Integrability and Vesture for Harmonic Maps into Symmetric Spaces

TL;DR

This paper establishes the integrability of axially symmetric harmonic maps into symmetric spaces, introduces a dressing method for solution generation, and explicitly constructs solutions related to important physical models like Kerr and Kerr-Newman.

Contribution

It provides the most general formulation of integrability for these harmonic maps and demonstrates the dressing method's effectiveness in generating solutions, including physically significant ones.

Findings

All axially symmetric harmonic maps under mild conditions are integrable.

The dressing method can generate N-solitonic solutions from any given solution.

Explicit 1-solitonic solutions include Kerr and Kerr-Newman families.

Abstract

After giving the most general formulation to date of the notion of integrability for axially symmetric harmonic maps from R^3 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 noncompact Grassmann manifold SU(p,q)/S(U(p) x 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 by explicitly computing a 1-solitonic harmonic map for the two cases (p = 1, q = 1) and (p = 2, q = 1); and we show 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.