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Hermitian-Yang-Mills equations and pseudo-holomorphic bundles on nearly Kaehler 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^6 which is the twistor space of an oriented Riemannian manifold M^4. Each solution of the HYM equations on such X^6 defines a pseudo-holomorphic structure on the bundle E. It is shown that the pull-back to X^6 of any anti-self-dual gauge field on M^4 is a solution of the HYM equations on X^6. This correspondence allows us to introduce new twistor actions for bosonic and supersymmetric Yang-Mills theories. As examples of X^6 we consider homogeneous nearly Kaehler and nearly Calabi-Yau manifolds which are twistor spaces of S^4, CP^2 and B_4, CB_2 (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.