Higher Representation Theory and Subfactors

这个项目将涉及对量子对称性的研究。对称性的概念是经典物理学的基本概念。艾美诺特的一个著名结果表明，对于自然规律的每一种对称性，都有一个由此产生的守恒物理量。例如，物理定律的时间不变性导致了能量守恒定律。在量子物理学的背景下，需要更一般的量子对称性概念来理解系统的行为。这个项目涉及研究量子对称性如何作用于某些系统，最终目标是完全理解这些行为并对其进行分类。我们把量子对称性的这些作用称为“更高表示理论”。特别强调的是与拓扑量子计算相关的例子。这个项目将为新汉普郡大学的本科生提供研究机会。更严格地说，量子对称性的概念是通过张量范畴来数学刻画的，而量子对称的作用是通过这些张量范畴上的模范畴来刻画的。这个项目将研究模块范畴的构造和分类的基本问题。1)对Wess-Zumino-Witten共形场理论中的张量范畴上的模范畴进行了构造和分类，2)构造了介于共形场理论范畴之间的张量范畴的新的连续族，3)利用Jones的图平面代数技巧研究了Izumi的近群张量范畴，和4)调查与1)中的模块类别相关的高级范畴对象。该项目由代数和数论计划和既定的激励竞争研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.