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Holographic complexity of the Klebanov-Strassler background

Andrew R. Frey*

Michael P. Grehan†

Prakriti Singh‡,§

  • *Contact author: a.frey@uwinnipeg.ca
  • †Contact author: michael.grehan@mail.utoronto.ca
  • ‡Contact author: psingh39@syr.edu
  • §Present address: Department of Physics, Syracuse University, Syracuse, New York 13244, USA.
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Phys. Rev. D 113, 026008 – Published 13 January, 2026

DOI: https://doi.org/10.1103/6ft7-phhs

Abstract

We study the complexity of the gravity dual to the confining SU(N)×SU(N+M) Klebanov-Strassler gauge theory, which is an important test case for holographic complexity in higher-dimensional and nonconformal gauge-gravity dualities. We emphasize the dependence of the complexity on parameters of the gauge theory, finding a common behavior with confinement scale for several complexity functionals. We also analyze how the complexity diverges with the UV cutoff, which is more complicated than in AdS backgrounds because the theory is nonconformal. Our results may provide new perspectives on questions in the holographic complexity program as well as a starting point for further studies of complexity in general gauge-gravity dualities.

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References (43)

  1. S. Chapman and G. Policastro, Quantum computational complexity from quantum information to black holes and back, Eur. Phys. J. C 82, 128 (2022).
  2. M. A. Nielsen, A geometric approach to quantum circuit lower bounds, Quantum Inf. Comput. 6, 213 (2006).
  3. M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Quantum computation as geometry, Science 311, 1133 (2006).
  4. M. R. Dowling and M. A. Nielsen, The geometry of quantum computation, Quantum Inf. Comput. 8, 0861 (2008).
  5. R. Jefferson and R. C. Myers, Circuit complexity in quantum field theory, J. High Energy Phys. 10 (2017) 107.
  6. S. Chapman, M. P. Heller, H. Marrochio, and F. Pastawski, Toward a definition of complexity for quantum field theory states, Phys. Rev. Lett. 120, 121602 (2018).
  7. R. Khan, C. Krishnan, and S. Sharma, Circuit complexity in fermionic field theory, Phys. Rev. D 98, 126001 (2018).
  8. L. Hackl and R. C. Myers, Circuit complexity for free fermions, J. High Energy Phys. 07 (2018) 139.
  9. A. Belin, R. C. Myers, S.-M. Ruan, G. Sárosi, and A. J. Speranza, Does complexity equal anything?, Phys. Rev. Lett. 128, 081602 (2022).
  10. A. Belin, R. C. Myers, S.-M. Ruan, G. Sárosi, and A. J. Speranza, Complexity equals anything II, J. High Energy Phys. 01 (2023) 154.
  11. J. Yang and A. R. Frey, Complexity, scaling, and a phase transition, J. High Energy Phys. 09 (2023) 029.
  12. L. Susskind, Computational complexity and black hole horizons, Fortschr. Phys. 64, 24 (2016); 64, 44(A) (2016).
  13. D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90, 126007 (2014).
  14. J. Couch, W. Fischler, and P. H. Nguyen, Noether charge, black hole volume, and complexity, J. High Energy Phys. 03 (2017) 119.
  15. A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Holographic complexity equals bulk action?, Phys. Rev. Lett. 116, 191301 (2016).
  16. A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Complexity, action, and black holes, Phys. Rev. D 93, 086006 (2016).
  17. N. Engelhardt and Å. Folkestad, General bounds on holographic complexity, J. High Energy Phys. 01 (2022) 040.
  18. N. Engelhardt and Å. Folkestad, Negative complexity of formation: The compact dimensions strike back, J. High Energy Phys. 07 (2022) 031.
  19. A. R. Frey, Holographic complexity in string and M theory, Phys. Rev. D 111, 046011 (2025).
  20. I. R. Klebanov and M. J. Strassler, Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities, J. High Energy Phys. 08 (2000) 052.
  21. G. T. Horowitz and R. C. Myers, The AdS/CFT correspondence and a new positive energy conjecture for general relativity, Phys. Rev. D 59, 026005 (1998).
  22. E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2, 505 (1998).
  23. A. P. Reynolds and S. F. Ross, Complexity of the AdS soliton, Classical Quantum Gravity 35, 095006 (2018).
  24. O. Aharony, A. Buchel, and P. Kerner, The black hole in the throat: Thermodynamics of strongly coupled cascading gauge theories, Phys. Rev. D 76, 086005 (2007).
  25. A. Buchel, Klebanov-Strassler black hole, J. High Energy Phys. 01 (2019) 207.
  26. A. Buchel, A bestiary of black holes on the conifold with fluxes, J. High Energy Phys. 06 (2021) 102.
  27. S. Chapman, H. Marrochio, and R. C. Myers, Complexity of formation in holography, J. High Energy Phys. 01 (2017) 062.
  28. S. S. Gubser, C. P. Herzog, and I. R. Klebanov, Symmetry breaking and axionic strings in the warped deformed conifold, J. High Energy Phys. 09 (2004) 036.
  29. A. Butti, M. Grana, R. Minasian, M. Petrini, and A. Zaffaroni, The baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds, J. High Energy Phys. 03 (2005) 069.
  30. C. P. Herzog, I. R. Klebanov, and P. Ouyang, Remarks on the warped deformed conifold, in Modern Trends in String Theory: 2nd Lisbon School on g Theory Superstrings (2001), arXiv:hep-th/0108101.
  31. I. R. Klebanov and A. A. Tseytlin, Gravity duals of supersymmetric SU(N) x SU(N + M) gauge theories, Nucl. Phys. B578, 123 (2000).
  32. G. Papadopoulos and A. A. Tseytlin, Complex geometry of conifolds and five-brane wrapped on two sphere, Classical Quantum Gravity 18, 1333 (2001).
  33. F. G. Tricomi, Asymptotische Eigenschaften der unvollständigen Gammafunktion, Math. Z. 53, 136 (1950).
  34. F. G. Tricomi, Sulla funzione gamma incompleta, Ann. Mat. Pura Appl. 31, 263 (1950).
  35. Böhmer and Paul Eugen, Differenzengleichungen und Bestimmte Integrale (Koehler, Liepzig, 1939).
  36. L. Lehner, R. C. Myers, E. Poisson, and R. D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94, 084046 (2016).
  37. N. Agarwal and A. R. Frey (to be published).
  38. A. Bernamonti, F. Galli, J. Hernandez, R. C. Myers, S.-M. Ruan, and J. Simón, First law of holographic complexity, Phys. Rev. Lett. 123, 081601 (2019).
  39. A. Bernamonti, F. Galli, J. Hernandez, R. C. Myers, S.-M. Ruan, and J. Simón, Aspects of the first law of complexity, J. Phys. A 53, 29 (2020).
  40. A. Fatemiabhari and C. Nunez, From conformal to confining field theories using holography, J. High Energy Phys. 03 (2024) 160.
  41. E. Jørstad, R. C. Myers, and S.-M. Ruan, Holographic complexity in dSd+1, J. High Energy Phys. 05 (2022) 119.
  42. S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68, 046005 (2003).
  43. A. Frey, M. Grehan, and P. Singh, Data for “Holographic complexity of Klebanov-Strassler background”, 10.5683/SP3/XYWDDT (2025).

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Phys. Rev. D 113, 026008– Published 13 January, 2026
  • Received 28 October 2025
  • Accepted 5 December 2025
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