Properties of Gromov-Witten Invariants

AbstractAward：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。 第一个项目的动机是由两位物理学家Gopakumar和Vafa提出的一个猜想。该猜想暗示了对Calabi-Yau ~ 3-fold的Gromov不变量的几个令人惊讶的限制。我们的目标是通过采用Taubes发展的一些分析技术来证明四维空间中的Seiberg-Witten和Gromov不变量的简化公式。 第二个项目的目标是获得一个标记的黎曼曲面上的复杂结构的模空间的上同调环的新关系。使用的技术介绍了在以前的文件中，PI发现了几个家庭的有趣的关系，其中之一证明了十岁的猜想的费伯。最后一个项目寻求扩展Gromov-Witten不变量的求和公式到更一般的变形，然后从symplecticsum出现。在一般情况下，似乎已经出现了几种有趣的新现象，这项工作正处于弦理论和辛拓扑学的交叉点。弦理论是作为宇宙统一理论的潜在候选者而发展起来的，它扩展了相对论的范围。它是基于这样的想法，即基本粒子（如电子，光子）不应该被认为是点，而是小的振动环。研究这个理论的细节被证明是相当微妙的，并反过来激发了数学中许多显着的成果。但数学的基本结果也启发了物理学的许多新发现。我们希望这个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。

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.