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 \(\mathrm{SU}(2)_{10}\times \mathrm{Spin}(11)_1\) 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 \(N\subset M \) 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 \(\mathrm{SU}(2)_k\) and use the well-known fact that all subfactors with index \([M:N]<4\) arise by \(\alpha\)-induction from \(\mathrm{SU}(2)_k\).