An inverse scattering formalism for STU supergravity

Abstract

STU supergravity becomes an integrable system for solutions that effectively only depend on two variables. This class of solutions includes the Kerr solution and its charged generalizations that have been studied in the literature. We here present an inverse scattering method that allows to systematically construct solutions of this integrable system. The method is similar to the one of Belinski and Zakharov for pure gravity but uses a different linear system due to Breitenlohner and Maison and here requires some technical modifications. We illustrate this method by constructing a four-charge rotating solution from flat space. A generalization to other set-ups is also discussed.

A Riemann-Hilbert approach to rotating attractors

“…In the limit ρ → 0 + (with v = 0), we have 8) in agreement with (10.2). The terms proportional to k 1 and to k 2 are thus projected out and do not appear in the monodromy matrix M(ω).…”

“…Most of the literature is focussed on existence, and employs factorization algorithms that quickly become very cumbersome when applied in practice. Moreover [5] and the subsequent literature, including [6][7][8][9], impose a particular ansatz which requires the spacetime to be asymptotically flat and the monodromy matrix to have only first order poles in ω. These are severe limitations as they exclude extremal solutions, and the attractors solutions which are their near-horizon limits.…”

“…Given M (x), we seek a matrix M(ω) whose canonical factorization yields back M (x), 8) JHEP06 (2017)123…”

“…The following paragraphs follow closely section 4 of [6], which we translate into our notation. 8 The manifest symmetries acting on coset representatives V (x) are…”

“…Therefore one needs to explore alternative systematic methods for constructing rotating solutions. One such method was pioneered in the seminal work of Breitenlohner and Maison [5], and has since then been explored further by various authors, including [6][7][8][9]. It is based on the observation that the stationary axisymmetric sector of four-dimensional gravity is integrable, and that the problem of solving the Einstein equations can be replaced by solving a linear system depending on an additional variable, the 'spectral parameter.'…”