Аннотация:
Harmonic spheres conjecture establishes a correspondence between
Yang–Mills $G$-fields on $\mathbb R^4$ and harmonic maps of
the Riemann sphere $S^2$ into the loop space $\Omega G$ of the group
$G$. It is an extension to general Yang–Mills
$G$-fields of the Atiyah–Donaldson theorem establishing a
correspondence between the moduli space of $G$-instantons on
$\mathbb R^4$ and holomorphic maps $S^2\to\Omega G$.
In our talk we present an approach to the proof of this conjecture based on the
adiabatic limit construction proposed by Popov. His
construction uses a nice parametrization of the sphere $S^4\setminus
S^1$ with one deleted circle found by Jarvis and Norbury.
With the help of this construction one can associate
in a natural way with arbitrary Yang–Mills $G$-field on $S^4$ a
harmonic map of the sphere $S^2$ to the loop space $\Omega G$.