For arbitrary spin and to all orders in an external source, the boundary limit of conformally smeared bulk operators equals the CFT generating functional, so the GKP-Witten relation follows from symmetry and OPE alone.
Special flow equation and GKP-Witten relation
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abstract
We develop a framework for the reconstruction of the bulk theory dual to conformal field theory (CFT) without any assumption by means of a flow equation. To this end we investigate a minimal extension of the free flow equation and find that at a special parametrization the conformal transformation for a normalized smeared operator exactly becomes the isometry of anti-de Sitter space (AdS). By employing this special flow equation to O$(N)$ vector models, we explicitly show that the AdS geometry as well as the scalar field satisfying the GKP-Witten relation concurrently emerge in this framework.
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Derivation of the GKP-Witten relation by symmetry without Lagrangian
For arbitrary spin and to all orders in an external source, the boundary limit of conformally smeared bulk operators equals the CFT generating functional, so the GKP-Witten relation follows from symmetry and OPE alone.