Quantum Symmetries: Subfactors, Topological Phases, and Higher Categories

在数学和物理科学中，对称性出现在许多地方。通常，系统的对称性是由群的数学概念来描述的，群是具有单位和乘法的集合，其中每个运算都有一个逆。组通过保持结构的贴图作用于对象。近几十年来，我们看到了量子数学对象的出现，比如冯·诺伊曼代数和物质的拓扑相，它们自然地生活在“更高的范畴”中，因此它们的对称性可以更好地用张量范畴来描述。这个项目的目的是研究von Neumann代数和物质的拓扑相中出现的更高范畴，以更好地理解这些量子系统及其更高的量子对称性。该项目为本科生和研究生提供研究培训机会。该项目主要有三个重点。首先，它继续研究小指标子因子和酉合范畴，以寻找奇异量子对称性的新例子。其次，研究了类似于共形场理论的共形网络描述的(2+1)维物质拓扑相的算符代数网络。第三，它调查了更高类别的统一性及其与(2+1)维拓扑阶段的边界和相变的关系。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Symmetry arises in many places in mathematics and the physical sciences. Typically, symmetries of a system are described by the mathematical notion of a group, which is a set with a unit and multiplication, where every operation has an inverse. Groups act on objects by structure-preserving maps. In recent decades, we have seen the emergence of quantum mathematical objects, like von Neumann algebras and topological phases of matter, which naturally live in "higher categories," and so their symmetries are better described by tensor categories. This project aims to study the higher categories arising in the study of von Neumann algebras and topological phases of matter to better understand these quantum systems and their higher quantum symmetries. The project provides research training opportunities for undergraduate and graduate students.This project has three main focuses. First, it continues the investigation of small index subfactors and unitary fusion categories to search for new examples of exotic quantum symmetries. Second, it investigates a nets of operator algebras approach to (2+1)D topological phases of matter, analogous to the conformal net description of conformal field theory. Third, it investigate unitarity for higher categories and its relationship to boundaries and phase transitions for (2+1)D topological phases of matter.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.