The Modified Tetrahedron Equation and Its Solutions

Abstract

A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator R 1,2,3 in the space of a triple Weyl algebra. R 1,2,3 is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for R 1,2,3 follows without further calculation. If the Weyl parameter is taken to be a root of unity, R 1,2,3 decomposes into a matrix conjugation operator R 1,2,3 and a c-number functional mapping R (f ) 1,2,3 . The operator R 1,2,3 satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equations. R 1,2,3 can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.

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

“…as basic theta-function, and E(X, Y ) ∼ θ 1 (X − Y) as the prime form. This formulas simplify the definitions (25). Periodicity conditions (65) may be chosen…”