Global anti-self-dual Yang-Mills fields in split signature and their scattering

Abstract

This article concerns solutions to the anti-self-dual Yang Mills (ASDYM) equations in split signature that are global on the double cover of the appropriate conformally compactified Minkowski space = S2 × S2. Ward's ASDYM twistor construction is adapted to this geometry using a correspondence between points of and holomorphic discs in , twistor space, with boundary on the real slice ℝℙ3. Smooth global U(n) solutions to the ASDYM equations on are shown to be in 1 : 1 correspondence with pairs consisting of an arbitrary holomorphic vector bundle E over together with a positive definite Hermitian metric H on E|ℝℙ3. There are no topological or other restrictions on the bundle E. In ultrahyperbolic signature solutions are generically non-analytic or only finitely differentiable and such solutions arise from a corresponding choice of regularity for H. When E is trivial, the twistor data consists of the Hermitian matrix function H on ℝℙ3 up to constants and the correspondence provides a nonlinear generalisation of the X-ray transform. In general it provides a higher-dimensional analogue of the (inverse) scattering transform in which H plays the role of the reflection coefficient and E the algebraic data.