The PENROSE Transform for Bundles Non‐Trivial on the General Line

Abstract

AbstractLet L0 be a fixed projective line in CP3 and let M ≅ C4 be the complexified MINKOWSKI space interpreted as the manifold of all projective lines L ⫅ CP3 with L ∩ L0 ∅︁ Ø. Let D ⫅ M, D′ ⫅ CP3/L0 be open sets such that \documentclass{article}\pagestyle{empty}\begin{document}$ D' = \mathop \cup \limits_{L \in D} $\end{document}. Under certain topological conditions on D, R. S. WARD'S PENROSE transform sets up an 1–1 correspondence between holomorphic vector bundles over D′ trivial over each L ϵ D and holomorphic connections with anti‐self‐dual curvature over D (anti‐self‐dual YANG‐MILLIS fields). In the present paper WARD'S construction is generalized to holomorphic vector bundles E over D′ satisfying the condition that \documentclass{article}\pagestyle{empty}\begin{document}$ E|_L \cong E|_{\tilde L} $\end{document} for all \documentclass{article}\pagestyle{empty}\begin{document}$ L,\tilde L \in D $\end{document}.