Holomorphic maps from compact manifolds into loop groups as Blaschke products (Q1313228)

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.