Functor of points and height functions for noncommutative Arakelov geometry
Functor of points and height functions for noncommutative Arakelov geometry
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非交换 Arakelov 几何的点函子和高度函数
DOI：
10.1016/j.geomphys.2021.104337
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发表时间：
2021
影响因子：
1.5
通讯作者：
Marcolli, Matilde
中科院分区：
文献类型：
--
作者：
Lima, Alicia;Marcolli, Matilde
我们提出了一个非交换空间的点函子的概念，在双模的范畴中赋值，并赋予由厄米结构和高度函数的概念决定的作用泛函，以经典点函子作为物理sigma模型的解释为模型。我们讨论了基于不同体积和轨迹概念的这种高度函数的不同选择，包括基于服事-斯托林斯等级的选择。我们证明了高度函数决定了与我们的点函子相关的可观测值代数上的动态时间演化。我们特别关注非交换算术曲线的情况，其中相关代数是矩阵代数在数域上的除法代数的和，并且我们讨论了高维非交换算术空间的更一般的概念，其中我们的方法建议将琼斯指数解释为高度函数。
We propose a notion of functor of points for noncommutative spaces, valued in categories of bimodules, and endowed with an action functional determined by a notion of hermitian structures and height functions, modeled on an interpretation of the classical functor of points as a physical sigma model. We discuss different choices of such height functions, based on different notions of volumes and traces, including one based on the Hattori-Stallings rank. We show that the height function determines a dynamical time evolution on an algebra of observables associated to our functor of points. We focus in particular the case of noncommutative arithmetic curves, where the relevant algebras are sums of matrix algebras over division algebras over number fields, and we discuss a more general notion of noncommutative arithmetic spaces in higher dimensions, where our approach suggests an interpretation of the Jones index as a height function.