Mathematical Aspects of String Duality

摘要：项目负责人：刘克峰，我们计划通过局部化方法深入研究弦理论的各种猜想，特别是弦对偶性的猜想。这包括从泛函定位中得到的最一般的高格的镜像原理、gopakumar - vafa完整性猜想、椭圆格与Gromov-Witten不变量之间的关系。我们已经朝着我们的目标取得了重大进展，其中在证明镜像原理过程中发现的功能定位公式是关键技术之一。弦理论与数学的相互作用是近二十年来数学界为数不多的最活跃、最令人兴奋的领域之一。弦的对偶性创造了许多令人惊奇的美丽的数学猜想。我们的定位方法已经非常成功地证明了它们，包括镜像猜想、Marino-Vafa猜想、hori - vafa猜想和拓扑顶点的数学构造。这种相互作用不仅产生了数学理论和结果，而且为物理理论提供了有力的证据。

AbstractAward: DMS-0405117Principal Investigator: Kefeng LiuBy using localization methods we plan to intensively studyvarious conjectures from string theory, in particular thosearised from string duality. These include the most general mirrorprinciple for higher genus from functorial localization, theGopakumar-Vafa integrality conjecture, the relationship betweenelliptic genus and Gromov-Witten invariants, both fromequivariant index theory. We have made significant progressestowards our goals, for which the functorial localization formulawe discovered during our proof of the mirror principle is one ofthe key techniques.The interactions of string theory and mathematics has been one ofthe very few most active and most exciting fields in mathematicsfor the past twenty years. String duality has created manyamazingly beautiful mathematical conjectures. Our localizationmethods have been very successful in proving them, inlcuding themirror conjecture, the Marino-Vafa conjecture, the Hori-Vafaconjecture and the mathematical construction of the topologicalvertex. Such interactions not only produce mathematical theoriesand results, but also give strong evidences to the physicaltheories.