Abstract: One can place a BV theory on a cylinder I x Sigma and, under some assumptions, integrate out the modes along the cylinder to obtain an effective theory along Sigma. This theory depends on the choice of a polarization, and couples to the boundary conditions. I will briefly recall the setup and some older examples published in joint work with Schiavina, Mnev, and Cattaneo. Then I will discuss recent work with Cabrera and Cueca to apply this to split Chern-Simons theory with topological boundary conditions, where one recovers the Poisson Sigma Model for the Poisson-Lie group integrating the Lie bialgebra. I will comment on possible applications, and generalizations, of this observation.