Constructing CFTs from AdS flows

TL;DR

This paper develops a method to derive conformal field theory correlation functions from AdS space flows, providing explicit solutions and recursion relations, and applies it to compute one-loop corrections in an O(N) model.

Contribution

It introduces a flow equation approach for CFT correlators from AdS, solving for low-point functions and deriving Mellin amplitude recursion relations at tree and one-loop levels.

Findings

Derived flow equations for correlation functions in AdS/CFT.

Solved for one- and two-point functions and obtained conformal dimension corrections.

Established recursion relations for Mellin amplitudes at tree and one-loop levels.

Abstract

We study the renormalization group flow equations for correlation functions of weakly coupled quantum field theories in AdS. Taking the limit where the external points approach the conformal boundary, we obtain a flow of conformally invariant correlation functions. We solve the flow for one- and two-point functions and show that the corrections to the conformal dimensions can be obtained as an integral over the Mellin amplitude of the four-point function. We also derive the flow of the Mellin amplitude for higher $n$-point functions. We then consider the flows at tree level and one loop (in AdS), and show that one obtains exactly the recursion relations for the corresponding Mellin amplitudes derived earlier by Fitzpatrick et al. [arXiv:1107.1499] at tree level and Yuan [arXiv:1710.01361,arXiv:1801.07283] at one loop. As an application, we furthermore compute one-loop corrections to the conformal dimensions for some operators in the CFT dual to an $\mathrm{O}(N)$ scalar model in AdS.