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

Marcel Bischoff
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 $\mathrm{SU}(2)_k$.
open access link Lett Math Phys 106(3), 341-363 (2016)
@article{Bischoff:2015lda, author = "Bischoff, Marcel", title = "{A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index ${ < 4}$}", journal = "Lett. Math. Phys.", volume = "106", year = "2016", number = "3", pages = "341-363", doi = "10.1007/s11005-016-0816-z", eprint = "1506.02606", archivePrefix = "arXiv", primaryClass = "math-ph", SLACcitation = "%%CITATION = ARXIV:1506.02606;%%" }

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