An elliptic parameterisation of the Zamolodchikov model

Vladimir V. Bazhanov, Vladimir V. Mangazeev, Yuichiro kada, Sergey Sergeev

    Research output: Contribution to journalArticle

    2 Citations (Scopus)

    Abstract

    The Zamolodchikov model describes an exact relativistic factorized scattering theory of straight strings in (2+1)-dimensional space–time. It also defines an integrable 3D lattice model of statistical mechanics and quantum field theory. The three-string S-matrix satisfies the tetrahedron equation which is a 3D analog of the Yang–Baxter equation. Each S-matrix depends on three dihedral angles formed by three intersecting planes, whereas the tetrahedron equation contains five independent spectral parameters, associated with angles of an Euclidean tetrahedron. The vertex weights are given by rather complicated expressions involving square roots of trigonometric function of the spectral parameters, which is quite unusual from the point of view of 2D solvable lattice models. In this paper we consider a particular four-parameter specialisation of the tetrahedron equation when one of its vertices goes to infinity and the tetrahedron itself degenerates into an infinite prism. We show that in this limit all the vertex weights in the tetrahedron equation can be represented as meromorphic functions on an elliptic curve. Moreover we show that a special reduction of the tetrahedron equation in this case leads precisely to an example of the tetrahedral Zamolodchikov algebra, previously constructed by Korepanov. This algebra plays important role for a “layered” construction of the Shastryʼs R-matrix and the 2D S-matrix appearing in the problem of the ADS/CFT correspondence for N=4 SUSY Yang–Mills theory in four dimensions. Possible applications of our results in this field are briefly discussed
    Original languageEnglish
    Pages (from-to)127-144
    Number of pages18
    JournalNuclear Physics B
    Volume871
    Issue number1
    DOIs
    Publication statusPublished - 1 Jun 2013

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    parameterization
    tetrahedrons
    apexes
    algebra
    strings
    matrices
    meromorphic functions
    trigonometric functions
    statistical mechanics
    infinity
    prisms
    dihedral angle
    analogs
    curves
    scattering

    Cite this

    Bazhanov, V. V., Mangazeev, V. V., kada, Y., & Sergeev, S. (2013). An elliptic parameterisation of the Zamolodchikov model. Nuclear Physics B, 871(1), 127-144. https://doi.org/10.1016/j.nuclphysb.2013.02.011
    Bazhanov, Vladimir V. ; Mangazeev, Vladimir V. ; kada, Yuichiro ; Sergeev, Sergey. / An elliptic parameterisation of the Zamolodchikov model. In: Nuclear Physics B. 2013 ; Vol. 871, No. 1. pp. 127-144.
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    Bazhanov, VV, Mangazeev, VV, kada, Y & Sergeev, S 2013, 'An elliptic parameterisation of the Zamolodchikov model', Nuclear Physics B, vol. 871, no. 1, pp. 127-144. https://doi.org/10.1016/j.nuclphysb.2013.02.011

    An elliptic parameterisation of the Zamolodchikov model. / Bazhanov, Vladimir V.; Mangazeev, Vladimir V.; kada, Yuichiro; Sergeev, Sergey.

    In: Nuclear Physics B, Vol. 871, No. 1, 01.06.2013, p. 127-144.

    Research output: Contribution to journalArticle

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