TOPOLOGY OF MODULI SPACES AND REPRESENTATION THEORY

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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