Dual branes in topological sigma models over Lie groups. BF-theory and non-factorizable Lie bialgebras

TL;DR

This paper completes the analysis of Poisson-Sigma models over Poisson-Lie groups, showing duality relations, classifying D-branes, and demonstrating the naturalness of non-coisotropic branes as boundary conditions.

Contribution

It provides a comprehensive study of dual branes in Poisson-Sigma models, including the equivalence of models to BF-theory and the classification of dual branes beyond coisotropic submanifolds.

Findings

Model over G* is equivalent to BF-theory.

Identifies dual branes as subgroups of G and G* not necessarily coisotropic.

Duality transformations connect coisotropic and non-coisotropic branes.

Abstract

We complete the study of the Poisson-Sigma model over Poisson-Lie groups. Firstly, we solve the models with targets $G$ and $G^*$ (the dual group of the Poisson-Lie group $G$) corresponding to a triangular $r$-matrix and show that the model over $G^*$ is always equivalent to BF-theory. Then, given an arbitrary $r$-matrix, we address the problem of finding D-branes preserving the duality between the models. We identify a broad class of dual branes which are subgroups of $G$ and $G^*$, but not necessarily Poisson-Lie subgroups. In particular, they are not coisotropic submanifolds in the general case and what is more, we show that by means of duality transformations one can go from coisotropic to non-coisotropic branes. This fact makes clear that non-coisotropic branes are natural boundary conditions for the Poisson-Sigma model.