Locality, bulk equations of motion and the conformal bootstrap

TL;DR

The authors present a framework to construct local bulk operators in a CFT up to order $1/N^2$ by enforcing locality across multiple OPE channels, a program they term the bulk bootstrap. This approach reduces the problem to a set of 3-point functions and their smearing into bulk operators, without requiring explicit conformal blocks, and shows how consistency across channels determines $1/N^2$ corrections to OPE coefficients or anomalous dimensions. They demonstrate the method with scalar and vector exchanges, deriving how CFT data maps directly to bulk equations of motion and interactions, including local bulk cubic and quartic vertices. The results illuminate the connection between bulk locality and bootstrap constraints in holographic theories and offer a practical route to extract CFT data from bulk equations of motion. The work also outlines extensions to higher spin exchanges and discusses limitations and future directions, such as loop effects and the need for a large spectral gap for a local bulk description.

Abstract

We develop an approach to construct local bulk operators in a CFT to order $1/N^2$. Since 4-point functions are not fixed by conformal invariance we use the OPE to categorize possible forms for a bulk operator. Using previous results on 3-point functions we construct a local bulk operator in each OPE channel. We then impose the condition that the bulk operators constructed in different channels agree, and hence give rise to a well-defined bulk operator. We refer to this condition as the "bulk bootstrap." We argue and explicitly show in some examples that the bulk bootstrap leads to some of the same results as the regular conformal bootstrap. In fact the bulk bootstrap provides an easier way to determine some CFT data, since it does not require knowing the form of the conformal blocks. This analysis clarifies previous results on the relation between bulk locality and the bootstrap for theories with a $1/N$ expansion, and it identifies a simple and direct way in which OPE coefficients and anomalous dimensions determine the bulk equations of motion to order $1/N^2$.