Grant: Discovery Development Grant

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.