A Stereoscopic Look into the Bulk

TL;DR

The paper introduces OPE blocks as a natural, diffeomorphism-friendly CFT basis whose bulk duals are geodesic (and minimal-surface) operators, establishing a gauge-invariant holographic dictionary via kinematic space. It shows that OPE blocks satisfy Klein-Gordon dynamics on kinematic space, while their bulk duals arise from Radon/X-ray transforms of bulk fields, with intertwining relations linking AdS and kinematic-space operators. This framework unifies the emergence of bulk locality, explains the geometric origin of conformal blocks as kinematic-space propagators, and reproduces key results (linearized Einstein equations, geodesic Witten diagrams) at leading order in 1/N. The discussion extends to higher dimensions, surfaces, and potential extensions to interacting bulk dynamics, excited states, and even de Sitter holography, offering a versatile, diffeomorphism-invariant organizing principle for AdS/CFT. Overall, the work furnishes a deep, geometrically flavored route to reconstruct bulk physics from boundary data using OPE blocks and their kinematic-space incarnation.

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.

Figures (21)

• Figure 1: Kinematic space for $\rm{AdS}_3$ has the $\rm{dS}_2 \times \rm{dS}_2$ metric of eq. (\ref{['eq:ksads3']}). It is both the space of causal diamonds in $\rm{CFT}_2$ and the space of space-like geodesics in $\rm{AdS}_3$. To account for complementary causal diamonds (such as the two shown in the left panel) which are associated with the same geodesic (the right panel), we work with the space of oriented geodesics. The middle panel shows the two images of the same bulk geodesic that differ in orientation.
• Figure 2: The coordinates (\ref{['eq:defkscoords']}) of kinematic space represent the center position and the half-width of the causal diamond in the left-moving light-like coordinate; analogous relations define $\bar{z}$ and $\bar{\ell}$.
• Figure 3: When the left-moving projection of one diamond contains the left-moving projection of another, they are time-like separated in the $z$ (left-moving) factor of kinematic space; similar relations apply in the $\bar{z}$ (right-moving) component. This leads to several possible causal relations between two intervals, e.g. the big blue causal diamond is in the $z$-future of diamonds A and C and in the $\bar{z}$-future of diamonds B and C. In the overarching causal structure, the blue diamond is preceded by C, but not related to A or B.
• Figure 4: Causality in each de Sitter component of kinematic space means that the OPE block at $(z_1, \bar{z}_1, z_2, \bar{z}_2)$ depends only on the initial data between $z_1$ and $z_2$ in the first component and between $\bar{z}_1$ and $\bar{z}_2$ in the second component. These loci span the CFT causal diamond with corners at $x_1$ and $x_2$.
• Figure 5: A product of scalar $\mathrm{CFT}_{2}$ operators inserted at $x_1$ and $x_2$ can be expanded in terms of OPE blocks, which consist of primary operators smeared over the causal diamond $\diamond_{12}$.