How Einstein's Equation Emerges From CFT$_2$

TL;DR

This paper demonstrates that the dynamical equation governing entanglement entropy in two-dimensional conformal field theories (CFT$_2$) directly leads to Einstein's equations in three dimensions, revealing a deep connection between entanglement and gravity.

Contribution

It shows that the entanglement entropy dynamics in CFT$_2$ encode Einstein's equations, establishing a direct link between entanglement and emergent gravitational geometry.

Findings

Entanglement entropy in CFT$_2$ satisfies Einstein's equations.

A relation between the cosmological constant and CFT$_2$ entanglement entropy is established.

The renormalization group equation is shown to be a geometric identity.

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.