The local information of a 2d rational conformal field theory (RCFT) is encoded in a vertex operator algebra, whose modules constitute a modular fusion category C. The collection of global observables of the theory is given by conformal blocks and carries actions of mapping class groups, which is described mathematically by a modular functor that assigns the Drinfeld center Z(C) to a circle. The string-net construction, first appeared in the study of topological phases of matter, not only provides such a modular functor but also supplies a graphical construction of correlators. A generalization of the string-net construction takes a pivotal bicategory as input. When such a bicategory is taken to be C (considered as a bicategory), it recovers the modular functor of conformal blocks. On the other hand, the modular functor associated with the Morita bicategory of separable symmetric Frobenius algebras internal to C classifies stratified worldsheets up to "categorical symmetries". In this talk we explain, using the framework of double categories, that RCFT correlators exhibit an equivalence between these modular functors. More generally, the modular functors associated with a pivotal bicategory and its orbifold completion are canonically equivalent.
