Fix the dual geometries of $T\bar{T}$ deformed CFT$_2$ and highly excited states of CFT$_2$

Abstract: In previous works, we have developed an approach to fix the leading behaviors of the pure AdS$3$ and BTZ black hole from the entanglement entropies of the free CFT$_2$ and finite temperature CFT$_2$, respectively. We exclusively use holographic principle only and make no restriction about the bulk geometry, not only the kinematics but also the dynamics. In order to verify the universality and correctness of our method, in this paper, we apply it to the $T\bar{T}$ deformed CFT$_2$, which breaks the conformal symmetry. In terms of the physical arguments of the $T\bar{T}$ deformed CFT$_2$, the derived metric is a deformed BTZ black hole. The requirement that the CFT$_2$ lives on a conformally flat boundary leads to $r{c}^{2}=\ 6R_{AdS}^{4}/(\pi c\mu)$ naturally, in perfect agreement with previous conjectures in literature. The energy spectum and propagation speed calculated with this deformed BTZ metric are the same as these derived from $T\bar{T}$ deformed CFT$_2$. We furthermore fix the dual geometry of highly excited states with our approach. The result contains the descriptions for the conical defect and BTZ black hole.