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

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Abstract

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 μ-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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Yasuyuki Kawahigashi

  2. Dipartimento di Matematica Università, di Roma ‘‘Tor Vergata’’ Via della Ricerca Scientifica, 1, 00133, Roma, Italy

    Roberto Longo

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  1. Yasuyuki Kawahigashi
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  2. Roberto Longo
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Correspondence to Roberto Longo.

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Communicated by A. Connes

Supported in part by JSPS.

Supported in part by GNAMPA and MIUR.

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Kawahigashi, Y., Longo, R. Classification of Two-Dimensional Local Conformal Nets with c < 1 and 2-Cohomology Vanishing for Tensor Categories. Commun. Math. Phys. 244, 63–97 (2004). https://doi.org/10.1007/s00220-003-0979-1

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  • DOI: https://doi.org/10.1007/s00220-003-0979-1

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