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.

该提案涉及弦理论和规范理论之间对偶性的数学研究。对偶性的数学表现之一是由 Maulik、Nekrasov、Okounkov 和 Pandharipande 推测的 Gromov-Witten/Donaldson-Thomas 对偶性。该猜想将三重稳定曲线模空间上积分的生成函数与三重曲线希尔伯特格式积分的生成函数联系起来。现在我们已经能够证明环面三重猜想了。为了证明一般三重猜想，需要解决对偶性的相对版本。这是我们研究的重点。我们还希望探索 Gromov-Witten/Donaldson-Thomas 理论与表面点的希尔伯特方案的量子上同调之间的联系。目前，这些关系仅针对作为简单奇点的解决方案的表面而被完全理解。 计算满足某些自然几何条件的曲线的数量是几何中的一个经典问题。例如，可以询问三维空间中有多少条线与给定的四条线相交。在这种情况下，答案是无穷大、二或一（取决于四条线的位置）。此类问题的首次进展归功于 19 世纪意大利代数几何学家。最近的进展是由物理学的新见解推动的。特别是弦理论和规范理论给出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的闭合弦的轨迹，对于第二个理论，曲线是通过两个方程的同时消失来定义的。这两种方法之间的相互作用是我们研究的中心。