Classifying subfactors and fusion categories

对子因素和融合类别进行分类

基本信息

批准号：
1655912
负责人：
David Penneys
金额：
$ 9.69万
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：
2016
资助国家：
美国
起止时间：
2016-07-01 至 2018-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1655912&HistoricalAwards=false
关键词：
Classifying subfactors fusion categories

项目摘要

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和物理科学中发挥着重要作用。例如，艾美·诺特的定理指出，物理系统的对称性（例如时间和空间平移）分别对应于守恒量（例如能量和动量）。冯·诺依曼在研究量子力学时发现，希尔伯特空间上的某些算子代数描述了量子系统的对称性。这些冯诺依曼代数是由称为因子的基本构建块构建的。子因子是因子的包含体，其表示理论编码量子对称性。在经典设置中，特定对象的对称性形成一个群，就像正方形或分子的对称性集合一样。当人们从经典设置过渡到量子设置时，这些群被所谓的量子群和张量类别所取代。酉张量类别在子因子的研究中自然出现，作为回报，子因子理论提供了丰富的分类和构造示例的技术。此外，酉融合范畴的量子双体是酉模范畴，这对于物质拓扑相和拓扑量子计算的研究至关重要。本项目的首要目标是子因子和融合范畴的分类。小指数子因子分类计划最近已成功分类到指数 5，主要研究者将把这个指数范围提高到略高于 5。为了进一步提高界限，达到六级，需要新的技术和障碍。该项目还将开发更多研究无限指数子因素的技术，但目前成果相对较少。第二个目标是在子因子和自由概率、C* 代数、非交换几何和共形场论 (CFT) 之间建立更深入的联系。 Guiionnet、Jones 和 Shlyakhtenko 最近的工作开发了子因子、随机矩阵和自由概率之间的联系。首席研究员与 Hartglass 一起发展了这种联系，通过 Pimsner 和 Voiculescu 的工作发现了 C* 代数和非交换几何的新联系。该项目将继续研究这些新进展。最后，圆上的共形网络与子因素和 CFT 密切相关。首席研究员将与 Henriques 和 Tener 合作研究共形平面代数，这是 Jones 次因子平面代数和属零 Segal CFT 的常见推广。 Tener 和主要研究者预计会根据模块类别对该 CFT 的表示类别进行分类。他们还推测子因子/平面代数对偶性扩展到共形平面代数和共形网络 3 类中的某些态射之间的对偶性。