Geometric Dirac operator on noncommutative torus and $M_2(\Bbb C)$

Abstract

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