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

TL;DR

This paper demonstrates that for subfactors with index less than 4, their quantum doubles can be realized as representation categories of conformal nets, linking subfactor theory with conformal field theory and vertex operator algebras.

Contribution

It establishes a method to realize quantum doubles of subfactors with index less than 4 as conformal nets, including explicit constructions for the $E_6$ case, connecting subfactor theory with conformal field theory.

Findings

Quantum doubles of subfactors with index < 4 are realized as conformal nets.

Constructed a vertex operator algebra for each such subfactor.

Showed that the quantum double of $E_6$ is a simple current extension of known models.

Abstract

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