TOPOLOGY OF MODULI SPACES AND REPRESENTATION THEORY

在投影前，我们利用紧李群的表示理论，成功地系统地构造了反自对偶(ASD)连通族，推广了四维球上的ADHM-构造和复射影平面上的Buchdahl的瞬子构造。将降维方法应用到我们的构造中，我们可以证明不同基空间上的ASD联络之间存在着一定的联系。这种方法有望给出一种寻找具有ASD连接的向量丛的新方法。我们ASD家庭的联系是否完整仍然是一个重要的问题。这个问题在压缩ASD连接的模空间时是至关重要的。我们可以成功地构造一个旋量截面理论，它是满足旋量方程的矢丛的一个截面。因此，我们在各种情况下都得到了对上述问题的肯定回答。这是因为一个扭曲截面对应于一个全纯…关于twistor空间的更多部分，我们可以应用同调代数方法来解决我们的问题。此外，当将扭曲截面理论应用于紧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之间存在双射。一个是由扭曲截面的零轨迹组成的集合，另一个是具有非平凡主迷向子群的单紧连通李群的实表示的集合，它们既不是撕裂群，也不是离散群。利用扭曲截面理论，我们还可以证明具有奇异集的奇异ASD联络和具有这种联络的向量丛之间存在着某种关系。在这里，当我们用单子理论紧致ASD联络的模空间时，一个奇异的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作为Poincare对偶的向量丛的特征类。在高维情况下，我们必然会遇到这样的困难，即在应用同调代数方法时，需要考虑twistor空间上太多的层上上同调群。虽然我们在项目之前得到了层上同调群的零化定理，但我们得到了更多的零化定理，可以作为最终的版本。将这些层上同调群的广义零化定理与扭曲截口理论相结合，可以在更多的情况下成功地构造出ASD联络的模空间。到目前为止，除了我们之外，任何系统的高维瞬时子模空间的具体例子都不能在任何地方看到。较少

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