Non-Hermitian Yang-Mills connections

Abstract

We study Yang-Mills connections on holomorphic bundles over complex Kahler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperkahler manifolds. We construct direct and inverse twistor transform from NHYM bundles on a hyperkahler manifold to holomorphic bundles over its twistor space. We study the stability and the modular properties of holomorphic bundles over twistor spaces, and prove that work of Li and Yau, giving the notion of stability for bundles over non-Kahler manifolds, can be applied to the twistors. We identify locally the following two spaces: the space of stable holomorphic bundles on a twistor space of a hyperkahler manifold and the space of rational curves in the twistor space of the ‘Mukai’ dual hyperkahler manifold.