| Abstract: | It is well known that the quantum double ({D(Nsubset M)}) of a finite depth subfactor ({Nsubset M}), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor ({Nsubset M}) with index ({[M:N] < 4}) the quantum double ({D(Nsubset M)}) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of ({E_6}) can be realized as a ({mathbb{Z}_2})-simple current extension of ({{{rm SU}(2)}_{10}times {{rm Spin}(11)}_1}) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor ({Nsubset M }) arises from ({alpha})-induction of completely rational nets ({mathcal{A}subset mathcal{B}}) and there is a net ({tilde{mathcal{A}}}) with the opposite braiding, then the quantum ({D(Nsubset M)}) is realized by completely rational net. We construct completely rational nets with the opposite braiding of ({{{rm SU}(2)}_k}) and use the well-known fact that all subfactors with index ({[M:N] < 4}) arise by ({alpha})-induction from ({{{rm SU}(2)}_k}). |
