Quantum three-dimensional de Sitter space

We compute the canonical partition function of (2+1)-dimensional de Sitter space using the Euclidean SU(2)×SU(2) Chern-Simons formulation of 3D gravity with a positive cosmological constant. First, 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 formulas 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 $c=3l/2G.\mathrm{}$