  We show that the Jones-Wassermann subfactors for disconnected intervals,
which are constructed from the representations of loop groups of type $A$, are
finite-depth subfactors. The index value and the dual principal graphs of these
subfactors are completely determined. The square root of the index value in the
case of two disjoint intervals for vacuum representation is the same as the
Quantum 3-manifold invariant of type $A$ evaluated on $S^1\times S^2$.
