Mathematical Sciences: Geometric Structure of Conformal Field Theory

本研究涉及共形场理论，特别是对顶点算子代数的研究。首席研究员将研究以下课题：顶点范畴理论；从顶点张量范畴重构顶点算子代数模顶点张量范畴；顶点群与顶点算子代数；顶点群与可积系统；怪物的几何构造；顶点算子代数的上同调理论和顶点流形。共形场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。除了在物理学上的重要性外，共形场论美丽而丰富的数学结构也引起了许多数学家的兴趣。在共形场论的研究中，揭示了无限维李代数与群的表示、黎曼曲面与代数曲线、怪物散群、模函数与模形式、椭圆属、结论等数学不同分支之间的新关系。人们相信，对共形场论的数学研究将导致新的数学结构，这在数学和理论物理中都是重要的。

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.