Conformally mapping HKLL to flat space

Abstract

We study two conformal images of the HKLL reconstruction problem for a conformally coupled scalar on AdS$_3$. The first mapping is to a Dirichlet problem on one half of the Einstein static universe. A further conformal transformation maps this half-ESU problem to a massless scalar in a region of $(2+1)$-dimensional Minkowski space bounded by an uniformly accelerated circular Dirichlet mirror. However, the global HKLL problem, the resulting flat-space problem is not a generic local mirror problem with arbitrary boundary data. Rather, the mirror response data must belong to a restricted spectral class inherited from normalizable global AdS modes. We formulate and solve this restricted mirror problem intrinsically in flat space. In the process, we find an example of a valid smearing function which itself does not satisfy the equations of motion. We further consider AdS-Rindler wedge reconstruction. By conformal mapping to an appropriate coordinate system in flat space, we find a simple way to see the blow-up of the smearing function.