Vertex Operator Algebras, Elliptic Genus and Conformal Nets

PI计划研究顶点算符代数的结构和表示理论，顶点算符代数与几何之间的联系，以及二维共形场论的代数方法和解析方法之间的联系。具体地说，我们将研究以下内容：(1)Frenkel-Lepowsky-Meurman关于顶点算子代数的唯一性猜想。其目的是完全证明这一猜想，并得到Griess代数(独立于怪物单群)的一个刻画。这是中心电荷24的全纯顶点算子代数分类程序的一部分。(2)由物理学家在研究超共形场论和超弦理论时定义的N=2酉点算符超代数和Calabi-Yau流形的椭圆亏格的手征环。目的是了解手性环在顶点算子超代数的结构和表示理论中的作用，并研究与Calabi-Yau流形相关的顶点算子超代数的手性环在多大程度上决定了流形的椭圆亏格。(3)顶点算子代数(共形场论的一种代数方法)与共形网(共形场论的一种解析方法)之间的联系。这是一个长期的计划。其目标是从彼此构造顶点算子代数和共形网。顶点算符代数理论为二维量子共形场理论提供了一个代数基础，也与表示论、群论、模形式、拓扑不变量、C*-代数等许多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和算子代数与拓扑学、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.