- 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

How Einstein’s equations emerge from CFT2

Xin Jiang*, Peng Wang†, Houwen Wu‡, and Haitang Yang§

  • *Contact author: domoki@stu.scu.edu.cn
  • †Contact author: pengw@scu.edu.cn
  • ‡Contact author: iverwu@scu.edu.cn
  • §Contact author: hyanga@scu.edu.cn
PDF

Phys. Rev. D 112, L081906 – Published 29 October, 2025

DOI: https://doi.org/10.1103/zg5x-34mn

Abstract

The finiteness of the entanglement entropies between disjoint subsystems enables us to show that the dynamical equation of the entanglement entropy in CFT2 is precisely the three-dimensional Einstein equation. We establish a profound relation between the cosmological constant and CFT2 entanglement entropy. Thus, entanglement entropies induce internal gravitational geometries in CFT2. Extracting the dual metric from an entanglement entropy becomes a straightforward procedure. Remarkably, we discover that the renormalization group equation is a geometric identity.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (24)

  1. Juan Martin Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  2. Gerard ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308, 284 (1993).
  3. Shinsei Ryu and Tadashi Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
  4. Shinsei Ryu and Tadashi Takayanagi, Aspects of holographic entanglement entropy, J. High Energy Phys. 08 (2006) 045.
  5. Barbara M. Terhal, Michal Horodecki, Debbie W. Leung, and David P. DiVincenzo, The entanglement of purification, J. Math. Phys. (N.Y.) 43, 4286 (2002).
  6. Tadashi Takayanagi and Koji Umemoto, Entanglement of purification through holographic duality, Nat. Phys. 14, 573 (2018).
  7. Xin Jiang, Peng Wang, Houwen Wu, and Haitang Yang, Alternative to purification in conformal field theory, Phys. Rev. D 111, L021902 (2025).
  8. Xin Jiang, Houwen Wu, Peng Wang, and Haitang Yang, Mixed state entanglement entropy in CFT, J. High Energy Phys. 09 (2025) 133.
  9. Xin Jiang, Haitang Yang, and Zilin Zhao, Entanglement entropy of mixed state in thermal CFT2, Phys. Rev. D 112, 046025 (2025).
  10. Peng Wang, Houwen Wu, and Haitang Yang, AdS3 metric from UV/IR entanglement entropies of CFT2, Chin. Phys. C 49, 045105 (2025).
  11. Peng Wang, Houwen Wu, and Haitang Yang, Fixing three dimensional geometries from entanglement entropies of CFT2, Chin. Phys. C 49, 025106 (2025).
  12. Peng Wang, Houwen Wu, and Haitang Yang, Fix the dual geometries of TT¯ deformed CFT2 and highly excited states of CFT2, Eur. Phys. J. C 80, 1117 (2020).
  13. Xin Jiang, Peng Wang, Houwen Wu, and Haitang Yang, Realization of “ER=EPR”, arXiv:2411.18485.
  14. J. Synge, Relativity: The General Theory (North-Holland, Amsterdam, The Netherlands, 1960).
  15. Peter Buser, Geometry and spectra of compact Riemann surfaces (Birkhäuser, Boston, 2010), 10.1007/978-0-8176-4992-0.
  16. J. David 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).
  17. Kotaro Tamaoka, Entanglement wedge cross section from the dual density matrix, Phys. Rev. Lett. 122, 141601 (2019).
  18. Souvik Dutta and Thomas Faulkner, A canonical purification for the entanglement wedge cross-section, J. High Energy Phys. 03 (2021) 178.
  19. Qiang Wen, Balanced partial entanglement and the entanglement wedge cross section, J. High Energy Phys. 04 (2021) 301.
  20. Takato Mori and Beni Yoshida, Does connected wedge imply distillable entanglement?, arXiv:2411.03426.
  21. H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B363, 486 (1991).
  22. Pasquale Calabrese and John L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  23. Xin Jiang, Peng Wang, Houwen Wu, and Haitang Yang, Timelike entanglement entropy in dS3/CFT2, J. High Energy Phys. 08 (2023) 216.
  24. Xin Jiang and Haitang Yang, Entanglement entropy of conformal field theory in all dimensions, arXiv:2506.02786.

Outline

Information

Phys. Rev. D 112, L081906– Published 29 October, 2025
  • Received 4 February 2025
  • Accepted 13 October 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