Derive Einstein equation from CFT entanglement entropy

TL;DR

The work tackles the problem of deriving the $(D+1)$-dimensional vacuum Einstein equation from the entanglement entropy of disjoint subregions in a $D$-dimensional CFT, addressing the divergence of adjacent entanglement and the lack of general higher-D entanglement formulas. It computes the finite disjoint entanglement entropy $S_{ m disj}(A:B)$ using a solid-torus replica construction and expresses it as a function of the conformal cross-ratio $ig|rac{|\xi_{12}||\xi_{34}|}{|\xi_{13}||\xi_{24}|}ig|$, with an explicit hypergeometric form for symmetric configurations. From $S_{ m disj}$, a $(D+1)$-dimensional metric is defined via $oldsymbol{ ext{χ}}(Y,Y') = rac{D-1}{2} ig(G^{(D+1)} S_{ m disj}(A:B)ig)^{rac{2}{D-1}}$ and $g_{\mu u} = -[oldsymbol{ ext{χ}}_{\mu u'}]$, and substituting yields the vacuum Einstein equation $R_{\mu u}-rac{1}{2}g_{\mu u}R + oldsymbol{} g_{\mu u} = 0$ with $oldsymbol{} = rac{D-1}{2(D+1)}R$, i.e., gravity in the bulk emerges from CFT entanglement data. This establishes an emergent, entanglement-driven perspective on gravity, clarifies the boundary/bulk nonlocality, and points to future work on matter-sourced Einstein equations.

Abstract

We explicitly show how to derive the $(D+1)$-dimensional Einstein equation from the entanglement entropy between codimension-one {\it disjoint} regions in $D$-dimensional conformal field theory.

Figures (6)

• Figure 1: The pure state density matrix $\rho=|\psi\rangle\langle\psi|$ of the solid torus CFT$_D$. Entangling regions $A$ and $B$ (colored) are disjoint, ensuring a finite entanglement entropy between $A$ and $B$.
• Figure 2: Two configurations are conformally equivalent to the solid torus. Left panel (the cavity configuration): A ($D-1$)-ball living in a spheric cavity is entangled with the region outside the cavity. Right panel (the juxtaposed configuration): Two disjoint ($D-1$)-balls with different radii are entangled.
• Figure 3: Two disjoint $(D-1)$-balls $A$ and $B$ on a spacelike hyperplane $\mathcal{P}$, determined by four collinear endpoints (red dots). The cross ratio $\eta$ (Eq. \ref{['eq:cross-ratio']}) parametrizes their relative positions, governing the entanglement entropy via conformal invariance.
• Figure 4: As $\epsilon\to 0$, the cavity configuration reduces to the usual adjacent configuration.
• Figure 5: In AdS$_3$/CFT$_2$, the RT surfaces are geodesics. The geodesic dual to $S_\text{disj}(A:B)$ is $L_{AB}$.