Quantum Symmetries: Subfactors, Topological Phases, and Higher Categories

量子对称性：子因子、拓扑相和更高类别

基本信息

批准号：
2154389
负责人：
David Penneys
金额：
$ 40.27万
依托单位：
Ohio State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154389&HistoricalAwards=false
关键词：
Quantum Symmetries Subfactors Topological Phases

项目摘要

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.

对称性在数学和物理科学的许多地方出现。通常，系统的对称性是通过组的数学概念来描述的，该组是一个带有单位和乘法的集合，每个操作都有逆。小组通过具有结构的图表作用于对象上。近几十年来，我们已经看到了量子数学对象的出现，例如von Neumann代数和物质的拓扑阶段，它们自然地生活在“更高类别”中，因此它们的对称性可以通过张量类别来更好地描述。该项目旨在研究von Neumann代数和物质拓扑阶段研究中产生的更高类别，以更好地了解这些量子系统及其更高的量子对称性。该项目为本科和研究生提供了研究培训机会。该项目具有三个主要重点。首先，它继续研究小型指数子因子和单一融合类别，以搜索新的量子量子对称性示例。其次，它研究了（2+1）d物质拓扑阶段的操作员代数方法的网，类似于保形场理论的保形净描述。第三，它调查了更高类别的单位性及其与物质的拓扑阶段（2+1）d的界限和相变的关系。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的审查标准通过评估来通过评估来支持的。