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