Factorizations for self-dual gauge fields

Abstract

For a particular class of patching matrices onP ₃(ℂ), including those for the complex instanton bundles with structure group Sp(k,ℂ) orO(2k,ℂ), we show that the associated Riemann-Hilbert problemG(x, λ)=G−(x, λ)·G ⁻¹₊ (x, λ) can be generically solved in the factored formG ₋=φ ₁ φ ₂.....φ _n . IfГ=Г _n is the potential generated in the usual way fromG ₋, and we setψ _i =φ ₁.....,φ _i withψ _n =G ₋, then eachψ _i also generates a selfdual gauge potentialΓ _i . The potentials are connected via the “dressing transformations”

$$\Gamma _\iota = \phi _i^{ - 1} \cdot \Gamma _{\iota - 1} \cdot \phi _i + \phi _i ^{ - 1} D\phi _i$$

of Zakharov-Shabat. The factorization is not unique; it depends on the (arbitrary) ordering of the poles of the patching matrix.