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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

17. 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 Scalar Field Theory in the EAdS/CFT 17 Belgrade, 9 September 2012

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

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

20. p1p1 p2p2 p3p3 p4p4 p5p5 Momentum-space Representation of CFT Correlators Figure 1: Four-point Witten diagram in cubic scalar theory Associated 4-point function is 20 Belgrade, 9 September 2012

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

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

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

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

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

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