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Topological Sectors and a Dichotomy in Conformal Field Theory

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Abstract

Let be a local conformal net of factors on S1 with the split property. We provide a topological construction of soliton representations of the n-fold tensor product that restrict to true representations of the cyclic orbifold We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then is not completely rational iff the symmetrized tensor product has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of have a conjugate sector then either is completely rational or has uncountably many different irreducible sectors. Thus is rational iff is completely rational. In particular, if the μ-index of is finite then turns out to be strongly additive. By [31], if is rational then the tensor category of representations of is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.

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Acknowledgments.

The first named author would like to thank S. Carpi, F. Fidaleo, Y. Kawahigashi and L. Zsido for comments. He also thanks Sorin Popa for the invitation and warm hospitality at UCLA in May 2003 while this work was in progress.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133, Roma, Italy

    Roberto Longo

  2. Department of Mathematics, University of California at Riverside, Riverside, CA, 92521, USA

    Feng Xu

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  1. Roberto Longo
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Correspondence to Roberto Longo.

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Communicated by Y. Kawahigashi

Supported in part by GNAMPA-INDAM and MIUR

Supported in part by NSF

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Longo, R., Xu, F. Topological Sectors and a Dichotomy in Conformal Field Theory. Commun. Math. Phys. 251, 321–364 (2004). https://doi.org/10.1007/s00220-004-1063-1

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