A Stereoscopic Look into the Bulk (1604.03110v2)

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.

• The paper introduces a novel framework using OPE blocks to map non-local CFT operators to local bulk operators in AdS.
• It leverages kinematic space to reinterpret the operator product expansion, linking geodesic integrals to conformal blocks.
• The work extends the Ryu-Takayanagi approach by employing inverse X-ray techniques for reconstructing local bulk fields and modular Hamiltonians.

A Stereoscopic Look into the Bulk: A Summary

Introduction

This paper presents a novel perspective on the AdS/CFT correspondence by introducing a new framework for understanding the bulk-boundary duality through conformal field theory (CFT) operators. The authors propose a holographic dictionary that emphasizes stereoscopic depth perception in the bulk, aiming to provide a richer understanding of the duality. Central to this discussion are the so-called "OPE blocks," which are segments of the operator product expansion (OPE) in conformal field theories and establish a connection between CFT operators and their dual bulk operators in anti-de Sitter (AdS) space.

Key Concepts

1. OPE Blocks: In CFTs, operator product expansions decompose into contributions from conformal families. The paper identifies a class of operators, OPE blocks, which are non-local in the CFT but correspond closely to local bulk operators when holographically mapped. At leading order in $1/N$, these blocks are dual to the integrals of bulk fields along geodesics or on minimal surfaces in AdS.
2. Kinematic Space: The authors emphasize the significance of kinematic space, which is framed as the set of all point pairs in a CFT. In this framework, OPE blocks can be treated as fields propagating within this kinematic space, presenting a new lens through which to consider local vs. non-local operator transformations in holographic theories.
3. Holographic Duality and Ryu-Takayanagi Generalization: The Ryu-Takayanagi proposal links minimal surface areas in the bulk to entanglement entropy in the boundary CFT. By establishing that OPE blocks can be seen as holographic duals related to minimal surfaces, this work sets the foundation for a more generalized framework beyond entanglement entropy, extending to other bulk fields.

Implications and Applications

Numerical and Theoretical Insights

The paper asserts that when CFT operators join in OPE blocks, they obtain a simple geometric realization within kinematic space. This realization enables derivations of many existing results in the AdS/CFT framework, such as:

• Derivation of linearized Einstein’s equations.
• Calculation methods for conformal blocks and their geometric correspondents, geodesic Witten diagrams.

These results imply strong numerical coherence between the calculated properties in kinematic spaces and their established theoretical counterparts.

Construction of Local Bulk Operators:

The inverse X-ray transform, typically a problem of reconstructing density functions from line integrals, is adapted here to reconstruct local bulk operators. This approach elucidates connections between extended geodesic operators and the much-discussed HKLL (Hamilton-Kabat-Lifschytz-Lowe) bulk reconstruction proposal.

Applications to Entanglement and Modular Hamiltonians:

The paper finds that the modular Hamiltonian of a subsystem's reduced density matrix can be viewed through the lens of an OPE block, corroborating its place within a broader perturbative framework under gravitational theories.

Future Directions in Higher Dimensions:

Though the focus is primarily on CFT$_2$ and AdS$_3$, extensions to higher dimensions bring about discussions of homogeneous spaces and Radon transforms, with novel findings regarding surface operator expansions in CFT$_d$.

Conclusion

This work bridges geometrical optics from traditional stereoscopic methods to quantum field learning within the holographic duality narrative. The novel engagement with kinematic spaces could lead to enhanced understanding of holographic principles, potentially stimulating further exploration of connections between entanglement, geometry, and deeper structural symmetries in dualities. Future applications may delve into refining our knowledge of gravitational theories and bringing forward enriched analytical tools for studying field theories with strong holographic properties.