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abstract: 'We consider the Hermitian-Yang-Mills (HYM) equations for gauge potentials on a complex vector bundle ${{\cal E}}$ 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 ${{\cal E}}$. 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 Kähler and nearly Calabi-Yau manifolds which are twistor spaces of $S^4$, ${\mathbb C}P^2$ and $B_4$, ${\mathbb C}B_2$ (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.'
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2.5cm

[**Hermitian-Yang-Mills equations and pseudo-holomorphic bundles\
on nearly Kähler and nearly Calabi-Yau twistor 6-manifolds**]{}

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[*Bogoliubov Laboratory of Theoretical Physics, JINR\
141980 Dubna, Moscow Region, Russia*]{}\
[Email: [popov@theor.jinr.ru]{}]{}

Introduction
============

In the recent paper [@Lust], the flux compactifications of type IIA string theory of the form AdS$_4\times X^6$ with nearly Kähler 