Advertisement

Lectures on Holographic Renormalization

  • Ioannis PapadimitriouEmail author
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 176)

Abstract

We provide a pedagogical introduction to the method of holographic renormalization, in its Hamiltonian incarnation. We begin by reviewing the description of local observables, global symmetries, and ultraviolet divergences in local quantum field theories, in a language that does not require a weak coupling Lagrangian description. In particular, we review the formulation of the Renormalization Group as a Hamiltonian flow, which allows us to present the holographic dictionary in a precise and suggestive language. The method of holographic renormalization is then introduced by first computing the renormalized two-point function of a scalar operator in conformal field theory and comparing with the holographic computation. We then proceed with the general method, formulating the bulk theory in a radial Hamiltonian language and deriving the Hamilton–Jacobi equation. Two methods for solving recursively the Hamilton–Jacobi equation are then presented, based on covariant expansions in eigenfunctions of certain functional operators on the space of field theory couplings. These algorithms constitute the core of the method of holographic renormalization and allow us to obtain the holographic Ward identities and the asymptotic expansions of the bulk fields.

Keywords

Ward Identity Dilatation Operator Canonical Momentum Holographic Renormalization Holographic Dictionary 
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.

Notes

Acknowledgments

I would like to thank the International Institute of Physics, Natal, for the hospitality and financial support during the School on Theoretical Frontiers in Black Holes and Cosmology, June 8–19, 2015, where these lectures were delivered.

References

  1. 1.
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113–1133 (1999). arXiv:hep-th/9711200 [Adv. Theor. Math. Phys. 2, 231(1998)]
  2. 2.
    E. Witten, Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998). arXiv:hep-th/9802150
  3. 3.
    S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from noncritical string theory. Phys. Lett. B 428, 105–114 (1998). arXiv:hep-th/9802109 ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Henningson, K. Skenderis, The holographic Weyl anomaly. J. High Energy Phys. 07, 023 (1998). arXiv:hep-th/9806087 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    V. Balasubramanian, P. Kraus, A stress tensor for Anti-de Sitter gravity. Commun. Math. Phys. 208, 413–428 (1999). arXiv:hep-th/9902121 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. de Boer, E.P. Verlinde, H.L. Verlinde, On the holographic renormalization group. J. High Energy Phys. 08, 003 (2000). arXiv:hep-th/9912012 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    P. Kraus, F. Larsen, R. Siebelink, The gravitational action in asymptotically AdS and flat space-times. Nucl. Phys. B 563, 259–278 (1999). arXiv:hep-th/9906127 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    S. de Haro, S.N. Solodukhin, K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence. Commun. Math. Phys. 217, 595–622 (2001). arXiv:hep-th/0002230 ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Bianchi, D.Z. Freedman, K. Skenderis, How to go with an RG flow. J. High Energy Phys. 08, 041 (2001). arXiv:hep-th/0105276 ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Bianchi, D.Z. Freedman, K. Skenderis, Holographic renormalization. Nucl. Phys. B 631, 159–194 (2002). arXiv:hep-th/0112119 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. Martelli, W. Mueck, Holographic renormalization and ward identities with the Hamilton-Jacobi method. Nucl. Phys. B 654, 248–276 (2003). arXiv:hep-th/0205061 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    K. Skenderis, Lecture notes on holographic renormalization. Class. Quantum Gravity 19, 5849–5876 (2002). arXiv:hep-th/0209067 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    I. Papadimitriou, K. Skenderis, AdS/CFT correspondence and geometry, AdS/CFT correspondence: Einstein metrics and their conformal boundaries. Proceedings, 73rd Meeting of Theoretical Physicists and Mathematicians, Strasbourg, France, 11–13 September 2003 (2004), pp. 73–101. arXiv:hep-th/0404176
  14. 14.
    O. Aharony, N. Seiberg, and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115, 1305.0318Google Scholar
  15. 15.
    H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories. Nucl. Phys. B 363, 486–526 (1991)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    B.P. Dolan, Symplectic geometry and Hamiltonian flow of the renormalization group equation. Int. J. Mod. Phys. A 10, 2703–2732 (1995). arXiv:hep-th/9406061 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    I. Papadimitriou, Holographic renormalization as a canonical transformation. J. High Energy Phys. 11, 014 (2010). arXiv:1007.4592 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    D.Z. Freedman, K. Johnson, J.I. Latorre, Differential regularization and renormalization: a new method of calculation in quantum field theory. Nucl. Phys. B 371, 353–414 (1992)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    I. Papadimitriou, Holographic renormalization made simple: an example. Subnucl. Ser. 41, 508–514 (2005)Google Scholar
  20. 20.
    G.W. Gibbons, S.W. Hawking, Action integrals and partition functions in quantum gravity. Phys. Rev. D 15, 2752–2756 (1977)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    I. Papadimitriou, K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes. J. High Energy Phys. 08, 004 (2005). arXiv:hep-th/0505190 ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    R. Arnowitt, S. Deser, C.W. Misner, Canonical variables for general relativity. Phys. Rev. 117, 1595–1602 (1960)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    I. Papadimitriou, Multi-trace deformations in AdS/CFT: exploring the vacuum structure of the deformed CFT. J. High Energy Phys. 05, 075 (2007). arXiv:hep-th/0703152 ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Henneaux, C. Teitelboim, Quantization of Gauge Systems (University of Princeton, Princeton, 1992)zbMATHGoogle Scholar
  25. 25.
    B. C. van Rees, Holographic renormalization for irrelevant operators and multi-trace counterterms. Physics 42 (2011). arXiv:1102.2239
  26. 26.
    B.C. van Rees, Irrelevant deformations and the holographic Callan-Symanzik equation. J. High Energy Phys. 10, 067 (2011). arXiv:1105.5396 CrossRefzbMATHGoogle Scholar
  27. 27.
    K.A. Intriligator, Maximally supersymmetric RG flows and AdS duality. Nucl. Phys. B 580, 99–120 (2000). arXiv:hep-th/9909082 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    K. Skenderis, M. Taylor, Kaluza-Klein holography. J High Energy Phys. 05, 057 (2006). arXiv:hep-th/0603016 ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    K. Skenderis, M. Taylor, Holographic Coulomb branch vevs. J. High Energy Phys. 08, 001 (2006). arXiv:hep-th/0604169 ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Y. Korovin, K. Skenderis, M. Taylor, Lifshitz as a deformation of Anti-de Sitter. J. High Energy Phys. 08, 026 (2013). arXiv:1304.7776 ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    W. Chemissany, I. Papadimitriou, Lifshitz holography: the whole shebang. J. High Energy Phys. 01, 052 (2015). arXiv:1408.0795 ADSCrossRefGoogle Scholar
  32. 32.
    A. O’Bannon, I. Papadimitriou, J. Probst, A Holographic Two-Impurity Kondo Model. arXiv:1510.08123
  33. 33.
    S.F. Ross, Holography for asymptotically locally Lifshitz spacetimes. Class. Quantum Gravity 28, 215019 (2011). arXiv:1107.4451 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    T. Griffin, P. Horava, C.M. Melby-Thompson, Conformal Lifshitz gravity from holography. J. High Energy Phys. 05, 010 (2012). arXiv:1112.5660 ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    I. Papadimitriou, Holographic renormalization of general dilaton-axion gravity. J High Energy Phys. 1108, 119 (2011). arXiv:1106.4826 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    W. Chemissany, I. Papadimitriou, Generalized dilatation operator method for non-relativistic holography. Phys. Lett. B 737, 272–276 (2014). arXiv:1405.3965 ADSCrossRefzbMATHGoogle Scholar
  37. 37.
    I. Kanitscheider, K. Skenderis, M. Taylor, Precision holography for non-conformal branes. J. High Energy Phys. 09, 094 (2008). arXiv:0807.3324 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    P. Breitenlohner, D.Z. Freedman, Positive energy in anti-De Sitter backgrounds and gauged extended supergravity. Phys. Lett. B 115, 197 (1982)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    I. Papadimitriou, K. Skenderis, Correlation functions in holographic RG flows. J. High Energy Phys. 0410, 075 (2004). arXiv:hep-th/0407071 ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    H.J. Rothe, K.D. Rother, Classical and Quantum Dynamics of Constrained Hamiltonian Systems (World Scientific, Singapore, 2010)Google Scholar
  41. 41.
    R. Abraham, J. Marsden, Foundations of Mechanics, 2nd edn. (Benjamin-Cumming, Reading, 1978)zbMATHGoogle Scholar
  42. 42.
    M. de Leon, J.C. Marrero, D.M. de Diego, A geometric Hamilton-Jacobi theory for classical field theories (2008). arXiv:0801.1181

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.SISSA and INFN - Sezione di TriesteTriesteItaly

Personalised recommendations