Theta function parameterization and fusion for 3-D integrable Boltzmann weights

Details Theta function parameterization and fusion for 3-D integrable Boltzmann weights Theta function parameterization and fusion for 3-D integrable Boltzmann weights

2003-05-20

arxiv.org/abs/nlin/0305031v2

J.Phys.A37:1159-1179,2004

Nonlinear Sciences

Exactly Solvable and Integrable Systems

24 pages, 5 figures, typos corrected

Scientific paper

10.1088/0305-4470/37/4/005

We report progress in constructing Boltzmann weights for integrable 3-dimensional lattice spin models. We show that a large class of vertex solutions to the modified tetrahedron equation can be conveniently parameterized in terms of N-th 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 parameterization allows to write a simple formula for fused Boltzmann weights R 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 R satisfy the normal tetrahedron equation. The scheme described contains the Zamolodchikov-Baxter-Bazhanov model and the Chessboard model as special cases.

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