Gauged Gromov-Witten theory and holomorphic quilts

计量格罗莫夫-维滕理论和全纯被子

• 批准号: 0904358
• 负责人: Christopher Woodward
• 金额: $ 40.19万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904358&HistoricalAwards=false
• 关键词: Gauged; Gromov; Witten; theory; holomorphic

AbstractAward: DMS-0904358Principal Investigator: Christopher WoodwardThe PI will carry out projects on functoriality for Lagrangiancorrespondences in Fukaya-Floer theory and functoriality for quotientsin Gromov-Witten theory. The first group of projects will haveapplications in Gromov-Witten theory and ``cohomological'' mirrorsymmetry, that is, in the sense of Givental etc. With F. Ziltener andhis former postdoctoral advisee E. Gonzalez he will investigatefunctoriality for Gromov-Witten invariants under the symplecticquotient construction. Potential applications include invariance ofGromov-Witten invariants under symplectic birational equivalence, tocohomological mirror symmetry for complete intersections of generaltype. With K. Wehrheim and his former student S. Mau the PI willstudy functoriality of Lagrangian correspondences in Floer-Fukayatheory. Applications include symplectic definitions of non-abelianFloer homology for tangles and arbitrary three-manifolds, possiblegeneralizations of Khovanov homology and categorification of quantumgroups. These projects will have applications in homological mirrorsymmetry and low-dimensional topology. Some of the projects have agraduate education component, and the PI also proposes severalundergraduate research projects.Overall the research carried out under this grant will advance theunderstanding of symplectic geometry, which is the mathematicallanguage for classical dynamical systems, and the relationship betweengauge theory, representation theory, and quantum physics. Gaugetheories arise naturally in a number of physical settings, such aselectromagnetism. The first part of the project concerns certaingauge theories with an addition "non-linear" field taking values in aclassical phase space, which have been substantially studied in thephysics literature in the linear case under the name "gauged sigmamodels". The second part of the project concerns the structuralproperties of "Floer-theoretical" invariants which have beenextensively studied in relation to dynamical systems in recent years.

AbstractAward：DMS-0904358首席研究员：Christopher Woodward PI将开展关于Lagrangiancorrespondences在Mesaya-Floer理论中的函性和Gromov-Witten理论中的函性的项目。 第一组项目将应用于Gromov-Witten理论和“上同调”镜像对称，即Givental等意义上的镜像对称。Ziltener和他的前博士后E. Gonzalez，他将研究Gromov-Witten不变量在辛商构造下的函性. 潜在的应用包括Gromov-Witten不变量在辛双有理等价下的不变性，一般类型完全相交的上同调镜像对称。 与K. Wehrheim和他以前的学生S。Mau博士将研究Floer-Escheraya理论中拉格朗日对应的泛函性。 应用包括非交换Floer同调的辛定义的缠结和任意三流形，可能的推广Khovanov同调和quantumgroups的分类。 这些项目将在同调镜像对称和低维拓扑中得到应用。 其中一些项目有本科教育部分，PI还提出了几个本科研究项目。总的来说，在该资助下进行的研究将促进对辛几何的理解，这是经典动力系统的语言，以及规范理论，表示论和量子物理之间的关系。 规范理论在许多物理环境中自然产生，例如电磁学。 该项目的第一部分涉及某些规范理论与一个额外的“非线性”字段采取的值在aclassical相空间，这已经在物理文献中大量研究的线性情况下的名称“规范sigmamodels”。 该项目的第二部分涉及的“Floer理论”不变量，近年来已被广泛研究与动力系统的结构性质。