Einstein gravity from ANEC correlators

TL;DR

This work shows that in large $N$ CFTs with a large gap to higher-spin operators, the OPE of a local operator with the ANEC can be written as a finite differential operator, enabling a resummed, finite-distance OPE and exact control of multi-ANEC correlators. The vanishing of ANEC commutators imposes strong constraints on CFT data, ultimately forcing $a=c$ in $d=4$ and ensuring a bulk dual described by semi-classical Einstein gravity with minimally coupled matter. The approach extends to higher-point ANEC correlators, where positivity further tightens collider bounds, effectively isolating minimal couplings in the holographic dual and highlighting a Virasoro-like structure in higher dimensions. Overall, the paper provides a practical, bootstrap-like framework connecting light-ray operators to bulk Einstein gravity under a large-gap, large-$N$ regime, with potential extensions to finite gaps and broader light-ray algebras.

Abstract

We study correlation functions with multiple averaged null energy (ANEC) operators in conformal field theories. For large $N$ CFTs with a large gap to higher spin operators, we show that the OPE between a local operator and the ANEC can be recast as a particularly simple differential operator acting on the local operator. This operator is simple enough that we can resum it and obtain the finite distance OPE. Under the large $N$ - large gap assumptions, the vanishing of the commutator of ANEC operators tightly constrains the OPE coefficients of the theory. An important example of this phenomenon is the conclusion that $a=c$ in $d=4$. This implies that the bulk dual of such a CFT is semi-classical Einstein-gravity with minimally coupled matter.

Figures (6)

• Figure 1: The Penrose diagram of Minkowski space in three dimensions. The ANEC operator has been sent to spatial infinity, at a given point on the celestial sphere (here a point on the circle). It is still integrated along a null direction represented by the red line. The vector $\vec{n}$ indicates the direction in which the operator is inserted.
• Figure 2: The OPE expansion of the four-point function in the channel we have picked. To compute the correlator to the first two orders in $1/N^2$, the sum over $O'$ is over all single-trace and double-trace operators.
• Figure 3: A bulk picture of a graviton scattering through a shockwave. At tree-level the particle simply gets shifted when it passes through the shockwave and particle number is conserved.
• Figure 4: A bulk picture of graviton production when passing through a shockwave. One can see that it is necessarily a loop effect.
• Figure 5: The sum over operators has reduced to a sum over only the single-trace operators.