Plane-parallel waves as Jacobi-Lie models

Abstract

T-duality and its generalizations are widely recognized either as symmetries or solution-generating techniques in string theory. Recently introduced Jacobi-Lie T-plurality is based on Leibniz algebras whose structure constants ${f_{ab}}^c, {f_c}^{ab}, Z_a, Z^a$ satisfy further conditions. Low dimensional Jacobi-Lie bialgebras were classified a few years ago. We study four- and six-dimensional algebras with structure constants ${f_b}^{ba} = Z^a = 0$ and show that there are several classes consisting of mutually isomorphic algebras. Using isomorphisms between Jacobi-Lie bialgebras we investigate three- and four-dimensional sigma models related by Jacobi-Lie T-plurality with and without spectators. In the Double Field Theory formulation constant generalized fluxes $F_A$ are used in the literature to transform dilaton field. We extend the procedure to non-constant fluxes and verify that obtained backgrounds and dilatons solve Supergravity Equations. Most of the resulting backgrounds have vanishing curvature scalars and, as can be seen by finding Brinkmann coordinates, represent plane-parallel waves solving Supergravity Equations.