de Sitter extremal surfaces

TL;DR

Narayan investigates de Sitter extremal surfaces in the Poincare upper patch anchored to strip subregions on the future boundary, uncovering real extremal surfaces that are boundaries of past lightcone wedges with zero area and a class of complex saddles whose areas are generally nonreal. The work illuminates how these surfaces relate to holographic notions via analytic continuation from AdS and connects to dS/CFT correlators through a central-charge-like structure, notably in $dS_4$ where complex extremal surfaces yield a real, negative area. Extensions to the $dS$ black brane and bluewall geometries reveal a finite, cutoff-independent extensive piece and surfaces crossing Cauchy horizons, respectively, suggesting rich but subtle entanglement-like physics in nonunitary Euclidean duals. Overall, the results challenge a direct entanglement interpretation in dS/CFT while offering a framework to explore complex saddles, their central charges, and potential holographic information measures in cosmological settings.

Abstract

We study extremal surfaces in de Sitter space in the Poincare slicing in the upper patch, anchored on spatial subregions at the future boundary ${\cal I}^+$, restricted to constant boundary Euclidean time slices (focussing on strip subregions). We find real extremal surfaces of minimal area as the boundaries of past lightcone wedges of the subregions in question: these are null surfaces with vanishing area. We also find complex extremal surfaces as complex extrema of the area functional, and the area is not always real-valued. In $dS_4$ the area is real. The area has structural resemblance with entanglement entropy in a dual $CFT$. There are parallels with analytic continuation from the Ryu-Takayanagi expressions for holographic entanglement entropy in $AdS$. We also discuss extremal surfaces in the $dS$ black brane and the de Sitter "bluewall" studied previously. The $dS_4$ black brane complex surfaces exhibit a real finite cutoff-independent extensive piece. In the bluewall geometry, there are real surfaces that go from one asymptotic universe to the other through the Cauchy horizons.

Figures (2)

• Figure 1: Extremal surfaces in de Sitter made of two half-extremal surfaces joined continuously but with a sharp cusp at $\tau_0$. As $B$ increases (till eventually $B\gg {1\over\epsilon^{d-1}}$), the surface approaches ${\dot x}^2\rightarrow 1$ (figure on the right).
• Figure 2: de Sitter "bluewall" Penrose diagram and some extremal surfaces with at least one end anchored at ${\cal I}^+$. The blue wedge is null, while the red timelike surface goes from ${\cal I}^+$ to ${\cal I}^-$.