On de Sitter future-past extremal surfaces and the "entanglement wedge"

TL;DR

This work extends the holographic-like program in de Sitter space by constructing codim-2 future-past extremal surfaces tying subregions on $I^+$ to equivalent subregions on $I^-$. A limiting surface emerges in $dS_4$ and higher, with turning points in the $N/S$ diamond and a width $\Delta w$ that diverges as the subregion fills the boundary, while the finite part of the area scales linearly with $\Delta w$, reminiscent of entanglement growth in thermal systems. For multiple disjoint subregions, the mutual information vanishes and strong subadditivity is saturated at leading order, suggesting a finite-temperature-like decoupling in the de Sitter context. The authors introduce a codim-1 envelope surface (an entanglement wedge analogue) formed from the family of codim-2 surfaces, discuss domain of dependence and Cauchy horizons, and propose a subregion-duality-like framework for de Sitter holography, including interpretations in a ghost-like dS/CFT and avenues for incorporating quantum corrections and bulk reconstruction.

Abstract

We develop further the codim-2 future-past extremal surfaces stretching between the future and past boundaries in de Sitter space, discussed in previous work. We first make more elaborate the construction of such surfaces anchored at more general subregions of the future boundary, and stretching to equivalent subregions at the past boundary. These top-bottom symmetric future-past extremal surfaces cannot penetrate beyond a certain limiting surface in the Northern/Southern diamond regions: the boundary subregions become the whole boundary for this limiting surface. For multiple disjoint subregions, this construction leads to mutual information vanishing and strong subadditivity being saturated. We then discuss an effective codim-1 envelope surface arising from these codim-2 surfaces. This leads to analogs of the entanglement wedge and subregion duality for these future-past extremal surfaces in de Sitter space.

Figures (8)

• Figure 1: Future-past extremal surfaces in de Sitter stretching between $I^\pm$. These are akin to rotated Hartman-Maldacena surfaces in the eternal $AdS$ black hole. The red curve is for generic subregion. The blue curve is a limiting curve obtained as the subregion becomes the whole space.
• Figure 2: Future-past extremal surfaces in de Sitter stretching between $I^\pm$ for a generic subregion: these lie on some $S^{d-1}$ equatorial plane and have endpoints $w_{L,0}$ at the left edge and $w_{R,0}$ at the right edge.
• Figure 3: Two disjoint subregions $A\equiv (w_{1},w_{2})$ and $B\equiv (w_{3},w_{4})$ at $I^{+}$ alongwith the equivalent ones at $I^{-}$, and the corresponding future-past extremal surfaces.
• Figure 4: Generic subregion, extremal surface and domain of dependence.
• Figure 5: Subregion becoming all $I^+$, limiting surface, domain of dependence.