论文标题
几何狄拉克运算符和$ m_2（\ bbb c）$
Geometric Dirac operator on noncommutative torus and $M_2(\Bbb C)$
论文作者
论文摘要
我们解决了量子 - 晶格实现的频谱三元组或“ dirac运算符”，以$ \ bbbc_θ[t^2] $和代数$ m_2（\ bbb c）$ 2 \ $ 2 \ $ 2 \ times times times times times times times timess 2 $矩阵及其标准的量子量子和相关的量子levi-civi-civita connection。对于$ \ bbbc_θ[t^2] $，我们获得了一个均匀的标准频谱三重，但现在由完整的几何真实性确定。对于$ m_2（\ bbb c）$，我们被迫进入平坦的量子Levi-civita连接，并再次获得自然的完全几何实现的频谱三重。在这两种情况下，对于符号参数的不同选择，也有一个奇的光谱三重。我们还考虑$ M_2（\ bbb c）$具有弯曲的量子Levi-Civita连接的替代量子度量标准，并找到一个自然的2参数几乎具有光谱三重的参数，因为$ d $不可能是反ihermitian。在所有情况下，我们将构造分为与量子几何形状相关的局部张力水平，在该水平上，我们对结果进行了更广泛的分类，以及与希尔伯特空间结构有关的更多要求。我们还为$ d^2 $的Lichnerowicz公式说明了，该公式适用于完整的几何实现。
We solve for quantum-geometrically realised spectral triples or `Dirac operators' on the noncommutative torus $\Bbb C_θ[T^2]$ 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 $\Bbb C_θ[T^2]$, 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.
