Vertex Operator Algebras and Their Automorphisms

主要研究者将研究顶点算子代数及其自同构群。研究全纯轨道理论和有理顶点算子代数。讨论了顶点算子代数之间的关系，顶点算子代数的自同构群，量子双精度，对偶对。确定一个有理顶点算子代数的自同构群。对某些有理顶点算子代数进行了表征和分类。顶点算子代数理论中的几个基本问题有望得到解决。这个项目的目标是为顶点算子代数及其表示的理论奠定一些进一步的基础。弦理论目前是唯一能结合所有基本相互作用的候选理论。二维共形场论是凝聚态物理中描述二维临界现象的重要物理理论，它为弦理论提供了一个框架。顶点算子代数是共形场论中的对称代数。物理理论，可能比数学理论更多，通常从特定的结构开始，称为模型。另一方面，数学理论从例子中抽象出来，预测一般情况下会发生什么。这是一项从代数领域开始的研究，但超越了这一领域，涉及到变量理论、量子群、模形式、共形和拓扑场论。对这个项目的成功研究将导致新的数学发现，这在数学和物理学中都是重要的。

The principal investigator will investigate vertex operator algebrasand their automorphism groups. Holomorphic orbifold theory, andrational vertex operator algebras will be studied. Relation amongvertex operator algebras, automorphism groups of vertex operatoralgebras, quantum doubles, dual pairs will be discussed. Theautomorphism group of a rational vertex operator algebra will bedetermined. Certain rational vertex operator algebras will becharacterized and classified. Several fundamental problems in the theory of vertex operator algebra are expected to be solved. The goal ofthis project is to lay some further foundations of the theory of vertexoperator algebras and their representations. String theory at present is the only candidate for a theory combining all the fundamental interactions. Two dimensional conformal field theorywhich is an important physical theory describing two-dimensionalcritical phenomena in condensed matter physics provides a framework for string theory. Vertex operator algebras are symmetry algebras in conformal field theory. Physical theories, perhaps more often than mathematical theories,typically start from particular structure, called the models. On theother hand, mathematical theories abstract from the examples and predictwhat happens in general. This is a research thatstarts out in the field of algebra but moves beyond that to touchinvariant theory, quantum groups, modular forms, and conformal and topological field theories. A successful study of this project will lead to new mathematical discoveries which will be important both in mathematics and physics.