Classifying subfactors and fusion categories

对称性在数学、生物和物理科学中扮演着重要的角色。例如，艾美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·诺伊曼在他的量子力学研究中发现，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von Neumann代数是由称为因子的基本构件构建而成的。子因子是因子的包含，其表示理论编码了量子对称性。在经典的背景下，特定物体的对称性形成一个群，就像正方形或分子的对称性的集合。当一个人从经典设置到量子设置时，这些群被所谓的量子群和张量范畴所取代。酉张量范畴在对子因子的研究中自然而然地产生，作为回报，子因子理论提供了丰富的分类和构造实例的技术。此外，酉化融合范畴的量子偶是酉模范畴，这对于物质拓扑相的研究和拓扑量子计算都是至关重要的。小指数子因素分类方案最近成功地将指数分类到指数5，首席调查员将把这个指数界限提高到略高于5的水平。为了进一步提高界限，甚至达到6个，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素，而这方面的结果相对较少。第二个目标是发展子因子与自由概率、C*-代数、非对易几何和共形场论(CFT)之间更深层次的联系。Guion net、Jones和Shlyakhtenko最近的工作发展了子因子、随机矩阵和自由概率之间的联系。与哈特格拉斯一起，首席研究员发展了这种联系，通过皮姆斯纳和沃库列斯库的工作，发现了与C*-代数和非交换几何的新联系。该项目将继续调查这些新的发展。最后，圆周上的保形网与子因子和CFT密切相关。在与Henrique和Tener的合作中，主要研究人员将研究共形平面代数，它是Jones的子因子平面代数和亏格为零的西格尔CFT的常见推广。Tener和首席调查员期望根据模块类别对此CFT的表示类别进行分类。他们还猜想，子因子/平面代数对偶可推广到共形平面代数与共形网3-范畴中的某些态射之间的对偶。

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.