CAREER: Representing and Classifying Enriched Quantum Symmetry

对称性在数学和科学中起着重要的作用。 经典上，数学对象的对称性形成了一个“群”，它是一个具有二元运算的集合，例如具有加法的整数。近几十年来，我们已经看到了量子数学对象的出现，其对称性形成了一个称为“张量类别”的类群结构，它具有一个具有二元融合操作的对象集合。张量范畴被认为是量子对称性的编码：它们描述了物理学中物质的拓扑相，它们给了我们结点和三维表面的量子不变量。 我们目前看到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。此时，我们有几个相互竞争的形式主义。这个项目旨在统一这些概念，并通过分类产生异国情调的例子。该项目的教育部分包括在俄亥俄州州立大学进行的关于子因子和量子对称性的本科生研究和暑期学校。 该项目将把主要研究人员目前的学习材料和那些为这些程序开发成一本关于子因子理论的书。 他还将与俄亥俄州州立大学的蒸汽工厂合作，提高社区的一般科学和数学素养。这个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。酉融合范畴的对象的维数不一定是整数，因此表示酉融合范畴需要冯诺依曼因子，其模有连续维数的概念。 在这个项目中，首席研究员将利用他以前在小指数子因子分类方面的经验来分类在小幺正带类别中丰富的量子对称性。这将研究一个丰富的算子代数理论，发展一个丰富的子因子理论。主要研究者还将发展理论的bicommutant类别，这是一个更高的范畴模拟冯诺依曼代数最初由于亨利克斯。这些双同变范畴与共形场论有着重要的联系，它们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.