Harmonic spheres and Yang-Mills fields. (English) Zbl 1275.81067

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 [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.


81T13 Yang-Mills and other gauge theories in quantum field theory
32Q15 Kähler manifolds
35J60 Nonlinear elliptic equations
22E70 Applications of Lie groups to the sciences; explicit representations
81T20 Quantum field theory on curved space or space-time backgrounds
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)


Zbl 0564.58040