Modular Invariants, Graphs and $\alpha$-Induction for Nets of Subfactors. III

TL;DR

This paper advances the theory of $eta$-induction for nets of subfactors, linking sector systems, braiding, and principal graphs, with applications to conformal embeddings of SU(n) WZW models and modular invariants.

Contribution

It develops a relative braiding between sector systems, expresses principal graphs in terms of induced sectors, and establishes a formula for modular invariants in conformal embeddings.

Findings

Constructed a relative braiding for sector systems.

Expressed principal graphs via induced sectors.

Derived a formula for modular invariants in specific models.

Abstract

In this paper we further develop the theory of $\alpha$-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two ``chiral'' induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU(n) WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices i.e. on the ``physical spectrum'' of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU(n) and many conformal inclusions, covering in particular all type I modular invariants of SU(2) and SU(3), and we conjecture that it holds also for any other conformal inclusion of SU(n) as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion $SU(3)_5 \subset SU(6)_1$ is computed for the first time.