Dirac operator associated to a quantum metric

TL;DR

This paper constructs a canonical Dirac operator from quantum metric data, providing a geometric spectral triple framework applicable to noncommutative spaces like graphs, lattices, and fuzzy spheres.

Contribution

It introduces a new method to build Dirac operators from quantum metrics and Levi-Civita connections, unifying geometric and algebraic approaches in noncommutative geometry.

Findings

Recovering Dirac operators on graphs and lattices as spectral triples

Application to fuzzy sphere and quantum Riemannian geometries

Proposal for minimal coupling to external potentials

Abstract

We construct a canonical geometrically realised Connes spectral triple or `Dirac operator' $D\!\!\!/$ from the data of a quantum metric $g\in \Omega^1\otimes_A\Omega^1$ and quantum Levi-Civita bimodule connection, at the pre-Hilbert space level. Here $A$ is a possibly noncommutative coordinate algebra, $\Omega^1$ a bimodule of 1-forms and the spinor bundle is $S=A\oplus\Omega^1$. When applied to graphs or lattices, we essentially recover a Dirac operator previously proposed by Bianconi but now as a geometrically realised spectral triple. We also apply the construction to the fuzzy sphere and to $2\times 2$ matrices with their standard quantum Riemannian geometries. We also propose how $D\!\!\!/$ can be minimally coupled to an external potential.