TOPOLOGY OF MODULI SPACES AND REPRESENTATION THEORY

模空间拓扑和表示论

基本信息

批准号：
14340025
负责人：
NAGATOMO Yasuyuki
金额：
$ 2.5万
依托单位：
Kyushu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14340025/
关键词：
moduli space vector bundle anti-self-dual connection quaternion manifold representation theory0 twistor space twistor operator symmetric space 四元数多様対

项目摘要

We succeeded systematic constructions of families of anti-self dual (ASD) connections using representation theory of compact Lie groups before the project, which is a generalization of the ADHM-construction on the 4-dimensional sphere and Buchdahl's construction of instantons on the complex projective plane. Applying a method of dimensional reduction to our constructions, we can show that there is a relation between ASD connections on different base spaces. This method is expected to give a new way of finding vector bundles with ASD connections. It remains an important question whether our families of ASD connections are complete or not. This problem would be crucial in compactifying moduli spaces of ASD connections. We can succeed to construct a theory of twistor sections which is a section of a vector bundle satisfying the twistor equation. As a result, we obtain affirmative answers to the above question in various cases. This is because a twistor section corresponds to a holomorphic … More section on the twistor space, and we can apply homological algebraic methods to our problems. Moreover, when a theory of twistor sections is applied to homogeneous vector bundles on compact quaternion symnmetric spaces, we can show that there exists a bijection between the two sets. One is a set consists of zero loci of twistor sections and the others is the set of the real representations of simple compact connected Lie groups with non-trivial principal isotropy subgroups which are neither torn nor discrete groups. Using a theory of twistor sections, we can also show that there exists a relation between a singular ASD connection with a singular set and a vector bundle with such a connection. Here, a singular ASD connection naturally appears when we compactify the moduli spaces of ASD connections using the theory of monads. In short, we can show the fact in many cases that the homology class represented by the singular set of the singular ASD connection has a characteristic lass of a vector bundle as a Poincare dual. In higher dimensional cases, we necessarily meet the difficulty such that we need to consider too many sheaf cohomology groups on the twistor spaces when applying homological algebraic methods. Though we obtained vanishing theorems of sheaf cohomology groups before the project., we got more vanishing theorems which can be regarded as final versions. Combined these generalized vanishing theorems of sheaf cohomology groups with a theory of twistor sections, we can succeed to construct moduli spaces of ASD connections in more cases. Up to now, any systematic concrete examples of moduli spaces of higher dimensional instantons can not been seen anywhere except ours. Less

在项目之前，我们使用紧凑型谎言组的代表理论对反自我双重（ASD）连接的家族进行了系统的结构，这是对4维球体上ADHM构建的概括，并且Buchdahl在复杂的投影平面上构建了instantons。将尺寸降低的方法应用于我们的构造方法，我们可以证明不同基本空间上的ASD连接之间存在关系。预计该方法将提供一种与ASD连接一起查找向量束的新方法。我们的ASD连接家属是否完整仍然是一个重要的问题。这个问题对于压实ASD连接的模量空间至关重要。我们可以成功构建一个扭曲部分的理论，这是满足旋风方程的向量束的一部分。结果，我们在各种情况下为上述问题获得了肯定的答案。这是因为扭曲器部分对应于全体形态……有关曲折空间的更多部分，我们可以将同源代数方法应用于我们的问题。此外，当将扭曲部分的理论应用于紧凑的四个症状空间上的均质矢量束时，我们可以证明两组之间存在两组之间的两组。一个是一个集合由零局部扭曲部分组成，而其他曲折部分是简单紧凑型连接的谎言组的真实表示的集合，其既不被撕裂也不是离散组的非平凡主体各向同性亚组。使用扭曲部分的理论，我们还可以证明，与单数集合的单数ASD连接与具有这种连接的矢量束之间存在关系。在这里，当我们使用MONAD理论压实ASD连接的模量空间时，自然会出现单数ASD连接。简而言之，在许多情况下，我们可以证明以奇异ASD连接为单一的同源类别的同源类具有一个矢量束的特征性下音，这是一个繁殖双重的。在较高维度的情况下，我们必须遇到困难，以便在应用同源代数方法时需要在扭转空间上考虑过多的捆捆扎组。尽管我们在项目之前获得了消失的捆捆同学组的定理。，我们得到了更多的消失定理，可以将其视为最终版本。将这些捆捆同学群体的普遍消失理论与扭曲部分理论相结合，我们可以成功地在更多情况下构建ASD连接的模量空间。到目前为止，除我们的任何地方都无法在任何地方看到任何较高instantons的模量空间的系统混凝土示例。较少的