# Conformal correlators as simplex integrals in momentum space

2020-08-20

Authors: Adam Bzowski, Paul McFadden, Kostas Skenderis

Preprint number: UUITP-31/20

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.