Digital finite quantum Riemannian geometries

Abstract

We study bimodule quantum Riemannian geometries over the field ${\mathbb{F}}_2$ 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 $\Delta =(,)\nabla \mathrm{d}$ on our models and characterise when it has a massive eigenvector.

37 pages latex, one figure. Corrected some of the curvature formulae in line with final version in press

References in corpus (4)

• Complex Quantum Network Manifolds in Dimension $d>2$ are Scale-Free
• Evolution of Universes in Causal Set Cosmology
• Classification of digital affine noncommutative geometries
• Evolution of Gravitational Perturbations in Non-Commutative Inflation

Cited by in corpus (4)

• Quantum gravity on polygons and $\mathbb{R}\times {\mathbb{Z}}_n$ FLRW model
• Geometric Dirac operator on the fuzzy sphere
• Quantum Gravity and Riemannian Geometry on the Fuzzy Sphere
• Digital quantum groups