Abstract:
I would not really like to tell you the meaning of the two rational numbers in the title, I would rather like to keep them mysterious until the occasion of my talk. However, I must give some ingredients of my talk, otherwise you might not come to the seminar, maybe.
One of the main ingredients is conformal field theories. As you may know, conformal field theories involve many mathematical aspects, for instance, vertex operator algebras, moduli spaces of algebraic curves, modular forms, linear differential equations with regular singularities, and so on. In my research on classifying rational conformal field theories, these two numbers appeared naturally. On my way to deal with these numbers, I needed the notion of modular tensor categories which include the concepts of quantum dimensions and global dimensions.
What I did, and will explain in this talk, is that I first find solutions of lin- ear differential equations with regular singularities whose spaces of solutions are invariant under the slash 0 action, and get the exact form of S-matrices corresponding to the S-transformation. Then the theory of modular tensor categories allows us to compute quantum dimensions and then global dimen- sions. By contrast, the theory of vertex operator algebra gives us different global dimensions, which means such conformal field theories do not exist.
Because I use many mathematical tools including modular forms on the congruence groups (which you know very well), I decided to use slides to give you enough information and make the talk self-contained. In fact, every notion which appears in my talk will be explained. A prior knowledge is not required.