Four-point functions of different-weight operators in the AdS/CFT correspondence

TL;DR

The paper computes the first non-trivial four-point function of two weight-2 and two weight-3 half-BPS operators in N=4 SYM at strong coupling via AdS/CFT. The authors derive the five-dimensional supergravity Lagrangian, show that four-derivative quartic terms cancel, and evaluate exchange diagrams including massive vectors and symmetric tensors, expressing the result in terms of D-functions. A key result is that the quantum part of the amplitude is fixed by the insertion procedure and depends on a single function F(s,t) of the cross-ratios, with F obeying the crossing relation F(s/t,1/t) = t F(s,t). Comparison with CFT predictions confirms the gravity result matches the insertion-derived constraints up to free-field constants, extending tests to mixed-weight BPS operators. The work reinforces AdS/CFT in a new sector and suggests a sigma-model like structure underlies the AdS5 supergravity description.

Abstract

We calculate four-point correlation functions of two weight-2 and two weight-3 1/2-BPS operators in \mathcal{N}=4 SYM in the large N limit in supergravity approximation. By the AdS/CFT conjecture, these operators are dual to AdS supergravity scalar fields s_2 and s_3 with mass m^2 = -4 and m^2 = -3 respectively. This is the first non-trivial four-point function of mixed-weight operators of lowest conformal dimensions. We show that the supergravity-induced four-point function splits into a "free" and a "quantum" part, where the quantum contribution obeys non-trivial constraints coming from the insertion procedure in the gauge theory, in particular, it depends on only one function of the conformal cross-ratios.

Figures (2)

• Figure 1: Propagator structures. The diagrams are divided into four equivalence classes. The elements in an equivalence class are related by crossing symmetry.
• Figure 2: Witten diagrams contributing to the four-point function.