  We study moduli spaces of flat connections on surfaces with boundary, with
boundary conditions given by Lagrangian Lie subalgebras. The resulting
symplectic manifolds are closely related with Poisson-Lie groups and their
algebraic structure (such as symplectic groupoid structure) gets a geometrical
explanation via 3-dimensional cobordisms. We give a formula for the symplectic
form in terms of holonomies, based on a central extension of the gauge group by
closed 2-forms. This construction is finally used for a certain extension of
the Morita equivalence of quantum tori to the world of Poisson-Lie groups.

Symplectic Geometry