# Boundary-to-bulk maps for AdS causal wedges and RG flow

## TL;DR

This paper constructs boundary-to-bulk propagators in AdS space with Robin boundary conditions, connecting boundary deformations to bulk fields and ensuring the mathematical consistency of the propagators.

## Contribution

It introduces explicit expressions for boundary-to-bulk kernels with Robin boundary conditions, bridging boundary deformations and bulk fields in AdS/CFT.

## Key findings

- Explicit boundary-to-bulk propagators with Robin boundary conditions derived.
- Demonstrated the consistent mapping from boundary Wightman functions to bulk.
- Proved the microlocal spectrum condition for boundary two-point functions.

## Abstract

We consider the problem of defining spacelike-supported boundary-to-bulk propagators in AdS$_{d+1}$ down to the unitary bound $\Delta=(d-2)/2$. That is to say, we construct the `smearing functions' $K$ of HKLL but with different boundary conditions where both dimensions $\Delta_+$ and $\Delta_-$ are taken into account. More precisely, we impose Robin boundary conditions, which interpolate between Dirichlet and Neumann boundary conditions and we give explicit expressions for the distributional kernel $K$ with spacelike support. This flow between boundary conditions is known to be captured in the boundary by adding a double-trace deformation to the CFT. Indeed, we explicitly show that using $K$ there is a consistent and explicit map from a Wightman function of the boundary QFT to a Wightman function of the bulk theory. In order to accomplish this we have to study first the microlocal properties of the boundary two-point function of the perturbed CFT and prove its wavefront set satisfies the microlocal spectrum condition. This permits to assert that $K$ and the boundary two-point function can be multiplied as distributions.