Fixing three dimensional geometries from entanglement entropies of CFT$_2$

TL;DR

This paper introduces a method to determine three-dimensional bulk geometries solely from dual CFT$_2$ entanglement entropies, without relying on the traditional AdS/CFT assumptions, applicable to various geometries including AdS$_3$ and BTZ black holes.

Contribution

The authors develop a novel approach to fix bulk geometries from entanglement entropies without assuming the AdS/CFT correspondence, applicable to multiple topologically distinct geometries.

Findings

Successfully fixed pure AdS$_3$ metric from free CFT$_2$ entanglement entropy.

Reconstructed BTZ black hole geometry from finite temperature CFT$_2$ entanglement entropy.

Method can, in principle, determine all topologically distinct 3D geometries from CFT$_2$ entanglement data.

Abstract

In this paper, we propose a method of fixing the leading behaviors of three dimensional geometries from the dual CFT$_2$ entanglement entropies. We employ only the holographic principle and do not use any assumption about the AdS/CFT correspondence and bulk geometry. Our strategy involves using both UV and IR-like CFT$_2$ entanglement entropies to fix the bulk geodesics. With a simple trick, the metric can be extracted from the geodesics. As examples, we fix the leading behaviors of the pure AdS$_3$ metric from the entanglement entropies of free CFT$_2$ and, more importantly, the BTZ black hole from the entanglement entropies of finite temperature CFT$_2$. Consequently, CFT$_2$ with finite size or topological defects can be determined through simple transformations. Following the same steps, in principle, the leading behaviors of all three dimensional (topologically distinct) holographic classical geometries from the dual CFT$_2$ entanglement entropies can be fixed.