Exploring de Sitter Space and Holography

TL;DR

This work investigates holography for de Sitter space with a positive cosmological constant by (i) introducing a nonlocal map from Lorentzian dS to Euclidean AdS that commutes with the isometries and maps the two dS boundaries to a single $EAdS$ boundary, (ii) analyzing the on-shell actions for free scalars and 3d gravity to expose the subtleties of a GKPW-like dictionary in dS, and (iii) developing a framework in which de Sitter holography requires two entangled CFTs with novel hermiticity conditions governed by $SL(2,\mathbb{C})$ symmetry. A concrete toy model of Euclidean Virasoro algebra illustrates how such duals could realize dS vacuum states and thermodynamics through entanglement between boundary theories. The evidence from free scalar fields supports a two-boundary, two-CFT picture and motivates further exploration of unconventional boundary theories and their string-theoretic realizations. Collectively, the results suggest a holographic description of de Sitter space in terms of two entangled boundary theories with unusual hermiticity, rather than a single conventional CFT, and point to rich avenues for connecting boundary data to bulk de Sitter physics.

Abstract

We explore aspects of the physics of de Sitter (dS) space that are relevant to holography with a positive cosmological constant. First we display a nonlocal map that commutes with the de Sitter isometries, transforms the bulk-boundary propagator and solutions of free wave equations in de Sitter onto the same quantities in Euclidean anti-de Sitter (EAdS), and takes the two boundaries of dS to the single EAdS boundary via an antipodal identification. Second we compute the action of scalar fields on dS as a functional of boundary data. Third, we display a family of solutions to 3d gravity with a positive cosmological constant in which the equal time sections are arbitrary genus Riemann surfaces, and compute the action of these spaces as a functional of boundary data from the Einstein gravity and Chern-Simons gravity points of view. These studies suggest that if de Sitter space is dual to a Euclidean conformal field theory (CFT), this theory should involve two disjoint, but possibly entangled factors. We argue that these CFTs would be of a novel form, with unusual hermiticity conditions relating left movers and right movers. After exploring these conditions in a toy model, we combine our observations to propose that a holographic dual description of de Sitter space would involve a pure entangled state in a product of two of our unconventional CFTs associated with the de Sitter boundaries. This state can be constructed to preserve the de Sitter symmetries and and its decomposition in a basis appropriate to antipodal inertial observers would lead to the thermal properties of static patch.