Modular Invariants from Subfactors:¶Type I Coupling Matrices and Intermediate Subfactors

Abstract:

A braided subfactor determines a coupling matrix Z which commutes with the S- and T-matrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a block-diagonal structure which can be reinterpreted as the diagonal coupling matrix with respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right maximal extensions and which determine “parent” coupling matrices Z ^± of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z ⁺=Z ⁻, then Z is related to Z ⁺ by an automorphism of the extended fusion rules. The intertwining relations of chiral branching coefficients between original and extended S- and T-matrices are also clarified. None of our results depends on non-degeneracy of the braiding, i.e. the S- and T-matrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z ⁺≠Z ⁻, and that Z need not be related to a type I invariant by such an automorphism.