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.

Abstractaward：DMS-0306299原理研究者：Eleny-Nicoleta Ionelthis提案的旨在通过结合来自不同研究领域的观念来理解Gromov-witten不变性的结构。 第一个项目是由Twophysicists，Gopakumar和Vafa做出的猜想所激发的。该猜想意味着对卡拉比（Calabi-Yau3）倍的gromov不变的限制进行了限制。目的是通过Taubes开发的自适应分析技术来证明猜想的公式，以在4维中与Theyberg-Witten和Gromov不变性相关联。 第二大项目的目标是在标记的riemannsurface上获得复杂结构的模量空间中的共同关系。 ThePI使用了上一篇论文中引入的技术，发现了几个有趣的关系家族，其中一种具有十年的Faber猜想。最后一个项目Seeksto扩展了Gromov-witten不变性的总和公式，而不是从Symplecticmum出现的总和。在一般情况下，似乎已经有几种有趣的新现象施加了。拟议的工作在于弦理论和隔核拓扑的交集。弦理论发展为宇宙统一理论的潜在范围，该理论扩展了相对论。它基于这样的想法，即应将元素颗粒（例如电子，光子）视为点，而应将其视为小的振动环。锻炼这一理论的尾能结果非常微妙，Hasin转向了许多在数学方面的显着结果。但是数学的Alsofundiental结果激发了许多物理学的新发现。希望这个项目能够归因于数学和高能物理学之间的相互作用的增加。同时，其中一个项目将研究生。