  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.
