Hermitian-Yang-Mills equations and pseudo-holomorphic bundles on nearly Kähler and nearly Calabi-Yau twistor 6-manifolds

We consider the Hermitian-Yang-Mills (HYM) equations for gauge potentials on a complex vector bundle E over an almost complex manifold X which is the twistor space of an oriented Riemannian manifold M. Each solution of the HYM equations on such X defines a pseudoholomorphic structure on the bundle E . It is shown that the pull-back to X of any anti-selfdual gauge field on M is a solution of the HYM equations on X. This correspondence allows us to introduce new twistor actions for bosonic and supersymmetric Yang-Mills theories. As examples of X we consider homogeneous nearly Kähler and nearly Calabi-Yau manifolds which are twistor spaces of S, CP 2 andB4, CB2 (real 4-ball and complex 2-ball), respectively. Various explicit examples of solutions to the HYM equations on these spaces are provided. Applications in flux compactifications of heterotic strings are briefly discussed.