  We construct a Chern-Simons type of theory using the $l_\infty$ algebra
encoded by a Poisson structure on arbitrary Riemann surfaces with boundaries. A
deformation quantization within the Batalin-Vilkovisky framework is performed
by constructing propagators with Dirichlet boundary condition on
Fulton-MacPherson compactified configuration space. Our results show that the
BV quantization is independent of several gauge choices in propagators, which
leads to global observables that are candidates for geometric invariants of
Poisson structure and topological invariants for the worldsheet structure. At
the level of local observables, a Swiss-Cheese algebra structure has been
identified. If the Poisson structure is symplectic, the two-dimensional theory
is homotopic to a boundary theory. This is known in the classical case, and we
confirm that the quantum homotopy exists as well.
