Further remarks on de Sitter space, extremal surfaces and time entanglement

TL;DR

This work extends holographic ideas from AdS to de Sitter space by studying extremal surfaces anchored at I^+ and their (generally complex) areas. Through analytic continuation from AdS RT/HRT surfaces and a no-boundary framework, it introduces a geometric time-entanglement (pseudo-entanglement) wedge and a replica-like interpretation for the dS Wavefunction, yielding pseudo-entropy as a bulk observable. The analysis covers IR maximal subregions, general subregions in dS_{d+1}, and entropy-like inequalities in dS_3, showing that complex entropies still encode AdS-positive structures via continuation. The paper also links future-past entangled states, antipodal observers, and time evolution to a dS/CFT-inspired picture, and points to future work connecting these ideas to modular flow, error correction, and cosmological implications.

Abstract

We develop further the investigations in arXiv:2210.12963 [hep-th] on de Sitter space, extremal surfaces and time entanglement. We discuss the no-boundary de Sitter extremal surface areas as certain analytic continuations from $AdS$ while also amounting to space-time rotations. The structure of the extremal surfaces suggests a geometric picture of the time-entanglement or pseudo-entanglement wedge. We also study some entropy relations for multiple subregions. The analytic continuation suggests a heuristic Lewkowycz-Maldacena formulation of the extremal surface areas. In the bulk, this is now a replica formulation on the Wavefunction which suggests interpretation as pseudo-entropy. Finally we also discuss aspects of future-past entangled states and time evolution.

Figures (7)

• Figure 1: (reproduced from Narayan:2022afv) Entirely Lorentzian $dS$, future-past extremal surfaces (left); No-boundary $dS$, no-boundary extremal surfaces (right).
• Figure 2: $dS$ no-boundary extremal surfaces in the $t=const$ slice in the static coordinates (right side picture; left side is the "top view" from $I^+$). The red curve is the IR extremal surface for maximal boundary subregion $[-{\pi\over 2}, {\pi\over 2}]$ (hemisphere in $S^{d-1}$). The blue curve is the extremal surface when the subregion $[-\theta_\infty, \theta_\infty]$ at $I^+$ is not maximal.
• Figure 3: $dS$ no-boundary extremal surfaces in the $t=const$ slice (middle picture; left side is the "top view" from $I^+$). The red curve is the extremal surface for maximal boundary subregion, and the blue curve when the subregion is smaller. The green and violet shaded regions are the bulk subregions defined by the time-entanglement or pseudo-entanglement wedges restricted to this $t=const$ slice. The right side figure is the bulk region in the top Lorentzian half in the full $dS$ Penrose diagram, including the $t$ direction: the IR surface lies on the $t=const$ slice which is depicted as the red vertical line.
• Figure 4: $dS$ no-boundary extremal surfaces for disjoint subregions at $I^+$ in the $t=const$ slice (static coordinatization): this is the "top view" from $I^+$ . These are the red, violet, and blue curves for the three boundary subregions at $I^+$. The green, violet and blue shaded regions are the corresponding bulk subregions. The bulk subregions overlap (represented by the different color shadings in the various overlaps; e.g. light blue overlapping with violet leads to darker blue-violet etc).
• Figure 5: $dS$ no-boundary extremal surfaces and alternative ways to define bulk subregions ("top view" from $I^+$). The bulk subregions for disjoint boundary subregions at $I^+$ do not overlap now: however the extremal surface now has a cusp at the no-boundary point (the earlier smooth extremal surface is the blue curve).