The possible values of the Jones indices of subfactors have been studied since 1980's and they are now completely determined. That is, the possible values are of the form 4cos2π/n, n=3,4,5,..., when they are below 4, and all the values larger than or equal to 4 are also realized. In the operator algebraic approach to conformal field theory, we study local conformal nets, which are certain families of von Neumann algebras, and we have studied the problem to determine the possible values of the Jones indices of local conformal nets of subfactors with Carpi and Longo. It is easy to see tha all positive integer values are realized. Besides these values, we have shown that the two smallest values are 4cos2π/10, and 3+31/2. In connection to this result, we have also classified possible braiding structures on the objects corresponding to the even vertices of the Dynkin diagrams for the A-D-E subfactors.
We have also continued our operator algebraic studies on boundary conformal field theory and superconformal field theory.