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

TL;DR

The paper analyzes two-dimensional quantum gravity coupled to conformal matter at fixed area in the large negative matter central charge limit, showing that a round $S^2$ saddle dominates genus-zero configurations and enabling a controlled semiclassical expansion to two loops. It performs a detailed fixed-area Liouville analysis, demonstrates cancellation of UV divergences at both one- and two-loop orders, and derives the fixed-area sphere partition function $\,\mathcal{Z}^{(0)}_{ ext{grav}}[\nu]$ with precise $\nu$-scaling and scheme-dependent UV factors. A parallel DOZZ-based computation of the sphere partition function under the fixed-area constraint is carried out, with careful matching to the path-integral result once the UV scheme is fixed. The work also discusses the spacelike vs timelike fixed-area scenarios, the role of the Fadeev–Popov determinant, and connections to multicritical matrix integrals, suggesting a nonperturbative bridge to microscopic models of 2D quantum gravity. Overall, the results illuminate the fixed-area sector of 2D gravity and its interplay with Liouville theory, DOZZ data, and matrix-model duals.

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.

Figures (1)

• Figure 1: Propagators of the small fluctuations.