论文标题
模块化光谱三元和变形的弗雷德霍尔姆模块
Modular spectral triples and deformed Fredholm modules
论文作者
论文摘要
在非类型$ \ ty {ii_1} $表示的设置中，我们提出了一个{\ it fredholm模块} $ \ big [d_ \ ct | d_ \ ct | d_ \ ct |^{ - 1} \ ,, \ ,, \，{\ bf \ cdot}的定义$ \ ct $，其中$ d_ \ ct $是变形的狄拉克操作员。为了简单起见，假定$ d_ \ ct $是可逆的，其域是“必需的”操作员系统$ \ ce_ \ ct $。根据这样的定义，我们获得了$ \ big [d_ \ ct | d_ \ ct |^{ - 1} \ ,, \，{\ bf \ cdot} \，\ big] _ \ ct = | d_ \ ct |^{ -1} d_ \ ct（\，{\ bf \ cdot} \，）+d_ \ ct（\，{\ bf \ cdot} \，）| d_ \ ct |^{ - 1} $，其中$ d_ \ ct $是与$ d_ \ ct $相关的变形推导。由于“量子微分” $ 1/| d_ \ ct | $出现在对称位置，因此即使在未经构造的情况下，在奇特的情况下，Fredholm模块的定义也与通常的定义不同。因此，它似乎更适合研究非交通歧管，其中非平凡的模块化结构可能起着至关重要的作用。我们表明，非型$ \ ty {ii_1} $的所有模型{fs}中的非交换2-tori表示的表示确实提供了模块化的光谱三元组，另外，根据本文中提出的定义，变形了弗雷德姆模块。由于对狄拉克操作员的光谱的详细知识在光谱几何学中起着基本作用，因此我们在特定特定类别的特定类别的特定类别的特定型解决方案的特定解决方案中，对变形的dirac运算符$ d_ \ ct $的特征值和特征向量进行了表征。
In the setting of non-type $\ty{II_1}$ representations, we propose a definition of {\it deformed Fredholm module} $\big[D_\ct|D_\ct|^{-1}\,,\,{\bf\cdot}\,\big]_\ct$ for a modular spectral triple $\ct$, where $D_\ct$ is the deformed Dirac operator. $D_\ct$ is assumed to be invertible for the sake of simplicity, and its domain is an "essential" operator system $\ce_\ct$. According to such a definition, we obtain $\big[D_\ct|D_\ct|^{-1}\,,\,{\bf\cdot}\,\big]_\ct=|D_\ct|^{-1}d_\ct(\,{\bf\cdot}\,)+d_\ct(\,{\bf\cdot}\,)|D_\ct|^{-1}$, where $d_\ct$ is the deformed derivation associated to $D_\ct$. Since the "quantum differential" $1/|D_\ct|$ appears in a symmetric position, such a definition of Fredholm module differs from the usual one even in the undeformed case, that is in the tracial case. Therefore, it seems to be more suitable for the investigation of noncommutative manifolds in which the nontrivial modular structure might play a crucial role. We show that all models in \cite{FS} of non-type $\ty{II_1}$ representations of noncommutative 2-tori indeed provide modular spectral triples, and in addition deformed Fredholm modules according to the definition proposed in the present paper. Since the detailed knowledge of the spectrum of the Dirac operator plays a fundamental role in spectral geometry, we provide a characterisation of eigenvalues and eigenvectors of the deformed Dirac operator $D_\ct$ in terms of the periodic solutions of a particular class of eigenvalue Hill equations.
