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Gauge theory on resolutions of simple singularities and simple Lie algebras. (English) Zbl 0832.58007

Let \(\Gamma\) be a non-trivial finite subgroup in SU(2). The quotient \(\mathbb{C}^2/\Gamma\) is called a simple singularity. Let \(X\) be a complex surface which is a fiber of the semiuniversal deformation of \(\mathbb{C}^2/\Gamma\) or its simultaneous non-singular resolution. By definition, \(X\) satisfies the ALE-condition if there exists a hyperkähler metric on \(X\).
The author considers the moduli space of ASD-connections on \(X\) and develops the gauge theory on it. This moduli space is discussed in detail. For instance, it is proved that the moduli space \(\mathfrak M\) of ASD connections is a hyperkähler manifold. \({\mathcal S}^1\)-actions on \(\mathfrak M\) are discussed. Some relations with irreducible representations of simple Lie algebras are established.
Proofs are outlined.

MSC:

58D27 Moduli problems for differential geometric structures
17B81 Applications of Lie (super)algebras to physics, etc.
17B20 Simple, semisimple, reductive (super)algebras
81T13 Yang-Mills and other gauge theories in quantum field theory
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