Exploring the BTZ bulk with boundary conformal blocks

Understand

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.