Classifying subfactors and fusion categories

对称在数学、生物和物理科学中扮演着重要的角色。例如，埃米·诺特的一个定理指出，物理系统的对称性，如时间和空间的平移，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。这些冯·诺伊曼代数是由称为因子的基本构件组成的。子因子是因子的包含，它的表示理论编码量子对称性。在经典环境中，一个特定物体的对称性形成一个群，就像一个正方形或一个分子的对称性的集合。当一个人从经典环境过渡到量子环境时，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，反过来，子因子理论为分类和构造示例提供了丰富的技术。此外，统一融合范畴的量子双元是统一模范畴，这对物质的拓扑相和拓扑量子计算的研究至关重要。该项目的第一个目标是子因子和融合类别的分类。小指数子因子分类程序最近已经成功地分类到指数5，首席研究员将把这个指数范围稍微提高到5以上。为了进一步提高上限，达到6，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，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、非交换几何和共形场论（CFT）之间的更深层次的联系。Guionnet、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Hartglass一起，首席研究员通过Pimsner和Voiculescu的工作发现了C*代数和非交换几何之间的新联系。该项目将继续调查这些新的发展。最后，圆上的共形网将子因子与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属零Segal CFT的共同推广。Tener和主要研究者期望在模块类别方面对该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与共形网3类中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.