Plane-parallel waves as duals of the flat background

TL;DR

This paper classifies non-Abelian T-duals of flat four-dimensional spacetime, revealing many dual models as conformal plane-parallel wave backgrounds with torsion, and provides solutions to their classical field equations.

Contribution

It offers a comprehensive classification of non-Abelian T-duals of flat space and identifies their structure as conformal sigma models in plane-parallel wave backgrounds.

Findings

Most dual models are conformal sigma models in plane-parallel wave backgrounds.

Explicit forms of dual backgrounds are given in Brinkmann coordinates.

Solutions to classical field equations are expressed via wave equation solutions.

Abstract

We give a classification of non-Abelian T-duals of the flat metric in D=4 dimensions with respect to the four-dimensional continuous subgroups of the Poincare group. After dualizing the flat background, we identify 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 solution of the classical field equations for all the sigma models in terms of d'Alembert solutions of the wave equation.