Constructing Space From Entanglement Entropy

Abstract

We explicitly reconstruct the metric of a gravity dual to field theories using known entanglement entropies using the Ryu-Takayanagi formula. We use for examples CFT's in $d = 1$, 2 and 3 as well as CFT on a circle of length $L$ and a thermal CFT at temperature $\beta^{-1}$. We also give the first several coefficients in the Taylor series of the metric for a general entanglement entropy in 1+1 dimensions as well as some examples (Appendix B). The beginnings of a dictionary between the dual theories appears naturally and does not need to be inserted by hand. For example, the dictionary entries $c=3R/2G_N$ for 1+1 dimensional CFT and $N^2 = \pi R^3/2G_N$ for $\mathcal{N}=4$ SYM in 3+1 dimensions are forced upon us. After uploading this paper I was made aware of (arXiv:1012.1812) which solves the same problem in a similar way.