Classifying subfactors and fusion categories

对子因素和融合类别进行分类

• 批准号: 1500387
• 负责人: David Penneys
• 金额: $ 14.48万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2016-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500387&HistoricalAwards=false
• 关键词: Classifying; subfactors; fusion; categories; Classifying; subfactors; fusion; categories

Symmetry plays an important role in mathematics and in the biological and physical sciences. For example, a theorem of Emmy Noether states that symmetries of physical systems, like time and space translation, correspond to conserved quantities, like energy and momentum, respectively. Von Neumann, in his study of quantum mechanics, discovered that certain operator algebras on Hilbert space describe symmetries of quantum systems. These von Neumann algebras are built from basic building blocks called factors. A subfactor is an inclusion of factors, and its representation theory encodes quantum symmetries. In the classical setting, the symmetries of a particular object form a group, like the collection of symmetries of a square or of a molecule. When one passes from the classical setting to the quantum setting, these groups are replaced by so-called quantum groups and tensor categories. Unitary tensor categories arise naturally in the study of subfactors, and in return, subfactor theory provides a wealth of techniques for classification and construction of examples. Moreover, the quantum doubles of unitary fusion categories are unitary modular categories, which are vital to research in topological phases of matter and topological quantum computation.The first aim of this project is the classification of subfactors and fusion categories. The small index subfactor classification program has seen recent success classifying up to index five, and the principal investigator will raise this index bound slightly above five. To raise the bound even further, up towards six, new techniques and obstructions are necessary. The project will also develop more techniques for studying infinite index subfactors, where there are relatively few results. The second aim is developing deeper connections between subfactors and free probability, C*-algebras, noncommutative geometry, and conformal field theory (CFT). Recent work of Guionnet, Jones, and Shlyakhtenko developed a connection between subfactors, random matrices, and free probability. With Hartglass, the principal investigator developed this connection, discovering new connections to C*-algebras and noncommutative geometry via work of Pimsner and Voiculescu. The project will continue to investigate these new developments. Finally, conformal nets on the circle intimately relate subfactors and CFT. In joint work with Henriques and Tener, the principal investigator will study conformal planar algebras, which are a common generalization of Jones's subfactor planar algebras and genus-zero Segal CFT. Tener and the principal investigator anticipate a classification in terms of module categories for the representation category of this CFT. They also conjecture the subfactor/planar algebra duality extends to a duality between conformal planar algebras and certain morphisms in the 3-category of conformal nets.

对称性在数学以及生物学和物理科学中起着重要作用。例如，艾米（Emmy）noether的定理表明，如时间和空间翻译（如时间和空间平移）的对称性分别对应于能量和动量等保守数量。冯·诺伊曼（Von Neumann）在量子力学的研究中发现，希尔伯特空间上的某些操作员代数描述了量子系统的对称性。这些von Neumann代数是由称为因素的基本构建块构建的。子因子是因素的包含，其表示理论编码量子对称性。在经典环境中，特定对象的对称性形成一个组，例如平方或分子的对称性的集合。当一个人从经典设置传递到量子设置时，这些组将被所谓的量子组和张量类别取代。统一张量类别自然出现在亚比例的研究中，作为回报，亚比例理论为示例分类和构建提供了丰富的技术。此外，统一融合类别的量子双重类别是单一模块化类别，对于在物质和拓扑量子计算的拓扑阶段进行研究至关重要。该项目的第一个目的是分类子因子和融合类别。小型指数子因子分类计划已将最近的成功分类为索引五，主要研究者将提出该指数略高于五的指数。为了进一步提高束缚，必须采取新的技术和障碍物。该项目还将开发更多用于研究无限指数子因子的技术，而结果相对较少。第二个目的是在子因子与自由概率，C* - 代数，非交通性几何学和保形场理论（CFT）之间建立更深的联系。 Guionnet，Jones和Shlyakhtenko的最新工作在子因子，随机矩阵和自由概率之间建立了联系。借助Hartglass，首席研究人员建立了这种联系，发现了Pimsner和Voiculescu的工作发现与C* - 代数和非共同几何形状的新连接。该项目将继续调查这些新发展。最后，圆上的保形网紧密地关联了子因子和CFT。 在与Henriques和Tener的联合合作中，主要研究者将研究共形平面代数，这是对Jones的亚比例平面代数和Zer-Zero-Zero Segal CFT的普遍概括。 Tener和主要研究人员预计该CFT的表示类别的模块类别将进行分类。他们还猜想了亚比例/平面代数二元性延伸至保形平面代数与三类结合网中的某些形态之间的双重性。