Area operator and fixed area states in conformal field theories

The fixed area states are constructed by gravitational path integrals in previous studies.In this paper we show the dual of conformal field theories (CFTs).These CFT using spectrum decomposition reduced density matrix $\rho_A$ for a subsystem $A$. For 2 dimensional CFTs directly construct bulk metric, which is consistent with expected geometry states. arbitrary pure geometric state $|\psi\rangle$ any dimension also find consistency gravity R\'enyi entropy. We give relation parameters and boundary state. can be expanded as superposition Motivated this, propose an operator $\hat A^\psi$. eigenstate A^\psi$, associated eigenvalue related to entropy $A$ Ryu-Takayanagi formula expressed expectation value $\langle \psi| {\hat A}^\psi|\psi\rangle$ divided $4G$, where $G$ Newton constant. fluctuation suppressed semiclassical limit $G\to0$.