Mellin Representation of Conformal Correlation Functions in the AdS/CFT G.A. Kerimov Trakya University Edirne, Turkey ` 7th Mathematical Physics Meeting.

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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

Outline ETH Zurich, 30 June 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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ETH Zurich, 30 June Belgrade, 9 September 2012