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

Abstract

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.

Bulk reconstruction for spinor fields in AdS/CFT

“…• As Z → 0 our fermion fields have the requisite behavior (17). To see this note that the region of integration becomes very small as Z → 0, so one can bring the CFT operators out of the integral and explicitly verify the Z → 0 behavior.…”

“…This condition also avoids poles in the gamma functions. If the condition m > d 2 − 1 2 is violated we expect that the smearing functions (27) should be replaced with distributions, as noted for scalar fields in [13,16,17]. But we leave an exploration of this issue to future work.…”

“…The Dirac action (6) respects this symmetry, with Ψ transforming with weight zero under the rescaling. On the boundary this transformation acts as a dilation, and from (17) we see that ψ transforms with…”

“…Just as before, for the kernels in (41) to be integrable against smooth test functions we must have m > d 2 − 1 2 . If this condition is violated we expect that the smearing functions should be replaced with distributions, as noted for scalar fields in [13,16,17].…”

HKLL bulk reconstruction for small ∆

“…It was found that in this case the support of the smearing function is the intersection of the light-cone of the bulk point and the boundary. In [21] the range of allowed ∆ was extended to d/2 ≤ ∆ ≤ d − 1 by analytic continuation. Our purpose here is to find a direct derivation of the generalized HKLL formula for ∆ values below the original lower bound d − 1.…”