CFT descriptions of bulk local states in the AdS black holes

TL;DR

The paper tackles the problem of representing bulk local excitations within AdS3/CFT2, including black hole interiors, entirely from boundary data. It introduces a geodesic-driven map between bulk points and boundary coordinates, and constructs bulk local states by dressing boundary primaries with Virasoro descendants via a localizing operator. The method yields explicit expressions for bulk local states in global AdS, AdS Rindler, BTZ black holes, and in backgrounds dual to heavy primaries, and reproduces the corresponding bulk propagators and thermality properties in the leading large-$c$ limit. It further demonstrates background independence and connects with HKLL in Poincaré AdS, while revealing a possible phase transition in the semiclassical regime for deep interior bulk insertions. The framework provides a CFT-centric tool to probe black hole interiors and bulk spacetime structure using boundary calculations, with caveats about semiclassical approximations and state-dependence in certain setups.

Abstract

We present a new method for reconstructing CFT duals of states excited by the bulk local operators in the three dimensional AdS black holes in the AdS/CFT context. As an important procedure for this, we introduce a map between the bulk points in AdS and those on the boundary where CFT lives. This gives a systematic and universal way to express bulk local states even inside black hole interiors. Our construction allows us to probe the interior structures of black holes purely from the CFT calculations. We analyze bulk local states in the single-sided black holes as well as the double-sided black holes.

Figures (8)

• Figure 1: The bulk position $\rho$ can be obtained as an intersection of the timeslice $\tau=0$ and a geodesic which connects the boundary points $(\phi,\tau_0)$ and $(\phi,-\tau_0)$.
• Figure 2: Rindler coordinate and Euclidean plane defined on its boundary
• Figure 3: We can construct the map between bulk points and boundary points. This map can be obtained by the geodesics just the same as the global coordinate.
• Figure 4: Euclidean boundaries of the Rindler-AdS and the BTZ black hole. BTZ black holes are locally equivalent with the Rindler-AdS coordinate and they can be constructed by identifying the spatial coordinate $\phi$ in the Rindler coordinate as $-\pi L<\phi<\pi L$.
• Figure 5: The bulk local state in BTZ black hole is defined as a CFT primary operator dressed by the localizing operator $\hat{K}^{\boldsymbol{x}}$ which is inserted at the point $(\phi,\gamma)$ on a half torus. Two point functions of the bulk local states are calculated on the torus. It can be expressed as a sum of correlators of mirror images. Each contribution comes from the direct path (red dashed line) or an incontractible winding path which goes around the circle along the $\phi$ direction (green dashed line).