# Para-Hermitian Geometries for Poisson-Lie Symmetric $\sigma$-models

## TL;DR

This paper develops a para-Hermitian geometric framework for Poisson-Lie symmetric sigma-models, enabling new insights into T-duality, coset spaces, and integrable deformations of spheres.

## Contribution

It introduces a para-Hermitian geometric approach to Poisson-Lie sigma-models, extending to coset spaces and providing explicit constructions for integrable deformations.

## Key findings

- Formulated Poisson-Lie sigma-models using para-Hermitian geometry.
- Extended the framework to general coset spaces via dressing cosets.
- Constructed explicit examples of integrable deformations on spheres.

## Abstract

The doubled target space of the fundamental closed string is identified with its phase space and described by an almost para-Hermitian geometry. We explore this setup in the context of group manifolds which admit a maximally isotropic subgroup. This leads to a formulation of the Poisson-Lie $\sigma$-model and Poisson-Lie T-duality in terms of para-Hermitian geometry. The emphasis is put on so called half-integrable setups where only one of the Lagrangian subspaces of the doubled space has to be integrable. Using the dressing coset construction in Poisson-Lie T-duality, we extend our construction to more general coset spaces. This allows to explicitly obtain a huge class of para-Hermitian geometries. Each of them is automatically equipped which a generalized frame field, required for consistent generalized Scherk-Schwarz reductions. As examples we present integrable $\lambda$- and $\eta$-deformations on the three- and two-sphere.