Mathematical Sciences: Representation Theory of Vertex Operator Algebra

顶点算子代数理论及其 表示法是一个非常重要的研究课题， 在数学和物理学中的许多应用。 的第一区域 本项目是关于顶点表示理论的研究。 算子代数 这项研究将导致建设 不可约扭模，总是假设存在于 物理学文献。 该项目的第二个领域是 有限循环轨道理论的研究 群G对于全纯顶点代数V。 项目是研究月光模顶点算子代数 并证明了Frenkel-Lepowsky-Meurman猜想， 月光模顶点算子代数 共形场论是一种重要的物理理论 描述了凝聚态中的二维临界现象 物质物理学和弦理论中弦的经典运动。 除了在物理学上的重要性， 共形场论的数学结构也有 吸引了许多数学家。 不同国家之间的新关系 数学的分支，例如 无限维李代数和群，黎曼曲面 和代数曲线，Monster零星群，模 函数和模形式，椭圆属，纽结理论，是 在共形场论的研究中发现的。 据信 对共形场的数学研究 理论将导致新的数学结构， 在数学和理论物理学中都很重要。

The theory of vertex operator algebras and their representations is a very important subject of study, which has many applications in mathematics and physics. The first area of this project is the study of the representation theory of vertex operator algebras. This study will lead to the construction of irreducible twisted modules which are always assumed to exist in the physics literature. The second area of the project is the study of the orbifold theory associated with a finite cyclic group G for a holomorphic vertex algebra V. The last area of the project is to study the moonshine module vertex operator algebra and to prove the Frenkel-Lepowsky-Meurman conjecture on the moonshine module vertex operator algebra. 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.