Abstract
We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective correspondence between based harmonic spheres in the loop space \(\Omega G\) of a compact Lie group \(G\) and the moduli space of Yang–Mills \(G\)-fields on \(\mathbb R^4\).
Highlights
In the first part of this paper, we consider the twistor interpretation of harmonic maps of the Riemann sphere into Kähler manifolds.Harmonic maps of the Riemann sphere S := P1 into a given Riemannian manifold M are the extrema of the energy functional, given by the Dirichlet-type integral
In particular, M coincides with the loop space ΩG of a compact Lie group, we have an infinite-dimensional
The twistor description of Yang–Mills fields was proposed in the papers by Manin [5], Witten [6] and Isenberg-Green-Yasskin [7]
Summary
In the first part of this paper, we consider the twistor interpretation of harmonic maps of the Riemann sphere into Kähler manifolds. The local minima of this functional are given by instantons and anti-instantons Their twistor description was proposed by Atiyah–Ward [3] with the help of the Hopf bundle π : P3 → S 4 over the compactified Euclidean four-space, coinciding with the sphere. The twistor description of Yang–Mills fields was proposed in the papers by Manin [5], Witten [6] and Isenberg-Green-Yasskin [7] They are interpreted as holomorphic vector bundles over the incidence quadric in P3 × (P3 )∗ with some special properties described below. We formulate here the harmonic spheres conjecture relating Yang–Mills G-fields on R4 with harmonic spheres in ΩG and overview the idea of its proof
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