Modular Invariants, Graphs and $α$-Induction for Nets of Subfactors I
J. Böckenhauer, D. E. Evans
TL;DR
The paper develops a general α-induction framework for nets of subfactors in the Longo–Rehren setting, establishing a precise formula that connects α-induced sectors to the DHR sectors of the smaller net and the canonical endomorphisms, along with a reciprocity between α-induction and σ-restriction. It proves a homomorphism property of α-induction and shows how α-induced sectors interact with restricted sectors, including an inverse braiding analysis. The framework is then connected to conformal field theory applications, notably SU(n) WZW models, conformal embeddings, and orbifold constructions, providing a graph-based view that aligns with modular invariants and the A–D–E classification. The results offer a robust sector-algebra perspective and pave the way for labeling modular invariants by fusion graphs in broader CFT settings.
Abstract
We analyze the induction and restriction of sectors for nets of subfactors defined by Longo and Rehren. Picking a local subfactor we derive a formula which specifies the structure of the induced sectors in terms of the original DHR sectors of the smaller net and canonical endomorphisms. We also obtain a reciprocity formula for induction and restriction of sectors, and we prove a certain homomorphism property of the induction mapping. Developing further some ideas of F. Xu we will apply this theory in a forthcoming paper to nets of subfactors arising from conformal field theory, in particular those coming from conformal embeddings or orbifold inclusions of SU(n) WZW models. This will provide a better understanding of the labeling of modular invariants by certain graphs, in particular of the A-D-E classification of SU(2) modular invariants.