The principal investigator proposes to study various topics on vertex operator algebras. In particular, the following topics will be studied: (1) Induced modules: The PI will define an induced functor from the twisted module category for a vertex operator algebra to the twisted module category for a bigger vertex operator algebra. This will establish a basic tool for studying the representation theory for vertex operator algebras and lead to solving well known conjectures in orbifold conformal field theory.(2) Modular invariance: The PI expects to prove that twisted trace functions or partition functions are modular functions over congruence subgroups of the modular group for a rational vertex operator algebra. In particular, the graded dimension of any irreducible twisted module or twisted sector is a modular function. It is also expected to obtain that the sum of the squares of the absolute values of the graded dimensions of the irreducible modules is invariant under the full modular group. This will have immediate applications in classification of vertex operator algebras. (3) Classification of rational vertex operator algebras with small central charges: Rational vertex operator algebras which play the fundamental roles in rational conformal field theory form the most important class of vertex operator algebras. The PI will classify the rational vertex operator algebras with the central charge less than or equal to 1. This can be regarded as the first step in classification of rational vertex operator algebras.
Vertex operator algebra theory is a new area of mathematics. It provides an algebraic foundation for the 2 dimensional quantum conformal field theory in physics and is also deeply related to many important areas of mathematics such as representation theory, group theory, modular forms, topology invariants, and C*-algebras. The proposed research studies some fundamental problems in the field and has important applications in both mathematics and physics.
Vertex operator algebra theory is a new area of mathematics. It provides an algebraic foundation for the 2 dimensional quantum conformal field theory in physics and is also deeply related to many important areas of mathematics such as representation theory, group theory, modular forms, topology invariants, and C*-algebras. The PI studied the structure and representation theory of vertex operator algebras. Two important results on modular invariance of irreducible characters of the rational vertex operator algebras and characterizations of the rational vertex operator algebras with central charge 1 were obtained. The research activities solved some fundamental problems on the rational vertex operator algebra such. The research activities will lead to important progress in the theory of vertex operator algebra and its connections with other branches of mathematics, and have applications in conformal field theory in physics. The research activities strengthen and improve the graduate education and mathematical training of young mathematicians. Graduate students, post-doctors and underrepresented mathematicians participated in the project. The PI always encourages the participation of women and minorities in the area of Lie algebra and vertex operator algebra. The PI involves actively with organizing conferences and workshops on vertex operator algebras and related topics. The PI also gives talks in conferences, workshops and introduces mathematical results and ideas to physicists and young mathematicians.
