Gromov-Witten invariants and mirror symmetry

AbstractAward：DMS-1105871首席研究员：Mark Gross首席研究员计划研究Calabi-Yau流形的镜像对称几何。这将从与B共同开发的程序的角度来完成。Siebert，它“tropicalizes”Calabi-Yau流形，使用代数几何版本的Strominger-丘-Zaslow猜想。 特别是，主要研究者提出解决这个程序的一些剩余的关键步骤;这些步骤应该是亏格为零的镜像对称的完整概念证明的组成部分。这些步骤包括对数Gromov-Witten不变量理论的一般胶合公式以及PI、Pandharipande和Siebert的热带顶点到所有维度的推广。此外，Siebert的程序还导致了Calabi-Yau流形（退化族）上的线丛的正则θ函数的构造。这种theta函数的存在预期有各种应用，例如K3曲面模空间的几何紧化（由P. Hacking和S.龙骨）和热带方法同调镜像对称（正在追求与Siebert和M。在这个提议中描述的工作是弦理论和几何学的交叉点。弦理论用一个在时空中运动的弦环取代了传统的点粒子概念。 为了使弦论与量子力学相容，时空必须是一维的。既然时空看起来是四维的，那么我们可以预期其中的六个维度是一个非常小的“卷曲”的几何物体。 这些几何物体被称为Calabi-Yaumanifolds。在20世纪90年代早期，弦理论家提出了完全不同的卡-丘流形之间的显著联系：某些在一个卡-丘流形上极难执行的计算可以通过在另一个卡-丘流形上执行完全不同的、容易得多的计算来完成。这一发现被称为镜像对称。 从那时起，许多几何学家一直试图理解这一奇迹背后的数学原理。对这一现象的新的见解和解释正在迅速发展，这些见解正在导致代数几何中老问题的新方法。这项工作的目的是进一步理解镜像对称性，并将这种理解应用于代数几何的新发展。

AbstractAward: DMS-1105871Principal Investigator: Mark GrossThe principal investigator plans to study the geometry of mirrorsymmetry for Calabi-Yau manifolds. This will be done from theperspective of a program developed jointly with B. Siebert, which"tropicalizes" Calabi-Yau manifolds, using an algebro-geometricversion of the Strominger-Yau-Zaslow conjecture. In particular,the principal investigator proposes to resolve a number of theremaining key steps of this program; these steps should be thecomponent pieces of a complete conceptual proof of mirrorsymmetry at genus zero. These steps include a general gluingformula for the theory of logarithmic Gromov-Witten invariantsand a generalization of the tropical vertex of the PI,Pandharipande and Siebert to all dimensions. In addition, theprogram with Siebert has led to the construction of canonicaltheta functions for lines bundles on (degenerating families) ofCalabi-Yau manifolds. The existence of such theta functions isexpected to have various applications, such as geometriccompactifications of the moduli space of K3 surfaces (beingpursued with P. Hacking and S. Keel) and a tropical approach tohomological mirror symmetry (being pursued with Siebert andM. Abouzaid).The work described in this proposal lies at the intersection ofstring theory and geometry. String theory replaces thetraditional notion of the point particle with a small loop ofstring, moving through space-time. To make string theorycompatible with quantum mechanics, space-time must beten-dimensional. Since space-time appears four-dimensional, oneexpects six of these dimensions to be a very small "curled up"geometric object. These geometric objects are called Calabi-Yaumanifolds. In the early 1990s, string theorists proposed aremarkable association between completely different Calabi-Yaumanifolds: certain calculations extremely difficult to perform onone Calabi-Yau manifold could be completed by performingcompletely different, and much easier, calculations on adifferent Calabi-Yau manifold. This discovery was known as mirrorsymmetry. Since this time, many geometers have been trying tounderstand the mathematics behind this miraculousobservation. New insights and explanations for this phenomenonare developing rapidly, and these insights are leading to newapproaches to old problems in algebraic geometry. The proposedwork aims to both further understanding of mirror symmetry andapply this understanding to produce new developments in algebraicgeometry.