Correlation functions in the CFT(d)/AdS(d+1) correpondence

TL;DR

The paper develops a conformal-technique framework to compute AdS$_{d+1}$ integrals that define CFT$_d$ correlators, applying it to explicit 3-point amplitudes for scalars and gauge fields. A central methodological advance is an inversion-based approach that reduces bulk integrals to manageable forms, enabling closed-form results for scalar 3-point functions with coefficients $a_1$ and $a_2$. It then analyzes flavor-current correlators in ${ m CFT}_d$, detailing the conformal tensor structure and Ward identities, and computes the corresponding AdS$_5$ amplitudes from Yang-Mills and Chern-Simons terms. The results show that the normal-parity part of $igra J J J igra$ matches the expected field-theory structure and non-renormalization expectations, while the abnormal-parity piece is captured by the boundary anomaly via the bulk CS term, fixing the CS coupling to $k = N^2-1$ and demonstrating consistency with 1-loop results in the boundary theory. A subtle normalization issue for two-point functions with $oldsymbol{ riangle} eq d$ is discussed, highlighting subtleties in the Witten prescription for 2-point functions at the boundary.

Abstract

Conformal techniques are applied to the calculation of integrals on AdS(d+1) space which define correlators of composite operators in the superconformal field theory on the d-dimensional boundary. The 3-point amplitudes for scalar fields of arbitrary mass and gauge fields in the AdS supergravity are calculated explicitly. For 3 gauge fields we compare in detail with the known conformal structure of the SU(4) flavor current correlator <J_i^a J_j^b J_k^c> of the N=4, d=4 SU(N) SYM theory. Results agree with the free field approximation as would be expected from superconformal non-renormalization theorems. In studying the Ward identity relating <J_i^a O^I O^J> to <O^I O^J> for (non-marginal) scalar composite operators O^I, we find that there is a subtlety in obtaining the normalization of <O^I O^J> from the supergravity action integral.

Figures (1)

• Figure 1: Witten diagrams.