Yang-Baxter $σ$-models and dS/AdS T-duality

TL;DR

Yang-Baxter sigma-models on group manifolds exploit simultaneous left-invariance and right Poisson-Lie symmetry to reveal a rich T-duality structure encoded by Drinfeld doubles. The paper develops a first-order, double-based formalism with a self-adjoint operator ${\cal E}$ that generates a Yang-Baxter deformation via ${\tilde R}=({\cal J}{\cal E}{\tilde{\cal J}})^{-1}({\cal J}+{\cal J}{\cal E}{\cal J})$, and applies it to two key examples: the anisotropic principal chiral model and the SL(2,C)/SU(2) WZW model. It uncovers dual pairs for each model, including a left dual on the Abelianized dual and a right Poisson-Lie dual on $AN$, and shows a remarkable $dS_3$–$AdS_3$ duality emerging from the SL(2,C) double with explicit backgrounds and conformal data. The work clarifies how Poisson-Lie duality operates through Drinfeld doubles, highlights the role of $r$-matrices in encoding the non-Abelian structure, and points to potential implications for quantum properties and D-brane configurations in these geometries.

Abstract

We point out the existence of nonlinear $σ$-models on group manifolds which are left symmetric and right Poisson-Lie symmetric. We discuss the corresponding rich T-duality story with particular emphasis on two examples: the anisotropic principal chiral model and the $SL(2,C)/SU(2)$ WZW model. The latter has the de Sitter space as its (conformal) non-Abelian dual.