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 X6 which is the twistor space of an oriented Riemannian manifold M4. Each solution of the HYM equations on such X6 defines a pseudo-holomorphic structure on the bundle E. It is shown that the pull-back to X6 of any anti-self-dual gauge field on M4 is a solution of the HYM equations on X6. This correspondence allows us to introduce new twistor actions for bosonic and supersymmetric Yang-Mills theories. As examples of X6 we consider homogeneous nearly Kähler and nearly Calabi-Yau manifolds which are twistor spaces of S4, CP2 and B4, 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.