Poisson-Lie T-duality: Open strings and D-branes

Global issues of the Poisson-Lie T-duality are addressed. It is shown that oriented open strings propagating on a group manifold G are dual to D-brane-anti-D-brane pairs propagating on the dual group manifold G. The D-branes coincide with the symplectic leaves of the standard Poisson structure induced on the dual group G by the dressing action of the group G. The whole picture is then extended to the full modular space M(D) of the Poisson-Lie equivalent σ-models which is the space of all Manin triples of a given Drinfeld double. In this more general case the D-branes living on group targets from M(D) are preimages of symplectic leaves in certain Poisson homogeneous spaces of their targets.