AI 中文总结
该研究探讨整体de Sitter空间中基本补集与区域算子代数的关系，在手征共形网中推导连续核包含判据，引入依赖表示的区域代数交换子模型，揭示间隙对代数结构的保护作用及相关不连续变化。
AI 中文摘要
我们研究整体de Sitter空间中基本补集与区域算子代数之间的关系。对于二维整体dS₂中时间对称圆上的有限个开弧的并集，我们证明基本补集由最大的互补间隙决定：当间隙的角长度至少为π时，该间隙会产生贡献，且最多仅有一个间隙能产生贡献。在高维整体de Sitter空间中，对应的判据是包含一个开半球。随后，我们在抽象的Möbius协变手征共形网中研究算子代数的相关结果：每个区域代数通过共享的辅助时钟与其真空模流交叉，我们推导了所得连续核之间包含关系的精确判据，并证明当较大区域包含额外的类空弧时，真空关联会阻碍该判据。不过，通过使用分裂乘积态，可在由固定的一组分离弧生成的有限布尔代数上得到兼容族。受Bousso和Penington的全息图方案启发，我们引入了一种依赖表示的区域代数交换子模型：当受保护的间隙与固定参考弧重叠时，所分配的代数是参考II₁型因子的真冯·诺依曼子代数；对于显式的双弧族，添加任意短的对跖分量会消除基本补集，并使所分配的代数不连续地变为完整参考代数。
英文摘要
We investigate the relation between fundamental complements and regional operator algebras in global de Sitter space. For a finite union of open arcs on the time-symmetric circle of global dS$_2$, we show that the fundamental complement is determined by the largest complementary gap: a gap contributes precisely when its angular length is at least $π$, and at most one gap can do so. In higher-dimensional global de Sitter space, the corresponding criterion is containment of an open hemisphere. We then study the operator-algebraic consequences in an abstract Möbius-covariant chiral conformal net. Each regional algebra is crossed with its vacuum modular flow using a shared auxiliary clock. We derive an exact criterion for inclusions between the resulting continuous cores and show that vacuum correlations obstruct this criterion when the larger region contains an additional spacelike arc. A compatible family can nevertheless be obtained on the finite Boolean algebra generated by a fixed collection of separated arcs by using a split product state. Motivated by the holograms prescription of Bousso and Penington, we introduce a representation-dependent commutant model for regional algebras. When the protected gap overlaps a fixed reference arc, the assigned algebra is a proper von Neumann subalgebra of the reference Type-II$_1$ factor. For an explicit two-arc family, adding an arbitrarily short antipodal component removes the fundamental complement and changes the assigned algebra discontinuously to the full reference algebra.
