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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.

对称在数学和物理科学的许多地方都有出现。通常，系统的对称性是用群的数学概念来描述的，群是一个有单位和乘法的集合，其中每个操作都有一个逆。群通过保持结构的映射作用于对象。近几十年来，我们看到了量子数学对象的出现，比如冯·诺伊曼代数和物质的拓扑相，它们自然存在于“更高的类别”中，因此它们的对称性可以更好地用张量类别来描述。本项目旨在研究冯·诺依曼代数和物质拓扑相研究中出现的更高类别，以更好地理解这些量子系统及其更高的量子对称性。本项目为本科生和研究生提供研究训练机会。这个项目有三个重点。首先，继续研究小指数子因子和酉融合范畴，寻找奇异量子对称性的新例子。其次，它研究了一种接近物质（2+1）D拓扑相的算子代数网络，类似于共形场论的共形网络描述。第三，研究了高范畴的统一性及其与物质（2+1）D拓扑相的边界和相变的关系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。