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

TL;DR

This paper explores solutions to Hermitian-Yang-Mills equations on twistor spaces of four-manifolds, establishing a link with anti-self-dual gauge fields and introducing new twistor actions relevant for string theory compactifications.

Contribution

It demonstrates that anti-self-dual gauge fields on four-manifolds lift to solutions of HYM equations on twistor spaces, and develops new twistor actions for Yang-Mills theories on nearly Kähler and nearly Calabi-Yau manifolds.

Findings

Anti-self-dual gauge fields lift to HYM solutions on twistor spaces.

Explicit solutions of HYM equations on specific twistor spaces.

Applications to flux compactifications in heterotic string theory.

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 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.