Modular invariants, graphs and alpha-induction for nets of subfactors. II

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