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Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center

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Abstract

The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π into a ``pointwise'' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of .

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Authors and Affiliations

  1. Landau Institute for Theoretical Physics, Kosygina Str., 2, Moscow, 119334, Russia

    B.L. Feigin

  2. Physics Department, Moscow State University, Moscow, Russia

    A.M. Gainutdinov

  3. Lebedev Physics Institute, 53 Leninsky prospect, Moscow, 119991, Russia

    A.M. Semikhatov & I.Yu. Tipunin

Authors
  1. B.L. Feigin
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  2. A.M. Gainutdinov
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  3. A.M. Semikhatov
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  4. I.Yu. Tipunin
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Correspondence to B.L. Feigin.

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Communicated by L. Takhtajan

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Feigin, B., Gainutdinov, A., Semikhatov, A. et al. Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center. Commun. Math. Phys. 265, 47–93 (2006). https://doi.org/10.1007/s00220-006-1551-6

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  • DOI: https://doi.org/10.1007/s00220-006-1551-6

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