Mathematical Sciences: Representation Theory of Vertex Operator Algebra

顶点算子代数及其表示理论是一门非常重要的研究课题，在数学和物理中有着广泛的应用。这个项目的第一个领域是顶点算子代数的表示理论的研究。这项研究将导致不可约扭曲模的构造，这种模在物理文献中总是被假设存在。项目的第二个领域是研究全纯顶点代数v的有限循环群G的轨道理论。项目的最后一个领域是研究moonshine模顶点算子代数，并证明了moonshine模顶点算子代数上的Frenkel-Lepowsky-Meurman猜想。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。

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.