de Sitter extremal surfaces, time contours, complexifications and pseudo-entropies

TL;DR

This paper investigates extremal surfaces and pseudo-entropy in de Sitter space, exploring their geometric interpretations, complexifications, and implications for entropy inequalities and analytic continuations from AdS.

Contribution

It introduces a comprehensive analysis of de Sitter extremal surfaces, including complex and timelike cases, and explores their implications for entropy and geometric interpretations.

Findings

Existence of timelike, Euclidean, and complex extremal surfaces in de Sitter space.

Multiple extremal surfaces can have indistinguishable areas and are related via complex time contour deformations.

Proposes a connection between de Sitter entropy inequalities and analytic continuation from Anti-de Sitter space.

Abstract

We study no-boundary de Sitter extremal surfaces and their pseudo-entropy areas for generic subregions at the future boundary, building on previous work. For large subregions, timelike+Euclidean extremal surfaces exist with transparent geometric interpretations, as do complex ones. The situation for small subregions is analogous to Poincare $dS$ and only complex extremal surfaces exist. In general, the extremal surface area integrals are defined via time contours in the complex time plane. We find multiple extremal surfaces with indistinguishable areas whose time contours can be deformed into each other in the complex time plane without obstruction, which are equivalent for these purposes. This also suggests equivalences between complex $dS$ replica geometries. We discuss $dS_3$ as a simple example at length. This suggests a picture for multiple subregions and entropy inequalities in de Sitter, as encoding $AdS$ ones via analytic continuation. We also discuss mapping future boundary subregions and those on constant time slices in the static patch via lightrays.