Starting from an extensive description of rational D4-branes in toric Calabi-Yau hypersurfaces, we compute the dimension of their moduli space both in geometry and in conformal field theory. The result yields compelling evidence for the decoupling conjecture stating that this dimension is independent of the Kähler structure of the Calabi-Yau space. By looking at particular cases, we also find which boundary states have an interpretation in terms of complexes of sheaves, supporting the picture advocated by Douglas.