On extremal surfaces and de Sitter entropy

Abstract: We study extremal surfaces in the static patch coordinatization of de Sitter space, focussing on the future and past universes. We find connected timelike codim-2 surfaces on a boundary Euclidean time slice stretching from the future boundary $I^+$ to the past boundary $I^-$. In a limit, these surfaces pass through the bifurcation region and have minimal area with a divergent piece alone, whose coefficient is de Sitter entropy in 4-dimensions. These are reminiscent of rotated versions of certain surfaces in the $AdS$ black hole. We close with some speculations on a possible $dS/CFT$ interpretation of 4-dim de Sitter space as dual to two copies of ghost-CFTs in an entangled state. For a simple toy model of two copies of ghost-spin chains, we argue that similar entangled states always have positive norm and positive entanglement.