Dirac operator associated to a quantum metric

We construct a canonical geometrically realised Connes spectral triple or ‘Dirac operator’ D /\, from the data of a quantum metric g ^1₀^1 and a bimodule connection on ^1, at the pre-Hilbert space level. Here A is a possibly noncommutative coordinate algebra, ^1 a bimodule of 1 -forms and the spinor bundle is S=A^1. When applied to graphs or lattices, we essentially recover a previously proposed Dirac operator but now as a geometrically realised spectral triple. We also apply the construction to the fuzzy sphere and to 2 2 matrices with their standard quantum Riemannian geometries. We propose how D /\, can be extended to an external bundle with bimodule connection.