Liouville theory beyond the cosmological horizon

TL;DR

The paper investigates Liouville theory as a holographic dual to 3D de Sitter gravity beyond the cosmological horizon. It demonstrates that Euclidean Liouville theory on the future boundary is simultaneously obtainable from a Dirichlet problem on a fixed timelike slice inside the static patch, with the Liouville time identified with the static observer's time. Using a Chern-Simons formulation and two complementary slicings, the authors show that the bulk supports two Virasoro algebras with central charge c = 3ℓ/(2G), and perform a Hamiltonian reduction to Liouville theory on the boundary. This provides a concrete bulk realization of dS/CFT for pure gravity and extends the holographic dictionary to regions inside the static patch, offering a route to higher-spin generalizations and deeper insights into de Sitter holography.

Abstract

The dS/CFT correspondence postulates the existence of a Euclidean CFT dual to a suitable gravity theory with Dirichlet boundary conditions asymptotic to de Sitter spacetime. A semi-classical model of such a correspondence consists of Einstein gravity with positive cosmological constant and without matter which is dual to Euclidean Liouville theory defined at the future conformal boundary. Here we show that Euclidean Liouville theory is also dual to Einstein gravity with Dirichlet boundary conditions on a fixed timelike slice in the static patch. Intriguingly, the spacetime interpretation of Euclidean Liouville time is the physical time of the static observer. As a prerequisite of this correspondence, we show that the asymptotic symmetry algebra which consists of two copies of the Virasoro algebra extends everywhere into the bulk.

Figures (4)

• Figure 1: Fefferman-Graham coordinates
• Figure 2: Eddington-Finkelstein coordinates
• Figure 3: Bulk region bounded by $\Sigma_r^+$ and $\Sigma_r^-$
• Figure 4: Bulk region bounded by $\Sigma_r$