Quiver varieties, moduli spaces and representation theory

我与Kota Yoshioka和Lothar Gottsche一起研究了Donaldson不变量与Nekrasov配分函数的关系。此外，Takuro Mochizuki也加入了我们的团队，继续研究。Donaldson不变量被定义为4流形上的实例模空间上的自然上同调类的积分。当底层的4流形具有b_+ =1时，不变量取决于黎曼度规的选择。过墙公式给出了两个黎曼度量的唐纳森不变量的差值。当向量束的秩为2时，我们用Nekrasov配分函数来表示过壁公式。通过对模空间上的环面作用的研究，证明了这一点。本研究是2004年开始的一项研究的扩展，我们证明了同样的公式适用于任意投影曲面，而不一定是环面。我们进一步研究了k理论版本的瞬态计数。我们研究了与Seiberg-Witten曲线相关的θ函数，它是k理论的实例计数的镜像，并证明了Seiberg-Witten曲线可以从模空间的坐标环重构。这个结果是涅克拉索夫猜想的k理论推广。我们也证明了类似于上面的过壁公式，但是由于奇点的存在，我们不明白如何使用瞬时模空间来定义不变量。因此，我们将注意力限制在射影曲面的情况下，并将不变量定义为模空间的代数-几何紧化上自然线束的全纯欧拉特征。在Takuro Mochizuki的帮助下，我们研究了高阶（>2）情况，证明了过壁公式的递推表达式，并证明了它再次由Nekrasov配分函数给出。

Together with Kota Yoshioka and Lothar Gottsche I have studied relation between Donaldson invariants and Nekrasov's partition function. Further more, Takuro Mochizuki have joined our group to continue the research.Donaldson invariants are defined as integration of natural cohomology classes over moduli spaces of instantons on 4-manifolds.When the underlying 4-manifold has b_+ =1, the invariants depends on the choice of a Riemannian metric. The wall-crossing formula gives the difference of Donaldson invariants with respect to two Riemannian metrics. We express the wall-crossing formua in terms of Nekrasov's partition function, when the rank of vector bundles is 2. This was proved via a study of the torus action on the moduli space when the underlying 4-manifold is a toric surface. This research is an expansion of one started in 2004, we give the proof that the same formula holds for arbitrary projective surfaces, not necessarily toric surfaces.We further study the K-theoretic version of instanton counting. We study the theta function associated with the Seiberg-Witten curve which is a mirror of K-theoretic instanton couting and show that the Seiberg-Witten curve can be reconstructed from the coordinate ring of moduli spaces. This result is the K-theoretic generalization of the Nekrasov's conjecture. We also prove the wall-crossing formula similar to above, but we do not understand how to define the invariants using the instanton moduli spaces, due to singularieties. So we restrict our attention to the case of projective surfaces, and define invariants as holomorphic Euler characteristic of natural line bundles over the algebra-geometric compactification of the moduli spaces.With helps of Takuro Mochizuki, we study higher rank (>2) cases, and we prove a recursive expression of the wall-crossing formula, and proved that it is again given by the Nekrasov's partition function.