Vertex Operator Algebras, Elliptic Genus and Conformal Nets

PI计划研究顶点算子代数的结构和表示理论，顶点算子代数和几何之间的联系，以及二维共形场论的代数和分析方法之间的联系。 本文主要研究了以下几个问题：（1）月光顶点算子代数的Frenkel-Lepowsky-Meurman唯一性猜想。我们的目标是完全证明这个猜想，并得到Griess代数的一个特征（与Monster单群无关）。这是中心荷全纯顶点算子代数分类程序的一部分. (2)N=2酉顶点算子超代数的手征环（物理学家在研究超共形场论和超弦理论时定义的）和Calabi-Yau流形的椭圆亏格。目的是了解手征环在顶点算子超代数的结构和表示理论中所起的作用，并研究与Calabi-Yau流形相关联的顶点算子超代数的手征环在何种程度上决定该流形的椭圆亏格. (3)顶点算子代数（共形场论的代数方法）和共形网（共形场论的解析方法）之间的联系。这是一个长期的计划。我们的目标是从对方构造顶点算子代数和共形网。顶点算子代数理论不仅为二维量子共形场论提供了代数基础，而且与表示论、群论、模形式、拓扑变式、C*-代数等重要的数学领域有着密切的联系。计划中的研究将顶点代数算子代数与拓扑学、C*-代数和量子场论联系起来，在数学和物理学中都有重要的应用。

The PI plans to study the structure and representation theory forvertex operator algebras, the connection between vertex operatoralgebras and geometry, and the connection between algebraic andanalytic approaches to 2 dimensional conformal field theory. Inparticular, the following topics will be studied: (1) TheFrenkel-Lepowsky-Meurman's uniqueness conjecture of the moonshinevertex operator algebra. The goal is to prove the conjecture completelyand to obtain a characterization of the Griess algebra (independent ofthe monster simple group). This is a part of program of classificationof holomorphic vertex operator algebras of central charge 24. (2) Thechiral ring (which was defined by physicists in the study of superconformal field theory and super string theory) of a N=2 unitary vertexoperator superalgebra and elliptic genus of a Calabi-Yau manifold. Thepurpose is to understand the role that the chiral ring plays in thestructure and representation theory of vertex operator superalgebra,and to investigate to what extend the chiral ring of the vertexoperator superalgebra associated to a Calabi-Yau manifold determinesthe elliptic genus of the manifold. (3) The connection between vertexoperator algebra (an algebraic approach to conformal field theory) andconformal nets (an analytic approach to conformal field theory). Thisis a long term program. The goal is to construct vertex operatoralgebras and conformal nets from each other. This will lead to aunification of the algebraic and analytic approaches to 2 dimensionalconformal field theory.The theory of vertex operator algebra provides an algebraic foundationfor the 2 dimensional quantum conformal field theory in physics and isalso deeply related to many important areas of mathematics such asrepresentation theory, group theory, modular forms, topologyinvariants, and C*-algebras. The planned research links vertex algebraoperator algebras with topology, C*-algebras and quantum field theory,and has important applications in both mathematics and physics.