XXV Workshop on Geometric Methods in Physics 2-8.07.2006

Armen Sergeev

Harmonic spheres in loop spaces and Yang-Mills fields

Atiyah has found a relation between $G$-instantons on the Euclidean 4-space and holomorphic spheres in the loop space $\Omega G$ of the group $G$. We conjecture that in an analogous way there exists a 1-1 correspondence between Yang-Mills $G$-fields on $R^4$ and harmonic spheres in $\Omega G$. We discuss this conjecture and propose a general method, based on twistor considerations, how to construct harmonic maps of compact Riemann surfaces into $\Omega G$.