Mathematical Sciences: Geometric Structure of Conformal Field Theory

数学科学：共形场论的几何结构

• 批准号: 9301020
• 负责人: Yi-Zhi Huang
• 金额: --
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9301020&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Structure; Conformal

This research is concerned with conformal field theory and specifically with the study of vertex operator algebras. The principal investigator will study the following topics: vertex category theory; reconstruction of vertex operator algebras from vertex tensor categories; modular vertex tensor categories; vertex groups and vertex operator algebras; vertex groups and integrable systems; geometric construction of the Monster; cohomology theory for vertex operator algebras; and vertex manifolds. Conformal field theory is an important physical theory describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory also has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera, and knot theory, is revealed in the study of conformal field theory. It is believed that the study of the mathematics involved in conformal field theory will lead to new mathematical structures which will be important both in mathematics and theoretical physics.

本文主要研究共形场理论，特别是顶点算子代数的研究。主要研究者将研究以下主题：顶点范畴理论；由顶点张量范畴重构顶点算子代数；模顶点张量范畴；顶点群和顶点算子代数；顶点群和可积系统；怪物的几何构造；顶点算子代数的上同调理论；以及顶点流形。共形场理论是描述凝聚态物理中的二维临界现象和弦理论中弦的经典运动的重要物理理论。保形场理论除了在物理学中具有重要意义外，其优美而丰富的数学结构也引起了许多数学家的兴趣。在共形场理论的研究中，揭示了无穷维李代数和群的表示、Riemann曲面和代数曲线、Monster零星群、模函数和模形式、椭圆亏格和纽结理论等数学分支之间的新关系。人们相信，对保形场理论所涉及的数学的研究将导致新的数学结构，这在数学和理论物理中都将是重要的。