S-matrix in orbifold theory

Let V be a simple vertex operator algebra and G a finite automorphism group of V such that V^G is regular. It is proved that every irreducible V^G module occurs in an irreducible g-twisted V-module for some g in G and the global dimension of V is independent of any finite order automorphisim of V. Moreover, the S-matrix associated to a rational orbifold theory is determined. In the case of permutation orbifolds, the S-matrix is given explicitly in terms of the S matrices of the original vertex operator algebra. This is a joint work with Chongying Dong and Feng Xu.