de Sitter space and extremal surfaces for spheres

TL;DR

The paper analyzes complex extremal surfaces for spherical subregions in de Sitter space and shows their areas exhibit a leading area-law divergence and, in even boundary dimensions, a universal logarithmic term whose coefficient mirrors the conformal anomaly in the dual Euclidean CFT on a sphere. Using the dS/CFT dictionary with the wavefunction of the universe, the author computes the corresponding sphere free energy in global de Sitter space and finds exact agreement (including numerical factors) with the logarithmic coefficients obtained from the complex extremal-surface area, supporting an entanglement-entropy interpretation in dS/CFT via analytic continuation from AdS. The work extends previous strip analyses to spheres, demonstrating a consistent bridge between bulk complex geometry and boundary CFT anomalies, while highlighting subtleties due to nonunitarity and complex central charges in the de Sitter context. Overall, the results strengthen the view that complex extremal surfaces in $dS_{d+1}$ encode entanglement-like data for the dual Euclidean CFT and that their logarithmic terms play the role of $a$-type central charges in this holographic setting.

Abstract

Following arXiv:1501.03019 [hep-th], we study de Sitter space and spherical subregions on a constant boundary Euclidean time slice of the future boundary in the Poincare slicing. We show that as in that case, complex extremal surfaces exist here as well: for even boundary dimensions, we isolate the universal coefficient of the logarithmically divergent term in the area of these surfaces. There are parallels with analytic continuation of the Ryu-Takayanagi expressions for holographic entanglement entropy in $AdS/CFT$. We then study the free energy of the dual Euclidean CFT on a sphere holographically using the $dS/CFT$ dictionary with a dual de Sitter space in global coordinates, and a classical approximation for the wavefunction of the universe. For even dimensions, we again isolate the coefficient of the logarithmically divergent term which is expected to be related to the conformal anomaly. We find agreement including numerical factors between these coefficients.