2D quantum gravity partition function on the fluctuating sphere

Abstract

Motivated by recent works on the connection between 2D quantum gravity and timelike Liouville theory, we revisit the latter and clarify some aspects of the computation of its partition function: We present a detailed computation of the Liouville partition function on the fluctuating sphere at finite values of the central charge. The results for both the spacelike theory and the timelike theory are given, and their properties analyzed. We discuss the derivation of the partition function from the DOZZ formula, its derivation using the Coulomb gas approach, a semiclassical computation of it using the fixed area saddle point, and, finally, we arrive to an exact expression for the timelike partition function whose expansion can be compared with the 3-loop perturbative calculations reported in the literature. We also discuss the connection to the 2D black hole and other related topics.

Finite features of quantum de Sitter space

“…In particular, in this regime no microscopic completion is known. At genus zero, the path integral (6) for c m ≫ 1 can be computed in a sensible way using a semiclassical path integral representation of timelike Liouville theory [29][30][31]. In contrast to c m < 1, the gravitational path integral of timelike Liouville theory on the two-sphere is finite for small physical area upon dividing by a residual volume of PSL(2, C).…”