CAREER: Representing and Classifying Enriched Quantum Symmetry

职业：丰富的量子对称性的表示和分类

基本信息

批准号：
1654159
负责人：
David Penneys
金额：
$ 42万
依托单位：
Ohio State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1654159&HistoricalAwards=false
关键词：
CAREER Representing Classifying Enriched Quantum

项目摘要

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.

对称性在数学和科学中起着重要作用。传统上，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，例如带有加法的整数。近几十年来，我们看到了量子数学对象的出现，它们的对称性形成了一个称为“张量范畴”的类群结构，它具有具有二元融合操作的对象的集合。据说张量类别编码量子对称性：它们描述了物理学中物质的拓扑相，并为我们提供了结和 3 维表面的量子不变量。我们目前正在看到新的数学对象的出现，它们编码“丰富的”量子对称性，描述了 3 维和 2 维量子系统之间的接口。此时，我们有几种相互竞争的形式主义。该项目旨在统一这些概念并通过分类产生奇异的例子。该项目的教育部分包括俄亥俄州立大学关于子因素和量子对称性的本科研究和暑期学校。该项目将把主要研究人员当前的学习材料和为这些项目开发的材料合并成一本关于次因素理论的书。他还将与俄亥俄州立大学的 STEAM Factory 合作，提高社区的科学和数学素养。该项目有两个主要重点：这些新的丰富量子对称性的表示和分类。酉融合类别的对象的维度不一定是整数，因此表示酉融合类别需要冯诺依曼因子，其模块具有连续维度的概念。在这个项目中，首席研究员将利用他之前在小指数子因子分类方面的经验，对小单一带类别中丰富的量子对称性进行分类。这将研究丰富的算子代数理论以发展丰富的子因子理论。首席研究员还将发展双交换范畴理论，这是最初由 Henriques 提出的冯诺依曼代数的更高范畴模拟。这些双交换类别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。