How Einstein's Equation Emerges From CFT${}_2$

Abstract

The {\it finiteness} of the entanglement entropies between disjoint subsystems enables us to show that, the dynamical equation of the entanglement entropy in CFT${}_2$ is precisely three dimensional Einstein's equation. We establish a profound relation between the cosmological constant and CFT${}_2$ entanglement entropy. Thus entanglement entropies induce internal gravitational geometries in CFT${}_2$. Extracting the dual metric from an entanglement entropy becomes a straightforward procedure. Remarkably, we discover that the renormalization group equation is a geometric identity.

V5: Accepted for publication in PRD Letter. Used an entirely different logic from V4 to emphasize the purely CFT-based derivation of Einstein's equation. Added an appendix to show the equivalence between EoP and EE of annular CFT. But derivations of V4 are easier to follow