Dirac operator associated to a quantum metric

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.