Inverse Scattering and the Geroch Group

TL;DR

This paper explores the integrability of gravity-matter systems in two dimensions using inverse scattering methods, revealing the Geroch group's structure and generating solutions like the Kerr-NUT spacetime.

Contribution

It clarifies the relation between the Geroch group and inverse scattering methods, and demonstrates algebraic solution generation including the Kerr-NUT solution.

Findings

Established the connection between Geroch group and inverse scattering

Developed algebraic solution methods for gravity-matter systems

Constructed the Kerr-NUT solution via Riemann-Hilbert problem

Abstract

We study the integrability of gravity-matter systems in D=2 spatial dimensions with matter related to a symmetric space G/K using the well-known linear systems of Belinski-Zakharov (BZ) and Breitenlohner-Maison (BM). The linear system of BM makes the group structure of the Geroch group manifest and we analyse the relation of this group structure to the inverse scattering method of the BZ approach in general. Concrete solution generating methods are exhibited in the BM approach in the so-called soliton transformation sector where the analysis becomes purely algebraic. As a novel example we construct the Kerr-NUT solution by solving the appropriate purely algebraic Riemann-Hilbert problem in the BM approach.