2D quantum gravity partition function on the fluctuating sphere

TL;DR

This paper provides a comprehensive analysis of the 2D quantum gravity partition function on a fluctuating sphere, comparing spacelike and timelike Liouville theories, and deriving exact and perturbative results.

Contribution

It offers a detailed computation of the Liouville partition function at finite central charge, clarifies derivations from multiple approaches, and connects to 2D black hole physics.

Findings

Derived exact partition function for timelike Liouville theory

Compared semiclassical and perturbative expansions

Connected results to 2D black hole models

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.