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。从经典上讲，数学对象的对称性形成了一个“组”，该集合具有带有二进制操作的集合，例如整数和添加。近几十年来，我们看到了量子数学对象的出现，其对称性形成了类似组的结构，称为“张量类别”，该结构具有带有二进制融合操作的对象的集合。据说张量的类别编码量子对称性：它们描述了物理中物质的拓扑阶段，它们为我们提供了结和3维表面的量子不变性。我们目前正在看到编码“富集”量子对称性的新数学对象的出现，该对象描述了三维和二维量子系统之间的接口。目前，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，并通过分类产生异国情调的例子。该项目的教育组成部分包括俄亥俄州立大学的亚因子和量子对称性的本科生研究和暑期学校。该项目将将主要研究人员当前的学习材料以及这些程序开发的材料纳入有关子因子理论的书。他还将与俄亥俄州立大学的Steam Factory合作，以增强社区中的一般科学和数学素养。该项目有两个主要重点：这些新富集的量子对称性的代表性和分类。单一融合类别具有其尺寸不一定是整数的对象，因此代表单一融合类别的尺寸需要von Neumann因素，其模块具有连续尺寸的概念。在这个项目中，主要研究人员将利用他以前在小索引子因子分类中的经验来对富含小型单一色带类别的量子对称性进行分类。这将研究一个丰富的操作员代数理论，以发展丰富的亚比例理论。首席研究者还将发展双学士类别的理论，该理论是最初由于Henriques造成的von Neumann代数的更高分类类似物。这些双学位类别与保形场理论具有重要的联系，并且预计它们将成为富集量子对称性分类的重要工具。