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

Understand

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.