Subfactor Realisation of Modular Invariants

Abstract: We study the problem of realising modular invariants by braided subfactors and the related problem of classifying nimreps. We develop the fusion rule structure of these modular invariants. This structure is a useful tool in the analysis of modular data from quantum double subfactors, particularly those of the double of cyclic groups, the symmetric group on 3 letters and the double of the subfactors with principal graph the extended Dynkin diagram D5(1). In particular for the double of S3, 14 of the 48 modular modular invariants are nimless, and only 28 of the remaining 34 nimble invariants can be realised by subfactors.