On Harmonic Maps from $CP^1$ to $\Omega SU(n)$

Abstract

Using the geometry of based loop group ΩG and some results on harmonic maps, we shall show that when $n\geq 3$ there exist harmonic but not±holomorphic maps from CP$F^1$ to ΩSU (n) for each degree. Further we shall discuss what our result suggests from the point of view of the difference between Yang-Mills connections and instantons on S$F^4$ in particular Atiyah-Jones conjecture.