Random Disks of Constant Curvature: the Lattice Story

TL;DR

This paper develops a discretized, UV-complete formulation of 2D quantum gravity models with finite boundary lengths, revealing new qualitative behaviors across different curvatures and deriving the Schwarzian theory in a specific limit.

Contribution

It introduces a matrix model for counting self-overlapping curves in 2D quantum gravity with finite boundaries, unifying descriptions across curvature regimes.

Findings

Same UV description for all curvatures, different macroscopic properties.

Schwarzian theory emerges in the negative curvature limit.

Large geometries are prevalent in positive curvature models.

Abstract

We lay the groundwork for a UV-complete formulation of the Euclidean Jackiw-Teitelboim two-dimensional models of quantum gravity when the boundary lengths are finite, emphasizing the discretized approach. The picture that emerges is qualitatively new. For the disk topology, the problem reduces to counting so-called self-overlapping curves, that are closed loops that bound a distorted disk, with an appropriate multiplicity. We build a matrix model that does the correct counting. The theories in negative, zero and positive curvatures have the same UV description but drastically different macroscopic properties. The Schwarzian theory emerges in the limit of very large and negative cosmological constant in the negative curvature model, as an effective theory valid on distance scales much larger than the curvature length scale. In positive curvature, we argue that large geometries are ubiquitous and that the theory exists only for positive cosmological constant. Our discussion is pedagogical and includes a review of several relevant topics.