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Exploring the BTZ bulk with boundary conformal blocks

Bruno Carneiro da Cunha, Monica Guica

TL;DR

This work shows that a classical bulk field in BTZ can be extracted from boundary data by leveraging the Ferrara–Gatto–Grillo–Parisi relation between the bulk field and the boundary OPE, recast as an inverse Mellin transform of the OPE. In BTZ, heavy operators create a black hole state, and heavy-light Virasoro blocks reduce to global blocks in a state-dependent conformal frame $w$, enabling the bulk field to be represented as the inverse Mellin transform of the boundary block in this frame. The radial AdSBTZ depth then corresponds to boundary conformal ratios in the $w$-plane, and monodromies of the bulk wave equation map to monodromies of the transformed conformal block, offering a CFT interpretation of radial monodromy and horizon physics. This framework connects geodesic Witten diagrams, Mellin transforms, and heavy-light blocks to provide a boundary-driven, quasi-local bulk reconstruction in nontrivial backgrounds and suggests avenues for generalization to higher dimensions and deeper analyses of monodromy structures.

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.

Exploring the BTZ bulk with boundary conformal blocks

TL;DR

This work shows that a classical bulk field in BTZ can be extracted from boundary data by leveraging the Ferrara–Gatto–Grillo–Parisi relation between the bulk field and the boundary OPE, recast as an inverse Mellin transform of the OPE. In BTZ, heavy operators create a black hole state, and heavy-light Virasoro blocks reduce to global blocks in a state-dependent conformal frame , enabling the bulk field to be represented as the inverse Mellin transform of the boundary block in this frame. The radial AdSBTZ depth then corresponds to boundary conformal ratios in the -plane, and monodromies of the bulk wave equation map to monodromies of the transformed conformal block, offering a CFT interpretation of radial monodromy and horizon physics. This framework connects geodesic Witten diagrams, Mellin transforms, and heavy-light blocks to provide a boundary-driven, quasi-local bulk reconstruction in nontrivial backgrounds and suggests avenues for generalization to higher dimensions and deeper analyses of monodromy structures.

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.

Paper Structure

This paper contains 12 sections, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Bulk fields and conformal blocks
  3. Bulk fields from the boundary OPE
  4. Correlators and geodesic Witten diagrams
  5. Bulk fields in BTZ
  6. Bulk fields and heavy-light conformal blocks
  7. Properties of the bulk solution in BTZ
  8. CFT analysis of the bulk solution
  9. Mapping the singular points
  10. Mapping the monodromies
  11. Discussion
  12. Details of the conformal block calculations

Figures (3)

  • Figure 1: Bulk geodesic prescription for computing the $\langle \Phi^{(0)} A B \rangle$ correlator.
  • Figure 2: Geodesic Witten diagram prescription for computing the conformal block, which follows from applying $(1.3)$ to the OPEs of the external operators.
  • Figure 3: The CFT $(t,\phi)$ coordinates only cover the right Rindler wedge of the $w$ plane. The shaded region represents a particular fundamental region in the $w$ plane covered by the $\phi, t$ cooordinates.