Topological Sectors and a Dichotomy in Conformal Field Theory

Let A be a local conformal net of factors on S1 with the split property. We provide a topological construction of soliton representations of the n-fold tensor product A⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ 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 ⊗ 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 μ-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 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. Supported in part by GNAMPA-INDAM and MIUR. Supported in part by NSF.