Gauged Gromov-Witten theory and holomorphic quilts

项目负责人：Christopher woodwardpi将开展Fukaya-Floer理论中拉格朗日对应的泛函性和Gromov-Witten理论中商的泛函性的研究。第一组项目将应用于Gromov-Witten理论和“上同”镜像对称，即给定意义上的镜像对称等。与F. Ziltener和他的前博士后导师E. Gonzalez一起，他将在辛商构造下研究Gromov-Witten不变量的泛函性。潜在的应用包括：辛二元等价下gromov - witten不变量的不变性，一般型完全交的上同调镜像对称。与K. wehheim和他以前的学生S. Mau一起，PI将研究flower - fukayatheory中拉格朗日对应的泛函性。应用包括纠缠和任意三流形的非阿贝尔花同调的辛定义，Khovanov同调的可能推广和量子群的分类。这些项目将在同调镜像对称和低维拓扑中得到应用。一些项目有研究生教育的组成部分，并且PI还提出了几个本科生研究项目。总的来说，在这项资助下进行的研究将促进对辛几何的理解，辛几何是经典动力系统的数学语言，以及engage理论，表示理论和量子物理学之间的关系。量规理论自然地出现在许多物理环境中，例如电磁学。该项目的第一部分涉及某些具有附加的“非线性”场的测量理论，这些理论在经典相空间中取值，这在物理文献中已经在线性情况下进行了大量的研究，名为“测量sigmammodel”。该项目的第二部分涉及近年来在动力系统中被广泛研究的“花理论”不变量的结构性质。

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.