Superconformal Invariants or How to Relate Four-point AdS Amplitudes

TL;DR

The paper derives an explicit eighth-order differential operator $\Delta^{(8)}$ that maps the N=2 superconformal bottom component of the four-point correlator of stress-tensor multiplets in N=4 SYM to its top component, using the underlying N=2 superconformal invariants. This operator, with $\Delta^{(8)}=(\Delta^{(2)})^2 u^2 v^2 (\Delta^{(2)})^2$ and $\Delta^{(2)}=u\partial_u^2+v\partial_v^2+(u+v-1)\partial_u\partial_v+2(\partial_u+\partial_v)$, acts on the function $F(u,v)$ of cross-ratios to produce the top component, and has a compact form in $x,\bar{x}$ variables. In the supergravity limit, the authors relate $F$ to the dilaton/axion amplitude via $F_{SG}$ and show that $H_{SG}=\Delta^{(8)}F_{SG}$ reproduces the known AdS$_5$ results (via the auxiliary function $\Phi^{(1)}(u,v)$) up to a normalization, providing a non-trivial consistency check with D'Hoker and Arutyunov. The framework thus offers a robust tool to connect CFT data to AdS amplitudes and may extend to higher-weight correlators and corrections beyond leading $1/N^2$.

Abstract

Using the form of N=2 superconformal invariants we derive the explicit relation between the bottom and top components of the correlator of four stress-tensor multiplets in N=4 Super Yang-Mills. The result is given in terms of an eighth order differential operator acting on the function of two variables which characterises these correlators. It allows us to show a non-trivial consistency relation between the known results for the corresponding supergravity amplitudes on AdS5.