A Stereoscopic Look into the Bulk

TL;DR

This paper develops a holographic dictionary linking CFT operators called OPE blocks to bulk fields in AdS space, enabling geometric and computational insights into holography and gravity.

Contribution

It introduces OPE blocks as duals to bulk integrals along geodesics, generalizes the Ryu-Takayanagi relation, and provides tools for constructing local bulk operators from non-local CFT objects.

Findings

OPE blocks are dual to integrals of bulk fields along geodesics.

OPE blocks provide a geometric description in kinematic space.

Derived linearized Einstein's equations from CFT data.

Abstract

We present the foundation for a holographic dictionary with depth perception. The dictionary consists of natural CFT operators whose duals are simple, diffeomorphism-invariant bulk operators. The CFT operators of interest are the "OPE blocks," contributions to the OPE from a single conformal family. In holographic theories, we show that the OPE blocks are dual at leading order in 1/N to integrals of effective bulk fields along geodesics or homogeneous minimal surfaces in anti-de Sitter space. One widely studied example of an OPE block is the modular Hamiltonian, which is dual to the fluctuation in the area of a minimal surface. Thus, our operators pave the way for generalizing the Ryu-Takayanagi relation to other bulk fields. Although the OPE blocks are non-local operators in the CFT, they admit a simple geometric description as fields in kinematic space--the space of pairs of CFT points. We develop the tools for constructing local bulk operators in terms of these non-local objects. The OPE blocks also allow for conceptually clean and technically simple derivations of many results known in the literature, including linearized Einstein's equations and the relation between conformal blocks and geodesic Witten diagrams.