TAILIEUCHUNG - Đề tài " Classification of local conformal nets. Case c

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-D2n-E6,8 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 | Annals of Mathematics Classification of local conformal nets. Case c 1 By Yasuyuki Kawahigashi and Roberto Longo Annals of Mathematics 160 2004 493 522 Classification of local conformal nets. Case c 1 By YASuyuKi Kawahigashi and Roberto Longo Dedicated to Masamichi Takesaki on the occasion of his seventieth birthday 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-D2n-E6 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 o-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. 1. Introduction Conformal field theory on S1 has been extensively studied in recent years by different methods with important motivation coming from various branches of theoretical physics two-dimensional critical phenomena holography . and mathematics quantum groups subfactors topological invariants in three dimensions . . In various approaches to the subject it is unclear whether different models are to be regarded as equivalent or to contain the same physical information. This becomes clearer by considering the operator algebra generated by smeared fields localized in a given interval I of S1 and taking its closure A I in the weak operator topology. The relative positions of the various von Neumann The first author was supported in part by the Grants-in-Aid for Scientific Research JSPS. The second author was supported in part by the Italian MIUR and .

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