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

Abstract

We consider 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 $\widetilde\M$. Ward's ASDYM twistor construction is adapted to this geometry by using a correspondence between points of $\widetilde\M$ and holomorphic discs in $\CP^3$, twistor space, with boundary on the real slice $\RP^3$. A 1-1 correspondence is obtained between smooth global $\U(n)$ solutions to the ASDYM equations on $\widetilde\M$ and pairs consisting of an arbitrary holomorphic vector bundle $E$ over $\CP^3$ together with a smooth positive definite hermitian metric $H$ on $E|_{\RP^3}$. There are no topological or other restrictions on the bundle $E$. The description generalises the result of the scattering transform for 1+1 dimensional integrable systems in which solutions are encoded into a combination of algebraic data, here $E$, and a reflection coefficient, here $H:\RP^3\to E\otimes\bar E$. For trivial $E$, the twistor data consists of the smooth Hermitian matrix function $H$ on $\RP^3$ up to constants; the correspondence then provides a nonlinear generalisation of the X-ray transform. Explicit examples are given with different topologies of $E$. A scattering problem for ASDYM fields in split signature is set up and it is shown that sufficiently small data at past infinity uniquely determines data at future infinity by taking a family of holonomies followed by a sequence of two Birkhoff factorizations. The scattering map is simple on the holonomies, but non-trivial at the level of the connection in the non-abelian case.