Classification of Local Conformal Nets. Case c < 1

TL;DR

This paper provides a complete classification of local conformal nets with central charge less than 1, linking them to specific Dynkin diagrams and using advanced algebraic methods.

Contribution

It introduces a full classification of irreducible conformal nets for c<1, connecting them to Dynkin diagrams and coset constructions, advancing the understanding of conformal field theories.

Findings

Classification of nets corresponds to pairs of Dynkin diagrams with Coxeter number difference of 1.

Identification of nets generated by Virasoro algebra representations with coset nets.

Explicit classification of all local irreducible extensions for c<1.

Abstract

We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of A-D_{2n}-E_{6,8} Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c<1 with certain coset nets. Then, by using the classification of modular invariants for the minimal models by Cappelli-Itzykson-Zuber and the method of alpha-induction in subfactor theory, we classify all local irreducible extensions of the Virasoro nets for c<1 and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the literature.