Geometry of Mirror Symmetry

项目编号：dms -0405859首席研究员：Eric zaslow该项目研究镜面对称的几何方面。该研究所将研究斯特罗明格-尤-扎斯洛猜想的几何形状，方法是观察在大型复杂结构极限附近的卡拉比-尤斯雷褶皱的猜想图，在那里，特殊的拉格朗日环面颤振会崩溃。所得到的流形是一个具有奇异点的实整数仿射流形。第一个项目将在y形奇异轨迹外的一个开放球上产生一个真正的蒙日-安培度规，然后研究它的单性。了解这一区域对于获得对卡拉比-丘的全面了解至关重要，包括光盘瞬时校正的作用。相关课题是研究一般里奇平面流形中平面或里奇平面环面振动的几何性质，以及通过改变里奇平面度规来研究特殊拉格朗日和其他标定子流形的计数问题。最后的课题是研究拓扑顶点下的可积层次结构。该项目研究镜像对称物理现象背后的数学原理。镜像对称是指两种不同的物理理论对自然产生相同的预测。在数学上，等效性背后有很深的联系，这个项目着眼于等效性的几何结构。理解镜面对称的几何结构将导致对数学结构的更广泛的理解，并且可以想象物理。例如，镜像对称性有助于拓扑场论的计算，拓扑场论已被证明与粒子物理学基础上的传统测量理论具有计算相关性。在更基本的数学层面上，理解镜面对称的几何结构可以导致对特殊结构的具体描述，这种结构由卡拉比在1957年推测存在，并由尤恩在1979年证明存在。然而，我们对几何学的这个重要方面所知甚少。数学和物理之间丰富的相互作用可以教给我们很多东西。

AbstractAward: DMS-0405859Principal Investigator: Eric ZaslowThis project addresses geometric aspects of mirror symmetry. ThePI will study the geometry of the Strominger-Yau-Zaslowconjecture by looking at the conjectural picture of a Calabi-Yauthreefold near the large complex structure limit, where thespecial Lagrangian torus fibration collapses. The resultingmanifold is a real integer affine manifold with singularities.The first project will produce a real Monge-Ampere metric on anopen ball outside a Y-shaped singular locus, then study itsmonodromy. A knowledge of this region is essential to gaining aglobal understanding of the Calabi-Yau, including the role ofdisc instanton corrections. Related projects are to study thegeometry of flat or Ricci-flat torus fibrations in Ricci-flatmanifolds in general, and to study the counting problem ofspecial Lagrangian and other calibrated submanifolds by varyingthe metric away from the Ricci flat one. The final project isinvestigate integrable hierarchies underlying the topologicalvertex.The project studies the mathematics behind the physicalphenomenon of mirror symmetry. Mirror symmetry is when twodifferent physical theories give rise to the same predictionsabout nature. Mathematically, there are deep connections behindthe equivalence, and this project looks at the geometry of theequivalence. Understanding the geometry of mirror symmetry willlead to a broader comprehension of the structure of mathematics,and conceivably physics. For instance, mirror symmetry aids inthe calculation of topological field theories, which have beenshown to have computational relevance to the conventional gaugetheories that underly particle physics. At a more basic,mathematical level, understanding the geometry of mirror symmetrycan lead to a concrete description of the special structuresconjectured by Calabi to exist in 1957 and proved to exist by Yauin 1979. Still, we know very little about this important aspectof geometry. The rich interplay of mathematics and physics hasmuch to teach us.