Timelike entanglement entropy in dS$_3$/CFT$_2$

TL;DR

The paper defines a timelike entanglement entropy in $dS_{3}$/CFT$_{2}$ via RG-flow using the Callan–Symanzik equation and shows that the IR EE is $S_A^{IR} = -\frac{i c_{dS}}{6} \log\frac{\xi}{\epsilon}$, which exactly matches the holographic entropy $S_A = L/(4G_N^{(3)})$ of a timelike geodesic connecting two spacelike bulk slices, with $L = \ell_{dS}\arccos\left(\frac{\xi^{2}+\epsilon^{2}}{2\xi\epsilon}\right) = -i\ell_{dS}\log\frac{\xi}{\epsilon}$. The AdS$_{3}$/CFT$_{2}$ counterpart is spacelike and related by a double Wick rotation, and the framework implies there are precisely three independent entanglement entropies (UV, IR/timelike, and a bulk-penetrating one) in both dS$_{3}$/CFT$_{2}$ and AdS$_{3}$/CFT$_{2}$ that suffice to reconstruct the bulk geometry. The results suggest connections to holographic screens for the inflation patch and point toward $T\bar{T}$ deformations as a holographic entangling surface, with potential extensions to more general asymptotic spacetimes.

Abstract

In the context of dS$_3$/CFT$_2$, we propose a timelike entanglement entropy defined by the renormalization group flow. This timelike entanglement entropy is calculated in CFT by using the Callan-Symanzik equation. We find an exact match between this entanglement entropy and the length of a timelike geodesic connecting two different spacelike surfaces in dS$_3$.The counterpart of this entanglement entropy in AdS$_3$ is a spacelike one, also induced by RG flow and extends all the way into the bulk of AdS$_3$. As a result, in both AdS$_3$/CFT$_2$ and dS$_3$/CFT$_2$, there exist exactly three entanglement entropies, providing precisely sufficient information to reconstruct the three-dimensional bulk geometry.

Figures (1)

• Figure 1: Penrose diagram of de Sitter spacetime. $\mathcal{I}^{\pm}$ is the global past and future spheres. The two vertical boundaries $\theta=0,\pi$ are the north pole and south pole respectively. Each point in the interior represents an $\mathbb{S}^{1}$. A horizontal slice is an $\mathbb{S}^{2}$. The planar coordinate (\ref{['eq:flat_metric']}) covers the shadow region $\mathcal{O}^{+}$, comprising the causal future of the south pole. The green dashed lines are constant $r=\sqrt{x^{2}+y^{2}}$. Orange lines of constant $t$ are shown. The violet line $\mathcal{I}^{+}$ denotes the future boundary $t\to \infty$. The red line indicates the past horizon $t\to -\infty$.