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We construct a Chern-Simons type of theory using the $l_\infty$ algebraencoded by a Poisson structure on arbitrary Riemann surfaces with boundaries. Adeformation quantization within the Batalin-Vilkovisky framework is performedby constructing propagators with Dirichlet boundary condition onFulton-MacPherson compactified configuration space. Our results show that theBV quantization is independent of several gauge choices in propagators, whichleads to global observables that are candidates for geometric invariants ofPoisson structure and topological invariants for the worldsheet structure. Atthe level of local observables, a Swiss-Cheese algebra structure has beenidentified. If the Poisson structure is symplectic, the two-dimensional theoryis homotopic to a boundary theory. This is known in the classical case, and weconfirm that the quantum homotopy exists as well.

A holography theory of Poisson sigma model and deformation quantization