Algebraic Holography

TL;DR

The paper develops a rigorous algebraic framework for holography between anti-deSitter space QFTs and conformal boundary QFTs by proving a canonical wedge–double-cone correspondence that yields a 1:1 map between bulk AdS nets and boundary conformal nets. It shows that vacuum states, positive-energy representations, and modular data are preserved under this mapping, and analyzes how compact bulk localization translates into boundary additivity properties, with detailed treatment in 1+1 dimensions where charge transporters encode superselection structure. The results offer a semi-classical realization of AdS/CFT within the Haag–Kastler net framework and illuminate when bulk locality aligns with boundary gauge-like behavior. The work also discusses flat-space limits, the Longo–Rehren net connection in 1+1D, and potential extensions to higher dimensions and covering spaces, highlighting implications for gauge theories and quantum gravity-inspired holography.

Abstract

A rigorous (and simple) proof is given that there is a one-to-one correspondence between causal anti-deSitter covariant quantum field theories on anti-deSitter space and causal conformally covariant quantum field theories on its conformal boundary. The correspondence is given by the explicit identification of observables localized in wedge regions in anti-deSitter space and observables localized in double-cone regions in its boundary. It takes vacuum states into vacuum states, and positive-energy representations into positive-energy representations.