Finite Features of Quantum De Sitter Space

We consider degrees of freedom for a quantum de Sitter spacetime. The problem is studied from both a Lorentzian and a Euclidean perspective. From a Lorentzian perspective, we compute dynamical properties of the static patch de Sitter horizon. These are compared to dynamical features of black holes. We point out differences suggestive of non-standard thermal behaviour for the de Sitter horizon. We establish that geometries interpolating between an asymptotically AdS$_2 \times S^2$ space and a dS$_4$ interior are compatible with the null energy condition, albeit with a non-standard decreasing radial size of $S^2$. The putative holographic dual of an asymptotic AdS$_2$ spacetime is comprised of a finite number of degrees of freedom. From a Euclidean perspective we consider the gravitational path integral for fields over compact manifolds. In two-dimensions, we review Polchinski's BRST localisation of Liouville theory and propose a supersymmetric extension of timelike Liouville theory which exhibits supersymmetric localisation. We speculate that localisation of the Euclidean gravitational path integral is a reflection of a finite number of degrees of freedom in a quantum de Sitter universe.