On the Jones index values for conformal subnets

We consider the smallest values taken by the Jones index for an inclusion of local conformal nets of von Neumann algebras on S^1 and show that these values are quite more restricted than for an arbitrary inclusion of factors. Below 4, the only non-integer admissible value is 4\cos^2 \pi/10, which is known to be attained by a certain coset model. Then no index value is possible in the interval between 4 and 3 +\sqrt{3}. The proof of this result based on \alpha-induction arguments. In the case of values below 4 we also give a second proof of the result. In the course of the latter proof we classify all possible unitary braiding symmetries on the A D E tensor categories, namely the ones associated with the even vertices of the A_n, D_{2n}, E_6, E_8 Dynkin diagrams.