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Gauge Fixing and Classical Dynamical r-Matrices in ISO(2, 1)-Chern–Simons Theory

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Abstract

We apply the Dirac gauge fixing procedure to Chern–Simons theory with gauge group ISO(2, 1) on manifolds \({\mathbb{R} \times S}\), where S is a punctured oriented surface of general genus. For all gauge fixing conditions that satisfy certain structural requirements, this yields an explicit description of the Poisson structure on the moduli space of flat ISO(2, 1)-connections on S in terms of classical dynamical r-matrices for \({\mathfrak{iso}}\) (2, 1). We show that the Poisson structures and classical dynamical r-matrices arising from different gauge fixing conditions are related by dynamical ISO(2, 1)-valued transformations that generalise the usual gauge transformations of classical dynamical r-matrices. By means of these transformations, it is possible to classify all Poisson structures and classical dynamical r-matrices obtained from such gauge fixings. Generically these Poisson structures combine classical dynamical r-matrices for non-conjugate Cartan subalgebras of \({\mathfrak{iso}}\)(2, 1).

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  1. Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstraße 11, 91058, Erlangen, Germany

    C. Meusburger & T. Schönfeld

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  1. C. Meusburger
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  2. T. Schönfeld
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Correspondence to C. Meusburger.

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Communicated by N. Reshetikhin

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Meusburger, C., Schönfeld, T. Gauge Fixing and Classical Dynamical r-Matrices in ISO(2, 1)-Chern–Simons Theory. Commun. Math. Phys. 327, 443–479 (2014). https://doi.org/10.1007/s00220-014-1938-8

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