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 a bimodule connection on ${\Omega }^1$, 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 $\mathcal{S}=A\oplus {\Omega }^1$. 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 $2\times 2$ matrices with their standard quantum Riemannian geometries. We propose how $D/$ can be extended to an external bundle with bimodule connection.