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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

我们成功地利用紧李群的表示理论系统地构造了反自对偶（ASD）联络族，这是四维球面上的ADHM构造和复射影平面上的Buchdahl瞬子构造的推广。应用降维的方法，我们可以证明在不同的基空间上的ASD连接之间存在关系。该方法有望为寻找ASD联络向量丛提供一种新的途径。我们的ASD连接家族是否完整仍然是一个重要的问题。这个问题对于ASD联络的模空间的紧化是至关重要的。我们可以成功地建立扭量截面理论，它是满足扭量方程的向量丛的截面。结果，我们在各种情况下对上述问题都得到了肯定的回答。这是因为扭量截面对应于全纯的 ...更多信息在扭量空间的第一节中，我们可以应用同调代数方法来解决我们的问题。此外，当扭量截面理论应用于紧致四元数对称空间上的齐次向量丛时，我们可以证明这两个集合之间存在一个双射。一类是扭量截面的零轨迹构成的集合，另一类是具有非平凡主迷向子群的单紧连通李群的真实的表示的集合，这些子群既不是撕裂群也不是离散群.利用扭量截面的理论，我们还可以证明具有奇异集的奇异ASD联络与具有这种联络的向量丛之间存在关系。这里，当我们使用单子理论紧化ASD联络的模空间时，奇异ASD联络自然出现。简而言之，我们可以证明，在许多情况下，由奇异ASD联络的奇异集表示的同调类具有向量丛作为Poincare对偶的特征类。在高维情形下，我们在应用同调代数方法时必然会遇到这样的困难，即我们需要考虑扭量空间上太多的层上同调群。虽然在此之前我们已经得到了层上同调群的消失定理，我们得到了更多的消失定理，这些定理可以看作是最终的版本。将层上同调群的这些广义消失定理与扭量截面理论相结合，我们可以在更多的情况下成功地构造ASD联络的模空间。到目前为止，除了我们的例子外，还没有任何系统的高维瞬子模空间的具体例子。少