Anti-Self-Dual Metrics and Kähler Geometry

The fact that the Lie group SO(4) is nonsimple gives 4-dimensional geometry an extremely distinctive flavor. Indeed, the choice of a Riemannian metric g on an oriented 4-manifold M splits the bundle of 2-forms 1 $${\Lambda ^2} = {\Lambda ^ + } \oplus {\Lambda ^ - }$$ Λ 2 = Λ + ⊕ Λ − into the rank-3 bundles of self-dual and anti-self dual 2-forms, respectively defined as the ±1-eigenspaces of the Hodge star operator ⋆ : ⋀2 → ⋀2; this just reflects the fact that the adjoint representation of SO(4) on the skew (4 x 4)-matrices is the sum of two 3-dimensional representations, as indicated by the Lie algebra isomorphism so(4) ≅ so(3)⊕so(3). The decomposition (1) is conformally invariant, in the sense that it is unchanged if g is replaced by ug for any positive function u; but reversing the orientation of M interchanges the bundles ⋀±.