AI 中文总结
本文研究模糊环面及带共形扭曲的模糊环面上的Connes谱距离，通过构造Dirac算子等探究其性质，发现倒数勾股定理等规律并计算了部分态的谱距离。
AI 中文摘要
本文研究模糊环面上态之间的Connes谱距离，通过对易子和反对易子构造Dirac算子，基于该Dirac算子构建模糊环面的谱三元组，探究模糊环面上谱距离的若干性质，发现谱距离之间存在倒数勾股定理；构造最优元的条件期望函数，该函数可使对应Lipschitz半范数收缩，发现任意对角态对应的谱距离最优元也为对角型；明确计算了若干简单态的谱距离，包括基态和部分简单混合态，发现对角态的最优元与谱距离中存在若干类循环对称性；此外，还构造了具有共形扭曲的模糊环面，研究共形参数与谱距离之间的关系。
英文摘要
In this paper, we study the Connes spectral distance between states on the fuzzy torus. We construct a Dirac operator by commutators and anticommutators. Based on this Dirac operator, we construct a spectral triple of the fuzzy torus. We study some properties of the spectral distance on the fuzzy torus. We find that there is a reciprocal Pythagorean theorem between the spectral distances. We construct a conditional expectation function of the optimal element which can lead to a contraction of the corresponding Lipschitz seminorm. We find that for any diagonal states, the corresponding optimal elements of spectral distances are also diagonal. We explicitly calculate the spectral distances of some simple states, including basic states and some simple mixed states. We find that there are some kinds of cyclic symmetry in both the optimal elements and the spectral distances between the diagonal states. Furthermore, we also construct a fuzzy torus with some type of conformal twist, and study the relation between conformal parameters and spectral distances.
