Instantons in six dimensions and twistors

TL;DR

This paper explores the extension of twistor methods to six-dimensional conformal field theories and investigates the correspondence between instantons and holomorphic bundles, revealing limitations compared to four-dimensional cases.

Contribution

It demonstrates a partial twistor correspondence for Yang-Mills instantons in six dimensions, highlighting differences from the well-understood four-dimensional scenario.

Findings

Twistor transform in six dimensions does not fully parametrize instantons.

Existence of a correspondence between instantons on CP^3 and holomorphic bundles.

Limitations of twistor methods in higher dimensions compared to four dimensions.

Abstract

Recently, conformal field theories in six dimensions were discussed from the twistorial point of view. In particular, it was demonstrated that the twistor transform between chiral zero-rest-mass fields and cohomology classes on twistor space can be generalized from four to six dimensions. On the other hand, the possibility of generalizing the correspondence between instanton gauge fields and holomorphic bundles over twistor space is questionable. It was shown by Saemann and Wolf that holomorphic line bundles over the canonical twistor space Tw(X) (defined as a bundle of almost complex structures over the six-dimensional manifold X) correspond to pure-gauge Maxwell potentials, i.e. the twistor transform fails. On the example of X=CP^3 we show that there exists a twistor correspondence between Abelian or non-Abelian Yang-Mills instantons on CP^3 and holomorphic bundles over complex submanifolds of Tw(CP^3), but it is not so efficient as in the four-dimensional case because the twistor transform does not parametrize instantons by unconstrained holomorphic data as it does in four dimensions.