Decoding the hologram: Scalar fields interacting with gravity

TL;DR

The paper develops a CFT-based construction of bulk observables for a scalar field interacting with gravity in AdS by enforcing bulk micro-causality through carefully chosen three-point functions with boundary Weyl tensors and scalars. It shows that adding smeared, non-primary, higher-dimension (multi-trace) operators cancels spacelike singularities and yields bulk operators that transform correctly under AdS isometries, at least to leading orders in 1/N. The approach extends to gauge fields, where a tower of operators ensures micro-causality and a Wilson-line–like bulk transformation in holographic gauge, and to gravity, where the boundary Weyl tensor plays the role of the gravitational observable. The results imply that, at large N, bulk locality can be systematically realized in higher-point functions, supporting a holographic map from boundary CFT data to semi-classical bulk physics, albeit with limitations at finite N due to the non-existence of the required operator towers.

Abstract

We construct smeared CFT operators which represent a scalar field in AdS interacting with gravity. The guiding principle is micro-causality: scalar fields should commute with themselves at spacelike separation. To O(1/N) we show that a correct and convenient criterion for constructing the appropriate CFT operators is to demand micro-causality in a three-point function with a boundary Weyl tensor and another boundary scalar. The resulting bulk observables transform in the correct way under AdS isometries and commute with boundary scalar operators at spacelike separation, even in the presence of metric perturbations.