Non-geometric States in a Holographic Conformal Field Theory

TL;DR

This paper investigates which states in a 2D conformal field theory have a classical bulk dual in AdS$_3$, identifying conditions for geometric states and providing examples of both geometric and non-geometric states.

Contribution

It establishes criteria for CFT states to be geometric in the AdS$_3$/CFT$_2$ correspondence and introduces the concept of non-geometric states, including superpositions and descendants.

Findings

Geometric states satisfy Bohr's correspondence principle.

Non-geometric states include superpositions and descendant states.

Constraints relate entanglement entropy to the classical limit.

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.