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

Abstract

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 Kähler and nearly Calabi–Yau manifolds which are twistor spaces of S 4 , C P 2 and B 4 , 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.