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.

Explicit examples are constructed for different choices of the topology of E.

A scattering problem for ASDYM fields on affine Minkowski space in split signature is set up and it is shown that sufficiently small data at past null infinity uniquely determines data at future null infinity by taking a family of holonomies associated to the initial data followed by a sequence of two Birkhoff factorizations. The scattering map is simple at the level of the holonomies, but non-trivial at the level of the connection in the non-abelian case.