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.

Figures (4)

• Figure 1: The Euclidean pure state density matrix $\rho=|\psi\rangle\langle\psi|$ for the annular CFT$_2$. $\xi_i = \tilde{x}_i + i\tilde{t}_i$ with $\tilde{t}_i = - i\tau_i$.
• Figure 2: In AdS$_3$, geodesic $\gamma_1$ is completely fixed by $\xi_1$ and $\xi_4$, geodesic $\gamma_2$ is completely fixed by $\xi_2$ and $\xi_3$. $L_{AB}$ is a geodesic and unique.
• Figure 3: After subtracting segments $C$ and $D$ with two discs in the infinite system, we obtain an asymmetric annular region in which $A$ and $B$ are in a pure entangled state $\psi_{AB}$, which is identical to the annular CFT$_2$.
• Figure 4: Two configurations for adding the auxiliary system $\bar{A}$. Each black point denotes the origin.