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

June 08, 2015

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 $\mathrm{SU}(2)_k$.

Keywords:

rational conformal field theories, chiral conformal field theory, subfactors, conformal nets