Integrability and vesture for harmonic maps into symmetric spaces

Abstract

After formulating the notion of integrability for axially symmetric harmonic maps from R3 into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular, we show that the problem of finding N-solitonic harmonic maps into a non-compact Grassmann manifold SU(p,q)/S(U(p)×U(q)) is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method, and establish its agreement with previously known special cases, by explicitly computing a 1-solitonic harmonic map for the two cases (p=1,q=1) and (p=2,q=1) and showing that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr–Newman family of solutions to the Einstein–Maxwell equations in the hyperextreme sector of the corresponding parameters.