Twistor approach to harmonic 2-spheres in the loop space

Atiyah's theorem establishes a bijection between the moduli space of G-instantons on
R^{4} and the space of based holomorphic 2-spheres in the loop space \Omega G of a
gauge group G. There is a conjecture (due to A. Sergeev) which asserts that there should
exist a bijection between the moduli space of Yang--Mills G-fields on R^{4} and the space
of based harmonic 2-spheres in the loop space \Omega G. In this talk we will discuss this
conjecture and a twistor approach to harmonic 2-spheres in the loop space \Omega G.