A vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and string theory in physics. In addition to physical applications, vertex operator algebras have connections with many branches of mathematics. Rational vertex operator algebras, which are closely related to classical objects of symmetry known as Lie algebras, form the most important class of vertex operator algebras. The proposed research will solve some fundamental problems concerning rational vertex operator algebras. The research will lead to important progress in the theory of vertex operator algebras and its connections with other branches of mathematics and should have applications in conformal field theory in physics.
This proposal deals with various topics on rational vertex operator algebras. The proposal consists of four parts. The first part is dedicated to the study of quantum dimensions. The quantum dimensions, which are analogues of dimensions of vector spaces, are important invariants of vertex operator algebras. The PI will use the quantum dimensions and global dimensions to study orbifold theory and classify the irreducible modules for orbifold vertex operator algebras. The quantum dimensions will also be used to give a characterization of rational vertex operator algebras. The second part of the proposal investigates the relation between the modularity of trace functions and rationality for vertex operator algebras. The PI will establish that a vertex operator algebra is rational if and only if the q-characters of the irreducible modules are modular functions. The third part on mirror extensions suggests a new way to construct vertex operator algebras which are not simple current extensions in general. Finally, in the fourth part the PI will give characterizations of the E-series of the rational vertex operator algebras with central charge 1 and complete the classification of rational vertex operator algebras with central charge 1.
