2022/08/16 by Lira-Torres, E., Majid, S.
#FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2208.07821
We solve for quantum-geometrically realised spectral triples or `Dirac operators' on the noncommutative torus \Bbb Cθ[T2] and on the algebra M2(\Bbb C) of 2× 2 matrices with their standard quantum metrics and associated quantum Levi-Civita connections. For \Bbb Cθ[T2], we obtain an even standard spectral triple but now uniquely determined by full geometric realisability. For M2(\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 M2(\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 D2 which applies in the case of a full geometric realisation.