# Einstein gravity from ANEC correlators

## TL;DR

This paper demonstrates that in certain large N conformal field theories, the properties of ANEC correlators impose strong constraints on the theory, leading to the conclusion that the dual gravity theory is semi-classical Einstein gravity with minimal matter coupling.

## Contribution

It introduces a simplified differential operator for the ANEC OPE in large N, large gap CFTs and derives constraints on OPE coefficients, notably showing that a=c in four dimensions.

## Key findings

- ANEC correlators lead to constraints on OPE coefficients.
- The OPE between local operators and ANEC can be resummed into a simple differential operator.
- In d=4, the theory implies a=c, indicating semi-classical Einstein gravity dual.

## 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.