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Seminar on Complex Analysis (Gonchar Seminar)
September 25, 2017 17:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)

Adiabatic limit in Yang–Mills equations on $\mathbb R^4$

A. G. Sergeev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: The harmonic spheres conjecture relates Yang–Mills fields on $\mathbb R^4$ with gauge group $G$ with harmonic maps of the Riemann sphere $S^2$ into the loop space $\Omega G$ of the group $G$. This conjecture is a generalization to arbitrary Yang–Mills $G$-fields of Atiyah–Donaldson theorem establishing a $1$$1$ correspondence between the moduli space of $G$-instantons on $\mathbb R^4$ and the space of based holomorphic maps $S^2\to\Omega G$.
In our talk we shall consider a possible way to prove the harmonic spheres conjecture using the adiabatic limit construction for the Yang–Mills equations on $S^4$ proposed by A. D. Popov. The Popov construction employs a nice parameterization of the sphere $S^4\setminus S^1$ without a circle, found by Jarvis and Norbury. Using this parameterization it is possible to associate in a natural way with arbitrary Yang–Mills $G$-field on $S^4$ a harmonic map of the sphere $S^2$ into the loop space $\Omega G$.