Topics in Vertex Operator Algebras

摘要：拟研究顶点算子代数的结构与表示理论、顶点算子代数与几何的关系、数学中的数论与物理中的共形场论。他计划研究以下内容：(1)巨单群、巨月光、排列轨道和椭圆属之间的联系。目标是理解在巨大的月光中排列轨道的一种新的神秘外观，并从几何上理解月光。(2)共集构造。协集构造是由给定的共形场论构造新的共形场论的重要方法。目的是证明Schur-Weyl型对偶理论，并确定一个协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。本研究对轨道理论具有基础性的应用。(4)有限生成顶点算子代数的刻画。本文研究了具有小中心电荷的有理顶点算子代数的分类。顶点算子代数理论是一个新兴的、发展迅速的数学领域。顶点算子代数本质上是物理中的共形场论和弦论（数学物理中“万有理论”的主要候选）中的手性代数。本文的研究为共形场理论奠定了一些代数基础。本研究在群论、拓扑学、几何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