超有限冯·诺依曼代数中的模理论与对称性分解
Modular theory and symmetry resolution in hyperfinite von Neumann algebras
AI总结：
该研究从子区域荷扇区分解视角研究超有限冯·诺依曼代数的模理论，通过有限维代数无限张量积与直积分方法得到对称性分解的相关模量，为相关领域研究奠基并具应用价值。
摘要：
我们从子区域荷扇区分解的视角，研究超有限冯·诺依曼代数（即II型或III型代数）中的模理论。我们通过考虑具有固定子区域荷值的有限维代数的无限张量积，来处理这一对称性分解问题。一个关键要素是利用直积分来组合这些代数，这使我们能够得到超有限代数的对称性分解模算子、模流和模关联函数。我们的方法为共形场论中对称性分解与模理论的近期研究成果奠定了数学基础。我们的分析既适用于定义在连续区间上的荷，也适用于离散集合上的荷，后者在凝聚态理论中具有研究价值。此外，在AdS/CFT对应中，我们预期我们的发现可作为体时空重构的新要素，纳入来自不同边界荷扇区的信息。
英文摘要：
We study modular theory in hyperfinite von Neumann algebras, i.e. in those of type II or type III, from the viewpoint of a subregion charge sector decomposition. We address this symmetry resolution by considering infinite tensor products of finite-dimensional algebras with fixed subregion charge values. An important ingredient is the combination of these algebras using direct integrals. This allows us to obtain the symmetry-resolved modular operator, modular flow, and modular correlation functions for hyperfinite algebras. Our approach establishes a mathematical foundation for recent results on symmetry resolution and modular theory in conformal field theory. Our analysis applies both to charges defined on a continuous range, or on a discrete set. The latter is of interest for condensed matter theory. Moreover, within the AdS/CFT correspondence we expect our findings to be relevant as a new ingredient for bulk spacetime reconstruction, including information from different boundary charge sectors.
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