Exploring the BTZ bulk with boundary conformal blocks (1604.07383v1)

Abstract: We point out a simple relation between the bulk field at an arbitrary radial position and the boundary OPE, by placing some old work by Ferrara, Gatto, Grillo and Parisi in the AdS/CFT context. This gives us, in principle, a prescription for extracting the classical bulk field from the boundary conformal block, and also clarifies why the latter is computed by a geodesic Witten diagram. We apply this prescription to the BTZ black hole - viewed as a pure state created by the insertion of a heavy operator in the boundary CFT_2 - and use it to relate a classical field in the bulk to a heavy-light Virasoro conformal block in the boundary. In particular, we obtain a relation between the radial bulk position and the conformal ratios in the boundary CFT. We use this to show that the singular points of the radial bulk equation occur when the dual boundary operators approach each other and that the associated bulk monodromies map to monodromies of the (appropriately transformed) conformal block, thus providing a CFT interpretation of the radial monodromy.

• The paper establishes a clear relation linking BTZ bulk fields to the boundary OPE via geodesic Witten diagrams.
• It applies this framework to BTZ black holes, showing that heavy-light Virasoro blocks on the boundary encode classical bulk data.
• It reveals a precise mapping between the BTZ radial position and boundary conformal ratios, clarifying corresponding singular behaviors.

Exploring the BTZ Bulk with Boundary Conformal Blocks

This paper investigates the relationship between bulk fields in the BTZ (Banados-Teitelboim-Zanelli) black hole and boundary conformal blocks within the framework of AdS/CFT correspondence. It builds upon earlier work by Ferrara, Gatto, Grillo, and Parisi, recasting it in this holographic context to further elucidate the extraction of classical bulk fields through boundary conformal blocks.

Main Contributions

The paper makes several key contributions:

1. Relation Between Bulk Fields and Boundary OPE: It establishes a straightforward relationship linking bulk fields at any radial position to the boundary Operator Product Expansion (OPE). This link enables the extraction of bulk fields from boundary conformal blocks by reinterpreting classical results as being computed by geodesic Witten diagrams.
2. Application to BTZ Black Holes: The authors apply this framework to the BTZ black hole scenario, conceptualized as a pure state generated via the insertion of a heavy operator in the boundary CFT$_2$. They discover a relationship between classical fields in the bulk and heavy-light Virasoro conformal blocks on the boundary.
3. Radial Position and Conformal Ratios: A significant find is the relationship between the radial bulk position in BTZ and conformal ratios within the boundary CFT. This aids in understanding singular points in the radial bulk equation as corresponding points in the boundary theory, such as when operators approach one another.

Theoretical Implications

The paper offers new insights into the AdS/CFT duality, especially in translating properties from the classical bulk scenario to boundary terms. Its implications are particularly relevant for elucidating gravitational dynamics in lower-dimensional settings and exploring the nature of black hole horizons through holography. Central to these endeavors is the mapping of singular radial points in the bulk to critical configurations of operators in the boundary CFT, providing a CFT interpretation of radial monodromies.

Practical Implications and Future Directions

The implications of this research are manifold, potentially advancing the understanding of black hole microstates and their holographic duals. Additionally, the work raises interesting questions about extending this framework to more complex or higher-dimensional contexts, where similar dualities might offer comparable insights into gravitational physics.

For future developments:

• Extension to Higher Dimensions: The methodology and results suggest potential for generalization to higher-dimensional AdS spaces, which could yield deeper understanding of universal CFT properties across diverse gravitational backgrounds.
• Numerical and Analytical Studies: Further exploration could involve numerical methods to test the robustness of these relations under various perturbative or non-perturbative settings within CFT and bulk dynamics.
• Quantum Corrections and Non-Classical Regimes: Evaluating how quantum corrections could affect the postulated relationships between bulk fields and boundary configurations might illuminate subtleties in the boundary-to-bulk map, especially near the horizon or in scenarios with entangled states.

Overall, the paper represents an important step in refining the conceptual and mathematical toolkit necessary for probing the depths of the AdS/CFT correspondence, particularly concerning black hole mechanics and conformal field theories.