Integrable models based on non-semi-simple groups and plane wave target spacetimes

Abstract

Abstract We initiate the construction of integrable λ-deformed WZW models based on non-semisimple groups. We focus on the four-dimensional case whose underlying symmetries are based on the non-semisimple group $$ {E}_2^c $$ E 2 c . The corresponding gravitational backgrounds of Lorentzian signature are plane waves which can be obtained as Penrose limits of the λ-deformed SU(2) background times a timelike coordinate for appropriate choices of the λ-matrix. We construct two such deformations which we demonstrate to be integrable. They both deform the Nappi-Witten plane wave and are inequivalent. Nevertheless, they have the same underlying symmetry algebra which is a Saletan-type contraction of that for the λ-deformed SU(2) background with a timelike direction. We also construct a plane wave from the Penrose limit of the λ-deformation of the $$ \frac{\textrm{SU}(2)}{\textrm{U}(1)} $$ SU 2 U 1 coset CFT times a timelike coordinate which represents the deformation of a logarithmic CFT constructed in the past. Finally, we briefly consider contractions based on the simplest Yang-baxter σ-models.