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 Meeting9-19 September 2012 Belgrade Serbia

http://www.photoshopmagazin.com/paylasim/2315

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

Introducton

Mellin amplitudes:

Mack (2009)

3 Belgrade, 9 September 2012

Introducton

with

Mellin amplitudes:

Mack (2009)


Introducton

At large

Mellin amplitudes are simple meromorphic functions of their arguments

Mack (2009)


the Mellin amplitudes are related to the S-matrix in flat space

Penedones (2011)

Scalar Field Theory in the EAdS/CFT

EAdS:



Kerimov (2007)with

EAdS:



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Belgrade, 9 September 2012

where : : denotes Wick ordering


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Horospherical coordinates:


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Horospherical coordinates:


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CFT Correlators:

CFT correlators in perturbation theory can be calculated from this by using Wick's theorem. The result is a set of rules, analogousto Feynman rules, namely


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Wigner coefficients of SO(d+1,1) :

Kerimov (1985)

where

Momentum-space Representation of CFT Correlators

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Feynman rules:

Vertex :

External line :

Enternal line

with or

Integrate over and

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p1

p2 p3

p4

5

1

2

3

4

p5


Figure 1: Four-point Witten diagram in cubic scalar theory

Associated 4-point function is

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Mellin amplitudes via EAdS/CFT

where

Inserting the integral representation

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where

with

Inserting the integral representation

one finds

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Symanziks n-star formulaSymanzik (1972)

with

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where

4-point Mellin amplitude:

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4-point Mellin amplitude:

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ETH Zurich, 30 June 2010 26


Mellin Representation of Conformal Correlation Functions in the AdS/CFT

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