Geroch Group Description of Bubbling Geometries

TL;DR

This paper explores the monodromy matrices associated with multi-center bubbling geometries in five-dimensional supergravity, revealing their mathematical properties and suggesting a potential inverse scattering construction for these solutions.

Contribution

It systematically studies monodromy matrices for a class of bubbling geometries, clarifies their properties, and connects spectral flow transformations to Harrison transformations in supergravity.

Findings

Monodromy matrices have simple poles with rank-two, nilpotent residues.

Properties suggest inverse scattering methods can be applied to these solutions.

Spectral flow transformations are equivalent to Harrison transformations in this context.

Abstract

The Riemann-Hilbert approach to studying solutions of supergravity theories allows us to associate spacetime independent monodromy matrices (matrices in the Geroch group) with solutions that effectively only depend on two spacetime coordinates. This offers insights into symmetries of supergravity theories, and in the classification of their solutions. In this paper, we initiate a systematic study of monodromy matrices for multi-center solutions of five-dimensional U(1)$^3$ supergravity. We obtain monodromy matrices for a class of collinear Bena-Warner bubbling geometries. We show that for this class of solutions, monodromy matrices in the vector representation of SO(4,4) have only simple poles with residues of rank two and nilpotency degree two. These properties strongly suggest that an inverse scattering construction along the lines of [arXiv:1311.7018 [hep-th]] can be given for this class of solutions, though it is not attempted in this work. Along the way, we clarify a technical point in the existing literature: we show that the so-called "spectral flow transformations" of Bena, Bobev, and Warner are precisely a class of Harrison transformations when restricted to the situation of two commuting Killing symmetries in five-dimensions.