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 Sämann 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=CP3 we show that there exists a twistor correspondence between Abelian or non-Abelian Yang–Mills instantons on CP3 and holomorphic bundles over complex submanifolds of Tw(CP3), 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.
Highlights
Introduction and summaryLet us consider an oriented real four-manifold X4 with a Riemannian metric g and the principal bundle P (X4, SO(4)) of orthonormal frames over X4
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
The possibility of generalizing the correspondence between instanton gauge fields and holomorphic bundles over twistor space is questionable. It was shown by Samann and Wolf that holomorphic line bundles over the canonical twistor space Tw(X) correspond to pure-gauge Maxwell potentials, i.e. the twistor transform fails
Summary
Let us consider an oriented real four-manifold X4 with a Riemannian metric g and the principal bundle P (X4, SO(4)) of orthonormal frames over X4. Samann and Wolf have shown [6] that holomorphic line bundles over Tw(R6) trivial on all CPx3 ֒→ Tw(R6) correspond to pure-gauge Maxwell potentials on R6, i.e. the twistor transform fails for the metric twistor space Tw(R6) This was partially cured in [15] where it was shown that instantons on the six-sphere S6 = R6 ∪ {∞} correspond to complex vector bundles over the reduced twistor space Z = G2/U(2) ֒→ Tw(S6) with flat partial connections, where. The DUY equations are well defined on six-dimensional Kahler manifolds X (as well as on nearly Kahler spaces [24, 25, 26]), and their solutions are natural connections A on holomorphic vector bundles E → X [17].
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