Topological sectors and a dichotomy in conformal field theory

Let A be a local conformal net of factors on S-1 with the split property. We provide a topological construction of soliton representations of the n-fold tensor product Ax ... x A, that restrict to true representations of the cyclic orbifold (Ax ... x A)(Zn). We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the symmetrized tensor product (A x A)(flip) has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the mu-index of A is finite then A turns out to be strongly additive. By [31], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is nondegenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.