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.
Variational Methods in AdS/CFT
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abstract
We prove that the AdS/CFT calculation of 1-point functions can be drastically simplified by using variational arguments. We give a simple universal proof, valid for any theory that can be derived from a Lagrangian, that the large radius divergencies in 1-point functions can always be renormalized away (at least in the semiclassical approximation). The renormalized 1-point functions then follow by a simple variational problem involving only finite quantities. Several examples, a massive scalar, gravity, and renormalization flows, are discussed. Our results are general and can thus be used for dualities beyond AdS/CFT.
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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.