Inverse scattering and the Geroch group

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 socalled 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.The best developed technique for generating solutions is that of Belinski and Zakharov [2,4], henceforth BZ for short and often called the inverse scattering method. It has been very successful in constructing novel solutions in both four-and five-dimensional vacuum gravity. The method involves some rather special adjustments to certain quantities before new physical solutions can be obtained. However, there are problems when applying this method to different gravity-matter systems like those of interest in supergravity where the implementation of the same adjustments fails and it is not guaranteed that an inverse scattering transformation preserves all features required of a solution to the gravity-matter equations [15]. In the group theoretical framework of Breitenlohner and Maison (BM) [3] this problem does not arise since the solution generating transformations form the so-called Geroch group (an affine group) and by the group property any transformation will generate a new solution. The drawback of the BM method is that it is not as easy to implement and does not always operate directly on the physical quantities. Despite these shortcomings, the promise of the BM approach is that it can be applied to various general settings of interest.In order to illustrate this point, we consider for concreteness D = 4 gravity with a spacelike and a time-like Killing vector such that the system is effectively two-dimensional and we are in the realm of so-called stationary axisymmetric solutions. The infinite-dimensional affine symmetry group can be viewed as the closure of two finite-dimensional symmetry groups that act on the space of solutions [6,3]. The first one is typically called the Matzner-Misner group SL(2, R) MM and consists of area preserving diffeomorphisms of the orbit space of the two Killing vectors (the 'torus' that one reduced on, the Killing vector orbits are not necessarily compact). The other group is called the Ehlers group SL(2, R) E and is a hidden symmetry already of the three-dimensional model. The fields that it acts on are partly formed from dualising a Kaluza-Klein vector field (in D = 3) to a scalar field; therefore it does not act directly on the metric components. Combining SL(2, R) MM and SL(2, R) E yields the infinite-dimensional affine Geroch group [6,3].The linear systems of BZ and BM d...

An inverse scattering formalism for STU supergravity

“…The relation between the two linear systems was studied in [11][12][13]. The BM linear system has not been used extensively for solution generation although in [14] it was shown how to implement a BZ like inverse scattering for SL(n, R).…”

“…In the algebraic case considered in the next section this is also easy to accomplish. As in our previous work [13], in this article we always work with flat space…”

“…with A + (t) containing only positive powers of t [12,14] and where the matrices A ± satisfy the relation [13,14] 5) and M # (x) = M (x). We also require matrices A ± (t, x) to be in SO(4, 4).…”

“…Unlike the case of SL(n, R) considered in [13,14] where the residue matrices A k are taken to be of rank one, in the present analysis we take the residue matrices A k to be of rank two. In the following, in particular in the next section, it will become clear that the rank-two case corresponds to the simple solutions of physical interest.…”

“…We therefore make the ansätze generalizing the ones used in [13,14] A + (t) = 1 1 − 22) with the parametrization of matrices C k as follows…”