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arXiv:2510.27250

Derive Einstein equation from CFT entanglement entropy

Xin Jiang, Haitang Yang

Published Oct 31, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2510.27250

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We explicitly show how to derive the (D+1)(D+1)(D+1)-dimensional Einstein equation from the entanglement entropy between codimension-one {\it disjoint} regions in DDD-dimensional conformal field theory.
We study the dynamics of entanglement entropy for weakly excited states in conformal field theories by using anti-de Sitter/conformal field theory (AdS/CFT). This is aimed at a first step to finding a counterpart of the Einstein equation in CFT language. In particular, we point out that the entanglement entropy satisfies differential equations that directly correspond to the Einstein equation in several setups of AdS/CFT. We also define a quantity called entanglement density in higher dimensional field theories and study its dynamical property for weakly excited states in conformal field theories.
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The purpose of these lecture notes is to give an introduction to entanglement entropy with an emphasis on holography. Apart from the standard definition and properties of entanglement entropy, we explicitly derive the relation between the coefficients of the conformal anomaly and the universal term in the entropy of a d-dimensional CFT. We also discuss the prescription for calculating the entanglement entropy holographically in theories dual to Einstein-Hilbert and Lovelock gravity and present several examples in detail.
A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS(3). We also compare the entropy computed in AdS(5)XS(5) with that of the free N=4 super Yang-Mills theory.
For a conformal field theory (CFT) deformed by a relevant operator, the entanglement entropy of a ball-shaped region may be computed as a perturbative expansion in the coupling. A similar perturbative expansion exists for excited states near the vacuum. Using these expansions, this work investigates the behavior of excited state entanglement entropies of small, ball-shaped regions. The motivation for these calculations is Jacobson’s recent work on the equivalence of the Einstein equation and the hypothesis of maximal vacuum entropy [ arXiv:1505.04753 ], which relies on a conjecture stating that the behavior of these entropies is sufficiently similar to a CFT. In addition to the expected type of terms which scale with the ball radius as R d , the entanglement entropy calculation gives rise to terms scaling as R 2Δ, where Δ is the dimension of the deforming operator. When Δ≤d2 \varDelta \le \frac{d}{2} Δ≤2d​ , the latter terms dominate the former, and suggest that a modification to the conjecture is needed.
This is an extended version of our short report hep-th/0603001, where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results: We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS3/CFT2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.