A Remark on CFT Realization of Quantum Doubles of Subfactors. Case Index < 4

It is well-known that the quantum double $D(N\subset M)$ of a finite depth subfactor $N\subset M$, or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. Thus should arise in conformal field theory. We show that for every subfactor $N\subset M$ with index $[M:N]<4$ the quantum double $D(N\subset M)$ 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 $\mathrm{S}\mathrm{U}(2{)}_{10}\times \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(11{)}_1$ 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 $N\subset M$ 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 $D(N\subset M)$ is realized by completely rational net. We construct completely rational nets with the opposite braiding of $\mathrm{S}\mathrm{U}(2{)}_k$ and use the well-known fact that all subfactors with index $[M:N]<4$ arise by $\alpha$-induction from $\mathrm{S}\mathrm{U}(2{)}_k$.