Uniqueness of open/closed rational CFT with given algebra of open states

We study the sewing constraints for rational two-dimensional conformal field theory on oriented surfaces with possibly nonempty boundary. The boundary condition is taken to be the same on all segments of the boundary. The following uniqueness result is established: for a solution to the sewing constraints with nondegenerate closed state vacuum and nondegenerate two-point correlators of boundary fields on the disk and of bulk fields on the sphere, up to equivalence all correlators are uniquely determined by the one-, two-and three-point correlators on the disk.