Poisson-Lie T-duality for quasitriangular Lie bialgebras

TL;DR

This paper introduces a new family of sigma models with Poisson-Lie T-duality on quasitriangular groups, featuring a simplified Lagrangian form and extending the theoretical framework to more general group factorizations.

Contribution

It develops a two-parameter family of models with Poisson-Lie T-duality, including a novel class with a constant generalized metric, and extends the Hamiltonian formulation to broader group factorizations.

Findings

Explicit models on SU(2) and its dual are computed.

The Hamiltonian formulation and reduction are developed for constant loops.

T-duality is generalized to non-dual group factorizations.

Abstract

We introduce a new 2-parameter family of sigma models exhibiting Poisson-Lie T-duality on a quasitriangular Poisson-Lie group $G$. The models contain previously known models as well as a new 1-parameter line of models having the novel feature that the Lagrangian takes the simple form $L=E(u^{-1}u_+,u^{-1}u_-)$ where the generalised metric $E$ is constant (not dependent on the field $u$ as in previous models). We characterise these models in terms of a global conserved $G$-invariance. The models on $G=SU_2$ and its dual $G^\star$ are computed explicitly. The general theory of Poisson-Lie T-duality is also extended; we develop the Hamiltonian formulation and the reduction for constant loops to integrable motion on the group manifold. Finally, we generalise T-duality in the Hamiltonian formulation to group factorisations $D=G\dcross M$ where the subgroups need not be dual or even have the same dimension and need not be connected to the Drinfeld double or to Poisson structures.