Plane-parallel waves as duals of the flat background

Abstract

We give a classification of non-Abelian T-duals (‘T’ standing for topological or toroidal) of the flat metric in D = 4 dimensions with respect to the four-dimensional continuous subgroups of the Poincaré group. After dualizing the flat background, we identify the majority of dual models as conformal sigma models in plane-parallel wave backgrounds, most of them having torsion. We give their form in Brinkmann coordinates. We find, besides the plane-parallel waves, several diagonalizable curved metrics with nontrivial scalar curvature and torsion. Using the non-Abelian T-duality, we find general solutions of the classical field equations for all the sigma models in terms of dʼAlembert solutions of the wave equation.