Classification of Local Conformal Nets. Case c < 1

2002/01/31 by Yasuyuki Kawahigashi, Roberto Longo · 2 citations
Physics and Astronomy · Mathematics · #math-ph #hep-th #math.MP #math.OA #msc:81T40 #msc:46L37 #msc:17B68 #msc:81T08 #msc:46L60 #msc:22E67 #msc:81R10

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published as Annals of Mathematics 160 (2004), 493-522 · 30 pages, LaTeX2e

arxiv created 2003/04/02 · arxiv updated 2016/09/07

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.

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• Twisted representations of conformal nets and crossed balanced tensor categories
• Balanced tensor categories of representations of fixed-points conformal nets