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

TL;DR

This paper develops a mathematical framework for understanding how sectors in nets of subfactors behave under induction and restriction, linking algebraic structures to conformal field theory and modular invariants.

Contribution

It introduces formulas for induced sectors, reciprocity relations, and homomorphism properties, advancing the algebraic understanding of nets of subfactors and their applications in conformal field theory.

Findings

Derived a formula for induced sectors in terms of original sectors

Established a reciprocity formula for induction and restriction

Proved a homomorphism property of the induction mapping

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.