Donaldson-Thomas, Gromov-Witten invariants and representation theory

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

• 批准号: 1042567
• 负责人: Alexei Oblomkov
• 金额: $ 2.63万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-01-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1042567&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 threefolds to the generating function for the integrals over Hilbert scheme of curves in the threefold. 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 the four 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 simultaneous vanishing of two equations.Interplay between this two approaches is in the center of our research.

该提议涉及到弦理论和规范理论之间的对偶性的数学研究。由Maulik、Nekrasov、Okounkov和Pandharipande提出的Gromov-Witten/Donaldson-Thomas对偶的数学表述之一是猜想的Gromov-Witten/Donaldson-Thomas对偶。该猜想将三重稳定曲线模空间上积分的母函数与三重Hilbert格式上积分的母函数联系起来。目前，我们能够证明三重环的猜想。为了证明关于一般三重的猜想，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理论与曲面上点的Hilbert方案的量子上同调之间的联系。目前，这些关系只对作为简单奇点的解的曲面有充分的理解。计算满足某些自然几何条件的曲线个数是一个经典的几何问题。例如，有人可以问三维空间中有多少条线与四条给定线相交。在这种情况下，答案要么是无穷大，要么是2，要么是1(取决于四条线的位置)。最早提出这类问题的是19世纪的代数几何学家。最近的进展是由来自物理学的新见解推动的。特别值得一提的是，弦理论和规范理论给出了在三维空间中观察曲线的两种方法。第一种理论将曲线视为飞行的闭弦的迹，第二种理论将曲线定义为两个方程同时消失，这两种方法之间的相互作用是我们研究的中心。