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Bulk geometry from entanglement entropy of CFT

In this paper, we compute the exact form of the bulk geometry emerging from a $(1+1)$-dimensional conformal field theory using the holographic principle. We first consider the $(2+1)$-dimensional asymptotic $AdS$ metric in Poincare coordinates and compute the area functional corresponding to the static minimal surface $γ_A$ and obtain the entanglement entropy making use of the holographic entanglement entropy proposal. We then use the results of the entanglement entropy for $(1+1)$-dimensional conformal field theory on an infinite line, on an infinite line at a finite temperature and on a circle. Comparing these results with the holographic entanglement entropy, we are able to extract the proper structure of the bulk metric. Finally, we also carry out our analysis in the case of $\mathcal{N}=4$ super Yang-Mills theory and obtain the exact form of the dual bulk geometry corresponding to this theory. The analysis reveals the behavior of the bulk metric in both the near boundary region and deep inside the bulk. The results also show the influence of the boundary UV cut-off "$a$" on the bulk metric. It is observed that the reconstructed metrics match exactly with the known results in the literature when one moves deep inside the bulk or towards the turning point.