Near-extremal correlators and vanishing supergravity couplings in AdS/CFT

TL;DR

The paper investigates near-extremal $n$-point correlators in ${\cal N}=4$ SYM and their dual description in type IIB supergravity on $AdS_5\times S^5$. It demonstrates that, to order $g^2$, field-theory correlators factor into sums of products of lower-point functions, and shows that AdS exchange diagrams replicate this factored structure, while bulk contact terms would spoil it unless the corresponding couplings vanish. The authors establish a precise match for the $E_5^2$ case and conjecture a general vanishing pattern for near-extremal bulk couplings $E_n^m$ with $m\le n-3$, consistent with consistent KK truncation. They provide an inductive argument and discuss the implications for the AdS/CFT correspondence, highlighting the role of KK reduction on $S^5$ in enforcing these vanishing couplings.

Abstract

We study near-extremal n-point correlation functions of chiral primary operators, in which the maximal scale dimension k is related to the others by k=\sum_i k_i-m with m equal to or smaller than n-3. Through order g^2 in field theory, we show that these correlators are simple sums of terms each of which factors into products of lower-point correlators. Terms which contain only factors of two- and three-point functions are not renormalized, but other terms have non-vanishing order g^2 corrections. We then show that the contributing AdS exchange diagrams neatly match this factored structure. In particular, for n=4,5 precise agreement in form and coefficient is established between supergravity and the non-renormalized factored terms from field theory. On the other hand, contact diagrams in supergravity would produce a non-factored structure. This leads us to conjecture that the corresponding bulk couplings vanish, so as to achieve full agreement between the structure of these correlators in supergravity and weak-coupling field theory.

Figures (14)

• Figure 1: Feynman graphs contributing to the correlator $\langle {\cal O}_4 {\cal O}_2 {\cal O}_2 {\cal O}_2 {\cal O}_2 \rangle$ at the free-field level.
• Figure 2: Graphs contributing to the correlator $\langle {\cal O}_4 {\cal O}_2 {\cal O}_2 {\cal O}_2 {\cal O}_2 \rangle$ at order $g^2$.
• Figure 3: Witten diagrams contributing to $E_5^2$ correlators in the AdS/CFT correspondence.
• Figure 4: Non-vanishing contributions to $E_5^2$ functions in AdS supergravity. Diagram $d'$ does not appear for $\langle {\cal O}_4 {\cal O}_2 {\cal O}_2 {\cal O}_2 {\cal O}_2 \rangle$.
• Figure 5: Graphs contributing to $E_5^2$ functions at the free-field level. Diagram $c$ is disconnected in the particular case $k=4$, $k_i=2$ and therefore was not considered in Section 2.1.