The Gromov-Witten invariants of curves, surfaces, and 3-folds

摘要奖：DMS-0072492主要研究者：Jim A. Bryan这个项目主要关注Gromov-Witteninvariants。 研究者将通过确定X中适当刚性曲线C的局部贡献来研究3重X的Gromov-Witten不变量。特别是，该计算器将研究这些贡献的完整性特性以及它们与M理论中某些BPS态的数量的关系，这些BPS态是通过Gopakumar和Vafa公式定义的。对于节点曲线C，研究者将继续他与Katzand Leung的工作;对于光滑的高亏格曲线C，研究者将继续他与R开始的工作。Pandharipande在与Leung的合作中，研究人员还将寻求定义辛4-流形的一个新的变形，该变形将专门针对Behrend和Fantech为具有正几何亏格的代数曲面定义的修改的Gromov-Witten不变量（这反过来又推广了研究人员和Leung为K3和Abel曲面定义的修改的不变量）。这样的不变量比普通的Gromov-Witten不变量更适合于研究无理面的计数几何。在弦论物理学中，粒子被一维物体（“弦”）所取代，因此物体随时间的轨迹就变成了时空中的一个面（“世界面”）。弦理论的方程告诉我们，这个表面应该是一个以全纯方式映射到时空中的全纯表面（黎曼表面）。这导致了纯数学概念的Gromov-Witteninvariants。Gromov-Witten不变量研究黎曼曲面到高维几何对象（例如，射影流形）的全纯映射。它们已经成为几何学、拓扑学和代数几何学中的重要不变式，也是弦理论的中心。Bryan的项目具体解决了投射流形的代数几何如何编码在Gromov-Witten不变量中以及它们如何与弦理论联系在一起的问题。 代数几何是纯数学中的一门经典学科，最近在机器人学和编码理论等不同学科中得到了应用;弦理论是“万物理论”的主要候选者，即描述所有已知物理现象的单一物理理论。

AbstractAward: DMS-0072492Principal Investigator: Jim A. BryanThis project is primarily concerned with Gromov-Witteninvariants. The investigator will study the Gromov-Witteninvariants of a 3-fold X by determining the local contributionsof a suitably rigid curve C in X. In particular, the investigatorwill study the integrality properties of such contributions andtheir relationship to the number of certain BPS states inM-theory as defined via the formula of Gopakumar and Vafa. Fornodal curves C, the investigator will continue his work with Katzand Leung; for smooth higher genus curves C, the investigatorwill continue work begun with R. Pandharipande. In collaborationswith Leung, the investigator will also seek to define a newinvariant of symplectic 4-manifolds which would specialize to themodified Gromov-Witten invariants defined by Behrend and Fantechifor algebraic surfaces with positive geometric genus (which inturn generalized the modified invariants defined by theinvestigator and Leung for K3 and Abelian surfaces). Such aninvariant would be better suited to study the enumerativegeometry of irrational surfaces than the ordinary Gromov-Witteninvariants.In the physics of string theory, particles are replaced withone-dimensional objects (``strings'') and so the path that aparticle traces out over time becomes a surface in space-time (a``world-sheet''). The equations of string theory then tell usthat the surface should be a holomorphic surface (a Riemannsurface) mapped into space-time in a holomorphic manner. This hasled to the purely mathematical notion of Gromov-Witteninvariants. Gromov-Witten invariants study holomorphic mappingsof Riemann surfaces into higher dimensional geometric objects(for example, projective manifolds). They have become importantinvariants in geometry, topology, and algebraic geometry as wellas being central in string theory. Bryan's project specificallyaddresses the problem of how the algebraic geometry of aprojective manifold is encoded in the Gromov-Witten invariantsand how they are tied to string theory. Algebraic geometry is aclassical subject in pure mathematics that has recently foundapplication in such diverse subjects as robotics and codingtheory; string theory is the leading candidate for a "theory ofeverything", i.e. a single physical theory that describes allknown physical phenomenon.