A UV-Finite Ryu-Takayanagi Relation from Relative Entropy in AdS$_3$/CFT$_2$

Albert Much; Philipp Dorau; Leonardo Sangaletti; Rainer Verch

A UV-Finite Ryu-Takayanagi Relation from Relative Entropy in AdS$_3$/CFT$_2$

High Energy Physics - Theory 2026-06-26 v1 General Relativity and Quantum Cosmology Mathematical Physics

Authors: Albert Much , Philipp Dorau , Leonardo Sangaletti , Rainer Verch

Abstract

We establish a Ryu-Takayanagi (RT) relation in AdS $_3$ /CFT $_2$ using \emph{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 Ba\~nados 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 $4G_N$ . 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.

Keywords

ryu-takayanagi relation relative entropy ads/cft uv finiteness operator algebra

Cite

@article{arxiv.2606.27915,
  title  = {A UV-Finite Ryu-Takayanagi Relation from Relative Entropy in AdS$_3$/CFT$_2$},
  author = {Albert Much and Philipp Dorau and Leonardo Sangaletti and Rainer Verch},
  journal= {arXiv preprint arXiv:2606.27915},
  year   = {2026}
}