Analysis Seminar: Armen Sergeev
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Speaker: Armen Sergeev, Professor, Steklov Mathematical Institute, Moscow, Russia
Title: Harmonic maps and Yang-Mills fields.
Abstract: We consider a connection between harmonic maps of Riemann surfaces and Yang–Mills fields on R^4. Harmonic map from a Riemann surface into a Riemannian manifold is the extremal of the energy functional given by the Dirichlet integral. Such maps satisfy nonlinear elliptic equations of 2-nd order, generalizing Laplace–Beltrami equation. In the case when the target Riemannian manifold is Kaehler, i.e. provided with a complex structure compatible with Riemannian metric, the holomorphic and anti-holomorphic maps realize local minima of the energy. We are especially interested in harmonic maps of the Riemann sphere called briefly harmonic spheres. The Yang–Mills fields on R^4 are the extremals of the Yang–Mills action functional. Local minima of this functional are given by instantons and anti-instantons. There is an evident formal similarity between the 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 for any compact Lie group G there is a bijective correspondence between the gauge classes of G-instantons on R^4 and based holomorphic spheres in the loop space ΩG of G. This theorem motivates the harmonic spheres conjecture stating that it should exist a bijective correspondence between the gauge classes of Yang–Mills G-fields on R^4 and based harmonic spheres in ΩG. In our talk we discuss this conjecture and possible ways of its proof.
Title: Harmonic maps and Yang-Mills fields.
Abstract: We consider a connection between harmonic maps of Riemann surfaces and Yang–Mills fields on R^4. Harmonic map from a Riemann surface into a Riemannian manifold is the extremal of the energy functional given by the Dirichlet integral. Such maps satisfy nonlinear elliptic equations of 2-nd order, generalizing Laplace–Beltrami equation. In the case when the target Riemannian manifold is Kaehler, i.e. provided with a complex structure compatible with Riemannian metric, the holomorphic and anti-holomorphic maps realize local minima of the energy. We are especially interested in harmonic maps of the Riemann sphere called briefly harmonic spheres. The Yang–Mills fields on R^4 are the extremals of the Yang–Mills action functional. Local minima of this functional are given by instantons and anti-instantons. There is an evident formal similarity between the 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 for any compact Lie group G there is a bijective correspondence between the gauge classes of G-instantons on R^4 and based holomorphic spheres in the loop space ΩG of G. This theorem motivates the harmonic spheres conjecture stating that it should exist a bijective correspondence between the gauge classes of Yang–Mills G-fields on R^4 and based harmonic spheres in ΩG. In our talk we discuss this conjecture and possible ways of its proof.
05.05.2015