Modular Data and Vector-Valued Modular Forms

我的研究涉及代数及其与数学物理的联系。一个共同的主题是模数据的研究——有理共形场理论（RCFTs），包括Wess-Zumino-Witten （WZW）模型、有限群和顶点算子代数（VOAs）。共形场论（CFT）是一种高度对称的量子场论，其中RCFT表现得尤为优异。一个CFT由两个voa组成，它们是由量子场自然形成的结构。CFT是1953年由蒂林首次提出的，但这个概念真正被接受是在80年代，在90年代，几乎一半的菲尔德奖都是用CFT颁发的，分别是德林菲尔德、琼斯、威滕、博尔切兹和康茨维奇。受CFT影响的数学领域包括几何、拓扑和代数，最引人注目的是Borcherds对VOA的定义。目前，CFT和voa都是数学和物理学的主要研究领域。每个RCFT都有一对矩阵（S和T）与之相关联。它们被称为模块化数据。s矩阵更为重要，它是物理学家进行测量的机制。RCFTs的丰富来源是WZW模型，它与代数有着密切的联系。最近，我一直在研究WZW模型的模块化数据，找到s矩阵的简化，称为不动点分解（FPF），并将其用于计算弦理论中的电荷，弦理论是量子引力的一种物理理论。有这样的简化是很重要的，因为不动点在数学和物理中都因引起复杂性而臭名昭著（例如，在cft的分类中）。在接下来的五年里，我计划继续研究WZW模型，但我也计划扩展到两个相关的方向。第一个新方向是将FPF放入数学和物理环境中，并确定这种简化是否存在于WZW模型之外。这可能表明了RCFTs的一种新的深层性质。第二个新方向是与阿尔伯塔大学（University of Alberta）合作，将涉及向量值模块形式，这是在VOAs环境中自然出现的形式。我计划让研究生和本科生都参与我的工作。通过这项研究，我计划大大推进我的长期目标，即理解各种数学和物理对象的模块化数据，从而更好地了解它们之间的联系。此外，这些知识将直接影响弦理论的研究，因此可以更好地理解我们宇宙的基本结构。

My research involves algebra and its connections with mathematical physics. A common theme is the study of modular data -- of rational conformal field theories (RCFTs), including Wess-Zumino-Witten (WZW) models, finite groups, and vertex operator algebras (VOAs). A conformal field theory (CFT) is a highly symmetric type of quantum field theory, and an RCFT is especially well-behaved. A CFT consists of two VOAs, which are the structures naturally formed by quantum fields. The first occurrence of CFT was in 1953 by Thirring, but the idea truly gained acceptance in the 80's, and in the 90's, nearly half of all Field's Medals awarded were in CFT, to Drinfel'd, Jones, Witten, Borcherds, and Kontsevich. Areas of mathematics impacted by CFT include geometry, topology and algebra, most notably the definition of a VOA due to Borcherds. Today, both CFT and VOAs are major research areas in mathematics and physics. Every RCFT has a pair of matrices (S and T) associated with it. They are called modular data. The S-matrix is more important and is the mechanism through which physicists can take measurements. A rich source of RCFTs are WZW models, which have close connections with algebra. Recently, I have been working with modular data of WZW models, finding a simplification of the S-matrix, called fixed point factorization (FPF) and using this towards computing charges in string theory, which is a physical theory of quantum gravity. It is important to have such simplifications as fixed points are notorious for causing complications in both mathematics and physics (for example, in the classification of CFTs). Over the next five years, I plan to continue working with the WZW models, but I also plan to expand into two related directions. The first new direction is putting FPF into mathematical and physical context and determining whether such simplifications exist outside of the WZW models. This may indicate a new deep property of RCFTs. The second new direction is joint work with the University of Alberta, and will involve working with vector-valued modular forms, which are forms that occur naturally in the context of VOAs. I plan to involve both graduate and undergraduate students in my work. With this research, I plan to significantly advance my long-term goal of understanding modular data of various mathematical and physical objects and thereby contribute to a better knowledge of the connections between them.  In addition, this knowledge would directly impact research in string theory and therefore could lead to a better understanding of the fundamental structure of our universe.