88 research outputs found

    Classification of Two-dimensional Local Conformal Nets with c<1 and 2-cohomology Vanishing for Tensor Categories

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    We classify two-dimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of A-D-E Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of one-dimensional local conformal nets, Dynkin diagrams D_{2n+1} and E_7 do not appear, but now they do appear in this classification of two-dimensional local conformal nets. Such nets are also characterized as two-dimensional local conformal nets with mu-index equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for one-dimensional nets, is 2-cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.Comment: 40 pages, LaTeX 2

    Classification of Local Conformal Nets. Case c < 1

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    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.Comment: 30 pages, LaTeX2

    α\alpha-induction for bi-unitary connections

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    The tensor functor called α\alpha-induction arises from a Frobenius algebra object, or a Q-system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of NN to MM arising from a subfactor N⊂MN\subset M of finite index and finite depth giving a braided fusion category of endomorpshisms of NN. It is also understood in terms of Ocneanu's graphical calculus. We study this α\alpha-induction for bi-unitary connections, which give a characterization of finite-dimensional nondegenerate commuting squares and gives certain 4-tensors appearing in recent studies of 2-dimensional topological order. We show that the resulting α\alpha-induced bi-unitary connections are flat if we have a commutative Frobenius algebra, or a local Q-system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.Comment: 32 pages, more explanations have been adde
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