Exploring de Sitter space and holography

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.

Elliptic de Sitter space:dS/Z2

“…Such a hermiticity condition has also been proposed in [28]. Consider a primary field with complex conjugate weights h ± .…”

Holography in three-dimensional Kerr–de Sitter space with a gravitational Chern–Simons term

“…With a unitary representation of the Virasoro algebras of L m ,L m in the standard form, which are defined on the plane, one can use the Cardy formula for the entropy of a CFT [26,63,64,65,66,67,68] 8) which is real-valued and positive semi-definite, by construction. Of course, a priori justification of this formula for the complex-valued c,c and L 0 ,L 0 which make the corresponding CFT non-unitary, is still missing, but I will just assume this formula and see what happen in our non-trivial circumstance.…”

q-deformed de Sitter/conformal field theory correspondence

“…= −β dE dm n r (E, m)e −βE (16) where E is the energy in static patch coordinates, m is the azimuthal quantum number, and n r (E, m) radial quantum number fixed by…”