- 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
  • Letter
  • Open Access

Exact low-temperature Green’s functions in AdS/CFT: From the Heun equation to the confluent Heun equation

Paolo Arnaudo* and Benjamin Withers†

  • *Contact author: p.arnaudo@soton.ac.uk
  • †Contact author: b.s.withers@soton.ac.uk
PDF

Phys. Rev. D 111, L121903 – Published 27 June, 2025

DOI: https://doi.org/10.1103/8n3f-2d33

Abstract

We obtain exact expressions for correlation functions of charged scalar operators at finite density and low temperature in CFT4 dual to the RN−AdS5 black-brane. We use recent developments in the Heun connection problem in black hole perturbation theory arising from Liouville conformal field theory (CFT) and the Alday-Gaiotto-Tachikawa correspondence. The connection problem is solved perturbatively in an instanton counting parameter, which is controlled in a double-scaling limit where ω,T→0 holding ω/T fixed. This provides analytic control over the emergence of the zero temperature branch cut as a confluent limit of the Heun equation. From the Green’s function we extract analytic results for the critical temperature of the holographic superconductor, as well as dispersion relations for both gapped and gapless low temperature quasinormal modes. We demonstrate precise agreement with numerics.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (57)

  1. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).
  2. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998).
  3. S. A. Hartnoll, Classical Quantum Gravity 26, 224002 (2009).
  4. J. Zaanen, Y. Liu, Y.-W. Sun, and K. Schalm, Holographic Duality in Condensed Matter Physics (Cambridge University Press, Cambridge, England, 2015).
  5. G. Policastro, D. T. Son, and A. O. Starinets, Phys. Rev. Lett. 87, 081601 (2001).
  6. P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2019).
  7. J. Berges, M. P. Heller, A. Mazeliauskas, and R. Venugopalan, Rev. Mod. Phys. 93, 035003 (2021).
  8. N. Kovensky, A. Poole, and A. Schmitt, Phys. Rev. D 105, 034022 (2022).
  9. C. Hoyos, N. Jokela, and A. Vuorinen, Prog. Part. Nucl. Phys. 126, 103972 (2022).
  10. G. Aminov, A. Grassi, and Y. Hatsuda, Ann. Henri Poincare 23, 1951 (2022).
  11. G. Bonelli, C. Iossa, D. P. Lichtig, and A. Tanzini, Phys. Rev. D 105, 044047 (2022).
  12. M. Bianchi, D. Consoli, A. Grillo, and J. F. Morales, Phys. Lett. B 824, 136837 (2022).
  13. G. Bonelli, C. Iossa, D. Panea Lichtig, and A. Tanzini, Commun. Math. Phys. 397, 635 (2023).
  14. O. Lisovyy and A. Naidiuk, J. Phys. A 55, 434005 (2022).
  15. J. B. Amado, B. C. da Cunha, and E. Pallante, Phys. Rev. D 105, 044028 (2022).
  16. M. Dodelson, A. Grassi, C. Iossa, D. Panea Lichtig, and A. Zhiboedov, SciPost Phys. 14, 116 (2023).
  17. H. F. Jia and M. Rangamani, J. High Energy Phys. 12 (2024) 047.
  18. J. Barragán Amado, S. Chakrabortty, and A. Maurya, J. High Energy Phys. 11 (2024) 070.
  19. See the Supplemental Material at http://link.aps.org/supplemental/10.1103/8n3f-2d33 for full expressions and for the gauge theory conventions used, which includes Refs. [20–24].
  20. J. Teschner, Classical Quantum Gravity 18, R153 (2001).
  21. Y. Nakayama, Int. J. Mod. Phys. A 19, 2771 (2004).
  22. L. F. Alday, D. Gaiotto, and Y. Tachikawa, Lett. Math. Phys. 91, 167 (2010).
  23. N. Seiberg and E. Witten, Nucl. Phys. B426, 19 (1994).
  24. N. Seiberg and E. Witten, Nucl. Phys. B431, 484 (1994).
  25. T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, Phys. Rev. D 83, 125002 (2011).
  26. T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. Vegh, Phil. Trans. R. Soc. A 369, 1640 (2011).
  27. S. A. Hartnoll and D. M. Hofman, Phys. Rev. Lett. 108, 241601 (2012).
  28. J. Ren, arXiv:1210.2722.
  29. R. A. Davison and A. Parnachev, J. High Energy Phys. 06 (2013) 100.
  30. D. Arean, R. A. Davison, B. Goutéraux, and K. Suzuki, Phys. Rev. X 11, 031024 (2021).
  31. J. Ren, J. High Energy Phys. 05 (2021) 080.
  32. N. Wu, M. Baggioli, and W.-J. Li, J. High Energy Phys. 05 (2021) 014.
  33. J. Ren and W. Zheng, Phys. Rev. D 105, 066013 (2022).
  34. R. A. Davison, B. Goutéraux, and E. Mefford, J. High Energy Phys. 12 (2023) 040.
  35. M. Edalati, J. I. Jottar, and R. G. Leigh, J. High Energy Phys. 01 (2010) 018.
  36. M. Edalati, J. I. Jottar, and R. G. Leigh, J. High Energy Phys. 04 (2010) 075.
  37. M. Edalati, J. I. Jottar, and R. G. Leigh, J. High Energy Phys. 10 (2010) 058.
  38. S. S. Gubser, Phys. Rev. D 78, 065034 (2008).
  39. S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Phys. Rev. Lett. 101, 031601 (2008).
  40. S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, J. High Energy Phys. 12 (2008) 015.
  41. I. Amado, M. Kaminski, and K. Landsteiner, J. High Energy Phys. 05 (2009) 021.
  42. S. S. Gubser, C. P. Herzog, S. S. Pufu, and T. Tesileanu, Phys. Rev. Lett. 103, 141601 (2009).
  43. M. Minucci and R. Panosso Macedo, Gen. Relativ. Gravit. 57, 33 (2025).
  44. K. Heun, Math. Ann. 33, 161 (1888).
  45. A. Ronveaux and F. Arscott, Heun’s Differential Equations, Oxford Science Publications (Oxford University Press, New York, 1995).
  46. D. T. Son and A. O. Starinets, J. High Energy Phys. 09 (2002) 042.
  47. C. P. Herzog and D. T. Son, J. High Energy Phys. 03 (2003) 046.
  48. B. C. van Rees, Nucl. Phys. B, Proc. Suppl. 192–193, 193 (2009).
  49. K. Skenderis, Classical Quantum Gravity 19, 5849 (2002).
  50. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Nucl. Phys. B241, 333 (1984).
  51. M. Matone, Phys. Lett. B 357, 342 (1995).
  52. J. P. Gauntlett, J. Sonner, and T. Wiseman, Phys. Rev. Lett. 103, 151601 (2009).
  53. J. P. Gauntlett, J. Sonner, and T. Wiseman, J. High Energy Phys. 02 (2010) 060.
  54. P. Arnaudo, A. Grassi, and Q. Hao, arXiv:2502.01526.
  55. H. Kodama and A. Ishibashi, Prog. Theor. Phys. 111, 29 (2004).
  56. M. P. Heller, A. Serantes, M. Spaliński, and B. Withers, Phys. Rev. Lett. 130, 261601 (2023).
  57. M. P. Heller, A. Serantes, M. Spaliński, and B. Withers, Nat. Phys. 20, 1948 (2024).

Outline

Information

Phys. Rev. D 111, L121903– Published 27 June, 2025
  • Received 29 January 2025
  • Accepted 10 June 2025
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