Quantum three-dimensional de Sitter space

TL;DR

This paper calculates the quantum partition function of 2+1 dimensional de Sitter space using Chern-Simons theory, linking it to the semiclassical entropy and Virasoro algebra degeneracy, thus providing a quantum gravitational perspective.

Contribution

It explicitly computes the partition function of 3D de Sitter space via Chern-Simons theory and connects it to the entropy through Virasoro algebra representations.

Findings

Partition function matches semiclassical results at large level k.

De Sitter entropy derived from state degeneracy of Virasoro representations.

Unitary Chern-Simons theory with integer level k used for quantization.

Abstract

We compute the canonical partition function of 2+1 dimensional de Sitter space using the Euclidean $SU(2)\times SU(2)$ Chern-Simons formulation of 3d gravity with a positive cosmological constant. Firstly, we point out that one can work with a Chern-Simons theory with level $k=l/4G$, and its representations are therefore unitary for integer values of $k$. We then compute explicitly the partition function using the standard character formulae for SU(2) WZW theory and find agreement, in the large $k$ limit, with the semiclassical result. Finally, we note that the de Sitter entropy can also be obtained as the degeneracy of states of representations of a Virasoro algebra with $c=3l/2G$.