CONFORMAL SUBNETS AND INTERMEDIATE SUBFACTORS

ROBERTO LONGO

Abstract. Given an irreducible local conformal net A of von Neumann algebras on S^{1} and a finite-index conformal subnet B ⊂ A, we show that A is completely rational iff 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 [?] hold. Our proofs are based on an analysis of the net inclusion B ⊂ A; among other things we show that, for a fixed interval I, every von Neumann algebra R intermediate between B(I) and A(I) comes from an intermediate conformal net L between B and A with L(I) = 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 ℑ(N , M) of intermediate subfactors in an irreducible inclusion of factors N ⊂ M with finite Jones index [M : N]. We provide an explicit bound for the cardinality of ℑ(N , M) which depends only on [M : N].