Multi-interval subfactors and modularity of representations in conformal field theory

We describe the structure of the inclusions of factors A (E) subset of A (E ')' associated with multi-intervals E subset of R for a local irreducible net A of von Neumann algebras on the real line satisfying the split propert y and Haag duality. In particular, if the net is conformal and the subfacto r has finite index, the inclusion associated with two separated intervals i s isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tenser category of superselection sectors of A. As a c onsequence, the index of A (E) subset of A(E ')' coincides with the global index associated with all irreducible sectors, the braiding symmetry associ ated with all sectors is non-degenerate, namely the representations of A fo rm a modular tenser category, and every sector is a direct sum of sectors w ith finite dimension. The superselection structure is generated by local da ta. The same results hold true if conformal invariance is replaced by stron g additivity and there exists a modular PCT symmetry.