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

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、生物学和物理学中起着重要的作用。例如，埃米·诺特的一个定理指出，物理系统的对称性，如时间和空间平移，分别对应于守恒量，如能量和动量。冯·诺依曼在量子力学的研究中发现，希尔伯特空间中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，它的表示理论编码了量子对称性。在经典环境中，特定物体的对称性形成一个组，就像正方形或分子的对称性的集合一样。当我们从经典环境转到量子环境时，这些群被所谓的量子群和张量范畴所取代。酉张量范畴在子因子的研究中自然产生，而子因子理论则为分类和构造例子提供了丰富的技术。此外，幺正融合范畴的量子偶是幺正模范畴，这对于研究物质的拓扑相和拓扑量子计算是至关重要的。小指数子因子分类程序最近成功地分类到指数5，首席研究员将把这个指数界限提高到略高于5。为了进一步提高界限，直到六，新的技术和障碍是必要的。该项目还将开发更多的技术来研究无限指数子因子，其中结果相对较少。第二个目标是发展子因子和自由概率、C*-代数、非交换几何和共形场论（CFT）之间更深层次的联系。 最近的工作Guionnet，琼斯和Shlyakhtenko开发子因子之间的联系，随机矩阵，和自由概率。与哈特格拉斯，首席研究员开发了这种连接，发现新的连接到C*-代数和非交换几何通过工作的皮姆斯纳和Voiculescu。本项目将继续研究这些新的发展。最后，圆上的共形网与子因子和CFT密切相关。 在与Henriques和Tener的联合工作中，首席研究员将研究共形平面代数，这是Jones的子因子平面代数和零属Segal CFT的共同推广。Tener和主要研究者预计，该CFT的代表类别将按照模块类别进行分类。他们还猜想子因子/平面代数对偶扩展到共形平面代数和共形网的3-范畴中的某些态射之间的对偶。

项目成果

The Classification of Subfactors with Index at Most 5\frac{1}4

索引至多为 5frac{1}4 的子因子的分类

• DOI: 10.1090/memo/1405
• 发表时间: 2023
• 期刊: Memoirs of the American Mathematical Society
• 影响因子: 1.9
• 作者: Afzaly, Narjess;Morrison, Scott;Penneys, David
• 通讯作者: Penneys, David