran, Caged black holes: Black holes in compactified space-times. 2. 5-d numerical implementation, Phys. Rev. D 69, 064032 (2004).
  39. H. Kudoh and T. Wiseman, Connecting black holes and black strings, Phys. Rev. Lett. 94, 161102 (2005).
  40. T. Harmark and N. A. Obers, Black holes on cylinders, J. High Energy Phys. 05 (2002) 032.
  41. T. Wiseman, Static axisymmetric vacuum solutions and nonuniform black strings, Classical Quantum Gravity 20, 1137 (2003).
  42. T. Harmark and N. A. Obers, New phase diagram for black holes and strings on cylinders, Classical Quantum Gravity 21, 1709 (2004).
  43. R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70, 2837 (1993).
  44. R. Gregory, The Gregory–Laflamme instability, in Black Holes in Higher Dimensions, edited by G. T. Horowitz (Cambridge University Press, Cambridge, 2012), pp. 29–43.
  45. A. Buchel and L. Lehner, Small black holes in AdS5×S5, Classical Quantum Gravity 32, 145003 (2015).
  46. B. Cardona and P. Figueras, Critical lumpy black holes in AdSp×Sq, J. High Energy Phys. 05 (2020) 265.
  47. E. Y. Urbach, The black hole/string transition in AdS3 and confining backgrounds, J. High Energy Phys. 09 (2023) 156.
  48. M. R. Gaberdiel and K. Naderi, The physical states of the hybrid formalism, J. High Energy Phys. 10 (2021) 168.
  49. L. Eberhardt, M. R. Gaberdiel, and R. Gopakumar, The worldsheet dual of the symmetric product CFT, J. High Energy Phys. 04 (2019) 103.
  50. M. R. Gaberdiel and S. Gerigk, The massless string spectrum on AdS3×S3 from the supergroup, J. High Energy Phys. 10 (2011) 045.
  51. N. Berkovits, C. Vafa, and E. Witten, Conformal field theory of AdS background with Ramond-Ramond flux, J. High Energy Phys. 03 (1999) 018.
  52. N. Seiberg and E. Witten, The D1/D5 system and singular CFT, J. High Energy Phys. 04 (1999) 017.
  53. J. M. Maldacena and H. Ooguri, Strings in AdS(3) and SL(2,R) WZW model 1.: The Spectrum, J. Math. Phys. (N.Y.) 42, 2929 (2001).
  54. E. Sorkin, A critical dimension in the black string phase transition, Phys. Rev. Lett. 93, 031601 (2004).
  55. B. Kol and E. Sorkin, LG (Landau-Ginzburg) in GL (Gregory-Laflamme), Classical Quantum Gravity 23, 4563 (2006).
  56. P. K. Townsend and P. van Nieuwenhuizen, Gauged seven-dimensional supergravity, Phys. Lett. 125B, 41 (1983).
  57. J. P. Gauntlett and J. B. Gutowski, Concentric black rings, Phys. Rev. D 71, 025013 (2005).
  58. F. Denef and G. W. Moore, Split states, entropy enigmas, holes and halos, J. High Energy Phys. 11 (2011) 129.
  59. J. de Boer, F. Denef, S. El-Showk, I. Messamah, and D. Van den Bleeken, Black hole bound states in AdS3×S2, J. High Energy Phys. 11 (2008) 050.
  60. I. Bena, B. D. Chowdhury, J. de Boer, S. El-Showk, and M. Shigemori, Moulting black holes, J. High Energy Phys. 03 (2012) 094.
  61. E. Bentivegna, T. Clifton, J. Durk, M. Korzyński, and K. Rosquist, Black-hole lattices as cosmological models, Classical Quantum Gravity 35, 175004 (2018).
  62. T. Harmark, Small black holes on cylinders, Phys. Rev. D 69, 104015 (2004).
  63. O. J. C. Dias, T. Harmark, R. C. Myers, and N. A. Obers, Multi-black hole configurations on the cylinder, Phys. Rev. D 76, 104025 (2007).
  64. Ó. J. C. Dias, J. E. Santos, and B. Way, Lattice black branes: Sphere packing in general relativity, J. High Energy Phys. 05 (2018) 111.
  65. S. Choi, D. Jain, S. Kim, V. Krishna, E. Lee, S. Minwalla, and C. Patel, Dual dressed black holes as the end point of the charged superradiant instability in N=4 Yang Mills, SciPost Phys. 18, 137 (2025).
  66. K. Bajaj, V. Kumar, S. Minwalla, J. Mukherjee, and A. Rahaman, Grey galaxies in AdS5, arXiv:2412.06904.
  67. H. Bateman, Tables of Integral Transforms. Volume I, Bateman Manuscript Project (McGraw–Hill, New York, 1954).
  68. H. Bateman, Tables of Integral Transforms. Volume II, Bateman Manuscript Project (McGraw–Hill, New York, 1954).
  69. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, New York, 2007), 10.1016/C2009-0-22516-5.

Outline

Information

Phys. Rev. D 113, 126024– Published 22 June, 2026
  • Received 26 March 2026
  • Accepted 3 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation