Fix the dual geometries of TT deformed CFT2 and highly excited states of CFT2

In previous works, we have developed an approach to fix the leading behaviors of the pure AdS3 and BTZ black hole from the entanglement entropies of the free CFT2 and finite temperature CFT2, 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 TT deformed CFT2, which breaks the conformal symmetry. In terms of the physical arguments of the TT deformed CFT2, the derived metric is a deformed BTZ black hole. The requirement that the CFT2 lives on a conformally flat boundary leads to rc2=6RAdS4/(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 TT deformed CFT2. 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.