The two-sphere partition function in two-dimensional quantum gravity at fixed area

TL;DR

This paper analyzes the two-sphere partition function in two-dimensional quantum gravity with conformal matter at fixed area, demonstrating divergence cancellations and comparing semiclassical results with the DOZZ formula.

Contribution

It provides a two-loop order calculation of the partition function in a fixed-area setting, revealing divergence cancellations and connecting semiclassical and exact results.

Findings

Ultraviolet divergences cancel at two-loop order.

Partition function matches the DOZZ formula predictions.

Semiclassical approach is consistent with exact results.

Abstract

We discuss two-dimensional quantum gravity coupled to conformal matter and fixed area in a semiclassical large and negative matter central charge limit. In this setup the gravity theory -- otherwise highly fluctuating -- admits a round two-sphere saddle. We discuss the two-sphere partition function up to two-loop order from the path integral perspective. This amounts to studying Feynman diagrams incorporating the fixed area constraint on the round two-sphere. In particular we find that all ultraviolet divergences cancel to this order. We compare our results with the two-sphere partition function obtained from the DOZZ formula.