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

TL;DR

This paper develops integrable lambda-deformed WZW models based on non-semisimple groups, specifically focusing on four-dimensional cases with plane wave spacetimes, revealing new integrable deformations and their underlying symmetries.

Contribution

It introduces the first construction of integrable lambda-deformed models on non-semisimple groups and explores their gravitational backgrounds and symmetries.

Findings

Constructed two integrable lambda-deformations of Nappi-Witten plane waves.

Demonstrated these deformations are inequivalent but share the same symmetry algebra.

Connected deformations to Penrose limits of known backgrounds and CFTs.

Abstract

We initiate the construction of integrable $\lambda$-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 $E_2^c$. The corresponding gravitational backgrounds of Lorentzian signature are plane waves which can be obtained as Penrose limits of the $\lambda$-deformed $SU(2)$ background times a timelike coordinate for appropriate choices of the $\lambda$-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 $\lambda$-deformed $SU(2)$ background with a timelike direction. We also construct a plane wave from the Penrose limit of the $\lambda$-deformation of the $\nicefrac{SU(2)}{U(1)}$ 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 $\sigma$-models.