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

TL;DR

This work develops a semiclassical framework for the two-dimensional gravitational path integral on S^2 with positive cosmological constant coupled to large-central-charge matter, using the Weyl gauge to map to timelike Liouville theory. It performs a detailed two-loop analysis, addresses residual gauge freedoms, and demonstrates UV divergence cancellations, concluding with an all-orders conjecture for the genus-zero partition function informed by DOZZ. A parallel DOZZ-based analysis on timelike Liouville is used to cross-check the sphere partition function, revealing a two-saddle structure and scale-dependence reconciled by a UV scale relation. The results provide a concrete route to exact genus-zero gravity partition functions and offer insight into de Sitter-inspired quantum gravity in two dimensions, with extensions to higher genus and potential connections to horizon entanglement entropy. Overall, the paper strengthens the bridge between semiclassical Liouville dynamics, exact CFT data, and gravitational path integrals in a tractable setting.

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.

Figures (1)

• Figure 1: Double-tadpoles, melons and cactus diagrams.