Nongeometric states in a holographic conformal field theory

Wu-Zhong Guo,Feng-Li Lin,Jiaju Zhang

doi:10.1103/physrevd.99.106001

Abstract

In the AdS$_3$/CFT$_2$ correspondence, we find some conformal field theory (CFT) states that have no bulk description by the Ba\~nados geometry. We elaborate the constraints for a CFT state to be geometric, i.e., having a dual Ba\~nados metric, by comparing the order of central charge of the entanglement/R\'enyi entropy obtained respectively from the holographic method and the replica trick in CFT. We find that the geometric CFT states fulfill Bohr's correspondence principle by reducing the quantum KdV hierarchy to its classical counterpart. We call the CFT states that satisfy the geometric constraints geometric states, and otherwise non-geometric states. We give examples of both the geometric and non-geometric states, with the latter case including the superposition states and descendant states.

Highlights

The anti-de Sitter (AdS)/conformal field theory (CFT) correspondence conjectures that the bulk quantum gravity is equivalent to the boundary CFT [1]
In the AdS3=CFT2 correspondence, we find some conformal field theory (CFT) states that have no bulk description by the Bañados geometry
In AdS5=CFT4 correspondence, people know that the vacuum state of SUðNÞ gauge theory admits only planar correlators in the large N limit, which is dual to classical gravity in pure AdS5 space

Summary

INTRODUCTION

The anti-de Sitter (AdS)/conformal field theory (CFT) correspondence conjectures that the bulk quantum gravity is equivalent to the boundary CFT [1]. In AdS5=CFT4 correspondence, people know that the vacuum state of SUðNÞ gauge theory admits only planar correlators in the large N limit, which is dual to classical gravity in pure AdS5 space. In this case, the quantum fluctuation of nonplanar diagrams is suppressed, and a bulk geometry is emerging as the holographic dual. By short-interval expansion, we can turn this criterion into the constraints on the standard deviation of the stress tensors and its higher order cousins in terms of Korteweg-de Vries (KdV) charges This will tell precisely how much the quantum fluctuation should be suppressed for a state to be geometric.

CRITERION FOR GEOMETRIC CFT STATES IN BAÑADOS GEOMETRY

CONSTRAINTS FOR GEOMETRIC CFT STATES

QUANTUM TO CLASSICAL KDV EQUATION AND CHARGES FOR GEOMETRIC CFT STATES

EXAMPLES OF GEOMETRIC CFT STATES

EXAMPLES OF NONGEOMETRIC CFT STATES

CONCLUSIONS