Modular invariants and subfactors

TL;DR

This paper explores the deep connection between modular invariants in conformal field theory and braided subfactors in operator algebras, providing a rigorous framework and explaining classical classification results.

Contribution

It introduces a subfactor approach using braided sector induction to rigorously derive properties of modular invariants and explains their relation to graph classifications.

Findings

Rigorous derivation of properties of modular invariants

Explanation of the A-D-E classification via subfactors

Discussion on the equivalence of classifications in conformal field theory

Abstract

In this lecture we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. Our analysis is based on an approach to modular invariants using braided sector induction ("$\alpha$-induction") arising from the treatment of conformal field theory in the Doplicher-Haag-Roberts framework. Many properties of modular invariants which have so far been noticed empirically and considered mysterious can be rigorously derived in a very general setting in the subfactor context. For example, the connection between modular invariants and graphs (cf. the A-D-E classification for $SU(2)_k$) finds a natural explanation and interpretation. We try to give an overview on the current state of affairs concerning the expected equivalence between the classifications of braided subfactors and modular invariant two-dimensional conformal field theories.