A UV-Finite Ryu-Takayanagi Relation from Relative Entropy in AdS3/CFT2
Abstract
We establish a Ryu-Takayanagi (RT) relation in AdS3/CFT2 using relative entropy as the central object, in place of the ultraviolet-divergent von Neumann entanglement entropy. Adapting Hollands' exact result for the chiral relative entropy to a diamond region, we express the boundary relative entropy between the vacuum and a coherent state as a Schwarzian functional, which the Fefferman-Graham dictionary identifies with the asymptotic data of a Bañados geometry; the rigidity of three-dimensional gravity promotes this boundary identification to the bulk. To linear order in the metric perturbation, the relative entropy then equals the variation of the RT geodesic length divided by 4GN. The construction rests only on the Bisognano-Wichmann/Borchers theorem and the holographic dictionary, giving a UV-finite, operator-algebraic counterpart to the RT relation.
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