Donaldson-Thomas, Gromov-Witten invariants and representation theory

Donaldson-Thomas、Gromov-Witten 不变量和表示论

• 批准号: 0701367
• 负责人: Alexei Oblomkov
• 金额: $ 11.13万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701367&HistoricalAwards=false
• 关键词: Donaldson; Thomas; Gromov; Witten; invariants

The proposal is concern with the mathematical study of the duality between String Theory and Gauge theory. One of the mathematical manifrestation of the duality is the conjecural Gromov-Witten/Donaldson-Thomas duality due to Maulik, Nekrasov, Okounkov and Pandharipande. The conjecture relates the generating function for the integrals over the module space of stable curves in the thefolds to the generating function for the integrals over Hilbert scheme of curves in the threfold. At the moment we are able to prove the conjecture for the toric threefolds. To prove the conjecture for the general threefold the relative version of the duality need to be addressed. This is the focus of our research. We also hope to explore the connections between Gromov-Witten/Donaldson-Thomas theory and the quantum cohomology of the Hilbert scheme of points on the surface. At the moment these relations are fully understood only for the surfaces which are resolutions of the simple singularities. It is a classical problem in geometry to compute the number of curves satisfying some natural geometric conditions. For example, one can ask how many lines in three-dimensional space intersect four given lines. In this case the answer is either infinity, two or one (depending on the position of thefour lines). The first advances this type of problems are due to 19 centuryitalian algebraic geometers. Recent advances are motivated by the new insights coming from physics. In particular String theory and Gauge theory give two ways of looking at the curves in three-dimensional space. The first theory treats the curves as the traces of the flying closed strings, for the second theory the curve is defined by simultanious vanishing of two equations.Interplay between this two approaches is in the center of our research.

该提案是对弦理论与仪表理论之间二元性的数学研究的关注。双重性的数​​学典范之一是由于Maulik，Nekrasov，Okounkov和Pandharipande而引起的诉讼Gromov-Witten/Donaldson-Thomas二元性。该猜想将稳定曲线模块上的积分的生成函数与稳定曲线的模块空间相关联，与三分之一中曲线的Hilbert方案相比，构成曲线的生成函数。目前，我们能够证明三倍的感谢您的猜想。为了证明一般三倍的猜想，需要解决双重性的相对版本。这是我们研究的重点。我们还希望探索Gromov-Witten/Donaldson-Thomas理论与希尔伯特（Hilbert of Surface）方案的量子共同体之间的联系。目前，这些关系仅针对简单奇点的分辨率而完全理解。 在几何学中，计算满足某些自然几何条件的曲线数量是一个经典的问题。例如，一个人可以问三维空间中的几行与四个给定线相交。在这种情况下，答案是无穷大，两个或一个（取决于双线的位置）。这类问题的第一个进展是由于19世纪的代数几何图形。最近的进步是由物理学带来的新见解所激发的。特别地，弦理论和量规理论提供了两种观察三维空间中曲线的方法。第一个理论将曲线视为飞行封闭字符串的痕迹，对于第二个理论，曲线通过同时消失了两个方程式来定义。这两种方法之间的讲述是我们研究的中心。