Constructing Space From Entanglement Entropy

TL;DR

This work demonstrates a concrete, invertible procedure to extract the bulk spacetime metric from boundary entanglement entropies using the Ryu–Takayanagi formula. By assuming a general but simple bulk metric with a single function g(z) and applying a Taylor expansion, the authors relate EE data to g(z) through coupled integral equations, recovering known holographic geometries (AdS, BTZ, AdS soliton) across 1+1 and higher dimensions. The approach yields explicit holographic dictionaries (e.g., c = 3R/2G_N in 1+1D, N^2 relations in N=4 SYM) and proves the uniqueness of analytic metrics for analytic EE, while also providing a rigorous convergence framework for the iterative reconstruction. The results pave the way for constructing gravity duals from EE data in theories with known gravity duals and motivate numerical exploration for less-understood gauge theories.

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 $β^{-1}$. 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 $N^2 = πR^3/2G_N$ 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.

Figures (2)

• Figure 1: a) This figure shows the convergence of an iterated test function to the solution. The test function was taken to be $g_0 = 1$. b) For this figure the initial test function was $g_0 = 1+2z^2$. In both cases $g_2$, $g_6$, $g_{10}$, $g_{14}$ and $g_{18}$ are plotted, where those closer to the solution correspond to high iterations. As expected the convergence is faster for a better test function.
• Figure 2: a) This figure shows the convergence of the coefficients of $g$ to the solution. Here $\Delta a_n = (a_\infty-a_n)/a_\infty.$ This figure shows that convergence is slower for higher order terms in the Taylor series. The blue squares are the $z^2$ coefficient, red circles are $z^4$ coefficients and green triangles are $z^6$ coefficients.