Quantum Symmetries: Subfactors, Topological Phases, and Higher Categories

量子对称性：子因子、拓扑相和更高类别

• 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

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。

项目成果

Composing topological domain walls and anyon mobility

• DOI: 10.21468/scipostphys.15.3.076
• 发表时间: 2022-08
• 期刊: SciPost Physics
• 影响因子: 5.5
• 作者: Peter Huston;F. Burnell;Corey Jones;David Penneys

A lattice model for condensation in Levin-Wen systems

• DOI: 10.1007/jhep09(2023)055
• 发表时间: 2023-03
• 期刊: Journal of High Energy Physics
• 影响因子: 5.4
• 作者: Jessica M. Christian;David Green;Peter Huston;David Penneys

An algebraic quantum field theoretic approach to toric code with gapped boundary

• DOI: 10.1063/5.0149891
• 发表时间: 2022-12
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Daniel Wallick