The two-sphere partition function in two-dimensional quantum gravity

TL;DR

This paper computes the two-sphere partition function in two-dimensional quantum gravity with positive cosmological constant and conformal matter, using semiclassical expansion and two-loop calculations, revealing divergence cancellations and proposing an all-orders result.

Contribution

It introduces a detailed two-loop analysis of the Euclidean path integral in 2D quantum gravity coupled to conformal matter, leading to a conjecture for the all-orders partition function.

Findings

Ultraviolet divergence cancellations in the path integral.

Path integral computation of the timelike Liouville central charge.

Proposal of an all-orders formula for the 2D gravitational partition function.

Abstract

We study the Euclidean path integral of two-dimensional quantum gravity with positive cosmological constant coupled to conformal matter with large and positive central charge. The problem is considered in a semiclassical expansion about a round two-sphere saddle. We work in the Weyl gauge whereby the computation reduces to that for a (timelike) Liouville theory. We present results up to two-loops, including a discussion of contributions stemming from the gauge fixing procedure. We exhibit cancelations of ultraviolet divergences and provide a path integral computation of the central charge for timelike Liouville theory. Combining our analysis with insights from the DOZZ formula we are led to a proposal for an all orders result for the two-dimensional gravitational partition function on the two-sphere.