  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.
