CAREER: Representing and Classifying Enriched Quantum Symmetry

对称性在数学和科学中扮演着重要的角色。经典地说，一个数学对象的对称性形成一个“群”，它是一个具有二元运算的集合，例如整数与加法。近几十年来，我们看到了量子数学对象的出现，这些对象的对称性形成了一种类似群组的结构，称为“张量范畴”，它具有一组具有二元融合运算的对象。张量范畴被认为编码了量子对称性：它们描述了物理中物质的拓扑相，它们给我们提供了纽结和三维表面的量子不变量。我们目前看到了新的数学对象的出现，它们编码了“丰富的”量子对称性，描述了三维和二维量子系统之间的界面。在这个时候，我们有几个相互竞争的形式主义。这个项目的目的是统一这些概念，并通过分类产生奇异的例子。这个项目的教育部分包括俄亥俄州立大学关于子因子和量子对称性的本科生研究和暑期班。该项目将把主要研究人员目前的学习材料和为这些项目开发的材料编入一本关于次因素理论的书中。他还将与俄亥俄州立大学的STEAM工厂合作，提高社区的一般科学和数学素养。该项目有两个主要重点：这些新的丰富的量子对称性的表示和分类。酉合范畴具有的对象的维度不一定是整数，因此表示酉合范畴需要von Neumann因子，其模具有连续维的概念。在这个项目中，首席研究员将利用他以前在小指标子因子分类方面的经验，对丰富在小酉带范畴中的量子对称性进行分类。这将学习丰富的算子代数理论，以发展丰富的子因子理论。首席研究人员还将发展双变异范畴理论，这是最初由于Henrique而产生的von Neumann代数的更高范畴类比。这些双突变类与共形场理论有着重要的联系，有望成为丰富量子对称性分类的重要工具。

Symmetry plays an important role in mathematics and science. Classically, the symmetries of a mathematical object form a "group", which is a set with a binary operation such as the integers with addition. In recent decades, we have seen the emergence of quantum mathematical objects whose symmetries form a group-like structure called a "tensor category", which has a collection of objects with a binary fusion operation. Tensor categories are said to encode quantum symmetry: they describe topological phases of matter in physics, and they give us quantum invariants of knots and 3-dimensional surfaces. We are currently seeing the emergence of new mathematical objects which encode "enriched" quantum symmetry, which describe interfaces between 3-dimensional and 2-dimensional quantum systems. At this time, we have several competing formalisms. This project aims to unify these notions and produce exotic examples through classification. The educational component of this project includes undergraduate research and Summer schools on subfactors and quantum symmetry at the Ohio State University. The project will incorporate the principal investigators current learning materials and those developed for these programs into a book on subfactor theory. He will also collaborate with the STEAM Factory at Ohio State University to enhance general scientific and mathematical literacy in the community.This project has two main focuses: the representation and the classification of these new enriched quantum symmetries. Unitary fusion categories have objects whose dimensions are not necessarily integers, so representing unitary fusion categories requires von Neumann factors, whose modules have a notion of continuous dimension. In this project the principal investigator will use his previous experience in the classification of small index subfactors to classify quantum symmetries enriched in small unitary ribbon categories. This will study an enriched operator algebra theory to develop an enriched subfactor theory. The principal investigator will also develop the theory of bicommutant categories, which are a higher categorical analog of von Neumann algebras originally due to Henriques. These bicommutant categories have important connections to conformal field theory, and they are expected to be an important tool in the classification of enriched quantum symmetries.