# T-Dualities and Doubled Geometry of the Principal Chiral Model

## TL;DR

This paper explores the duality structures of the Principal Chiral Model on SU(2), revealing new dualities and geometric formulations within the framework of Doubled Geometry and Poisson-Lie groups.

## Contribution

It introduces a family of Hamiltonian descriptions, identifies new dualities between SU(2) and SB(2,C), and constructs a parent action unifying these models within Doubled Geometry.

## Key findings

- Dual models related by O(3,3) transformations are defined on SB(2,C).
- A new duality between SU(2) PCM and SB(2,C) models is discovered.
- A parent action with doubled degrees of freedom unifies the dual models.

## Abstract

The Principal Chiral Model (PCM) defined on the group manifold of SU(2) is here investigated with the aim of getting a further deepening of its relation with Generalized and Doubled Geometry. A one-parameter family of equivalent Hamiltonian descriptions is introduced, and cast into the form of Born geometries. Then O(3,3) duality transformations of the target phase space are performed and we show that the resulting dual models are defined on the group SB(2,C) which is the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of the Drinfel'd double SL(2, C). Moreover, starting from the Lagrangian approach, a new kind of duality is found between the SU(2) PCM and the natural one defined on SB(2,C) which is not an isometry of the target phase space. A parent action with doubled degrees of freedom and configuration space SL(2, C) is then defined that reduces to either one of the dually related models, once suitable constraints are implemented.