Instantons on \({\mathbb{C}}{\mathbb{P}}_ 2\). (English) Zbl 0586.32034

Buchdahl, N. P.

J. Differ. Geom. 24, 19-52 (1986).

For the groups \(G=SU(n)\), Sp(n), SO(n) and U(n) all G-instantons on \({\mathbb{C}}{\mathbb{P}}_ 2\) are described and classified. The description is an analogue of the Atiyah-Drinfeld-Hitchin-Manin construction for instantons on \(S^ 4\), utilizing the one-to-one correspondence between instantons and holomorphic vector bundles on the associated twistor space together with techniques from the classification theory of stable bundles on projective spaces. From this description the various moduli spaces of (topologically distinct) instantons are constructed, and precise conditions are given under which, and only under which, irreducible G- instantons of specified topological type exist on \({\mathbb{C}}{\mathbb{P}}_ 2\). The moduli space of SU(2)-instantons of second Chern class -1 is constructed explicitly as an example.

MSC:

32L05	Holomorphic bundles and generalizations
32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results
32L25	Twistor theory, double fibrations (complex-analytic aspects)
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
32G13	Complex-analytic moduli problems

Keywords:

Yang-Mills field; monad; moduli spaces of instantons; G-instantons; holomorphic vector bundles; twistor space; stable bundles on projective spaces