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  • Letter
  • Open Access

Undecidability in spacetime geometry via the AdS/CFT correspondence

Sameer Ahmad Mir1,2,*, Francesco Marino3,4,†, Arshid Shabir5,‡, Lawrence M. Krauss6,§, and Mir Faizal5,7,8,9,∥

  • *Contact author: sameerphst@gmail.com
  • †Contact author: francesco.marino@ino.cnr.it
  • ‡Contact author: aslone186@gmail.com
  • §Contact author: lawrence@originsproject.org
  • ∥Contact author: mirfaizalmir@googlemail.com
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Phys. Rev. D 113, L021904 – Published 28 January, 2026

DOI: https://doi.org/10.1103/zqgf-6qp7

Abstract

Undecidability, a hallmark of Gödel incompleteness theorems, has recently emerged in quantum many-body physics through the spectral gap problem. We demonstrate how this logical limitation can be holographically transmitted to a class of gravitational theories via the AdS/CFT correspondence. By embedding a translationally invariant spin Hamiltonian with undecidable gap status into a large-N gauge theory, we generate an anti–de Sitter (AdS) dual in which the selection of dominant bulk saddle (Poincaré AdS or AdS soliton) is itself undecidable. Consequently, under standard semiclassical holographic assumptions, even determining which smooth spacetime geometry emerges from quantum gravity can be beyond the limits of computability.

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References (81)

  1. G. ’t Hooft, Conf. Proc. C 930308, 284 (1993).
  2. L. Susskind, J. Math. Phys. (N.Y.) 36, 6377 (1995).
  3. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).
  4. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B 428, 105 (1998).
  5. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998).
  6. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Phys. Rep. 323, 183 (2000).
  7. E. D’Hoker and D. Z. Freedman, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2001): Strings, Branes and Extra Dimensions (TASI, Boulder, CO, 2002), pp. 3–158.
  8. H. Nastase, arXiv:0712.0689.
  9. S. S. Gubser and A. Karch, Annu. Rev. Nucl. Part. Sci. 59, 145 (2009).
  10. S. A. Hartnoll, Classical Quantum Gravity 26, 224002 (2009).
  11. J. McGreevy, Adv. High Energy Phys. 2010, 723105 (2010).
  12. J. Penedones, in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI, Boulder, CO, 2017), pp. 75–136.
  13. N. Kajuri, SciPost Phys. Lect. Notes 22, 1 (2021).
  14. S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983).
  15. E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998).
  16. G. T. Horowitz and R. C. Myers, Phys. Rev. D 59, 026005 (1998).
  17. A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, Phys. Rev. D 74, 066009 (2006).
  18. A. Almheiri, X. Dong, and D. Harlow, J. High Energy Phys. 04 (2015) 163.
  19. J. M. Maldacena, J. High Energy Phys. 04 (2003) 021.
  20. D. Kabat, G. Lifschytz, and D. A. Lowe, Phys. Rev. D 83, 106009 (2011).
  21. T. Cubitt, D. Perez-Garcia, and M. M. Wolf, Nature (London) 528, 207 (2015).
  22. A. M. Turing, Proc. London Math. Soc. s2-42, 230 (1937).
  23. K. Godel, Monatsh. Math. Phys. 38, 173 (1931).
  24. See Supplemental Material at http://link.aps.org/supplemental/10.1103/zqgf-6qp7 https://link.aps.org/supplemental/ for detailed discussion, including Refs. [25–36].
  25. S. Bravyi and M. B. Hastings, Commun. Math. Phys. 307, 609 (2011).
  26. R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
  27. R. P. Feynman, Found. Phys. 16, 507 (1986).
  28. D. Aharonov, D. Gottesman, S. Irani, and J. Kempe, Commun. Math. Phys. 287, 41 (2009).
  29. D. Aharonov, D. Gottesman, S. Irani, and J. Kempe, Commun. Math. Phys. 287, 41 (2009).
  30. E. H. Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, Cambridge, England, 2013).
  31. S. Sachdev, Phys. World 12, 33 (1999).
  32. D. V. Vassilevich, Phys. Rep. 388, 279 (2003).
  33. R. Camporesi and A. Higuchi, J. Math. Phys. (N.Y.) 35, 4217 (1994).
  34. G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
  35. S. M. Christensen and M. J. Duff, Nucl. Phys. B170, 480 (1980).
  36. I. G. Avramidi, Heat Kernel and Quantum Gravity (Springer, New York, 2000).
  37. M. B. Hastings and T. Koma, Commun. Math. Phys. 265, 781 (2006).
  38. B. Nachtergaele and R. Sims, arXiv:1004.2086.
  39. R. P. Feynman, Opt. News 11, 11 (1985).
  40. A. Y. Kitaev, Russ. Math. Surv. 52, 1191 (1997).
  41. A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical and Quantum Computation, Graduate Studies in Mathematics Vol. 47, (American Mathematical Society, Providence, RI, 2002).
  42. J. Kempe, A. Kitaev, and O. Regev, SIAM J. Comput. 35, 1070 (2006).
  43. G. S. Boolos, J. P. Burgess, and R. C. Jeffrey, Computability and Logic, 5th ed. (Cambridge University Press, Cambridge, England, 2007).
  44. R. M. Smullyan, Gödel’s Incompleteness Theorems (Oxford University Press, New York, 1992).
  45. H. Trotter, Proc. Am. Math. Soc. 10, 545 (1959).
  46. M. Suzuki, Commun. Math. Phys. 51, 183 (1976).
  47. H. De Raedt and B. De Raedt, Phys. Rev. A 28, 3575 (1983).
  48. E. H. Lieb and D. W. Robinson, Commun. Math. Phys. 28, 251 (1972).
  49. K. G. Wilson and J. B. Kogut, Phys. Rep. 12, 75 (1974).
  50. C. Bagnuls and C. Bervillier, Phys. Rep. 348, 91 (2001).
  51. G. ’t Hooft, Nucl. Phys. B72, 461 (1974).
  52. E. Witten, Nucl. Phys. B160, 57 (1979).
  53. S. Coleman, Aspects of Symmetry: Selected Erice Lectures, (Cambridge University Press, Cambridge, England, 1985).
  54. M. Henningson and K. Skenderis, Fortschr. Phys. 48, 125 (2000).
  55. K. Skenderis, Classical Quantum Gravity 19, 5849 (2002).
  56. J. Polchinski, String Theory. Vol. 1: An Introduction to the Bosonic String, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2007).
  57. J. Polchinski, String Theory. Vol. 2: Superstring Theory and Beyond, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2007).
  58. I. R. Klebanov, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 99): Strings, Branes, and Gravity (TASI, Boulder, CO, 2000), pp. 615–650.
  59. D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Nucl. Phys. B546, 96 (1999).
  60. E. Witten, arXiv:hep-th/0112258.
  61. P. Breitenlohner and D. Z. Freedman, Ann. Phys. (N.Y.) 144, 249 (1982).
  62. V. Balasubramanian, P. Kraus, and A. E. Lawrence, Phys. Rev. D 59, 046003 (1999).
  63. H. Osborn and A. C. Petkou, Ann. Phys. (N.Y.) 231, 311 (1994).
  64. E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994).
  65. E. Elizalde, J. Phys. A 41, 304040 (2008).
  66. A. Tarski, Stud. Phil. 1, 261 (1936) A. Tarski[English translation in Logic, Semantics, Metamathematics (Clarenton Press, Oxford, 1956)].
  67. M. Faizal, L. Krauss, A. Shabir, and F. Marino, J. Hologr. Appl. Phys. 5, 10 (2025).
  68. I. Heemskerk, J. Penedones, J. Polchinski, and J. Sully, J. High Energy Phys. 10 (2009) 079.
  69. I. Heemskerk and J. Sully, J. High Energy Phys. 09 (2010) 099.
  70. N. Afkhami-Jeddi, T. Hartman, S. Kundu, and A. Tajdini, J. High Energy Phys. 12 (2017) 049.
  71. M. Henningson and K. Skenderis, J. High Energy Phys. 07 (1998) 023.
  72. O. Aharony, O. Bergman, D. L. Jafferis, and J. M. Maldacena, J. High Energy Phys. 10 (2008) 091.
  73. T. Hartman and L. Rastelli, J. High Energy Phys. 01 (2008) 019.
  74. V. Pestun, Commun. Math. Phys. 313, 71 (2012).
  75. K. Pilch and N. P. Warner, Nucl. Phys. B594, 209 (2001).
  76. X. Chen-Lin, A. Dekel, and K. Zarembo, J. High Energy Phys. 02 (2016) 109.
  77. J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk, and H. Verlinde, J. High Energy Phys. 09 (2008) 113.
  78. T. Nosaka, K. Shimizu, and S. Terashima, J. High Energy Phys. 03 (2017) 121.
  79. D. Jang, Y. Kim, O.-K. Kwon, and D. D. Tolla, J. High Energy Phys. 04 (2017) 104.
  80. M. Faizal, L. M. Krauss, A. Shabir, F. Marino, and B. Pourhassan, Int. J. Mod. Phys. D 34, 2544017 (2025).
  81. M. Faizal, A. Shabir, and A. K. Khan, Nucl. Phys. B1010, 116774 (2025).

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Information

Phys. Rev. D 113, L021904– Published 28 January, 2026
  • Received 11 November 2025
  • Accepted 9 January 2026
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