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To gain insight into how bulk locality emerges from the holographic conformal field theory (CFT), we reformulate the bulk-to-boundary map in as local a way as possible. In previous work, we carried out this program for Lorentzian anti-de Sitter (AdS), and showed the support on the boundary could always be reduced to a compact region spacelike separated from the bulk point. In the present work the idea is extended to a complexified boundary, where spatial coordinates are continued to imaginary values. This continuation enables us to represent a local bulk operator as a CFT operator with support on a finite disc on the complexified boundary. We treat general AdS in Poincar\'e coordinates and AdS₃ in Rindler coordinates. We represent bulk operators inside the horizon of a Banados-Teitelboim-Zanelli (BTZ) black hole and we verify that the correct bulk two-point functions are reproduced, including the divergence when one point hits the BTZ singularity. We comment on the holographic description of black holes formed by collapse and discuss locality and holographic entropy counting at finite N.
Hamilton et al. (Wed,) studied this question.