Topological Poisson Sigma models on Poisson-Lie groups

TL;DR

This paper solves topological Poisson Sigma models on Poisson-Lie groups, revealing a duality between phase spaces and uncovering a symplectic structure connection via the Heisenberg double, with extensions to models involving r-matrices.

Contribution

It provides explicit solutions for Poisson Sigma models on Poisson-Lie groups and their duals, demonstrating a duality and symplectic structure relations, including a generalization with r-matrices.

Findings

Duality between phase spaces of models and boundary degrees of freedom.

Existence of a map linking reduced phase spaces to the symplectic leaf of the Heisenberg double.

Extension of the model to Poisson structures defined by r-matrices.

Abstract

We solve the topological Poisson Sigma model for a Poisson-Lie group $G$ and its dual $G^*$. We show that the gauge symmetry for each model is given by its dual group that acts by dressing transformations on the target. The resolution of both models in the open geometry reveals that there exists a map from the reduced phase of each model ($P$ and $P^*$) to the main symplectic leaf of the Heisenberg double ($D_{0}$) such that the symplectic forms on $P$, $P^{*}$ are obtained as the pull-back by those maps of the symplectic structure on $D_{0}$. This uncovers a duality between $P$ and $P^{*}$ under the exchange of bulk degrees of freedom of one model with boundary degrees of freedom of the other one. We finally solve the Poisson Sigma model for the Poisson structure on $G$ given by a pair of $r$-matrices that generalizes the Poisson-Lie case. The Hamiltonian analysis of the theory requires the introduction of a deformation of the Heisenberg double.