Three projects are being proposed. In the first part the PI will pursue a new direction in representation theory of vertex algebras, by studying (new) families of vertex algebras of finite representation type, which involve both indecomposable and logarithmic modules. It is expected that categories of modules obtained in this way are equivalent to categories of modules for certain quantum (super)groups at root of unity. In addition, an extension of this project to the setup of vertex superalgebras will be obtained. The structures investigated in this part are basic ingredients for building logarithmic conformal field theories, increasingly popular among physicists. The second part involves studies of characters of vertex algebra modules, closely related pseudocharacters and modular differential equations. From these considerations the PI proposes a whole array of results of interest to number theorists, including constant term identities and modular identities. Finally, the PI, jointly with his collaborators, will continue to work on combinatorial aspects of vertex algebra theory. In particular, considerable attention will be devoted to "principal subspaces" of standard modules for affine Lie algebras, by using primarily the theory of intertwining operators. It is expected that this will lead to new combinatorial bases of standard modules.
Conformal field theory and string theory have had major impact on modern mathematics. Two-dimensional conformal field theory has also important applications in condensed mater physics and statistical mechanics. The aim of this research is to use symmetries in physical theories (through the language of representation theory) to study analytic and combinatorial properties of correlation functions and partition functions. This will, on one hand, advance our understanding of processes in nature and, on the other, lead to new results and structures in mathematics.
