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，顶点算子代数与几何之间的联系，以及二维共形场理论的代数和解析方法之间的联系。特别地，将研究以下主题：(1)moonshinevertex算子代数的frenkel - lepowsky - meurman的唯一性猜想。目标是完全证明这个猜想，并获得Griess代数的一个表征（独立于怪物单群）。这是中心电荷24的全纯顶点算子代数的分类规划的一部分。(2) N=2幺正顶点算子超代数和Calabi-Yau流形的椭圆格的手性环（由物理学家在研究超共形场论和超弦论中定义）。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，并探讨与Calabi-Yau流形相关的顶点算子超代数的手性环在何种程度上决定了流形的椭圆属。(3)顶点算子代数（共形场论的代数方法）与共形网（共形场论的解析方法）之间的联系。这是一个长期项目。目标是相互构造顶点算子代数和共形网。这将导致二维共形场论的代数和解析方法的统一。顶点算子代数理论为物理学中的二维量子共形场论提供了一个代数基础，并且与表示论、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深刻的联系。该计划的研究将顶点代数、算子代数与拓扑、C*代数和量子场论联系起来，在数学和物理上都有重要的应用。

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.