Classical integrability of Schrödinger sigma models and q-deformed Poincaré symmetry

Abstract

We discuss classical integrable structure of two-dimensional sigma models which have three-dimensional Schrödinger spacetimes as target spaces. The Schrödinger spacetimes are regarded as null-like deformations of AdS3. The original AdS3 isometry SL(2, R)L × SL(2, R)R is broken to SL(2, R)L × U(1)R due to the deformation. According to this symmetry, there are two descriptions to describe the classical dynamics of the system, 1) the SL(2, R)L description and 2) the enhanced U(1)R description. In the former 1), we show that the Yangian symmetry is realized by improving the SL(2, R)L Noether current. Then a Lax pair is constructed with the improved current and the classical inte-grability is shown by deriving the r/s-matrix algebra. In the latter 2), we find a non-local current by using a scaling limit of warped AdS3 and that it enhances U(1)R to a q-deformed Poincaré algebra. Then another Lax pair is presented and the corresponding r/s-matrices are also computed. The two descriptions are equivalent via a non-local map.