Geometry of Mirror Symmetry

摘要奖：DMS-0405859首席研究员：Eric Zaslow这个项目涉及镜面对称性的几何方面。PI将研究Strominger-Yau-Zaslow猜想的几何学，方法是观察大型复杂结构极限附近的Calabi-Yaureeold猜想图片，在那里特殊的拉格朗日环面纤维坍塌。得到的流形是一个具有奇点的实整数仿射流形。第一个项目将在一个Y形奇异轨迹外的开球上产生一个实Monge-Ampere度量，然后研究它的单向性。对这个区域的了解对于获得全球对Calabi-Yau的理解，包括盘瞬子修正的作用，是至关重要的。相关的项目是研究一般的Ricci-平坦流形中平坦或Ricci-平坦环面的几何，并通过改变远离Ricci平坦流形的度量来研究特殊的拉格朗日流形和其他定标子流形的计数问题。最后一个项目是研究拓扑顶点背后的可积族。这个项目研究镜像对称物理现象背后的数学。镜像对称性是指两种不同的物理理论对自然产生相同的预测。从数学上讲，等价性背后有很深的联系，这个项目着眼于等价性的几何结构。了解镜像对称的几何学将有助于更广泛地理解数学的结构，甚至可能是物理学的结构。例如，镜面对称性有助于拓扑场理论的计算，拓扑场理论已被证明与粒子物理学基础上的传统规范理论具有计算相关性。在更基本的数学水平上，理解镜面对称的几何可以导致对1957年Calabi猜想并由Yauin 1979证明存在的特殊结构的具体描述。尽管如此，我们对几何学的这个重要方面知之甚少。数学和物理的丰富相互作用可以教会我们很多东西。

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.