Higher Representation Theory and Subfactors

这个项目将涉及对量子对称性的研究。对称的概念是经典物理学的基本概念。埃米·诺特（Emmy Noether）的一个著名结果表明，对于自然定律的每一个对称，都有一个守恒的物理量。例如，物理定律的时不变性导致了能量守恒定律。在量子物理的背景下，要理解系统的行为，就需要更一般的量子对称性的概念。这个项目涉及量子对称性如何作用于某些系统的研究，最终目标是完全理解和分类这些行为。我们把这些量子对称的行为称为“高级表示理论”。特别强调将放在与拓扑量子计算相关的例子上。该项目将为新罕布什尔大学的本科生提供研究机会。从技术上讲，量子对称的概念在数学上以张量范畴为特征，量子对称的行为以这些张量范畴上的模范畴为特征。本课题将研究模块类的构建与分类的基本问题。将解决以下研究问题：1)在Wess-Zumino-Witten共形场理论的张量范畴上构造模范畴并进行分类，2)在共形场理论的张量范畴之间插入新的连续张量范畴族，3)利用Jones的图平面代数技术研究Izumi的近群张量范畴，4)研究与1)中模范畴相关的更高范畴对象。该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will involve research into quantum symmetry. The notion of symmetry is fundamental in classical physics. A famed result of Emmy Noether shows that for each symmetry of the laws of nature, there is a resulting conserved physical quantity. For example, the time invariance of the laws of physics results in the law of conservation of energy. In the setting of quantum physics, the more general notion of quantum symmetries is required to understand the behavior of the system. This project concerns the study of how quantum symmetries act on certain systems, with the end goal being to fully understand and classify these actions. We refer to these actions of quantum symmetries as `higher representation theory’. Particular emphasis will be placed on the examples which are relevant to topological quantum computation. This project will involve research opportunities for undergraduate students at the University of New Hampshire.More technically, the notion of quantum symmetry is characterized mathematically by a tensor category, and the actions of quantum symmetries are characterized by module categories over these tensor categories. This project will study fundamental problems on the construction and classification of module categories. The following research problems will be addressed: 1) construct and classify the module categories over the tensor categories coming from the Wess-Zumino-Witten conformal field theories, 2) construct new continuous families of tensor categories which interpolate between the categories coming from conformal field theories, 3) use Jones’s graph planar algebra techniques to study Izumi’s near-group tensor categories, and 4) investigate the higher categorical objects related to the module categories in 1).This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.