Conformal correlators as simplex integrals in momentum space

Conformal correlators as simplex integrals in momentum space

We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression
for the function of cross ratios involves (n−2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations.

AdS-CFT Correspondence, Conformal Field Theory, Conformal and W Symmetry

Skenderis, Kostas

09f32871-ffb1-4f4a-83bc-df05f4d17a09

McFadden, Paul

4e7762ff-9b96-4516-b333-be01784fdbae

Bzowski, Adam

89bef061-563e-4b14-ac4d-0418692e9992

Skenderis, Kostas

09f32871-ffb1-4f4a-83bc-df05f4d17a09

McFadden, Paul

4e7762ff-9b96-4516-b333-be01784fdbae

Bzowski, Adam

89bef061-563e-4b14-ac4d-0418692e9992

Skenderis, Kostas, McFadden, Paul and Bzowski, Adam
(2021)
Conformal correlators as simplex integrals in momentum space.
*Journal of High Energy Physics*, 2021 (1), [192].
(doi:10.1007/JHEP01(2021)192).

## Abstract

We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression
for the function of cross ratios involves (n−2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations.

Text

** JHEP_298P_0820_Accepted
- Accepted Manuscript**
## More information

Accepted/In Press date: 19 December 2020

e-pub ahead of print date: 28 January 2021

Keywords:
AdS-CFT Correspondence, Conformal Field Theory, Conformal and W Symmetry

## Identifiers

Local EPrints ID: 446743

URI: http://eprints.soton.ac.uk/id/eprint/446743

ISSN: 1029-8479

PURE UUID: 98526170-ab00-4321-9d5a-7d063858f96c

## Catalogue record

Date deposited: 19 Feb 2021 17:32

Last modified: 20 Apr 2021 01:45

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## Contributors

Author:
Paul McFadden

Author:
Adam Bzowski

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