Exploring the BTZ bulk with boundary conformal blocks

TL;DR

This paper establishes a relation between bulk fields in AdS and boundary conformal blocks, applying it to BTZ black holes to connect bulk radial positions with boundary conformal ratios, enhancing understanding of AdS/CFT correspondence.

Contribution

It provides a prescription to derive bulk fields from boundary conformal blocks and clarifies the geometric interpretation via geodesic Witten diagrams, specifically applied to BTZ black holes.

Findings

Bulk fields relate to boundary OPE and conformal blocks.

Radial bulk positions correspond to boundary conformal ratios.

Bulk monodromies map to boundary conformal block monodromies.

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.