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Quiver varieties, moduli spaces and representation theory

箭袋簇、模空间和表示理论

基本信息

批准号：
17340005
负责人：
NAKAJIMA Hiraku
金额：
$ 2.5万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17340005/
关键词：
Instanton counting Donaldson invariants wall-crossing formula prepotential

项目摘要

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.

我与Kota Yoshioka和Lothar Gottsche一起研究了Donaldson不变量与Nekrasov配分函数之间的关系。Donaldson不变量被定义为4-流形上瞬子的模空间上的自然上同调类的积分.当4-流形上的4-流形具有b_+=1时，不变量依赖于黎曼度量的选择.跨墙公式给出了Donaldson不变量关于两个黎曼度量的差值。当向量丛的秩为2时，我们用Nekrasov的配分函数来表示穿墙公式。这一点通过研究模空间上的环面作用得到了证明。这项研究是2004年开始的一个研究的扩展，我们证明了同样的公式适用于任意射影曲面，而不一定是环面。我们进一步研究了K-理论版本的瞬子计数。我们研究了与K-理论瞬子计算的镜像--Seiberg-Witten曲线有关的theta函数，证明了Seiberg-Witten曲线可以由模空间的坐标环重构。这一结果是Nekrasov猜想的K理论推广。我们也证明了类似于上面的穿墙公式，但由于奇异性，我们不理解如何用瞬子模空间来定义不变量。因此，我们将注意力局限于射影曲面的情形，并将不变量定义为模空间的代数几何紧化上的自然线丛的全纯欧拉特征。借助于Takuro Mochizuki，我们研究了高阶(&gt；2)情形，证明了跨越墙公式的递推表达式，并证明了它再次由Nekrasov配分函数给出。