  Let A be a local conformal net of factors on the circle with the split
property. We provide a topological construction of soliton representations of
the tensor product of n copies of A, that restrict to true representations of
subnet inviant under cyclic permutations (cyclic orbifold). 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 the cyclic orbifold 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 finite then A turns out to be strongly additive. By [KLM], if A
is rational then the tensor category of representations of A is automatically
modular, namely the braiding symmetry is non-degenerate. In interesting cases,
we compute the fusion rules of the topological solitons and show that they
determine all twisted sectors of the cyclic orbifold.
