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Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral su(2) WZNW Model

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Abstract

A zero modes’ Fock space \(\mathcal{F}_{q}\) is constructed for the extended chiral \({su(2)}\) WZNW model. It gives room to a realization of the fusion ring of representations of the restricted quantum universal enveloping algebra \(\bar{U}_{q}=\bar{U}_{q} sl(2)\) at an even root of unity, \(q^h=-1,\) and of its infinite dimensional extension \(\tilde{U}_{q}\) by the Lusztig operators \(E^{(h)}, F^{(h)}.\) We provide a streamlined derivation of the characteristic equation for the Casimir invariant from the defining relations of \(\bar{U}_{q}.\) A central result is the characterization of the Grothendieck ring of both \(\bar{U}_{q}\) and \(\tilde{U}_{q}\) in Theorem 3.1. The properties of the \(\tilde{U}_{q}\) fusion ring in \(\mathcal{F}_{q}\) are related to the braiding properties of correlation functions of primary fields of the conformal \(\widehat{su}(2)_{h-2}\) current algebra model.

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Authors and Affiliations

  1. Dipartimento di Fisica Teorica dell’, Università degli Studi di Trieste, Strada Costiera 11, 34014, Trieste, Italy

    Paolo Furlan

  2. Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste, Trieste, Italy

    Paolo Furlan & Ludmil Hadjiivanov

  3. Theoretical and Mathematical Physics Division, Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, 1784, Sofia, Bulgaria

    Ludmil Hadjiivanov & Ivan Todorov

  4. International School for Advanced Studies (SISSA/ISAS), via Beirut 2-4, 34014, Trieste, Italy

    Ludmil Hadjiivanov

Authors
  1. Paolo Furlan
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  2. Ludmil Hadjiivanov
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  3. Ivan Todorov
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Correspondence to Ludmil Hadjiivanov.

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Furlan, P., Hadjiivanov, L. & Todorov, I. Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral su(2) WZNW Model. Lett Math Phys 82, 117–151 (2007). https://doi.org/10.1007/s11005-007-0209-4

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