Nongeometric states in a holographic conformal field theory

In the AdS3/CFT2 correspondence, we find some conformal field theory (CFT) states that have no bulk description by the Bañados geometry. We elaborate the constraints for a CFT state to be geometric, i.e., having a dual Bañados metric, by comparing the order of central charge of the entanglement/Rényi 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 Korteweg-de Vries hierarchy to its classical counterpart. We call the CFT states that satisfy the geometric constraints geometric states, and otherwise, we call them nongeometric states. We give examples of both the geometric and nongeometric states, with the latter case including the superposition states and descendant states.