整体双曲时空中偏序集的同伦、净上同调和超选择扇区
Homotopy of posets, net-cohomology and superselection sectors in globally hyperbolic spacetimes
摘要：
我们研究了任意全局双曲时空中维数为$\geq 3$的局部可观测网的急剧局部化扇区，称为DHR型扇区。我们表明，这些部门的定义，它发生在闵可夫斯基空间，一个$\mathrm{C}^*-$类别中的电荷结构表现出自己的张量积，置换对称性和共轭的存在。数学框架是根据J.E.罗伯茨局部可观测量的网络由偏序集索引，偏序集由包含下有序时空拓扑的基形成。扇形范畴等价于值在网中的偏序集的1-上圈范畴。我们成功地分析了这个范畴的结构，因为我们展示了时空的拓扑性质是如何在作为索引集的偏序集中编码的：引入了偏序集的第一同伦群，并证明了偏序集的基本群和基本时空群是同构的;任何1-上圈定义了这些基本群的酉表示。另一个重要结果是网的指数集适当变化时，网的上同调不变性。
英文摘要：
We study sharply localized sectors, known as sectors of DHR-type, of a net of local observables, in arbitrary globally hyperbolic spacetimes with dimension $\geq 3$. We show that these sectors define, has it happens in Minkowski space, a $\mathrm{C}^*-$category in which the charge structure manifests itself by the existence of a tensor product, a permutation symmetry and a conjugation. The mathematical framework is that of the net-cohomology of posets according to J.E. Roberts. The net of local observables is indexed by a poset formed by a basis for the topology of the spacetime ordered under inclusion. The category of sectors, is equivalent to the category of 1-cocycles of the poset with values in the net. We succeed to analyze the structure of this category because we show how topological properties of the spacetime are encoded in the poset used as index set: the first homotopy group of a poset is introduced and it is shown that the fundamental group of the poset and the one of the underlying spacetime are isomorphic; any 1-cocycle defines a unitary representation of these fundamental groups. Another important result is the invariance of the net-cohomology under a suitable change of index set of the net.
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