Gromov-Witten invariants and mirror symmetry

摘要奖：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.Siebert共同开发的一个程序的角度来完成，该程序使用Strominger-Yau-Zaslow猜想的代数几何版本来“热带化”Calabi-Yau流形。特别是，首席研究人员建议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性完整概念性证明的组成部分。这些步骤包括对数Gromov-Witten不变量理论的一般粘合公式以及PI、Pandharipande和Siebert的热带顶点在所有维度的推广。此外，Siebert的程序还导致了Calabi-Yau流形(退化族)上线丛的典型altheta函数的构造。这种theta函数的存在被认为具有广泛的应用，例如K3曲面的模空间的几何紧致化(P.Hking和S.Keel正在追求)，以及热带方法的同调镜像对称性(Siebert和M正在追求)。这项建议中描述的工作是弦理论和几何学的交集。弦理论用一小圈弦取代了点粒子的传统概念，在时空中移动。为了使弦理论与量子力学兼容，时空必须是二维的。由于时空看起来是四维的，人们预计其中的六个维度会是一个非常小的“卷曲”的几何物体。这些几何对象被称为Calabi-Yaumanifold。20世纪90年代初，弦理论家提出了完全不同的Calabi-Yauman流形之间的显著联系：在一个Calabi-Yau流形上执行某些极其困难的计算，可以通过在不同的Calabi-Yau流形上执行完全不同且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，许多几何学家一直在试图理解这一奇迹观测背后的数学原理。对这一现象的新见解和新解释正在迅速发展，这些见解正在导致解决代数几何中的旧问题的新方法。这项工作旨在加深对镜面对称性的理解，并应用这一理解产生代数几何的新发展。

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.