On a Large N Degeneracy in N=4 SYM and the AdS/CFT Correspondence
G. Arutyunov, E. Sokatchev
TL;DR
The authors analyze the four-point correlator of 1/2-BPS weight-4 operators in ${\cal N}=4$ SYM and its AdS/CFT dual, focusing on how partial non-renormalization constrains the amplitude to two cross-ratio functions ${\cal F}$ and ${\cal G}$. Using ${\cal N}=2$ harmonic superspace and the insertion procedure, they compute the one-loop large-${N}$ gauge-theory amplitude, finding ${\cal F}^{(1\text{loop})}_k(s,t)=-\frac{k^2}{8\pi^2}\frac{\lambda}{N^2}\Phi(s,t)$ and ${\cal F}^{(1\text{loop})}_4={\cal G}^{(1\text{loop})}_4$, i.e., a degeneracy ${\cal F}={\cal G}$ at this order. In parallel, they derive the weight-4 four-point amplitude from Type IIB supergravity on $AdS_5\times S^5$, expressing it through ${\overline D}$-functions as ${\cal F}(s,t)=-\frac{4}{N^2}[2\overline D_{2246}+2s\overline D_{3346}+s^2\overline D_{4446}]$ and a distinct ${\cal G}(s,t)=-\frac{16}{N^2}s[\overline D_{4222}+\cdots+\overline D_{4446}]$, showing ${\cal F}\neq{\cal G}$. This contrast highlights a perturbative degeneracy not evident in gravity and motivates further investigation into higher-loop and instanton effects within the AdS/CFT framework. The work reinforces the role of ${\cal F}$ and ${\cal G}$ in encoding dynamical information and clarifies how large-${N}$ limits interface with holographic predictions.
Abstract
We study the four-point correlator of 1/2-BPS operators of weight 4 in N=4 SYM, which are dual to massive KK modes in AdS_5 supergravity. General field-theoretic arguments lead to a partially non-renormalized form of the amplitude that depends on two a priori independent functions of the conformal cross-ratios. We explicitly compute the amplitude in the large N limit at one loop (order g^2) and in AdS_5 supergravity. Surprisingly, the one-loop result shows that the two functions determining the amplitude coincide while in the supergravity regime they are distinctly different. We discuss the possible implications of this perturbative degeneracy for the AdS/CFT correspondence.