Mapping from Exact RG to Holographic RG in Flat Space

Abstract: In earlier papers a method was given for constructing from first principles a holographic bulk dual action in Euclidean AdS space for a Euclidean CFT on the boundary. The starting point was an Exact RG for the boundary theory. The bulk action is obtained from the evolution operator for this ERG followed by a field redefinition. This procedure guarantees that the boundary correlators are all recovered correctly. In this paper we use the same method in an attempt to construct a holographic dual action for the free $O(N)$ model where the bulk is flat Euclidean space with a plane boundary wall. The scalar cubic interaction is found to be local (in $D=3$) but depends on the distance from the boundary - which can be interpreted as a non constant background dilaton field. The spin 2 - scalar - scalar interaction is found to be non local - in contrast to the AdS case. A field redefinition that makes the kinetic term quartic in derivatives can be done to mitigate (but not eliminate) this non locality. It is shown that, in spite of the non locality, the action can be obtained by gauge fixing an action that has the linearized gauge invariance associated with general coordinate invariance. Boundary correlators (two point and three point) are shown to be reproduced by bulk calculations - as expected in this approach to holography.