Vertex Operator Algebras and Mathematical Conformal Field Theory

0070800-Lepowsky/Huang研究人员研究了一系列与顶点算符代数理论和共形场理论有关的问题，以及它们与数学和物理的各个领域的关系和应用。顶点算符代数理论自然而然地产生于无限维李代数的表示理论和Monster有限单群的无穷维“月光模”的构造中，而顶点算符代数是共形场论数学构造的基本组成部分，这一理论既出现在凝聚态物理中，也出现在弦理论中。研究人员使用了代数、几何和分析的方法和思想。黄发展了保形场理论的解析和几何理论，并将所获得的结果应用于几何和拓扑学的研究。莱波斯基继续研究顶点算子、代数论和数论之间的关系以及其他话题。研究人员还研究了几何均衡化背后的代数问题。所研究的各种主题实际上是相互深刻联系的，其中某些问题的解决有望对其他问题的分析和解决有所帮助。研究人员的长期目标是促进数学理论的更深层次发展，以加强对数学许多分支之间许多已知和未知联系的概念性理解。“顶点算子代数”理论是在研究连续对称和被称为“怪物”的非常特殊的大型对称有限群时自然产生的。这一理论是许多数学分支和理论物理中广泛问题的基础。几年前，令数学家感到非常惊讶的是，怪物似乎与数论有很深的联系。同样值得注意的是，这种纯粹的数学联系，一开始只是推测，通过引入与另一种完全不同的理论有关的新数学思想而得到相当大的澄清-一种被称为“弦理论”的物理理论，其目的是统一自然界中的所有基本力，包括重力、电力、磁力和核力。这一数学进步的一个结果是顶点算子代数理论的丰富发展，这一理论继续以快速的速度找到新的令人惊讶的应用。一个非常重要的主题是，顶点算符代数是被称为共形量子场论的物理理论的数学结构的基本组成部分，这种理论既出现在固体和流体的性质研究中，也出现在弦理论中。保形场理论正在迅速发展成为一门丰富多彩的数学理论。这一发展预计将继续产生许多数学问题的解决方案，涉及对称性、几何、拓扑学、代数和数论，并将进一步应用于更深入地理解自然，并有望发展涉及固体和流体的技术。提出的项目使用了各种数学思想来加深对顶点算符代数理论和共形场理论的理解，并开发了一系列新的应用。

Abstract for 0070800 - Lepowsky / HuangThe investigators study a range of problems related to vertex operator algebra theory and conformal field theory, and their relations with and applications to a variety of areas of mathematics and physics. Vertex operator algebra theory arose naturally in the representation theory of infinite-dimensional Lie algebras and in the construction of the infinite-dimensional "moonshine module" for the Monster finite simple group, and vertex operator algebras are basic ingredients in the mathematical construction of conformal field theories, which arose in both condensed matter physics and in string theory. The investigators use algebraic, geometric and analytic methods and ideas. Huang develops an analytic and geometric theory underlying conformal field theory and applies the results obtained to the study of geometry and topology. Lepowsky continues his investigations into the relations between vertex operator algebra theory and number theory and other topics. The investigators also study algebraic problems underlying geometric uniformization. The various topics studied are in fact deeply connected with one another, and the solution of certain of these problems is expected to be useful in the analysis and solution of other problems. The long-term goal of the investigators is to contribute to the deeper development of a mathematical theory that will enhance the conceptual understanding of many known and still-unknown connections among many branches of mathematics.The theory of "vertex operator algebras" arose naturally in the study of both continuous symmetries and a very special large finite group of symmetries called the "Monster." This theory is fundamental to a wide range of problems in many branches of mathematics and in theoretical physics. Some years ago, it was very surprising to mathematicians that the Monster appeared to be deeply connected to number theory. Equally remarkably, this purely mathematical connection, at first only speculative, was considerably clarified by the introduction of new mathematical ideas related to a completely different theory---a physical theory called "string theory," which aims at unifying all the fundamental forces in nature, including gravity, electricity and magnetism, and nuclear forces. One result of this mathematical progress has been a richly developed theory of vertex operator algebras, a theory that continues to find new and surprising applications at a rapid rate. A very important theme is that vertex operator algebras are basic ingredients of the mathematical construction of physical theories called "conformal quantum field theories," which arose both in the study of the properties of solids and fluids and in string theory. Conformal field theory is in the process of being rapidly developed into a rich and beautiful mathematical theory. This development is expected to continue to yield solutions to many mathematical problems, involving symmetry, geometry, topology, algebra and number theory, and to yield further applications to the deeper understanding of nature and, it is hoped, the development of technology involving solids and fluids. The proposed project uses a variety of mathematical ideas to deepen the understanding of vertex operator algebra theory and of conformal field theory, and to develop a range of new applications.