Connections for vector bundles over quaternionic Kähler manifolds
For vector bundles over quaternionic Kähler manifolds, two special types of Yang-Mills connections (which the author calls \(A_ 2'\)- connections and \(B_ 2\)-connections) are introduced. These two notions generalize those of self-dual and anti-self-dual connections for Riemannian 4-manifolds. Let M be a quaternionic Kähler manifold with the associated twistor space p: \(Z\to M\) and the real structure \(\tau\). Furthermore, take a complex \(C^{\infty}\) vector bundle V over M and a Hermitian metric \(h_ V\) on V. Then the main result is the following: Theorem. Let \({\mathcal B}\) be the set of all Hermitian \(B_ 2\)-connections \(\nabla\) on \((V,h_ V)\), and \({\mathcal H}\) the set of all Hermitian (1,0)- connections on \((p^*V,p^*h_ V)\) which are flat on each fibre of p and compatible with \(\tau\). Then the mapping \({\mathcal B}\ni \nabla \mapsto p^*\nabla \in {\mathcal H}\) defines a bijection \({\mathcal B}\cong {\mathcal H}.\) This theorem is naturally regarded as a generalization of a result of Atiyah-Hitchin-Singer on the twistor theory, though the concept of \(B_ 2\)-connections had already been obtained by Salamon. Moreover, in the special case where \(M={\mathbb{P}}^ n({\mathbb{H}})\), a result of Okonek and Spindler which made an algebro-geometric study of the moduli space \({\mathcal H}\) is known.
scientific article; zbMATH DE number 4094295