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

bstract We consider the problem of defining spacelike-supported boundary-to-bulk propagators in AdS d +1 down to the unitary bound ∆ = ( d − 2) / 2. That is to say, we construct the ‘smearing functions’ K of HKLL but with different boundary conditions where both dimensions ∆ + and ∆ − 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.