Classical integrability of Schrodinger sigma models and q-deformed Poincare symmetry

TL;DR

This paper demonstrates the classical integrability of Schrodinger sigma models with deformed AdS_3 target spaces, revealing a q-deformed Poincare symmetry and establishing two equivalent descriptions via non-local maps.

Contribution

It introduces a novel integrable structure for Schrodinger sigma models, connecting Yangian symmetry and q-deformed Poincare algebra through two complementary descriptions.

Findings

Constructed Lax pairs for both descriptions.

Derived r/s-matrix algebra confirming integrability.

Established equivalence of the two descriptions via non-local map.

Abstract

We discuss classical integrable structure of two-dimensional sigma models which have three-dimensional Schrodinger spacetimes as target spaces. The Schrodinger spacetimes are regarded as null-like deformations of AdS_3. The original AdS_3 isometry SL(2,R)_L x SL(2,R)_R is broken to SL(2,R)_L x 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 integrability 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 AdS_3 and that it enhances U(1)_R to a q-deformed Poincare 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.