Communications in Mathematical Physics

, Volume 301, Issue 3, pp 583–626 | Cite as

Holography and Wormholes in 2+1 Dimensions

  • Kostas Skenderis
  • Balt C. van ReesEmail author
Open Access


We provide a holographic interpretation of a class of three-dimensional wormhole spacetimes. These spacetimes have multiple asymptotic regions which are separated from each other by horizons. Each such region is isometric to the BTZ black hole and there is non-trivial spacetime topology hidden behind the horizons. We show that application of the real-time gauge/gravity duality results in a complete holographic description of these spacetimes with the dual state capturing the non-trivial topology behind the horizons. We also show that these spacetimes are in correspondence with trivalent graphs and provide an explicit metric description with all physical parameters appearing in the metric.


Black Hole Riemann Surface Boundary Component Fundamental Domain Conformal Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank Alex Maloney, Jan Smit and Erik Verlinde for discussions. KS acknowledges support from NWO via a VICI grant.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Maldacena J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998)zbMATHMathSciNetADSGoogle Scholar
  2. 2.
    Gubser S.S., Klebanov I.R., Polyakov A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105–114 (1998)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Witten E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Skenderis K., van Rees B.C.: Real-time gauge/gravity duality. Phys. Rev. Lett. 101, 081601 (2008)CrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Skenderis, K., van Rees, B.C.: Real-time gauge/gravity duality: Prescription, Renormalization and Examples.   [hep-th], 2009
  6. 6.
    Schwinger J.S.: Brownian motion of a quantum oscillator. J. Math. Phys. 2, 407–432 (1961)zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Bakshi P.M., Mahanthappa K.T.: Expectation value formalism in quantum field theory. 1. J. Math. Phys. 4, 1–11 (1963)CrossRefMathSciNetADSzbMATHGoogle Scholar
  8. 8.
    Bakshi P.M., Mahanthappa K.T.: Expectation value formalism in quantum field theory. 2. J. Math. Phys. 4, 12–16 (1963)CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Keldysh L.V.: Diagram technique for nonequilibrium processes. Zh. Eksp. Teor. Fiz. 47, 1515–1527 (1964) [Sov.  Phys.  JETP 20, 1018 (1965)]Google Scholar
  10. 10.
    Hartle J.B., Hawking S.W.: Wave Function of the Universe. Phys. Rev. D 28, 2960–2975 (1983)CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Maldacena J.M.: Eternal black holes in Anti-de-Sitter.. JHEP 04, 021 (2003)CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Aminneborg S., Bengtsson I., Brill D., Holst S., Peldan P.: Black holes and wormholes in 2+1 dimensions. Class. Quant. Grav. 15, 627–644 (1998)zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Brill, D.: Black holes and wormholes in 2+1 dimensions., 1999
  14. 14.
    Skenderis K., Solodukhin S.N.: Quantum effective action from the AdS/CFT correspondence. Phys. Lett. B 472, 316–322 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Krasnov K.: Holography and Riemann surfaces. Adv. Theor. Math. Phys. 4, 929–979 (2000)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Henningson M., Skenderis K.: Holography and the Weyl anomaly. Fortsch. Phys. 48, 125–128 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Henningson M., Skenderis K.: The holographic Weyl anomaly. JHEP 07, 023 (1998)CrossRefMathSciNetADSGoogle Scholar
  18. 18.
    de Haro S., Solodukhin S.N., Skenderis K.: Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence. Commun. Math. Phys. 217, 595–622 (2001)zbMATHCrossRefADSGoogle Scholar
  19. 19.
    Skenderis K.: Asymptotically Anti-de Sitter spacetimes and their stress energy tensor. Int. J. Mod. Phys. A 16, 740–749 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Banados M., Teitelboim C., Zanelli J.: The Black hole in three-dimensional space-time. Phys. Rev. Lett. 69, 1849–1851 (1992)zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Banados M., Henneaux M., Teitelboim C., Zanelli J.: Geometry of the (2+1) black hole. Phys. Rev. D 48, 1506–1525 (1993)CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Barbot T.: Causal properties of AdS-isometry groups. I: Causal actions and limit sets. Adv. Theor. Math. Phys. 12, 1–66 (2008)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Barbot, T.: Causal properties of AdS-isometry groups. II: BTZ multi black-holes. [math, GT], 2006
  24. 24.
    van Rees, B.: Worm holes in 2+1 dimensions. Master’s thesis, June, 2006
  25. 25.
    Imayoshi, Y., Taniguchi, M.: An Introduction to Teichmnüller Spaces. Berlin-Heidelberg Newyork: Springer-Verlag, 1992Google Scholar
  26. 26.
    Lehto O.: Univalent Functions and Teichmüller Spaces. Springer-Verlag, Berlin-Heidelberg Newyork (1986)Google Scholar
  27. 27.
    Nag S.: The Complex Analytic Theory of Teichmüller spaces. Newyork, John Wiley & Sons (1988)zbMATHGoogle Scholar
  28. 28.
    Galloway G.J., Schleich K., Witt D.M., Woolgar E.: Topological Censorship and Higher Genus Black Holes. Phys. Rev. D60, 104039 (1999)MathSciNetADSGoogle Scholar
  29. 29.
    Carlip S., Teitelboim C.: Aspects of black hole quantum mechanics and thermodynamics in (2+1)-dimensions. Phys. Rev. D51, 622–631 (1995)MathSciNetADSGoogle Scholar
  30. 30.
    Krasnov K.: On holomorphic factorization in asymptotically AdS 3D gravity. Class. Quant. Grav. 20, 4015–4042 (2003)zbMATHCrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Krasnov K.: Black Hole Thermodynamics and Riemann Surfaces. Class. Quant. Grav. 20, 2235–2250 (2003)zbMATHCrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Takhtajan L., Zograf P.: On uniformization of Riemann surfaces and the Weyl-Peterson metric on Teichmuller and Schottky spaces. Math. USSR Sbornik 60, 297–313 (1988)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Imbimbo C., Schwimmer A., Theisen S., Yankielowicz S.: Diffeomorphisms and holographic anomalies. Class. Quant. Grav. 17, 1129–1138 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Maldacena J.M., Maoz L.: Wormholes in AdS. JHEP 02, 053 (2004)CrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Parlier H.: Fixed point free involutions on Riemann surfaces. Israel J. Math. 166, 297–311 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Anderson, M.T.: Geometric aspects of the AdS/CFT correspondence., 2004
  37. 37.
    Papadimitriou I., Skenderis K.: Thermodynamics of asymptotically locally AdS spacetimes. JHEP 08, 004 (2005)CrossRefMathSciNetADSGoogle Scholar
  38. 38.
    Freivogel B. et al.: Inflation in AdS/CFT. JHEP 03, 007 (2006)CrossRefMathSciNetADSGoogle Scholar
  39. 39.
    Louko J., Marolf D.: Single-exterior black holes and the AdS-CFT conjecture. Phys. Rev. D59, 066002 (1999)MathSciNetADSGoogle Scholar
  40. 40.
    Hawking S.W., Page D.N.: Thermodynamics of black holes in anti-de Sitter space. Commun. Math. Phys. 87, 577 (1983)CrossRefMathSciNetADSGoogle Scholar
  41. 41.
    Maldacena J.M., Strominger A.: AdS(3) black holes and a stringy exclusion principle. JHEP 12, 005 (1998)CrossRefMathSciNetADSGoogle Scholar
  42. 42.
    Yin, X.: Partition Functions of Three-Dimensional Pure Gravity. [hep-th] (2008)
  43. 43.
    Aminneborg S., Bengtsson I., Holst S.: A spinning Anti-de Sitter wormhole. Class. Quant. Grav. 16, 363–382 (1999)zbMATHCrossRefMathSciNetADSGoogle Scholar
  44. 44.
    Brill D.: 2+1-dimensional black holes with momentum and angular momentum. Annalen Phys. 9, 217–226 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  45. 45.
    Krasnov K.: Analytic continuation for asymptotically AdS 3D gravity. Class. Quant. Grav. 19, 2399–2424 (2002)zbMATHCrossRefMathSciNetADSGoogle Scholar
  46. 46.
    Maloney, A.: To appearGoogle Scholar
  47. 47.
    Skenderis K., Taylor M.: The fuzzball proposal for black holes. Phys. Rept. 467, 117–171 (2008)CrossRefMathSciNetADSGoogle Scholar
  48. 48.
    Crnkovic, C., Witten, E.: Covariant description of canonical formalism in geometrical theories. Print-86-1309 (Princeton)Google Scholar
  49. 49.
    Crnkovic C.: Symplectic geometry and (super)Poincare algebra in geometrical theories. Nucl. Phys. B 288, 419 (1987)CrossRefMathSciNetADSGoogle Scholar
  50. 50.
    Lee J., Wald R.M.: Local symmetries and constraints. J. Math. Phys. 31, 725–743 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  51. 51.
    van Rees, B.: Dynamics and the gauge/gravity duality. PhD thesis, 2010, to appear Google Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsAmsterdamThe Netherlands
  2. 2.Korteweg-de Vries Institute for MathematicsAmsterdamThe Netherlands
  3. 3.YITP, State University of New YorkStony BrookUSA

Personalised recommendations