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.
Article funded by SCOAP3
0$} \\ N &\mbox{\qquad if \qquad $A\,\, \text{mod}\,\, N=0$}, \end{array} \right.\qquad\qquad\qquad\qquad \a=1+\left[\frac{A-1}{N}\right]\,, \end{align} where $\left[\cdot\right]$ denotes the integer part (floor function). The matrix $\G$ is defined as the $rN\times rN$ block matrix with entries \begin{align} \G^{\a\b}_{kl} = \left\{ \begin{array}{ll} \frac{\gamma^{\a}_k}{t_k}\delta_{\a\b} &\mbox{\qquad for \qquad $k=l$} \\ \frac{p^{\a\,\#}_k q^{\b}_l}{t_k-t_l} &\mbox{\qquad for \qquad $k \neq l$,} \end{array} \right. \end{align} where the upper indices denote the block entry and the lower indices the entries of each block. It is a symmetric matrix under the condition $p^{\a\,\#}_{k} q^{\b}_{l}=-p^{\b\,\#}_{l} q^{\a}_{k}$ for $ k\neq l$ and all $\a,\b$ in $\left\lbrace 1,2,\ldots ,r\right\rbrace$. Moreover, when the condition $p^{\a\,\#}_{k} q^{\b}_{l}=0$ for $k \neq l$ and $\a \neq \b$ holds, the off-diagonal blocks of $\G$ vanish (this is the case in all examples we have worked with so far). Solving~(\ref{qeqn}) for the vectors $r_B$ we find \begin{align} r_B=\sum_{A=1}^{rN} q_A\left(\G^{-1}\right)_{AB}\,. \end{align} There is one more set of vectors that we need to determine and these are the $s_k^{\a}$ in~(\ref{invAplus}). The requirement that $\left(\cM(t,x) A_{+}^{-1}\right)^{\#}$ have no poles at $t=-\frac{1}{t_k}$ is fulfilled when \begin{align} p^{\a}_k=t^{-1}_k s^{\a}_k \g^{\a}_k+\sum_{l\neq k}^N \sum_{\b=1}^{r} \frac{1}{t_k-t_l}s^{\b}_{l}p^{\a\,\#}_{k} q^{\b}_{l} \quad\Longleftrightarrow \quad p_A =\sum_{B=1}^{rN} \G_{AB} s_B \end{align} and the equation for the vectors $s_A$ is\footnote{The matrix equation is now $p=s\,\G^{T}$, where $p$, $s$ are $n\times rN$ matrices whose columns are the vectors $p_k^{\a}$,$s_k^{\a}$ respectively and are defined similarly to matrices $q$ and $r$.} \begin{align} s_A=\sum_{B=1}^{rN}=\left(\G^{-1}\right)_{AB}p_B\,. \end{align} Finally, the matrix $M(x)$ is obtained by \begin{align} M=A^{-1}_{+} (\infty)=\id+\sum_{A,B=1}^{rN}q_A t_{A}^{-1}\left(\G^{-1}\right)_{AB}p_B^{\#}\,, \end{align} where $t_A=t_k^\a=t_k$ for all values of $\a$. \paragraph{Conformal factor.} The formula for the conformal factor in the multisoliton case with residues of rank $r$ is given by \begin{align} \label{CFr} f^4&=k_\rom{BM}\cdot \det\G\cdot \prod_{A=1}^{rN}\left(t_A \nu_A\right)\no\\ & =k_\rom{BM}\cdot \det\G\cdot \prod_{k=1}^{N}\left(t_k \nu_k\right)^{r}\,. \end{align} This follows by a straightforward application of the computation of appendix A of~\cite{KKV} since the expression for $M$ is formally the same except for the enlarged range for the indices of $\Gamma_{AB}$. The power on $f$ on the left-hand side of~(\ref{CFr}) is due the changed normalization mentioned in footnote~\ref{normfn}. We note that~(\ref{CFr}) is consistent with~(\ref{CF44}) since in the discussion of section~\ref{sec:RHfac} the vectors were assumed to satisfy~(\ref{blockvectors}). In that case the matrix $\Gamma_{AB}$ becomes block diagonal with $r$ repeated blocks of the matrix $\Gamma_{kl}$. Then $\det(\Gamma_{AB}) = \left(\det(\Gamma_{kl})\right)^r$ and this leads to the agreement between~(\ref{CF44}) and~(\ref{CFr}) when one takes into account the different powers on $f$. ]]>