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p-Adic AdS/CFT

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Abstract

We construct a p-adic analog to AdS/CFT, where an unramified extension of the p-adic numbers replaces Euclidean space as the boundary and a version of the Bruhat–Tits tree replaces the bulk. Correlation functions are computed in the simple case of a single massive scalar in the bulk, with results that are strikingly similar to ordinary holographic correlation functions when expressed in terms of local zeta functions. We give some brief discussion of the geometry of p-adic chordal distance and of Wilson loops. Our presentation includes an introduction to p-adic numbers.

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References

  1. Maldacena J.M.: The large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113–1133 (1999) arXiv:hep-th/9711200

    Article  MathSciNet  MATH  Google Scholar 

  2. Gubser S.S., Klebanov I.R., Polyakov A.M.: Gauge theory correlators from noncritical string theory. Phys. Lett. B 428, 105–114 (1998) arXiv:hep-th/9802109

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Witten E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998) arXiv:hep-th/9802150

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Aharony O., Gubser S.S., Maldacena J.M., Ooguri H., Oz Y.: Large N field theories, string theory and gravity. Phys. Rep. 323, 183–386 (2000) arXiv:hep-th/9905111

    Article  ADS  MathSciNet  Google Scholar 

  5. Swingle B.: Entanglement renormalization and holography. Phys. Rev. D 86, 065007 (2012) arXiv:0905.1317

    Article  ADS  Google Scholar 

  6. Swingle, B.: Constructing Holographic Spacetimes Using Entanglement Renormalization. arXiv:1209.3304

  7. Qi, X.-L.: Exact Holographic Mapping and Emergent Space-Time Geometry. arXiv:1309.6282

  8. Pastawski F., Yoshida B., Harlow D., Preskill J.: Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. JHEP 06, 149 (2015) arXiv:1503.06237

    Article  ADS  MathSciNet  Google Scholar 

  9. Vegh, D.: The Broken String in Anti-de Sitter Space. arXiv:1508.06637

  10. Callebaut N., Gubser S.S., Samberg A., Toldo C.: Segmented strings in AdS3. JHEP 11, 110 (2015) arXiv:1508.07311

    Article  ADS  Google Scholar 

  11. Vegh, D.: Colliding Waves on a String in AdS3. arXiv:1509.05033

  12. Gubser S.S.: Evolution of segmented strings. Phys. Rev. D 94, 106007 (2016) arXiv:1601.08209

    Article  ADS  MathSciNet  Google Scholar 

  13. Gubser S.S., Parikh S., Witaszczyk P.: Segmented strings and the McMillan map. JHEP 1607, 122 (2016) arXiv:1602.00679

    Article  ADS  MathSciNet  Google Scholar 

  14. Vegh, D.: Segmented Strings from a Different Angle. arXiv:1601.07571

  15. Vegh, D.: Segmented Strings Coupled to a B-Field. arXiv:1603.04504

  16. Freund P.G.O., Olson M.: Nonarchimedean strings. Phys. Lett. B 199, 186 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  17. Freund P.G.O., Witten E.: Adelic string amplitudes. Phys. Lett. B 199, 191 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  18. Zabrodin A.V.: Nonarchimedean strings and Bruhat–Tits trees. Commun. Math. Phys. 123, 463–483 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Bruhat F., Tits J.: Groupes réductifs sur un corps local: I. Données radicielles valuées. Publ. Math. l’IHÉS 41, 5–251 (1972)

    Article  MATH  Google Scholar 

  20. Harlow D., Shenker S.H., Stanford D., Susskind L.: Tree-like structure of eternal inflation: a solvable model. Phys. Rev. D 85, 063516 (2012) arXiv:1110.0496

    Article  ADS  Google Scholar 

  21. Knizhnik, V.G., Polyakov, A.M.: Unpublished (1987)

  22. Parisi G.: On p-adic functional integrals. Mod. Phys. Lett. A 3, 639–643 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  23. Spokoiny B.L.: Quantum geometry of nonarchimedean particles and strings. Phys. Lett. B 208, 401–406 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  24. Zhang R.-b.: Lagrangian formulation of open and closed P-adic strings. Phys. Lett. B 209, 229–232 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  25. Melzer E.: Nonarchimedean conformal field theories. Int. J. Mod. Phys. A 04(18), 4877–4908 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  26. Ghoshal D.: p-adic string theories provide lattice discretization to the ordinary string worldsheet. Phys. Rev. Lett. 97, 151601 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Gerasimov A.A., Shatashvili S.L.: On exact tachyon potential in open string field theory. JHEP 10, 034 (2000) arXiv:hep-th/0009103

    Article  ADS  MathSciNet  Google Scholar 

  28. Gervais J.-L.: p-adic analyticity and Virasoro algebras for conformal theories in more than two dimensions. Phys. Lett. B 201, 306 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  29. Chekhov L.O., Mironov A.D., Zabrodin A.V.: Multiloop calculations in P-adic string theory and Bruhat–Tits trees. Commun. Math. Phys. 125, 675 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Manin Y.I., Marcolli M.: Holography principle and arithmetic of algebraic curves. Adv. Theor. Math. Phys. 5, 617–650 (2002) arXiv:hep-th/0201036

    Article  MathSciNet  MATH  Google Scholar 

  31. Manin Y.I.: Three-dimensional hyperbolic geometry as ∞-adic Arakelov geometry. Invent. Math. 104(1), 223–243 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Brekke L., Freund P.G.O., Olson M., Witten E.: Non-archimedean string dynamics. Nucl. Phys. B 302, 365 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  33. Gouvêa F.Q.: p-Adic Numbers. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  34. Manin Y.I.: p-Adic automorphic functions. J. Math. Sci. 5(3), 279–333 (1976)

    Article  MATH  Google Scholar 

  35. Mueck W., Viswanathan K.S.: Conformal field theory correlators from classical scalar field theory on AdS(d+1). Phys. Rev. D 58, 041901 (1998) arXiv:hep-th/9804035

    Article  ADS  MathSciNet  Google Scholar 

  36. Freedman D.Z., Mathur S.D., Matusis A., Rastelli L.: Correlation functions in the CFT(d)/AdS(d+1) correspondence. Nucl. Phys. B 546, 96–118 (1999) arXiv:hep-th/9804058

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. D’Hoker E., Freedman D.Z., Mathur S.D., Matusis A., Rastelli L.: Graviton exchange and complete four point functions in the AdS/CFT correspondence. Nucl. Phys. B 562, 353–394 (1999) arXiv:hep-th/9903196

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Bao N., Cao C., Carroll S.M., Chatwin-Davies A., Hunter-Jones N., Pollack J., Remmen G.N.: Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence. Phys. Rev. D 91(12), 125036 (2015) arXiv:1504.06632

    Article  ADS  MathSciNet  Google Scholar 

  39. Maldacena J.M.: Wilson loops in large N field theories. Phys. Rev. Lett. 80, 4859–4862 (1998) arXiv:hep-th/9803002

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Rey S.-J., Yee J.-T.: Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity. Eur. Phys. J. C 22, 379–394 (2001) arXiv:hep-th/9803001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press Limited, 24-28 Oval Road, London NW1 7DX (1982)

  42. Zinoviev Y.M.: Ising model on the generalized Bruhat–Tits tree. Commun. Math. Phys. 130, 433–440 (1990)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Joseph Henry Laboratories, Princeton University, Princeton, NJ, 08544, USA

    Steven S. Gubser & Sarthak Parikh

  2. Institut für Theoretische Physik, TU Dresden, 01062, Dresden, Germany

    Johannes Knaute

  3. Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120, Heidelberg, Germany

    Andreas Samberg

  4. ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, 64291, Darmstadt, Germany

    Andreas Samberg

  5. Institute of Physics, Jagiellonian University, S. Lojasiewicza 11, 30-348, Krakow, Poland

    Przemek Witaszczyk

Authors
  1. Steven S. Gubser
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  2. Johannes Knaute
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  3. Sarthak Parikh
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  4. Andreas Samberg
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  5. Przemek Witaszczyk
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Correspondence to Steven S. Gubser.

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Communicated by X. Yin

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Gubser, S.S., Knaute, J., Parikh, S. et al. p-Adic AdS/CFT. Commun. Math. Phys. 352, 1019–1059 (2017). https://doi.org/10.1007/s00220-016-2813-6

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  • DOI: https://doi.org/10.1007/s00220-016-2813-6

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