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.

Authors

Yasuyuki Kawahigashi

Department of Mathematical Sciences
University of Tokyo
Komaba
Tokyo
Japan

Roberto Longo

Department of Mathematics
University of Rome "Tor Vergata"
00133 Rome
Italy