Topics in Vertex Operator Algebras

主要研究者：董崇英提案编号：DMS -0245548机构：加州大学圣克鲁斯分校摘要：主要研究者提出研究顶点算子代数的结构和表示理论，顶点算子代数与几何的联系，数学中的数论和物理学中的共形场论.他计划研究以下内容：（1）怪物单群、怪物月光、置换orbifolds和椭圆属之间的联系。我们的目标是了解一个新的神秘外观的排列orbifolds在可怕的月光和理解月光几何。(2)陪集构造。陪集构造是从一个已知的共形场论出发构造新的共形场论的重要途径。目的是证明Schur-Weyl型对偶理论，并确定陪集顶点算子代数的模范畴。(3)顶点算子代数的子代数。目标是理解顶点算子代数的合理性。这项研究对轨道理论有基本的应用。(4)某些非线性生成顶点算子代数的刻画。研究了中心荷小的有理顶点算子代数的分类问题。顶点算子代数理论是一个新兴的、发展非常迅速的数学领域。顶点算子代数本质上是物理学中共形场论和弦论（数学物理学中“万有理论”的主要候选者）中的手征代数。这些研究为共形场论奠定了一定的代数基础。该研究在群论、拓扑学、几何学等数学领域也有广泛的应用

Principal Investigator: Chongying Dong Proposal Number: DMS - 0245548Institution: University of California-Santa CruzAbstract:The principal investigator proposes to study the structure and representation theory of vertex operator algebras, connections between vertex operator algebras and geometry, number theory in mathematics and conformal field theory in physics. He plans to investigate the following: (1) Connections among the monster simple group, the monstrous moonshine, permutation orbifolds and elliptic genera. The goal is to understand a new mysterious appearance of the permutation orbifolds in the monstrous moonshine and to understand the moonshine geometrically. (2) Coset construction. The coset construction is an important way to construct new conformal field theory from a given one. The goal is to prove a duality theory of Schur-Weyl type and determine the module category of a coset vertex operator algebra. (3) Subalgebras of vertex operator algebras. The goal is to understand the rationality of vertex operator algebras. This research has fundamental applications to orbifold theory. (4) Characterizations of certain finitely generated vertex operator algebras. This research leads to the classification of rational vertex operator algebras with small central charges. The theory of vertex operator algebra is a new and very rapidly developing area of mathematics. The vertex operator algebra is essentially the chiral algebra in conformal field theory and string theory (which is the leading candidate for the "theory of everything" in mathematical physics) in physics. The proposed research lays some algebraic foundations of conformal field theory. The proposed research also has a lot of applications in many areas of mathematics, such as group theory, topology and geometr