Harmonic maps as models for physical theories

Harmonic maps are an aesthetically appealing class of nonlinear field equations of which only a few nontrivial examples have as yet appeared in physical theories. These fields appear well suited for describing broken symmetries either in conjunction with or instead of the Yang-Mills equations. The harmonic mapping equation is quite similar in many respects to the Einstein equations for gravitation, although simpler in structure, and can describe any gauge symmetry group $G$ broken to a subgroup $H$ in a sense parallel to the way the gravitational (metric) field breaks general covariance [local $\mathrm{GL}(4, R)$ invariance] down to local Lorentz invariance. This paper outlines the basis for a program of exploring harmonic mapping theories to see whether they may provide models of physical phenomena that either are not recognized, or are not well fitted to more familiar field theories.