Instantons and Harmonic Maps

Abstract

Differential equations arising in Physics often appear as the Euler-Lagrange equations for a certain functional on a space of maps; in this lecture we shall be concerned with two recent examples, namely Yang-Mills theory and the theory of σ-models. Two striking features here are (a) the existence of special solutions known as instantons, and (b) the possibility of a topological relation between the parameter space or moduli space of instantons and the space on which the functional is defined. Technical problems with the Yang-Mills functional have prompted comparison with a general situation where some results already exist, i.e. the study of the energy functional $$ E:f \mapsto \frac{1}{2}\int\limits_M {{{\left| {df} \right|}^2}} $$ defined on the space of smooth maps Map(M,N) between (compact) Riemannian manifolds M,N. The critical points for E are, by definition, harmonic maps. This actually includes the usual σ-model example [10,11,20] where M = CP1 and N = CPn. Recent work of M. F. Atiyah and S. K. Donaldson (see [2]) indicates that this is much more than a useful analogy, however: Yang-Mills instantons for a G-bundle over S4 may be identified with “σ-model instantons” if M = CP1 and N is replaced by the infinite dimensional Lie group ΩG consisting of loops on G.