Finite action solutions of the nonlinear sigma-model

The instanton and anti-instanton solutions of the two-dimensional O (3) sigma-model are special examples of harmonic maps, which have been studied extensively in the mathematical literature. We give an elementary and self-contained proof that these solutions are the only continuous maps for which the action is finite and stationary under variations, without assuming any additional boundary conditions at infinity. An element of the proof is the vanishing of the stress tensor for a finite action solution, which actually holds true for the general O (N) sigma-model. For the two-dimensional O (2l+1) sigma-model we exhibit explicit finite action solutions that do not lie in any lower dimensional sphere; the existence of such solutions has been pointed out in the mathematical literature. We also present a rigorous proof, based on Derrick's scaling argument, that there are no nonconstant finite action solutions in more than two dimensions