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Mellin Representation of Conformal Correlation Functions in the AdS/CFT

Mellin Representation of Conformal Correlation Functions in the AdS/CFT G.A. Kerimov Trakya University Edirne , Turkey. `. 7th Mathematical Physics Meeting 9-19 September 2012 Belgrade Serbia.

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Mellin Representation of Conformal Correlation Functions in the AdS/CFT

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  1. Mellin Representation of Conformal Correlation Functions in the AdS/CFT G.A. Kerimov Trakya University Edirne, Turkey ` 7th Mathematical Physics Meeting 9-19 September 2012 Belgrade Serbia

  2. Belgrade, 9 September 2012 2 Outline • I. Introduction • Scalar Field Theory in the EAdS/CFT • Momentum-space Representation of CFT Correlators • IV. Mellin Amlitudes in EAdS/CFT ETH Zurich, 30 June 2010

  3. Belgrade, 9 September 2012 3 Introducton Mellin amplitudes: Mack (2009)

  4. Belgrade, 9 September 2012 4 Introducton Mellin amplitudes: with Mack (2009)

  5. Belgrade, 9 September 2012 5 Introducton Mellin amplitudes: with Mack (2009)

  6. Belgrade, 9 September 2012 6 Introducton Mellin amplitudes: with Mack (2009)

  7. Belgrade, 9 September 2012 7 Introducton Mellin amplitudes are simple meromorphic functions of their arguments Mack (2009) At large the Mellin amplitudes are related to the S-matrix in flat space Penedones (2011)

  8. Belgrade, 9 September 2012 8 Scalar Field Theory in the EAdS/CFT EAdS:

  9. Belgrade, 9 September 2012 9 Scalar Field Theory in the EAdS/CFT EAdS: Kerimov (2007) with

  10. Belgrade, 9 September 2012 10 Scalar Field Theory in the EAdS/CFT

  11. Belgrade, 9 September 2012 11 Scalar Field Theory in the EAdS/CFT where : : denotes Wick ordering

  12. Belgrade, 9 September 2012 12 Scalar Field Theory in the EAdS/CFT Horospherical coordinates:

  13. Belgrade, 9 September 2012 13 Scalar Field Theory in the EAdS/CFT Horospherical coordinates:

  14. Belgrade, 9 September 2012 14 Scalar Field Theory in the EAdS/CFT

  15. Belgrade, 9 September 2012 15 Scalar Field Theory in the EAdS/CFT

  16. Belgrade, 9 September 2012 16 Scalar Field Theory in the EAdS/CFT

  17. Belgrade, 9 September 2012 17 Scalar Field Theory in the EAdS/CFT CFT Correlators: CFT correlators in perturbation theory can be calculated from this by using Wick's theorem. The result is a set of rules, analogous to Feynman rules, namely

  18. Belgrade, 9 September 2012 18 Momentum-space Representation of CFT Correlators Wigner coefficients of SO(d+1,1) : Kerimov (1985) where

  19. Belgrade, 9 September 2012 19 Momentum-space Representation of CFT Correlators Feynman rules: Vertex : or with External line : Enternal line Integrate over and

  20. Belgrade, 9 September 2012 20 Momentum-space Representation of CFT Correlators p1 p4 p5 p3 p2 Figure 1: Four-point Witten diagram in cubic scalar theory Associated 4-point function is

  21. Belgrade, 9 September 2012 21 Mellin amplitudes via EAdS/CFT Inserting the integral representation where

  22. Belgrade, 9 September 2012 22 Mellin amplitudes via EAdS/CFT Inserting the integral representation where one finds with

  23. Belgrade, 9 September 2012 23 Mellin amplitudes via EAdS/CFT Symanziks n-star formula Symanzik (1972) with

  24. Belgrade, 9 September 2012 24 Mellin amplitudes via EAdS/CFT 4-point Mellin amplitude: where

  25. Belgrade, 9 September 2012 25 Mellin amplitudes via EAdS/CFT 4-point Mellin amplitude:

  26. Belgrade, 9 September 2012 26 Thank you! ETH Zurich, 30 June 2010

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