Vector bundles on the flag manifold and the Ward correspondence.

*(English)*Zbl 0578.53023
Geometry today, Int. Conf., Rome 1984, Prog. Math. 60, 109-119 (1985).

[For the entire collection see Zbl 0563.00006.]

This paper uses twistor methods to construct instantons on \(CP^ 2\). A prime motivation is to describe explicitly in this case the moduli space of instantons which the author constructed and analysed for a large class of 4-manifolds [J. Differ. Geom. 18, 279-315 (1983; Zbl 0507.57010)]. In that paper he showed that the moduli space for a simply connected 4- manifold with positive definite intersection form could be deformed to a 5-manifold with boundary the original manifold and with a certain number of singularities which are cones on \(CP^ 2\). In this case one expects simply a cone and that is what the formula yields. The general SU(2) instanton of charge 1 is shown to pull back from quaternionic projective space \(HP^ 2\) by the map \[ (w_ 1,w_ 2)\mapsto (w_ 1,(tjw_ 1+w_ 2)(1-t^ 2)^{-1/2}. \] When \(t=0\), this gives a reducible connection, the singular point of the moduli space and one approaches the boundary as t tends to 1.

The method used is to construct monads over the flag manifold which is the twistor space for \(CP^ 2\). A longer and more general treatment of the same problem has been given by N. P. Buchdahl [”Instantons on \(CP^ 2\)”, J. Differ. Geom. (to appear)].

This paper uses twistor methods to construct instantons on \(CP^ 2\). A prime motivation is to describe explicitly in this case the moduli space of instantons which the author constructed and analysed for a large class of 4-manifolds [J. Differ. Geom. 18, 279-315 (1983; Zbl 0507.57010)]. In that paper he showed that the moduli space for a simply connected 4- manifold with positive definite intersection form could be deformed to a 5-manifold with boundary the original manifold and with a certain number of singularities which are cones on \(CP^ 2\). In this case one expects simply a cone and that is what the formula yields. The general SU(2) instanton of charge 1 is shown to pull back from quaternionic projective space \(HP^ 2\) by the map \[ (w_ 1,w_ 2)\mapsto (w_ 1,(tjw_ 1+w_ 2)(1-t^ 2)^{-1/2}. \] When \(t=0\), this gives a reducible connection, the singular point of the moduli space and one approaches the boundary as t tends to 1.

The method used is to construct monads over the flag manifold which is the twistor space for \(CP^ 2\). A longer and more general treatment of the same problem has been given by N. P. Buchdahl [”Instantons on \(CP^ 2\)”, J. Differ. Geom. (to appear)].

Reviewer: N.Hitchin

##### MSC:

53C10 | \(G\)-structures |

53C80 | Applications of global differential geometry to the sciences |

32L25 | Twistor theory, double fibrations (complex-analytic aspects) |