Digital finite quantum Riemannian geometries

TL;DR

This paper explores finite-field quantum Riemannian geometries over $F_2$, classifies all parallelisable cases up to dimension 3, and analyzes the quantum Laplacian's eigenvectors, revealing diverse non-flat models with noncommutative differentials.

Contribution

It provides a classification of all parallelisable finite-field quantum Riemannian geometries up to dimension 3 and analyzes their quantum Laplacian properties.

Findings

Rich moduli of non-flat geometries found for n=3

Coordinate algebras are commutative with noncommutative differentials

Quantum Laplacian has massive eigenvectors under certain conditions

Abstract

We study bimodule quantum Riemannian geometries over the field $\Bbb F_2$ of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for coordinate algebras up to vector space dimension $n\le 3$, finding a rich moduli of examples for $n=3$ and top form degree 2, including many that are not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian $\Delta=(\ ,\ )\nabla{\rm d}$ on our models and characterise when it has a massive eigenvector.