Geometric Dirac operator on noncommutative torus and \(M_2({\mathbb {C}})\)

Abstract

We solve for quantum geometrically realised pre-spectral triples or ‘Dirac operators’ on the noncommutative torus \({\mathbb {C}}_\theta [T^2]\) and on the algebra \(M_2({\mathbb {C}})\) of \(2\times 2\) matrices with their standard quantum metrics and associated quantum Riemannian geometry. For \({\mathbb {C}}_\theta [T^2]\), we obtain a standard even spectral triple but now uniquely determined by full geometric realisability. For \(M_2({\mathbb {C}})\), we are forced to a particular flat quantum Levi-Civita connection and again obtain a natural fully geometrically realised even spectral triple. In both cases there is an odd spectral triple for a different choice of a sign parameter. We also consider an alternate quantum metric on \(M_2({\mathbb {C}})\) with curved quantum Levi-Civita connection and find a natural 2-parameter family of Dirac operators which are almost spectral triples, where fails to be antihermitian. In all cases, we split the construction into a local tensorial level related to the quantum Riemannian geometry, where we classify the results more broadly, and the further requirements relating to the pre-Hilbert space structure. We also illustrate the Lichnerowicz formula for which applies in the case of a full geometric realisation.