What is the Curvature of 2D Euclidean Quantum Gravity?

TL;DR

This paper investigates the nonperturbative curvature properties of 2D Euclidean quantum gravity using dynamical triangulations, providing evidence that its curvature profile resembles that of a classical 4-sphere rather than a 5-sphere, indicating a well-defined quantum Ricci curvature.

Contribution

It offers new insights into the curvature profile of 2D Euclidean quantum gravity, showing it aligns with a classical 4-sphere and suggesting the existence of a quantum Ricci curvature in the scaling limit.

Findings

Curvature profile matches that of a classical 4-sphere.

Finite-size effects significantly influence curvature measurements.

Evidence supports a well-defined quantum Ricci curvature.

Abstract

We re-examine the nonperturbative curvature properties of two-dimensional Euclidean quantum gravity, obtained as the scaling limit of a path integral over dynamical triangulations of a two-sphere, which lies in the same universality class as Liouville quantum gravity. The diffeomorphism-invariant observable that allows us to compare the averaged curvature of highly quantum-fluctuating geometries with that of classical spaces is the so-called curvature profile. A Monte Carlo analysis on three geometric ensembles, which are physically equivalent but differ by the inclusion of local degeneracies, leads to new insights on the influence of finite-size effects. After eliminating them, we find strong evidence that the curvature profile of 2D Euclidean quantum gravity is best matched by that of a classical round four-sphere, rather than the five-sphere found in previous work. Our analysis suggests the existence of a well-defined quantum Ricci curvature in the scaling limit.