Monodromy transform and the integral equation method for solving the string gravity and supergravity equations in four and higher dimensions

Abstract

The monodromy transform and corresponding integral equation method described here give rise to a general systematic approach for solving integrable reductions of field equations for gravity coupled bosonic dynamics in string gravity and supergravity as well as for pure vacuum gravity in four and higher dimensions. For physically different types of fields in space-times of $D\ensuremath{\ge}4$ dimensions with $d=D\ensuremath{-}2$ commuting isometries---stationary fields with spatial symmetries, interacting waves or evolution of partially inhomogeneous cosmological models---the string gravity equations govern the dynamics of interacting gravitational, dilaton, antisymmetric tensor, and any number $n\ensuremath{\ge}0$ of Abelian vector gauge fields (all depending only on two coordinates). The equivalent spectral problem constructed earlier allows one to parametrize the entire infinite-dimensional space of (normalized) local solutions of these equations by two pairs of arbitrary coordinate-independent holomorphic $d\ifmmode\times\else\texttimes\fi{}d$- and $d\ifmmode\times\else\texttimes\fi{}n$- matrix functions ${{\mathbf{u}}_{\ifmmode\pm\else\textpm\fi{}}(w),{\mathbf{v}}_{\ifmmode\pm\else\textpm\fi{}}(w)}$ of a spectral parameter $w$ which constitute a complete set of monodromy data for a normalized fundamental solution of this spectral problem. The ``direct'' and ``inverse'' problems of such monodromy transform---calculating the monodromy data for any local solution and constructing the field configurations for any chosen monodromy data---always admit unique solutions. We construct the linear singular integral equations which solve this inverse problem. For any rational and analytically matched [i.e. ${\mathbf{u}}_{+}(w)\ensuremath{\equiv}{\mathbf{u}}_{\ensuremath{-}}(w)$ and ${\mathbf{v}}_{+}(w)\ensuremath{\equiv}{\mathbf{v}}_{\ensuremath{-}}(w)$] monodromy data the solution of these integral equations and corresponding solution for string gravity equations can be found explicitly. Simple reductions of the space of monodromy data leads to the similar constructions for solving of other integrable symmetry reduced gravity models, e.g. 5D minimal supergravity or vacuum gravity in $D\ensuremath{\ge}4$ dimensions.