The Partition Function of the Timelike Liouville Theory

2D quantum gravity partition function on the fluctuating sphere

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 coulombgas approach, a semiclassical computation of it using the fixed area saddlepoint, and, finally, we arrive to an exact expression for the timelikepartition function whose expansion can be compared with the 3-loop perturbativecalculations reported in the literature.