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

Abstract

We report progress in constructing Boltzmann weights for integrable 3dimensional 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.

Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

“…The other known solutions, previously found by Hietarinta [9] and Korepanov [10], were shown to be special cases of [8]. A review of the most recent activity related to the TE can be found in the work of von Gehlen, Pakulyak and Sergeev [11].…”

“…It is particularly useful for constructing integrable evolution systems on a 2 + 1-dimensional lattice and can be viewed as a three-dimensional lattice analog of the Lax-Zakharov-Shabat zero-curvature condition in two dimensions (see [44] for further details). Suppose now, that the algebra A has a "quasi-classical" limit when it degenerates into a Poisson algebra P. In this limit all the generators v i , i = a, b, c, become commutative formal variables, and the relation (11) becomes a c-number equation which very much resembles the usual quantum Yang-Baxter equation (but does not coincide with it). The quasi-classical limit of the map (7) defines a symplectic transformation of the tensor cube of the Poisson algebra P, which, of course, solves the same functional TE (9).…”

“…The approach we employ in this paper can be viewed as a "reverse engineering" of the above procedure. We start from the c-number equation (11) with a simple particular Ansatz for the matrices L 12 , L 13 and L 23 acting in the product of two-dimensional vector spaces…”

“…Surprisingly, only minor modifications of the above symplectic transformation and the form of the matrices L are required to obtain the quantum map (7) for the tensor cube of H q , which solves the quantum variant of the equation (11). Finally, we obtain the corresponding solution of the TE (3) by using an explicit form of the quantum map (7) to calculate matrix elements of the operator R abc for the Fock representation of the q-oscillator algebra (13).…”

“…The paper is organized as follows. The solution of the local Yang-Baxter equation (11) with commutative variables (classical case) is given in Sect.2. The quantum case is considered in Sect.3.…”