Canonical structures on anti-self-dual four-manifolds and the diffeomorphism group

Abstract

An unusual and attractive system is studied that arises from the anti-self-dual (ASD) Yang–Mills equations with maximal translational symmetry and with gauge group the volume preserving diffeomorphisms of an auxiliary four-manifold ℳ. The resulting equations lead to a system consisting of a volume form together with four independent vector fields on ℳ satisfying three simple Lie bracket relations. This structure is shown to give rise to a two-sphere’s worth of closed simple two-forms which in turn lead to the standard hyper-Kahler structure of an anti-self-dual metric on ℳ. The system determines not only the ASD metric, but also a frame that is proportional to an orthonormal frame. It is shown that the freedom in the choice of frame is related to a pair of solutions of a modified Laplacian and can always be chosen so that the proportionality factor is unity. The Plebanski first and second heavenly forms for general ASD metrics are written out in terms of these structures (the first being the standard description in terms of a complex structure and Kahler scalar). One of the scalars is interpreted as the generating function for the diffeomorphisms (symplectomorphism) in line with the origin of the system as the ASD Yang–Mills equations with the volume preserving diffeomorphisms as gauge group.