  We describe the structure of the inclusions of factors A(E) contained in
A(E')' associated with multi-intervals E of R for a local irreducible net A of
von Neumann algebras on the real line satisfying the split property and Haag
duality. In particular, if the net is conformal and the subfactor has finite
index, the inclusion associated with two separated intervals is isomorphic to
the Longo-Rehren inclusion, which provides a quantum double construction of the
tensor category of superselection sectors of A. As a consequence, the index of
A(E) in A(E')' coincides with the global index associated with all irreducible
sectors, the braiding symmetry associated with all sectors is non-degenerate,
namely the representations of A form a modular tensor category, and every
sector is a direct sum of sectors with finite dimension. The superselection
structure is generated by local data. The same results hold true if conformal
invariance is replaced by strong additivity and there exists a modular PCT
symmetry.
