Gravitational S-matrix from CFT dispersion relations

TL;DR

This work shows that the double-discontinuities of the four-point stress-tensor correlator in ${\cal N}=4$ SYM at large $N$ and strong coupling encapsulate the full CFT data up to one-loop (order ${\cal O}(1/c^2)$), enabling a dispersion-like reconstruction via two complementary methods: the Froissart-Gribov inversion and large-spin perturbation theory. The authors derive explicit tree-level and one-loop CFT data, including anomalous dimensions and OPE coefficients, and demonstrate their consistency across methods. In the flat-space limit, the CFT data reproduce the ten-dimensional IIB supergravity one-loop amplitude, confirming a precise link between CFT analyticity and S-matrix dispersion relations and highlighting the role of KK mixing in encoding higher-dimensional geometry. The analysis also clarifies reconstruction ambiguities and shows that supersymmetry constrains possible counter-terms, with implications for bulk higher-derivative interactions and nonperturbative aspects of the dual gravity theory.

Abstract

We analyse the double-discontinuities of the four-point correlator of the stress-tensor multiplet in N=4 SYM at large t' Hooft coupling and at order $1/N^4$, as a way to access one-loop effects in the dual supergravity theory. From these singularities we extract CFT-data by using two inversion procedures: one based on a recently proposed Froissart-Gribov inversion integral, and the other based on large spin perturbation theory. Both procedures lead to the same results and are shown to be equivalent more generally. Our computation parallels the standard S-matrix reconstruction via dispersion relations. In a suitable limit, the result of the conformal field theory calculation is compared with the one-loop graviton scattering amplitude in ten-dimensional IIB supergravity in flat space, finding perfect agreement.

Figures (7)

• Figure 1: Correlators in any large-$N$ theories can be fully reconstructed from singularities (denoted by the cut) that are saturated by single-trace operators. Theories with a gravity dual correspond to the case where the sum is effectively finite.
• Figure 2: At one-loop order in the $1/N$ expansion, the singularities caused by double-trace exchanges are equal to products of single-trace tree amplitudes.
• Figure 3: The inversion integral produces the full correlator, given on the left as a sum over Witten diagrams, from the double-discontinuity in a single channel (and more generally, the $t$ and $u$ channels).
• Figure 4: The full one-loop correlator, equal to a sum over many Witten diagrams, is reconstructed from the double-discontinuity in a single channel.
• Figure 5: (a) Four-point kinematics in the complex $\rho$-plane. (b) The kinematics on the Lorentzian cylinder. In the "bulk-point" limit $z\to{\bar{z}}$, particles are effectively beamed onto a point in the AdS interior of the cylinder.