Donaldson-Thomas, Gromov-Witten invariants and representation theory

本文讨论了弦理论与规范理论对偶性的数学研究。对偶的数学表现之一是由Maulik， Nekrasov， Okounkov和Pandharipande提出的猜想的Gromov-Witten/Donaldson-Thomas对偶。该猜想将三折线中稳定曲线模空间上积分的生成函数与三折线中曲线希尔伯特格式上积分的生成函数联系起来。目前，我们能够证明这个猜想的环三倍。为了证明一般三重猜想，需要解决对偶的相对版本。这是我们研究的重点。我们还希望探索Gromov-Witten/Donaldson-Thomas理论与表面上点的Hilbert格式的量子上同调之间的联系。目前，只有对简单奇点分解的曲面，才能完全理解这些关系。计算满足某些自然几何条件的曲线的数目是几何中的一个经典问题。例如，你可以问在三维空间中有多少条线与四条给定的线相交。在这种情况下，答案要么是无穷大，要么是2或1（取决于这四条线的位置）。第一个提出这类问题的人是19世纪的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观察三维空间曲线的两种方法。第一种理论将曲线视为飞行闭合弦的轨迹，第二种理论将曲线定义为两个方程的同时消失。这两种方法之间的相互作用是我们研究的中心。

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.