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 Poincaré 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 planeparallel 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.

We give the classification of T-duals of the flat background in four dimensions with respect to one-, two-, and three-dimensional subgroups of the Poincaré group using non-Abelian T-duality with spectators. As duals we find backgrounds for sigma models in the form of plane-parallel waves or diagonalizable curved metrics often with torsion. Among others, we find exactly solvable time-dependent isotropic pp-wave, singular ppwaves, or generalized plane wave (K-model).

Poisson-Lie T-duality and plurality are important solution generating techniques in string theory and (generalized) supergravity. Since duality/plurality does not preserve conformal invariance, the usual beta function equations are replaced by Generalized Supergravity Equations containing vector J . In this paper we apply Poisson-Lie Tplurality on Bianchi cosmologies. We present a formula for the vector J as well as transformation rule for dilaton, and show that plural backgrounds together with this dilaton and J satisfy the Generalized Supergravity Equations. The procedure is valid also for non-local dilaton and non-constant J . We also show that Div Θ of the noncommutative structure Θ used for non-Abelian T-duality or integrable deformations does not give correct J for Poisson-Lie T-plurality. * hlavaty@fjfi.cvut.cz † ivo.petr@fit.cvut.cz ContentsG andḠ that may depend on coordinates of the dual group leading to puzzle discussed in [23]. Inspired by [21] we show how to deal with this issue and present formulas that allow us to calculate vector field J appearing in Generalized Supergravity Equations. The plural backgrounds, dilatons and J then satisfy these equations.Authors of [24,25,26] consider Yang-Baxter deformations of AdS 5 × S 5 and calculate vector field J as DivΘ from non-commutative structure Θ in the open string picture. This works for "full" non-Abelian dualities, but not for "factorized" dualities (e.g. some backgrounds obtained in [16]) or more general pluralities.Plan of the paper is following. In next Section we summarize basics of Poisson-Lie T-plurality including transformation of dilaton and write down formulas for vector field J . In Sections 3.1, 3.2 we derive metrics, B-fields and dilatons obtained by Poisson-Lie T-plurality of the flat background, and in Sections 4.2, 4.1 we present metrics, B-fields and dilatons obtained by Poisson-Lie T-plurality of curved Bianchi cosmologies with nontrivial dilaton. It turns out that for appropriately defined vector field J all of them satisfy Generalized Supergravity Equations.

By addition of non-zero, but torsionless B-field, we expand the classification of (non-)Abelian T-duals of the flat background in four dimensions with respect to one-, two-, three-, and four-dimensional subgroups of Poincaré group. We discuss the influence of the additional B-field on the process of dualization and identify essential parts of the torsionless B-field that cannot be eliminated in general by coordinate or gauge transformation of the dual background. These effects are demonstrated using particular examples. Due to their physical importance, we focus on duals whose metrics represent plane-parallel waves. Besides the previously found metrics, we find new pp-waves depending on parameters originating from the torsionless B-field. These pp-waves are brought into their standard forms in Brinkmann and Rosen coordinates. *

Using a suitable extension of Drinfel’d double we derive formulas for the Poisson–Lie T-plurality transformation with spectators, i.e., for σ-models whose group of generalized isometries does not act transitively on the target manifold. These formulas are further applied to transformations of the gluing matrices that define the boundary conditions of the σ-models.