Analytic Continuation of Complex Gauge Fields

Abstract

The main result of this note treats the problem of unique extension of holomorphic gauge fields across closed subsets of complex Euclidean space, and is based on a corresponding extension theorem for holomorphic vector bundles due to N. P. Buchdahl and the author. Alternatively, let F be a unitary gauge field corresponding to a complex differential form of type (1, 1) (e.g., an anti self‐dual Yang–Mills field on a punctured ball in C2). As a corollary of the main theorem, it is seen that a unique extension of such F, which preserves the curvature type, is obtained if the contraction of F with a holomorphic vector field lies in the image of the ∂¯‐operator of the associated holomorphic vector bundle.