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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.

对称性出现在数学和物理科学的许多地方。通常，系统的对称性由群的数学概念来描述，群是一个具有单位和乘法的集合，其中每个运算都有逆运算。组通过结构保持映射作用于对象。近几十年来，我们已经看到量子数学对象的出现，如冯诺依曼代数和物质的拓扑相，它们自然地生活在“更高的范畴”中，因此它们的对称性可以用张量范畴更好地描述。该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。该项目为本科生和研究生提供研究培训机会。首先，它继续研究小指数子因子和幺正融合范畴，以寻找奇异量子对称性的新例子。其次，研究了一种算子代数网对（2+1）维物质拓扑相的描述，类似于共形场论的共形网描述。第三，它调查了更高类别的么正性及其与物质的（2+1）D拓扑相的边界和相变的关系。该奖项反映了NSF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。