Moduli of quantum Riemannian geometries on <= 4 points

TL;DR

This paper classifies and analyzes the moduli space of quantum Riemannian geometries on small finite sets, providing explicit descriptions for up to 4 points and insights into the quantum theory via functional integration.

Contribution

It offers a complete classification of parallelizable noncommutative manifold structures on small finite sets within a specific formalism, extending to moduli space analysis and quantum theory aspects.

Findings

Full moduli space for up to 3 points

Restricted moduli space for 4 points

Topological moduli space for up to 9 points

Abstract

We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously. The full moduli space is found for $\le 3$ points, and a restricted moduli space for 4 points. The topological part of the moduli space is found for $\le 9$ points based on the known atlas of regular graphs. We also discuss aspects of the quantum theory defined by functional integration.