### Abstract

We report progress in constructing Boltzmann weights for integrable three-dimensional lattice spin models. We show that a large class of vertex solutions to the modified tetrahedron equation (MTE) can be conveniently parametrized in terms of Nth roots of theta functions on the Jacobian of a compact algebraic curve. Fay's identity guarantees the Fermat relations and the classical equations of motion for the parameters determining the Boltzmann weights. Our parametrization allows us to write a simple formula for fused Boltzmann weights fractur R sign which describe the partition function of an arbitrary open box and which also obey the modified tetrahedron equation. Imposing periodic boundary conditions we observe that the fractur R sign satisfy the normal tetrahedron equation. The scheme described contains the Zamolodchikov-Baxter-Bazhanov model and the chessboard model as special cases.

Original language | English |
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Pages (from-to) | 1159-1179 |

Number of pages | 21 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 37 |

Issue number | 4 |

DOIs | |

Publication status | Published - 30 Jan 2004 |

Externally published | Yes |

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### Cite this

*Journal of Physics A: Mathematical and General*,

*37*(4), 1159-1179. https://doi.org/10.1088/0305-4470/37/4/005