Harmonic spheres and Yang-Mills fields
type: Item
instance of: scholarly article
title: Harmonic spheres and Yang-Mills fields
zbMATH Open document ID: 1275.81067
published in: Journal of Geometry and Symmetry in Physics
publication date: 2013-04-03T00:00:00Z
review text: In this paper, the author investigates a relation between two classes of objects which from the first glance has nothing to do with each other. The first class is formed by harmonic spheres, i.e., harmonic maps of the Riemann sphere into Riemannian manifold which coincides with the classical solution of the sigma-model theory in theoretical physics and the second class consists of Yang-Mills fields on the Euclidean four-space \(\mathbb R^4\). The author indicates in this paper that after Atiyah's paper (see [\textit{M. F. Atiyah}, Commun. Math. Phys. 93, 437--451 (1984; Zbl 0564.58040)]) it became clear that there is a deep reason for a formal similarity between Yang-Mills fields and harmonic maps. Atiyah proved that the moduli space of \(G\)-instantons in \(\mathbb R^4\) can be identified with the space of based holomorphic spheres in the loop space \(\Omega G\) of a compact Lie group \(G\). The author generalizes this result and pose a conjecture stating that it should exist a bijective correspondence between the moduli space of Yang-Mills \(G\)-fields on \(\mathbb R^4\) and the space of based harmonic spheres in the loop space \(\Omega G\). The author discusses this conjecture and offers an idea of its proof.
reviewed by: Saeid Jafari
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 81T20
Mathematics Subject Classification ID: 14D21
zbMATH DE Number: 6149654
zbMATH Keywords: harmonic maps
zbMATH Keywords: Yang-Mills fields
zbMATH Keywords: the Riemann sphere
zbMATH Keywords: almost complex manifolds
zbMATH Keywords: loop space
author: Armen Sergeev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6149654