Conformal Subnets and Intermediate Subfactors
Roberto Longo
Given an irreducible local conformal net ${\mathcal A}$ of von Neumann algebras on $S^1$ and a finite-index conformal subnet ${\mathcal B}\subset{\mathcal A}$, we show that ${\mathcal A}$ is completely rational iff ${\mathcal B}$ is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in (Kawahigashi, Longo and Müger) hold. Our proofs are based on an analysis of the net inclusion ${\mathcal B}\subset{\mathcal A}$; among other things we show that, for a fixed interval $I$, every von Neumann algebra ${\mathcal R}$ intermediate between ${\mathcal B}(I)$ and ${\mathcal A}(I)$ comes from an intermediate conformal net ${\mathcal L}$ between ${\mathcal B}$ and ${\mathcal A}$ with ${\mathcal L}(I)={\mathcal R}$. We make use of Watatani's result, extended for the purpose from the type $II$ case to arbitrary factors, on the finiteness of the set $\mathfrak I({\mathcal N},{\mathcal M})$ of intermediate subfactors in an irreducible inclusion of factors ${\mathcal N}\subset{\mathcal M}$ with finite Jones index $[{\mathcal M}:{\mathcal N}]$. We provide an explicit bound for the cardinality of $\mathfrak I({\mathcal N},{\mathcal M})$ which depends only on $[{\mathcal M}:{\mathcal N}]$.