Properties of Gromov-Witten Invariants

Gromov-Witten 不变量的性质

• 批准号: 0306299
• 负责人: Eleny-Nicoleta Ionel
• 金额: $ 24.21万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-15 至 2006-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306299&HistoricalAwards=false
• 关键词: Properties; Gromov; Witten; Invariants

AbstractAward: DMS-0306299Principal Investigator: Eleny-Nicoleta IonelThis proposal aims to understand the structure of theGromov-Witten invariants of symplectic manifolds by combiningtogether ideas coming from different fields of research. Thefirst project is motivated by a conjecture made by twophysicists, Gopakumar and Vafa. The conjecture implies severalsurprising restrictions on the Gromov invariants of a Calabi-Yau3-fold. The goal is to prove the conjectured formula by adaptingsome analytical techiques developed by Taubes to relate theSeiberg-Witten and Gromov invariants in 4-dimensions. The secondproject's goal is to obtain new relations in the cohomology ringof the moduli space of complex structures on a marked Riemannsurface. Using the techniques introduced in a previous paper, thePI found several families of interesting relations, one of whichproves a ten year old conjecture of Faber. The last project seeksto extend the sum formula for Gromov-Witten invariants todeformations more general then those appearing from a symplecticsum. There already seems to be several interesting new phenomenaappearing in the general case.The proposed work lies at the intersection of string theory andsymplectic topology. String theory developed as a potentialcandidate for a unifying theory of the universe, which extendsEistein relativity theory. It is based on the idea thatelementary particles (like electrons, photons) should be thoughtnot as points, but rather small vibrating loops. Working out thedetails of this theory turned out to be quite delicate, and hasin turn inspired many remarkable results in mathematics. But alsofundamental results in mathematics have inspired many newdiscoveries in physics. It is hoped that this project willcontribute to the increased interaction between mathematics andhigh energy physics. In the same time, one of the projects willinvolve graduate students.

摘要奖项：DMS-0306299 首席研究员：Eleny-Nicoleta Ionel 该提案旨在通过结合来自不同研究领域的想法来理解辛流形的 Gromov-Witten 不变量的结构。 第一个项目的灵感来自两位物理学家 Gopakumar 和 Vafa 的猜想。该猜想暗示了对 Calabi-Yau3 倍的 Gromov 不变量的几个令人惊讶的限制。目标是通过采用 Taubes 开发的一些分析技术来关联 4 维的 Seiberg-Witten 和 Gromov 不变量来证明猜想的公式。 第二个项目的目标是在标记黎曼曲面上的复结构模空间的上同调环中获得新的关系。使用之前论文中介绍的技术，PI 发现了几个有趣的关系族，其中一个证明了 Faber 十年前的猜想。最后一个项目旨在将 Gromov-Witten 不变量的求和公式扩展到比辛和中出现的变形更普遍的变形。在一般情况下似乎已经出现了一些有趣的新现象。所提出的工作位于弦理论和辛拓扑的交叉点。弦理论发展成为统一宇宙理论的潜在候选者，它扩展了爱因斯坦相对论。它基于这样的想法：基本粒子（如电子、光子）不应被视为点，而应被视为小的振动环。事实证明，研究这个理论的细节是相当微妙的，并反过来激发了数学上许多非凡的成果。但数学的基本成果也激发了物理学的许多新发现。希望该项目将有助于增加数学和高能物理之间的相互作用。同时，其中一个项目将涉及研究生。