Geometric Dirac operator on noncommutative torus and M₂(C)

We solve for quantum geometrically realised pre-spectral triples or ‘Dirac operators’ on the noncommutative torus C_θ [T²] C θ [ T 2 ] and on the algebra M₂(C) M 2 ( C ) of 2× 2 2 × 2 matrices with their standard quantum metrics and associated quantum Riemannian geometry. For C_θ [T²] C θ [ T 2 ] , we obtain a standard even spectral triple but now uniquely determined by full geometric realisability. For M₂(C) M 2 ( 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 M₂(C) M 2 ( C ) with curved quantum Levi-Civita connection and find a natural 2-parameter family of Dirac operators which are almost spectral triples, where "Equation missing" 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 "Equation missing" which applies in the case of a full geometric realisation.