We describe a factorization theorem for holomorphic maps from a compact manifold
M into the loop group of
U(
N). We prove that any such map is a finite Blaschke product of maps into Grassmann manifolds (unitons), satisfying recursive holomorphicity conditions; each map being attached to a point in the open unit disc. This factorization is essentially unique. Using a theorem of Atiyah and Donaldson, we construct a stratification of the moduli space of framed
SU(2) Yang-Mills instanton over the 4-sphere, in which the strata are iterated fibrations of spaces of polynomials, indexed by plane partitions; and the unique open stratum of "generic" instantons of charge
d, is the configuration space of
d distinct points in the disc, labelled with
d biholomorphisms of the 2-sphere.
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