Correlation Functions and Massive Kaluza-Klein Modes in the AdS/CFT Correspondence

TL;DR

The paper advances the AdS/CFT program by analyzing four-point functions of 1/2-BPS operators dual to massive KK modes in AdS$_5$ and demonstrating a partial non-renormalisation structure predicted by field theory insertions. It develops a concise, sigma-model–type five-dimensional supergravity action for the dimension-3 KK mode and computes the corresponding four-point amplitude, which cleanly splits into a free and an interacting part in exact agreement with the insertion-based expectations. The results validate the dynamical content encoded in the Lagrangian and the OPE structure, providing a nontrivial test of AdS/CFT beyond the massless sector. The OPE analysis yields explicit anomalous-dimension formulas for long multiplets at order ${1/N^2}$ and clarifies how semi-short multiplets organize into complete long multiplets, ensuring consistency with unitarity and superconformal symmetry.

Abstract

We study four-point correlation functions of 1/2-BPS operators in N=4 SYM which are dual to massive KK modes in AdS_5 supergravity. On the field theory side, the procedure of inserting the SYM action yields partial non-renormalisation of the four-point amplitude for such operators. In particular, if the BPS operators have dimensions equal to three or four, the corresponding four-point amplitude is determined by one or two independent functions of the two conformal cross-ratios, respectively. This restriction on the amplitude does not merely follow from the superconformal Ward identities, it also encodes dynamical information related to the structure of the gauge theory Lagrangian. The dimension 3 BPS operator is the AdS/CFT dual of the first non-trivial massive Kaluza-Klein mode of the compactified type IIB supergravity, whose interactions go beyond the level of the five-dimensional gauged N=8 supergravity. We show that the corresponding effective Lagrangian has a surprisingly simple sigma-model-type form with at most two derivatives. We then compute the supergravity-induced four-point amplitude for the dimension 3 operators. Remarkably, this amplitude splits into a "free" and an "interacting" parts in exact agreement with the structure predicted by the insertion procedure. The underlying OPE fulfills the requirements of superconformal symmetry and unitarity.

Figures (3)

• Figure 1: Basic contractions
• Figure 2: The case $k=2$
• Figure 3: The case $k=3$