Harmonic spheres and Yang–Mills fields

In our lectures we study a relation between harmonic spheres in loop spaces and Yang–Mills fields on the Euclidean 4-space. Harmonic spheres are given by smooth maps of the Riemann sphere into Riemannian manifolds being the extremals of the energy functional given by Dirichlet integral. They satisfy nonlinear elliptic equations, generalizing Laplace–Beltrami equation. If the target Riemannian manifold is Kähler then holomorphic and anti-holomorphic spheres realize local minima of the energy. However, this functional usually have also non-minimal extremals. On the other hand, Yang–Mills fields are the extremals of Yang–Mills action functional. Local minima of this functional are called instantons and anti-instantons. It was believed that they exhaust all critical points of Yang–Mills action on R4, until examples of nonminimal Yang–Mills fields were constructed.
There is an evident formal similarity between Yang–Mills fields and harmonic maps and after Atiyah's paper of 1984 it became clear that there is a deep reason for such a similarity. Namely, Atiyah has proved that the moduli space of $G$-instantons on R4 can be identified with the space of based holomorphic spheres in the loop space $G$ of a compact Lie group $G$. Generalizing this theorem, we formulate a conjecture that it should exist a bijective correspondence between the moduli space of Yang–Mills $G$-fields on R4 and the space of based harmonic spheres in the loop space $G$.
All necessary notions from harmonic map and gauge field theories will be introduced in the lectures. We assume only basic knowledge of differential geometry and Lie group theory.