De Sitter Gravity and Liouville Theory

TL;DR

This paper establishes a correspondence between conical defects in three-dimensional de Sitter space and Liouville conformal field theory, linking classical gravity to quantum properties of the CFT and exploring implications for de Sitter entropy.

Contribution

It demonstrates a novel duality between de Sitter gravity and Liouville theory, connecting classical geometries with quantum conformal dimensions and providing insights into de Sitter entropy.

Findings

Classical conical defects correspond to Liouville vertex operators.

Quantum dimensions match Kerr-dS_3 solution masses.

De Sitter entropy relates to Liouville momentum, lacking statistical interpretation.

Abstract

We show that the spectrum of conical defects in three-dimensional de Sitter space is in one-to-one correspondence with the spectrum of vertex operators in Liouville conformal field theory. The classical conformal dimensions of vertex operators are equal to the masses of the classical point particles in dS_3 that cause the conical defect. The quantum dimensions instead are shown to coincide with the mass of the Kerr-dS_3 solution computed with the Brown-York stress tensor. Therefore classical de Sitter gravity encodes the quantum properties of Liouville theory. The equality of the gravitational and the Liouville stress tensor provides a further check of this correspondence. The Seiberg bound for vertex operators translates on the bulk side into an upper mass bound for classical point particles. Bulk solutions with cosmological event horizons correspond to microscopic Liouville states, whereas those without horizons correspond to macroscopic (normalizable) states. We also comment on recent criticism by Dyson, Lindesay and Susskind, and point out that the contradictions found by these authors may be resolved if the dual CFT is not able to capture the thermal nature of de Sitter space. Indeed we find that on the CFT side, de Sitter entropy is merely Liouville momentum, and thus has no statistical interpretation in this approach.