On the initial condition for instanton solutions

Abstract

To each gauge equivalence class of both local and global framed (in the sense of Donaldson) self-dual solutions with the gauge group U(r) there is related the unique canonical initial condition (in the sense of Takasaki) and in this way the gauge freedom is eliminated. A geometric interpretation is given and consequently the complete transcription of the ADHM construction into the inverse scattering formalism is derived. As an application, an injection holomorphic mapping of the instanton moduli space into a finite-dimensional complex vector space is described and the loop group action on the transition functions is discussed. The results suggest the possibility of a new description of the framed instanton moduli spaces directly as algebraic sets.