Holographic de Sitter Geometry from Entanglement in Conformal Field Theory

Abstract

We demonstrate that for general conformal field theories (CFTs), the entanglement for small perturbations of the vacuum is organized in a novel holographic way. For spherical entangling regions in a constant time slice, perturbations in the entanglement entropy are solutions of a Klein-Gordon equation in an auxiliary de Sitter (dS) spacetime. The role of the emergent time-like direction in dS is played by the size of the entangling sphere. For CFTs with extra conserved charges, e.g., higher spin charges, we show that each charge gives rise to a separate dynamical scalar field in dS.

Figures (4)

• Figure 1: One-to-one mapping between points in dS geometry and balls on its future asymptotic boundary $\mathcal{I}^+$. We identify the latter with the constant time slice in a given CFT. The future lightcone of the bulk point intersects $\mathcal{I}^+$ on the boundary of the corresponding ball.
• Figure 2: Penrose diagram of dS. Points correspond to $\mathbb{S}^{d-2}$, whereas horizontal lines represent $\mathbb{S}^{d-1}$. $\partial B$ represents the spherical entangling surface and $B$ (orange) and $\bar{B}$ (cyan) its interior and exterior. Red lines represent corresponding lightcones and $x$ and $\bar{x}$ their tips, i.e., bulk points in dS corresponding to $B$ and $\bar{B}$ regions. Note that $\bar{x}$ is the antipode of $x$.
• Figure 3: The domain of dependence $D$ of a ball $B$ enclosed by a given spherical entangling surface. Blue lines are tangent to the conformal Killing vector field $K^{\mu}$, which gives rise to conserved charges \ref{['eq.HS']}.
• Figure 4: Top: Rescaled energy density $\langle T_{t t}\rangle$ for the state (\ref{['eq.phi']}) with $d=3$ plotted along the $x$-axis at $t = t_{0} + 5\, \tau$, see Eq. \ref{['eq.f1']}. Bottom: Corresponding change in the entanglement entropy $\delta S$ as a function of dS time $R$ and position along the $x$-axis. The causal nature of propagation is clearly visible.