  We apply the theory of $\alpha$-induction of sectors which we elaborated in
our previous paper to several nets of subfactors arising from conformal field
theory. The main application are conformal embeddings and orbifold inclusions
of SU(n) WZW models. For the latter, we construct the extended net of factors
by hand. Developing further some ideas of F. Xu, our treatment leads
canonically to certain fusion graphs, and in all our examples we rediscover the
graphs Di Francesco, Petkova and Zuber associated empirically to the
corresponding SU(n) modular invariants. We establish a connection between
exponents of these graphs and the appearance of characters in the
block-diagonal modular invariants, provided that the extended modular
S-matrices diagonalize the endomorphism fusion rules of the extended theories.
This is proven for many cases, and our results cover all the block-diagonal
SU(2) modular invariants, thus provide some explanation of the A-D-E
classification.
