Bulk equations of motion from CFT correlators

TL;DR

This work shows that bulk dynamics in AdS/CFT can be reconstructed from CFT data at ${O}(1/N)$ by explicitly building local bulk fields as smeared sums of CFT operators. By enforcing bulk locality (analyticity of correlators for spacelike separations), the authors derive explicit coefficients for an infinite tower of higher-dimension double-trace operators that, when summed, reproduce the bulk equations of motion for interacting scalars, scalars coupled to gauge fields, and scalars coupled to gravity. The results demonstrate that bulk locality fixes the bulk cubic couplings and reveal an intrinsic ambiguity corresponding to bulk field redefinitions, which aligns with CFT uplift ambiguities. The methodology extends across dimensions ($d=1,2,3$) and includes charged scalars and gravitational couplings, providing a robust, fully CFT-derived route to bulk EFT in holographic setups.

Abstract

To O(1/N) we derive, purely from CFT data, the bulk equations of motion for interacting scalar fields and for scalars coupled to gauge fields and gravity. We first uplift CFT operators to mimic local AdS fields by imposing bulk microcausality. This requires adding an infinite tower of smeared higher-dimension double-trace operators to the CFT definition of a bulk field, with coefficients that we explicitly compute. By summing the contribution of the higher-dimension operators we derive the equations of motion satisfied by these uplifted CFT operators and show that we precisely recover the expected bulk equations of motion. We exhibit the freedom in the CFT construction which corresponds to bulk field redefinitions.