Stable bundles on hypercomplex surfaces
This article is concerned with hypercomplex manifolds, which are manifolds equipped with three complex structures \(I\), \(J\), and \(K\), which satisfy the usual quaternionic relations. The authors consider a 4-dimensional compact smooth manifold \(M\) equipped with a hypercomplex structure, and a vector bundle \(E\) over \(M\). They prove that the moduli space of anti-self-dual connections on \(E\) is hypercomplex, and admits a strong HKT metric. They also study manifolds with \((4,4)\)-supersymmetry, i.e., Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsions. Finally, they show that the moduli space of anti-self-dual connections on \(M\) is a \((4,4)\)-manifold if \(M\) is equipped with a \((4,4)\)-structure. Contents includes: Introduction, (including instanton moduli, stable holomorphic bundles, the Bismut connection, and HKT-structures); Holomorphic structures and HKT-metrics, (including \((4,4)\)-symmetry); Instanton moduli spaces, (including Hermitian-Einstein connections, stable bundles, hypercomplex structures, and HKT-metrics); and References (thirty-one items).
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