an:03939718
Zbl 0586.32034
Buchdahl, N. P.
Instantons on \({\mathbb{C}}{\mathbb{P}}_ 2\)
EN
J. Differ. Geom. 24, 19-52 (1986).
00151369
1986
j
32L05 32L10 32L25 14F05 32G13
Yang-Mills field; monad; moduli spaces of instantons; G-instantons; holomorphic vector bundles; twistor space; stable bundles on projective spaces
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.