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Steklov Mathematical Institute Seminar
June 21, 2007 16:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)

Harmonic maps and Yang–Mills fields

A. G. Sergeev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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A. G. Sergeev
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Abstract: We study harmonic maps from compact Riemann surfaces into compact Kähler manifolds. Such maps are interesting both from the mathematical and physical points of view (in the field theory they are interpreted as classical solutions of sigma-models).
For the construction of harmonic maps we use the twistor approach, which allows to reduce the “real” problem of construction of harmonic maps into a given manifold to the “complex” problem of construction of pseudoholomorphic maps into the twistor space of this manifold. With help of the twistor approach, a complete description of harmonic maps of compact Riemann surfaces into complex Grassmann manifolds was obtained (J. C. Wood).
In our talk we consider an infinite-dimensional version of this theory. Namely, we study harmonic maps of compact Riemann surfaces into infinite-dimensional Kähler manifolds, which are the loop spaces $\Omega G$ of compact Lie groups $G$. The interest in such maps is motivated by an Atiyah–Donaldson theorem, establishing a 1–1 correspondence between the moduli space of $G$-instantons on the 4-dimensional Euclidean space $\mathbb R^4$ and (based) holomorphic maps of the Riemann sphere into $\Omega G$. In accordance with this theorem, we may conjecture that there exists also a 1–1 correspondence between the moduli space of Yang–Mills $G$-fields on $\mathbb R^4$ and (based) harmonic maps of the Riemann sphere into $\Omega G$.
To construct harmonic maps into the loop space $\Omega G$, we use an isometric embedding of $\Omega G$ into an infinite-dimensional Grassmann manifold, called the Hilbert–Schmidt Grassmannian of a complex Hilbert space. By this embedding the original problem is reduced to the problem of construction of harmonic amps into the Hilbert–Schmidt Grassmannian, which is solved with an infinite-dimensional generalization of the twistor construction of such maps in the finite-dimensional case.
All necessary background on harmonic maps will be given in the talk.

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