Locality, bulk equations of motion and the conformal bootstrap

TL;DR

This paper introduces a 'bulk bootstrap' method to construct local bulk operators in a conformal field theory, connecting bulk locality with conformal bootstrap techniques and simplifying the determination of CFT data.

Contribution

It develops a new approach called the bulk bootstrap that ensures bulk operator consistency across OPE channels, linking bulk locality with conformal bootstrap without needing conformal blocks.

Findings

Bulk bootstrap reproduces key results of the conformal bootstrap.

It simplifies the determination of CFT data like OPE coefficients and anomalous dimensions.

The method clarifies the relation between bulk locality and the bootstrap in theories with a 1/N expansion.

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 $1/N^2$.