Classification of local conformal nets. Case $c < 1$

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 $c\lt 1$ 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 $c\lt 1$ and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the literature.