ITEP-TH-26/03 BONN-TH-02-2003

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

G. von Gehlen^{a,}^{1}, S. Pakuliak^{b,c,2}, S. Sergeev^{b,d,e,3}

aPhysikalisches Institut der Universität Bonn, Nussallee 12, D-53115 Bonn, Germany bBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,

Dubna 141980, Moscow region, Russia c Institute of Theoretical and Experimental Physics, Moscow 117259, Russia dMax-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany e Research School of Physical Sciences and Engineering, The Australian National University, Oliphant Building 60, Canberra ACT 0200, Australia

Received: June 14, 2003

Abstract

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.

AMS subject classes: 82B23, 70H06; keywords: tetrahedron equation, 3D integrable models PACS: 05.45-a, 05.50+q

Introduction

The tetrahedron equation is the three dimensional generalization of the Yang-Baxter equation which guarantees the existence of commuting transfer matrices. The importance of Yang-Baxter equations for modern mathematics and for mathematical physics is well known. However, the nature of the tetrahedron equation is much less understood, this mainly because it is a much more complicated equation.

Given the physical interest to understand the nature of the singularities which give rise to 3-D phase transitions, any effort which gets us closer to analytic results for 3-D statistical systems seems worthwhile. What has been achieved recently, is to construct large classes of 3-D solvable models with ℤ_{N} spin variables and to streamline the otherwise complicated formalism. The goal of an analytic calculation of partition functions and order parameters is not yet close by. The only available result from Baxter [26] does not lend itself to generalizations.

1e-mail: gehlen@th.physik.uni-bonn.de; ^{2}pakuliak@thsun1.jinr.ru; ^{3}sergeev@thsun1.jinr.ru

The first solution of the tetrahedron equation was obtained in 1980 by A. Zamolodchikov [1, 2] and then generalized by R. Baxter and V. Bazhanov [3] and others [4]. These models have ℤ_{N}-spin variables and solve the IRC (Interaction Round a Cube) version of the tetrahedron equation. However, later also the solution of the dual equation, the vertex tetrahedron equation, was obtained [5], generalizing several vertex solutions known previously [6, 7]. Here we shall consider only vertex-type solutions, which are usually denoted by R, the symmetry will include a ℤ_{N}. In general these R-matrices obey the so called “simple modified tetrahedron equation” which recently has been investigated in [9]. The modified tetrahedron equation (MTE) allows us to obtain the ordinary tetrahedron equation for composite weights or vertices. In the IRC formulation this has been shown in [10, 11], while the most simple vertex case was considered in [12].

In this paper we shall introduce a new convenient theta-function parameterization of general R operators. This parameterization will allows us to define fused weights ℜ, which are partition functions of open cubes of size M^{3}, and which obey a MTE. In special cases the ℜ solve an ordinary tetrahedron equation.

The vertex matrix ℜ ∈ End (ℂ^{3 N M^{2}}) is parameterized in the terms of N-th roots of theta-functions on the Jacobian of a genus g = (M − 1)^{2} compact algebraic curve Γ_{g}. The divisors of three meromorphic functions on Γ_{g} play the role of the spectral parameters for ℜ. An additional parameter of ℜ is an arbitrary v ∈ Jac(Γ_{g}). The tetrahedron equation for ℜ holds due to M^{4} simple modified tetrahedron equations. In the case when M = 1 and therefore Γ_{g} = S_{2}, the solution of the simple tetrahedron equation of [5] is reproduced.

This paper is organized as follows. In Sec.1 we recall the definitions of the vertex formulation of 3-D integrable ZBB model and sketch the derivation of the matrix operator R_{ijk} from a current conservation principle and Z-invariance. It satisfies the MTE and can be parameterized by quadrangle line-sections. In Sec.2 we introduce the parameterization of R_{ijk} in terms of theta functions. The Fermat relation and the Hirota equations are written as Fay identities. In Sec.3 we show that a theta function parameterization allows a compact formulation of the fusion of many R_{ijk} to the Boltzmann weight ℜ of a whole open cube which satisfies a MTE. Finally, in Sec.4 we first consider the special case of vanishing Jacobi transforms, in which ℜ satisfies a simple TE. Then we discuss the rational case and the relation to Chessboard models.

1 The R-matrix and its parameterization

1.1 The R-matrix of the vertex ZBB model

We start recalling the vertex formulation of the BBZ-model [5]. We consider a 3-dimensio-nal lattice with the elementary cell defined by three non-coplanar vectors e_{1}, e_{2}, e_{3} and general vertices

n = n_{1} e_{1} + n_{2} e_{2} + n_{3} e_{3},  n_{1}, n_{2}, n_{3} ∈ ℤ.  (1)

We label the directed link along e_{j} starting from n by (j, n). On these links there are spin variables σ_{j, n} which take values in ℤ_{N}. The partition function is defined by

Z = ∑_{{σ}} ∏_{n} ⟨σ_{1,n}, σ_{2,n+e_{2}}, σ_{3,n}| R |σ_{1,n+e_{1}}, σ_{2,n}, σ_{3,n+e_{3}}⟩ .  (2)

where R is an operator (which in the ZBB model is independent of n) mapping the initial three spin variables to the the three final ones, so that

R_{σ_{1},σ_{2},σ_{3}}^{σ′_{1},σ′_{2},σ′_{3}} = ⟨σ_{1}, σ_{2}, σ_{3}|R|σ′_{1}, σ′_{2}, σ′_{3}⟩  (3)

is a N^{3} × N^{3} matrix independent of n.

For the vertex ZBB-model, (3) can be expressed as a kind of cross ratio of four cyclic functions W_{p}(n). Introduce a two component vector p = (x, y) which is restricted to the Fermat curve

x^{N} + y^{N} = 1 .  (4)

Then define the function W_{p}(n) by

W_{p}(0) = 1,  W_{p}(n) = ∏_{ν=1}^{n} \frac{y}{1 − q^{ν} x} for n > 0.  (5)

where q = e^{2π i/N}  (6)

is the primitive N−th root of unity. Because of the Fermat curve restriction, W_{p}(n) is cyclic in n:

W_{p}(n + N) = W_{p}(n).

Now R = R(p_{1}, p_{2}, p_{3}, p_{4}) is defined by the following matrix function of four Fermat points p_{1}, p_{2}, p_{3}, p_{4}

R_{σ_{1},σ_{2},σ_{3}}^{σ′_{1},σ′_{2},σ′_{3}} {def \atop =} δ_{σ_{2}+σ_{3},σ′_{2}+σ′_{3}} q^{(σ′_{1}−σ_{1}) σ′_{3}} \frac{W_{p_{1}}(σ_{2} − σ_{1}) W_{p_{2}}(σ′_{2} − σ′_{1})}{W_{p_{3}}(σ′_{2} − σ_{1}) W_{p_{4}}(σ_{2} − σ′_{1})} ,  (7)

where x-coordinates of four Fermat curve points in (7) are identically related by

x_{1} x_{2} = q x_{3} x_{4} .  (8)

So the matrix elements R_{σ_{1},σ_{2},σ_{3}}^{σ′_{1},σ′_{2},σ′_{3}} depend on three complex numbers. These correspond to Zamolodchikov’s spherical angles in the IRC-formulation of the BBZ-model [5].

The structure of the indices of the matrix (7) allows one to consider R as the operator acting in the tensor product of three vector spaces

V = ℂ^{N} , R ∈ End (V ⊗ V ⊗ V)  (9)

It is conventional to enumerate naturally the components of the tensor product of several vector spaces, so that (7) are the matrix elements of R = R_{123}. Of course, R_{123} acts trivially on the all other vector spaces if one considers V^{⊗Δ} for some arbitrary Δ.

(7) is known as the R-matrix of the Zamolodchikov - Bazhanov - Baxter model, see [5]. The proof that (7) satisfies the Tetrahedron Equation

∑_{j_{1}...j_{6}} R_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}}(p^{(1)}) R_{j_{1},i_{4},i_{5}}^{k_{1},j_{4},j_{5}}(p^{(2)}) R_{j_{2},j_{4},i_{6}}^{k_{2},k_{4},j_{6}}(p^{(3)}) R_{j_{3},j_{5},j_{6}}^{k_{3},k_{5},k_{6}}(p^{(4)})

= ∑_{j_{1}...j_{6}} R_{i_{3},i_{5},i_{6}}^{j_{3},j_{5},j_{6}}(p^{(4)}) R_{i_{2},i_{4},j_{6}}^{j_{2},j_{4},k_{6}}(p^{(3)}) R_{i_{1},j_{4},j_{5}}^{j_{1},k_{4},k_{5}}(p^{(2)}) R_{j_{1},j_{2},j_{3}}^{k_{1},k_{2},k_{3}}(p^{(1)}) .  (10)

is rather tedious [5]. In (10) the arguments p^{(j)} (j = 1, …, 4) stand for four Fermat curve points (p_{1}^{(j)}, p_{2}^{(j)}, p_{3}^{(j)}, p_{4}^{(j)}) each. These 16 points depend on five independent parameters expressible in terms of spherical angles, see [5]. Note that here on the left and right hand side the same p^{(j)} appear. This will not be the case in the generalizations which will be discussed soon.

Baxter and Forrester [19] have studied whether this model describes phase transitions. They used variational and numerical methods and found strong evidence that for the parameter values for which (10) is satisfied, the ZBB-model is just at criticality [19]. So, in order to get a chance to describe phase transitions while staying integrable (recall that also for the 2D Potts model this is a problem), one should enlarge the framework and define more general Boltzmann weights and introduce less restrictive Tetrahedron equations. Less restrictive and still powerful generalized equations can be used, as has been shown by Mangazeev and Stroganov [10]: they introduced Modified Tetrahedron Equations which guarantee commuting layer-to-layer transfer matrices. Further work along this line has been done in [11, 12].

1.2 R-matrix satisfying the Modified Tetrahedron Equation

Since in the above-mentioned work the proof that particular Boltzmann weights satisfy a particular MTE has been rather tedious, here we shall follow the approach introduced in [21] in which there is no need for an explicit check of the MTE. The Boltzmann weights are constructed from ”physical” principles which guarantee the validity of the MTE and nevertheless leave much freedom to obtain a broad class of integrable 3D-models. We give a short summary of the argument.

One starts with an oriented 3-D basic lattice. The dynamic variables living on the links i of this lattice are taken to be elements u_{i}, w_{i} ∈ 𝔴_{i} an ultralocal Weyl algebra 𝔴 = ⨂ 𝔴_{i} at the primitive N-th root of unity: u_{i} w_{j} = q^{δ_{ij}} w_{j} u_{i} , (q as in eq.(6)), which generalize the ℤ_{N} spin variables of (2) and (3). The Weyl elements are represented by standard N × N raising and diagonal matrices. The N-th powers of the Weyl variables are centers of the algebra and so are scalar variables.

The main object constructed is an invertible canonical mapping R_{ijk} in the space of a triple Weyl algebra. R_{ijk} operates at the vertices of the 3D lattice, mapping the three Weyl elements on the ”incoming” links onto those on the ”outgoing” links, see Fig. 1.

The construction of R_{ijk} is based on two postulates, a Kirchhoff-like current conservation and a Baxter Z-invariance, and gives a unique explicit result: a canonical and invertible rational mapping operator. Since q is a root of unity, R_{ijk} decomposes into a matrix conjugation R_{ijk}, and a purely functional mapping R_{ijk}^{(f)} which acts on the scalar parameters (the Weyl centers). So, for any rational function Φ on 𝔴:

R_{123} ∘ Φ = R_{123}(R_{123}^{(f)} ∘ Φ)R_{123}^{−1}.  (11)

It turns out that the matrix R_{ijk} has the form (7) where the four Fermat curve parameters, again constrained by (8), are rational functions of the scalar Weyl center parameters.

Next consider an auxiliary plane which cuts the three incoming links near a vertex, and a second auxiliary plane cutting through the outgoing links, see Fig. 1. We take the six Weyl dynamic variables to sit on the six intersection points of the auxiliary planes. R_{123} can be regarded as the mapping of the ingoing auxiliary plane to the parallel shifted outgoing auxiliary plane. Now consider the vertices of the basic lattice to be formed as the intersection points of three sets of non-parallel planes. The three planes which form the vertex A of Fig. 1 intersect the auxiliary planes in the lines X, Y, Z shown in Fig. 2. Seen from the moving auxiliary plane, R_{123} shifts the line X through the vertex with index 1 or Y through the vertex 2 etc. We attach variables b_{1}, b_{2}, …, d_{3} to each section of the lines X, Y, Z as shown in Fig. 2. It is convenient to parameterize the two scalar

𝔴_{1} 𝔴_{3}  𝔴_{2}  𝔴′_{1} 𝔴′_{2}  𝔴′_{3} A

Figure 1: The six links of the basic lattice intersecting in the vertex A, intersected by auxiliary planes (shaded) in two different positions: first passing through 𝔴_{1}, 𝔴_{2}, 𝔴_{3} and second through 𝔴′_{1}, 𝔴′_{2}, 𝔴′_{3}. The second position is obtained from the first by moving the auxiliary plane parallel through the vertex A. The Weyl variables, elements of 𝔴_{i}, 𝔴′_{i} live on the links of the basic lattice. R maps the left auxiliary triangle onto the upper right one.

variables associated with the incoming dynamic Weyl variable 𝔴_{1} (corresponding to u_{1}^{N} and w_{1}^{N} in usual notation) by the ratios c_{2}^{N}/c_{3}^{N} and d_{3}^{N}/d_{2}^{N}. Analogously e.g. those for 𝔴′_{2} are defined as b_{1}^{N}/b′_{2}^{N} and d′_{2}^{N}/d_{1}^{N} etc. Details of the rule to parameterize the scalar variables in terms of ”line-section” variables b_{1}, …, d_{3} etc. are explained in [9]. However, these will not be essential here, since one of the aims of this paper is to introduce and use another parameterization. Just observe that R_{123}^{(f)} changes only three of the line-section parameters: b_{2}, c_{2}, d_{2}. ¿From the explicit form of the canonical operator R_{123} (see [9]) one finds that the functional mapping is rational in the N-th powers of the line sections:

b_{2}^{′N} = \frac{b_{1}^{N}c_{3}^{N}d_{2}^{N} + b_{2}^{N}c_{3}^{N}d_{3}^{N} + κ_{1}^{N}b_{3}^{N}c_{2}^{N}d_{3}^{N}}{c_{2}^{N}d_{2}^{N}};
c_{2}^{′N} = \frac{κ_{1}^{N}b_{3}^{N}c_{1}^{N}d_{2}^{N} + κ_{3}^{N}b_{2}^{N}c_{3}^{N}d_{1}^{N} + κ_{1}^{N}κ_{3}^{N}b_{3}^{N}c_{2}^{N}d_{1}^{N}}{κ_{2}^{N}b_{2}^{N}d_{2}^{N}};

[figure: Z; Z; Y; Y; X; X; 𝔴_{1}; 𝔴′_{3}; 𝔴_{3}; 𝔴′_{1}; 𝔴_{2}; 𝔴′_{2}; c_{1}; c_{1}; c_{3}; c_{3}; b_{1}; d_{1}; d_{3}; b_{3}; c_{2}; c′_{2}; d′_{2}; b_{2}; d_{2}; b′_{2}; d_{1}; b_{1}; d_{3}; b_{3}; R_{1 2 3}; ⟶]

Figure 2: The canonical invertible mapping R_{123} shown in the auxiliary planes passing through the incoming (left) and outgoing (right) dynamic variables which are elements of 𝔴_{i} resp. 𝔴′_{i}. The directed lines X, Y, Z are the intersections of the three planes forming the vertex A of Fig. 1. Their sections are labeled by the line-section parameters b_{1}, , …, d_{3}. Note, that orientation of the lines is not unique. But the choosen one corresponds to the numbering (44) and (45) of the fused vertex considered in the Section 3.

d_{2}^{′N} = \frac{b_{2}^{N}c_{1}^{N}d_{3}^{N} + b_{1}^{N}c_{1}^{N}d_{2}^{N} + κ_{3}^{N}b_{1}^{N}c_{2}^{N}d_{1}^{N}}{b_{2}^{N}c_{2}^{N}} .  (12)

Here κ_{1}, κ_{2}, κ_{3} are fixed parameters (”coupling constants”) of the mapping R_{123}. In the line-section parameterization the three independent Fermat curve parameters of R_{123} are

x_{1} = \frac{b_{2}c_{3}}{κ_{1}b_{3}c_{2}} ;  x_{2} = \frac{κ_{2}b_{1}c′_{2}}{b′_{2}c_{1}} ;  x_{3} = \frac{b_{1}c_{3}}{\sqrt{q} b′_{2}c_{2}} .  (13)

The matrix elements of R_{ijk} will be used as the Boltzmann weights of integrable 3D lattice models of statistical mechanics with the partition function defined in analogy to (2).

Via the physical assumptions made in constructing R_{ijk} the validity of the TE is built in. Simply considering two different sequences of Z-invariance shifts in a geometric figure formed by four intersecting straight lines (”quadrangle”), one concludes that (see e.g. [9])

R_{123} · R_{145} · R_{246} · R_{356} ∼ R_{356} · R_{246} · R_{145} · R_{123} ,  (14)

i.e. that R_{ijk} satisfies the Tetrahedron equation. Inserting (11) into (14) and choosing various phases of N-th roots leads to the MTE for the matrix operator R_{ijk}:

R_{123} · (R_{123}^{(f)} ∘ R_{145})·(R_{123}^{(f)}R_{145}^{(f)} ∘ R_{246})·(R_{123}^{(f)}R_{145}^{(f)}R_{246}^{(f)} ∘ R_{356})
∼ R_{356} · (R_{356}^{(f)} ∘ R_{246})·(R_{356}^{(f)}R_{246}^{(f)} ∘ R_{145})·(R_{356}^{(f)}R_{246}^{(f)}R_{145}^{(f)} ∘ R_{123}) .

(15)

Via the Fermat points R_{ijk} depends on the scalar variables, see e.g. (13). The scalar variables which appear in the matrices R_{ijk} are to be transformed by the functional transformations R_{ijk}^{(f)}. Let us write shorthand

R^{(1)} = R_{123};  R^{(2)} = R_{123}^{(f)} ∘ R_{145};  R^{(3)} = R_{123}^{(f)}R_{145}^{(f)} ∘ R_{246};

R^{(4)} = R_{123}^{(f)}R_{145}^{(f)}R_{246}^{(f)} ∘ R_{356};  R^{(5)} = R_{356}^{(f)}R_{246}^{(f)}R_{145}^{(f)} ∘ R_{123};
R^{(6)} = R_{356}^{(f)}R_{246}^{(f)} ∘ R_{145};  R^{(7)} = R_{356}^{(f)} ∘ R_{246};  R^{(8)} = R_{356}. (16)

Then (20) becomes

R^{(1)} R^{(2)} R^{(3)} R^{(4)} = ρ R^{(8)} R^{(7)} R^{(6)} R^{(5)},  (17)

where each R^{(j)} acts non-trivially in only three of the six spaces V = ℂ^{N} . ρ is a scalar density factor which comes in when passing from mappings to matrix equations.

The parameters which determine the R^{(j)} are the corresponding Fermat curve coordinates. Taking into account the functional transformations in (16) in terms of the line-section parameters we find (for details see [9]): R^{(j)} = R(x_{1}^{(j)}, x_{2}^{(j)}, x_{3}^{(j)}) with

( \begin{matrix}x_{1}^{(1)} & x_{2}^{(1)} & x_{3}^{(1)} \\ x_{1}^{(2)} & x_{2}^{(2)} & x_{3}^{(2)} \\ ⋮ & ⋮ & ⋮ \\ x_{1}^{(7)} & x_{2}^{(7)} & x_{3}^{(7)} \\ x_{1}^{(8)} & x_{2}^{(8)} & x_{3}^{(8)} \end{matrix} ) = q^{−1/2} ( \begin{matrix}\frac{b_{2}c_{3}}{κ_{1}b_{3}c_{2}} & \frac{κ_{2}b_{1}c′_{2}}{b′_{2}c_{1}} & \frac{b_{1}c_{3}}{q^{1/2} b′_{2}c_{2}} \\ \frac{a_{2}c′_{2}}{κ_{1}a_{3}c_{1}} & \frac{κ_{4}a_{1}c′′_{1}}{a′′_{2}c_{0}} & \frac{a_{1}c′_{2}}{q^{1/2} a′′_{2}c_{1}} \\ ⋮ & ⋮ & ⋮ \\ \frac{a_{2}b_{3}}{κ_{2}a_{3}b_{2}} & \frac{κ_{4}a_{1}^{†}b_{2}^{††}}{a_{2}^{††}b_{1}^{†}} & \frac{a_{1}^{†}b_{3}}{q^{1/2} a_{2}^{††}b_{2}} \\ \frac{a_{1}b_{2}}{κ_{3}a_{2}b_{1}} & \frac{κ_{5}a_{0}b_{1}^{†}}{b_{0}a_{1}^{†}} & \frac{a_{0}b_{2}}{q^{1/2} a_{1}^{†}b_{1}} \end{matrix} ) .  (18)

The once or multiply transformed parameters like c′′_{1}, b_{1}^{†} follow from the iteration of equations like (12). Altogether, since there are eight matrices R^{(j)} appearing in the MTEs, and as seen in Fig. 2, each transformation changes three line-section parameters, we have 24 equations for 32 different line-section parameters (these parameters can seen in Table 1 below). This is a set of classical integrable equations which conveniently are written in Hirota form:

b_{2}^{′N}c_{2}^{N}d_{2}^{N} = b_{1}^{N}c_{3}^{N}d_{2}^{N} + b_{2}^{N}c_{3}^{N}d_{3}^{N} + κ_{1}^{N}b_{3}^{N}c_{2}^{N}d_{3}^{N};
κ_{2}^{N}b_{2}^{N}c_{2}^{′N}d_{2}^{N} = κ_{1}^{N}b_{3}^{N}c_{1}^{N}d_{2}^{N} + κ_{3}^{N}b_{2}^{N}c_{3}^{N}d_{1}^{N} + κ_{1}^{N}κ_{3}^{N}b_{3}^{N}c_{2}^{N}d_{1}^{N};
b_{2}^{N}c_{2}^{N}d_{2}^{′N} = b_{2}^{N}c_{1}^{N}d_{3}^{N} + b_{1}^{N}c_{1}^{N}d_{2}^{N} + κ_{3}^{N}b_{1}^{N}c_{2}^{N}d_{1}^{N};
a_{2}^{′′N}c_{1}^{N}d_{1}^{N} = a_{1}^{N}c_{2}^{′N}d_{1}^{N} + a_{2}^{N}c_{2}^{′N}d_{2}^{′N} + κ_{1}^{N}a_{3}^{N}c_{1}^{N}d_{2}^{′N};
κ_{4}^{N}a_{2}^{N}c_{1}^{′′N}d_{1}^{N} = κ_{1}^{N}a_{3}^{N}c_{0}^{N}d_{1}^{N} + κ_{5}^{N}a_{2}^{N}c_{2}^{′N}d_{0}^{N} + κ_{1}^{N}κ_{5}^{N}a_{3}^{N}c_{1}^{N}d_{0}^{N};
a_{2}^{N}c_{1}^{N}d_{1}^{′′N} = a_{2}^{N}c_{0}^{N}d_{2}^{′N} + a_{1}^{N}c_{0}^{N}d_{1}^{N} + κ_{5}^{N}a_{1}^{N}c_{1}^{N}d_{0}^{N};
a_{1}^{′′′N}b_{1}^{N}d_{2}^{′N} = a_{0}^{N}b_{2}^{′N}d_{2}^{′N} + d_{3}^{N}a_{1}^{N}b_{2}^{′N} + κ_{2}^{N}d_{3}^{N}b_{1}^{N}a_{2}^{′′N};
κ_{4}^{N}a_{1}^{N}b_{1}^{′′′N}d_{2}^{′N} = κ_{2}^{N}b_{0}^{N}a_{2}^{′′N}d_{2}^{′N} + κ_{6}^{N}a_{1}^{N}b_{2}^{′N}d_{1}^{′′N} + κ_{2}^{N}κ_{6}^{N}a_{2}^{′′N}b_{1}^{N}d_{1}^{′′N};
a_{1}^{N}b_{1}^{N}d_{2}^{′′′N} = b_{0}^{N}d_{3}^{N}a_{1}^{N} + a_{0}^{N}b_{0}^{N}d_{2}^{′N} + κ_{6}^{N}a_{0}^{N}b_{1}^{N}d_{1}^{′′N};
⋮  ⋮
b_{1}^{t}^{N}c_{1}^{†}^{N}d_{1}^{††}^{N} = b_{0}^{N}c_{2}^{†††}^{N}d_{1}^{††}^{N} + b_{1}^{†}^{N}c_{2}^{†††}^{N}d_{2}^{†††}^{N} + κ_{1}^{N}b_{2}^{††}^{N}c_{1}^{†}^{N}d_{2}^{†††}^{N};
κ_{2}^{N}b_{1}^{†}^{N}c_{1}^{t}^{N}d_{1}^{††}^{N} = κ_{1}^{N}c_{0}^{N}b_{2}^{††}^{N}d_{1}^{††}^{N} + κ_{3}^{N}d_{0}^{N}b_{1}^{†}^{N}c_{2}^{†††}^{N} + κ_{1}^{N}κ_{3}^{N}d_{0}^{N}b_{2}^{††}^{N}c_{1}^{†}^{N};
b_{1}^{†}^{N}c_{1}^{†}^{N}d_{1}^{t}^{N} = c_{0}^{N}b_{1}^{†}^{N}d_{2}^{†††}^{N} + b_{0}^{N}c_{0}^{N}d_{1}^{††}^{N} + κ_{3}^{N}b_{0}^{N}d_{0}^{N}c_{1}^{†}^{N}.  (19)

The first three of these equations are just (12), defining R_{123}^{(f)}, i.e. b′_{2}^{N}, c′_{2}^{N}, d′_{2}^{N} in terms of the unprimed b_{1}, …, d_{3} . The first six equations together (e.g. express in the fourth eq. on the right hand side c_{2}^{′N} and d_{2}^{′N} from the first and third equations) define R_{123}^{(f)} ∘ R_{145}^{(f)}, etc. The complete expressions for (18) and (19) can be found in [9].

Straightforward (best done by e.g. Maple) combination of the first twelve eqs. of (19) on one hand, and of the last twelve of eqs.(19) on the other hand, shows that the functional mappings given in (19) automatically satisfy the functional TE:

R_{123}^{(f)} · R_{145}^{(f)} · R_{246}^{(f)} · R_{356}^{(f)} = R_{356}^{(f)} · R_{246}^{(f)} · R_{145}^{(f)} · R_{123}^{(f)}  (20)

where for the superposition of two operators acting on a function Φ we use the notation ((A · B) · Φ) =^{def} (A · (B · Φ)). Of course, the validity of (20) is a consequence of the physical rules used when constructing R_{ijk}. In the line-section parameterization the relation between the first, second etc. lines in both (18) and in (19) is not transparent. Introducing a new parameterization in the next subsection will make these relations simple and explicit.

2 Parameterization using concepts of algebraic geometry

2.1 Theta functions

It is well-known [14, 15, 16, 17, 18] that Hirota-type equations can be identically satisfied by a parameterization in terms of theta functions on an algebraic curve. We shall now introduce such a parameterization in order to write (18) and eqs.(19) in a more systematic way which also allows us to formulate fusion in a transparent manner. For the notations of algebraic geometry see e.g. [13].

Let Γ_{g} be an abstract generic algebraic curve of the genus g with ω being the canonical g-dimensional vector of the homomorphic differentials. For any two points X, Y ∈ Γ_{g} let I_{Y}^{X} : Γ_{g}^{2} ↦ Jac(Γ_{g}) be

I_{Y}^{X} {def \atop =} ∫_{Y}^{X} ω .  (21)

Let further E(X, Y) = − E(Y, X) be the prime form on Γ_{g}^{2} , and Θ(v) be the thetafunction on Jac(Γ_{g}).

It is well known, the theta-functions on the Jacobian of an algebraic curve obey the Fay identity

Θ(v) Θ(v + I_{B}^{A} + I_{D}^{C}) = Θ(v + I_{D}^{A}) Θ(v + I_{B}^{C}) \frac{E(A, B) E(D, C)}{E(A, C) E(D, B)}
+ Θ(v + I_{B}^{A}) Θ(v + I_{D}^{C}) \frac{E(A, D) E(C, B)}{E(A, C) E(D, B)} ,  (22)

which involves four points A, B, C, D ∈ Γ_{g} and a v ∈ Jac(Γ_{g}). We shall show that in the parameterization to be introduced below, the Fermat relations become just Fay-identities. The Fay identity involves only cross ratios of prime forms, and these ratios have a simple expression in terms of non-singular odd characteristic theta functions:

[ \begin{matrix}X & X′ \\ Y & Y′ \end{matrix} ] {def \atop =} \frac{E(X, Y) E(X′, Y′)}{E(X, Y′) E(X′, Y)} = \frac{Θ_{ϵ_{odd}}( I_{X}^{Y} ) Θ_{ϵ_{odd}}( I_{X′}^{Y′} )}{Θ_{ϵ_{odd}}( I_{X}^{Y′} ) Θ_{ϵ_{odd}}( I_{X′}^{Y} )} .  (23)

For solving the 24 trilinear equations we shall need an identity with more arguments obtained by combining two Fay identities:

Θ(v + I_{X}^{Q}) Θ(v + I_{Y}^{Q} + I_{Z}^{Z′}) Θ(v + I_{Z}^{Q} + I_{Y}^{Y′})  (24)
− Θ(v + I_{X}^{Q} + I_{Z}^{Z′}) Θ(v + I_{Y}^{Q}) Θ(v + I_{Z}^{Q} + I_{Y}^{Y′}) [ \begin{matrix}X & Y \\ Z & Z′ \end{matrix} ]
− Θ(v + I_{X}^{Q} + I_{Y}^{Y′}) Θ(v + I_{Y}^{Q} + I_{Z}^{Z′}) Θ(v + I_{Z}^{Q}) [ \begin{matrix}X & Z \\ Y & Y′ \end{matrix} ]
+ Θ(v + I_{X}^{Q} + I_{Y}^{Y′} + I_{Z}^{Z′}) Θ(v + I_{Y}^{Q}) Θ(v + I_{Z}^{Q}) [ \begin{matrix}X & Y \\ Z & Z′ \end{matrix} ] [ \begin{matrix}X & Z′ \\ Y & Y′ \end{matrix} ] = 0 .

Further we will need the N-th roots of theta-functions and prime forms. Define e(X, Y) and θ(v) by

e(X, Y)^{N} = Θ_{ϵ_{odd}}(I_{Y}^{X}) ∼ E(X, Y) ;  θ(v)^{N} = Θ(v) .  (25)

Since in the following we shall have to write many equations involving theta-functions, it is convenient to introduce special abbreviations. For ( Q, A, B_{1}, B′_{1}, … ∈ Γ_{g}) we define:

(A, B_{1} + B_{2} + … + B_{n}) ≡ Θ(v + I_{A}^{Q} + ∑_{j=1}^{n} I_{B_{j}}^{B_{j}′}) ;  ⟨A, B⟩ ≡ E(A, B) ;

[A, B_{1} + B_{2} + … + B_{n} ] ≡ θ(v + I_{A}^{Q} + ∑_{j=1}^{n} I_{B_{j}}^{B_{j}′})
\{A, B\} ≡ − q^{−1/2} e (A, B′) / e(A, B) ;  \{\overline{A, B}\} ≡ −e(A′, B) / e(A, B) .  (26)

Note that the two brackets introduced here do not indicate explicitly the dependence on the variables v, Q, B′_{1}, … B′_{n} since these always come in the same form.

We also introduce, using these notations:

F (v; X, Y′, Y, Z′, Z) {def \atop =}

(X)(Y, Z)(Z, Y) ⟨Y, Z⟩ ⟨X, Z′⟩ ⟨X, Y′⟩ − (X, Z)(Y)(Z, Y) ⟨X, Z⟩ ⟨Y, Z′⟩ ⟨X, Y′⟩
− (X, Y)(Y, Z)(Z) ⟨X, Y ⟩ ⟨Y′, Z⟩ ⟨X, Z′⟩ + (X, Y + Z)(Y)(Z) ⟨X, Z⟩ ⟨X, Y ⟩ ⟨Y′, Z′⟩ , (27)

so that the Double-Fay-identity (24) is

F (v; X, Y′, Y, Z′, Z) = 0.  (28)

The dependence on Q is trivial since it appears only in the combination v + I_{...}^{Q} . So Q is not an independent variable.

2.2 Re-parameterization of R

Let us introduce the new parameterization of the matrix (7). As we illustrated in Fig. 2, in the auxiliary plane the mapping R_{123} can be considered as a relative shift of three directed lines X, Y, Z with respect to each other. Now, for the given algebraic curve Γ_{g} and v ∈ C^{g}, we introduce three pairs of points on Γ_{g}:

X′, X, Y′, Y, Z′, Z ∈ Γ_{g}.  (29)

Another point Q ∈ Γ_{g} will just serve as a trivial normalization. Then let

R = R(p_{1}, p_{2}, p_{3}, p_{4}) ⟺ R = R(v; X′, X; Y′, Y; Z′, Z)  (30)

with, using the shorthand notations (26) and p_{j} = (x_{j}, y_{j}):

x_{1} = \frac{1}{q} \frac{\{X, Z\}}{\{Y′, Z\}} \frac{[X, Y] [Y, Z]}{[X, Y + Z] [Y]};  y_{1} = \frac{e(Z, Z′) e(X, Y′)}{e(X, Z) e(Y′, Z′)} \frac{[Z, Y] θ(v + I_{Y}^{Q} + I_{X}^{Z′})}{[X, Y + Z] [Y]}

x_{2} = \frac{\{X′, Z\}}{\{Y, Z\}} \frac{[X] [Y, X + Z]}{[X, Z] [Y, X]};  y_{2} = q \frac{e(Z, Z′) e(X′, Y)}{e(X′, Z) e(Y, Z′)} \frac{[Z, X] θ(v + I_{Y}^{Q} + I_{X}^{Z′})}{[X, Z] [Y, X]}

x_{3} = \frac{1}{q} \frac{\{X, Z\}}{\{Y, Z\}} \frac{[X] [Y, Z]}{[X, Z] [Y]};  y_{3} = q \frac{e(Z, Z′) e(X, Y)}{e(X, Z) e(Y, Z′)} \frac{[Z] θ(v + I_{Y}^{Q} + I_{X}^{Z′})}{[X, Z] [Y]}

x_{4} = \frac{1}{q} \frac{\{X′, Z\}}{\{Y′, Z\}} \frac{[X, Y] [Y, X + Z]}{[X, Y + Z] [Y, X]}; y_{4} = \frac{e(Z, Z′) e(X′, Y′)}{e(X′, Z) e(Y′, Z′)} \frac{[Z, X +Y] θ(v + I_{Y}^{Q} + I_{X}^{Z′})}{[X, Y +Z] [Y, X]}.

(31)

Actually, see (5), for defining R_{123} we don’t need the y_{i} themselves but only their three ratios

\frac{y_{3}}{y_{1}} = q \frac{\{\overline{Y, Z′}\}}{\{X, Y \}} \frac{[Z] [X, Y + Z]}{[X, Z] [Z, Y]};  \frac{y_{3}}{y_{2}} = \frac{e(X, Y) e(X′, Z)}{e(X, Z) e(X′, Y)} \frac{[Z] [Y, X]}{[Y] [Z, X]} ;

\frac{y_{4}}{y_{1}} = \frac{\{\overline{X, Y′}\}}{\{\overline{X, Z}\}} \frac{[Z, X + Y] [Y]}{[Z, Y] [Y, X]} .  (32)

from which e(Z, Z′) and θ(v + I_{Y}^{Q} + I_{X}^{Z′}) drop out. These three ratios cannot be expressed all in the same form.

Note that for this parameterization we used a generic algebraic curve and generic points on it, and a generic point on its Jacobian, all in order to parameterize just three independent complex numbers x_{1}, x_{2}, x_{3}. In (31) all x_{k}, y_{k} are the periodical functions of v ∈ Jac(Γ_{g}) .

The parameterization (31) is suggested by a few assumptions: First: The prime forms shall appear in the x_{i} only in the form of N-th roots of (23):

\frac{\{X, Z\}}{\{Y, Z\}} = [ \begin{matrix}X & Y \\ Z & Z′ \end{matrix} ]^{1/N} .  (33)

Second, considering (13), we demand that the line-section parameters b_{1}, b_{2}, b_{3}, b′_{2} (sections of the line X in Fig. 2) should be proportional to N-th roots of theta functions of the form [X, …] defined in (26). Analogously, the sections c_{1}, …, c′_{2} of the line Y are assumed to be proportional to [Y, …] . Finally, we consider that we want to use Fay identities to satisfy the Fermat relations, and later, the Hirota equations.

The merit of this parameterization will be seen in several places: when we consider the transformed mappings R^{(2)}, …, R^{(6)} , when we re-write eqs.(19) and when we construct composite weights in Sec. 3.

We now must verify that (31), and its generalization to the other Fermat points in the MTE, give a consistent parameterization of the relevant equations (4), (18) and (19). We first check that (31) satisfies the Fermat relations

x_{j}^{N} + y_{j}^{N} = 1.  (34)

Indeed, these are true due to the Fay identity, which for (A, B, C, D ∈ Γ_{g}) we write as

− ⟨A, C⟩⟨D, B⟩ Θ(v) Θ(v + I_{B}^{A} + I_{D}^{C}) + ⟨A, B⟩⟨D, C⟩ Θ(v + I_{D}^{A}) Θ(v + I_{B}^{C})
+ ⟨A, D⟩⟨C, B⟩ Θ(v + I_{B}^{A}) Θ(v + I_{D}^{C}) = 0. (35)

For j = 1 put in (35) (A, B, C, D) → (Y′, X, Z′, Z) and v → v′ = v + I_{Y}^{Q} , giving

⟨Z, X⟩⟨Y′, Z′⟩(Y)(X, Y + Z) − ⟨Z, Z′⟩⟨Y′, X⟩(Z, Y)Θ(v′ + I_{X}^{Z′})
− ⟨Z′, X⟩⟨Y′, Z⟩(X, Y)(Y, Z) = 0,

for j = 2 put in (35) (A, B, C, D) → (X′, Y, Z′, Z), v → v′′ = v + I_{X}^{Q} , giving

⟨Z′, Y ⟩⟨X′, Z⟩(Y, X)(X, Z) − ⟨Z, Y ⟩⟨X′, Z′⟩(X)(Y, X + Z)
+ ⟨Z, Z′⟩⟨X′, Y ⟩(Z, X)Θ(v′′ + I_{Y}^{Z′}) = 0,

for j = 3 put in (35) (A, B, C, D) → (Z, X, Z′, Y ), v → v^{+} = v + I_{Z}^{Q} :

⟨X, Z′⟩⟨Y, Z⟩(X)(Y, Z) − ⟨X, Z⟩⟨Y, Z′⟩(Y)(X, Z) + ⟨Z, Z′⟩⟨X, Y ⟩(Z)Θ(v+I_{Y}^{Q}+I_{X}^{Z′}) = 0,

or

⟨X, Z′⟩⟨Y, Z⟩Θ(v^{+} + I_{X}^{Z})Θ(v^{+} + I_{Y}^{Z′}) − ⟨X, Z⟩⟨Y, Z′⟩Θ(v^{+} + I_{Y}^{Z})Θ(v^{+} + I_{X}^{Z′})
+ ⟨Z, Z′⟩⟨X, Y ⟩Θ(v^{+})Θ(v^{+} + I_{X}^{Z} + I_{Y}^{Z′}) = 0.

For j = 4 put in (35) (A, B, C, D) → (X′, Z, Y′, Z′), v → v^{∗} = v + I_{X}^{Q} + I_{Y}^{Z′} .

2.3 Line-section parameters and Hirota equations in terms of theta functions

For writing the MTE in our new parameterization and to check (18) and (19) we have to consider three more spaces V = ℂ^{N} , corresponding to the indices 4, 5, 6 . In Fig. 2 the first three spaces were located at the intersection points of the lines X, Y, Z . To include the other three spaces, consider the ”quadrangle” formed by four lines shown in Fig. 3. Corresponding to the new line U we introduce another pair of points U′, U ∈ Γ_{g}.

Looking at Fig. 3 we see that, instead of labeling the spaces by the vertices 1, …, 6 of the quadrangle, we can as well label them by the pair of lines which intersect in these vertices, so identifying

1 ∼ YZ ;  2 ∼ XZ ;  3 ∼ XY ;
4 ∼ UZ ;  5 ∼ UY ;  6 ∼ UX .  (36)

\begin{aligned}a_{0} & = [U]\{U, X\}\{U, Y \}\{U, Z\} & b_{1} & = [X]\{X, Y\}\{X, Z\}\{\overline{U, X}\} \\ a_{1} & = [ U, X]\{U, Y \}\{U, Z\} & b_{2} & = [X, Y]\{X, Z\}\{\overline{U, X}\} \\ a_{1}^{†} & = [ U, Y]\{U, Z\}\{U, X\} & b′_{2} & = [X, Z]\{X, Y \}\{\overline{U, X}\} \\ a′′′_{1} = a_{1}^{†††} & = [ U, Z]\{U, X\}\{U, Y \} & b_{0} & = [X, U]\{X, Y \}\{X, Z\} \\ a_{2} & = [ U, X + Y]\{U, Z\} & b_{3} & = [X, Y + Z]\{\overline{U, X}\} \\ a_{2}^{††} = a_{2}^{T} & = [ U, Y + Z]\{U, X\} & b′′′_{1} = b_{1}^{t} & = [X, U + Z]\{X, Y \} \\ a′′_{2} & = [ U, X + Z]\{U, Y \} & b_{1}^{†} & = [X, U + Y]\{X, Z\} \\ a_{3} & = [ U, X + Y + Z] & b_{2}^{††} = b_{2}^{T} & = [X, U + Y + Z] \\ c_{2} & = [Y]\{Y, Z\}\{\overline{U, Y} \} \{\overline{X, Y} \} & d_{3} & = [Z]\{\overline{U, Z}\} \{\overline{X, Z}\} \{\overline{Y, Z}\} \\ c_{1}^{†} & = [Y, U]\{Y, Z\}\{\overline{X, Y} \} & d′′′_{2} = d_{2}^{†††} & = [Z, U]\{\overline{X, Z}\} \{\overline{Y, Z}\} \\ c_{3} & = [Y, Z]\{\overline{U, Y} \} \{\overline{X, Y} \} & d′_{2} & = [Z, X]\{\overline{Y, Z}\} \{\overline{U, Z}\} \\ c_{1} & = [Y, X] \{Y, Z\}\{\overline{U, Y} \} & d_{2} & = [Z, Y]\{\overline{U, Z}\} \{\overline{X, Z}\} \\ c_{0} & = [Y, U + X]\{Y, Z\} & d′′_{1} = d_{1}^{t} & = [Z, U + X]\{\overline{Y, Z}\} \\ c_{2}^{†††} = c_{2}^{T} & = [Y, Z + U]\{\overline{X, Y} \} & d_{1} & = [Z, X + Y]\{\overline{U, Z}\} \\ c′_{2} & = [Y, X + Z]\{\overline{U, Y} \} & d_{1}^{††} & = [Z, U + Y]\{\overline{X, Z}\} \\ c′′_{1} = c_{1}^{t} & = [Y, Z + U + X] & d_{0} & = [Z, U + X + Y] \end{aligned}

Table 1: The 32 line section parameters appearing in eqs.(19), expressed in terms of theta functions and prime factor ratios, using the abbreviations (26). Observe that in the prime factor brackets, the points come always in the order U, X, Y, Z (without and with primes).

Note that ordering of lines pair in the identification (36) is important, since it distinguishes the orientation we used and mirror reflected one.

Next we assume that we can write the κ_{j} all in the form (33). Then from (36) it is suggestive to build e.g. κ_{1} from the points Y′, Y, Z′, Z only, etc. and put (factors q^{1/2} are inserted to produce correct signs when forming N-th powers for (19)):

κ_{1} = q^{1/2} \frac{\{Y′, Z\}}{\{Y, Z\}} ;  κ_{2} = q^{1/2} \frac{\{X′, Z\}}{\{X, Z\}} ;  κ_{3} = q^{1/2} \frac{\{X′, Y \}}{\{X, Y \}} ;
κ_{4} = q^{1/2} \frac{\{U′, Z\}}{\{U, Z\}} ;  κ_{5} = q^{1/2} \frac{\{U′, Y \}}{\{U, Y \}} ;  κ_{6} = q^{1/2} \frac{\{U′, X\}}{\{U, X\}} . (37)

Now we consider (31) and (37) and assume that the line-section parameters a_{0}, a_{1}, … and d_{0}, d_{1}, … follow the same scheme as postulated for b_{0}, …, c_{0}, … above after (33): a_{i} ∼ [U, …] , d_{i} ∼ [Z, …]. So eqs.(18) lead us to the expressions for all line-section parameters shown in Table 1. We make ample use of the short-hand notations (26).

Apart from Q which always comes with v, we use eight arbitrary points X′, X, Y′, Y,

U X Y Z  ^{1}  ^{2}^{3}  ^{4}^{5}^{6}

Figure 3: Quadrangle in the auxiliary plane formed by the directed intersection lines of four oriented lattice planes. The six spaces V in which the MTE operates are considered to be located at the six intersection points.

Z′, Z, U′, U ∈ Γ_{g}. The κ_{j} may as well be written in terms of the bared brackets using \{\overline{A, B′}\}\{A, B\} = \{\overline{A, B}\}\{A′, B\} . ¿From Table 1 we see that the a_{i} don’t depend on U′, the b_{i} not on X′ etc.

Finally, using the results of Table 1 and (37), we can re-write all the Hirota equations (19) in terms of theta functions on Γ_{g} and prime form cross ratios. Not very surprisingly in view of [14, 17, 18], it turns out that these all have the form of the double-Fay identity. Also, as expected from Fig. 3, and the meaning of the mappings as moving of lines within the quadrangle, the 24 equations follow from each other by a sequence of simple substitutions. Just inserting from Table 1 and (37), the first three eqs. of (19) (recall that these are eqs.(12) defining the functional mapping R_{123}^{(f)}) become

F (v; X, Y′, Y, Z′, Z) \frac{(\{\overline{U, X}\}\{\overline{U, Y} \}\{\overline{U, Z}\}\{\overline{X, Y} \}\{\overline{X, Z}\})^{N}}{E(X, Y) E(X, Z) E(Y, Z)} = 0

F (v + I_{Y}^{Y′}; Y′, X′, X, Z′, Z) \frac{(\{\overline{U, X}\}\{\overline{U, Y} \}\{\overline{U, Z}\})^{N}}{E(X, Z) E(X, Y′) E(Y′, Z)} = 0

F (v; Z, Y′, Y, X′, X) \frac{(\{\overline{U, X}\}\{\overline{U, Y} \}\{\overline{U, Z}\}\{X, Z\}\{Y, Z\})^{N}}{E(X, Y) E(X, Z) E(Y, Z)} = 0.  (38)

The dependence on U′, U appears only in the factors on the right, not in the F . Assuming generic points U′, U, … we conclude that the F must vanish and we combine the essential terms of (38) into

𝔉 (v; X′, X, Y′, Y, Z′, Z) {def \atop =} (\begin{matrix}F (v; X, Y′, Y, Z′, Z) \\ F (v + I_{Y}^{Y′}; Y′, X′, X, Z′, Z) \\ F (v; Z, Y′, Y, X′, X) \end{matrix}).  (39)

Then the 24 Hirota equations (19) which describe the functional mappings take the form

𝔉 (v; X′, X, Y′, Y, Z′, Z) = 0 ,  𝔉 (v + I_{U}^{U′}; X′, X, Y′, Y, Z′, Z) = 0 ,
𝔉 (v + I_{X}^{X′}; U′, U, Y′, Y, Z′, Z) = 0 ,  𝔉 (v; U′, U, Y′, Y, Z′, Z) = 0 ,
𝔉 (v; U′, U, X′, X, Z′, Z) = 0 ,  𝔉 (v + I_{Y}^{Y′}; U′, U, X′, X, Z′, Z) = 0 ,
𝔉 (v + I_{Z}^{Z′}; U′, U, X′, X, Y′, Y) = 0 ,  𝔉 (v; U′, U, X′, X, Y′, Y) = 0 .

(40)

To get the order in which the Hirota equations are written in eq.(68) of [9] and in (19), read these equations on the left hand side up to down and then continue on the right hand side from the bottom up. So e.g. the last line of (40) collects the 10th to 15th eqs. of (68)

[9] or (19). As already mentioned with (20), eqs. (40) incorporate the functional TE.

2.4 Theta-parameterization of the simple modified tetrahedron equation

Using the parameterization (31) for the MTE (17) with (18) we find that the functional mapping just produces a permutation of the four pairs of points X, X′, …, U, U′ ∈ Γ_{g}, together with shifts in the vector v. Of course, the result corresponds to (40) and is explicitly:

Theorem 1 The simple modified tetrahedron equation may be parameterized in the terms of Γ_{g}, v ∈ Jac(Γ_{g}) and four pairs X′, X, Y′, Y, Z′, Z, U′, U ∈ Γ_{g} by the definition (30),(31),(18) as follows:

R^{(1)} = R(v; X′, X; Y′, Y; Z′, Z) ,  R^{(5)} = R(v + I_{U}^{U′}; X′, X; Y′, Y; Z′, Z) ,
R^{(2)} = R(v + I_{X}^{X′}; U′, U; Y′, Y; Z′, Z) , R^{(6)} = R(v; U′, U; Y′, Y; Z′, Z) ,
R^{(3)} = R(v; U′, U; X′, X; Z′, Z) ,  R^{(7)} = R(v + I_{Y}^{Y′}; U′, U; X′, X; Z′, Z) ,
R^{(4)} = R(v + I_{Z}^{Z′}; U′, U; X′, X; Y′, Y) , R^{(8)} = R(v; U′, U; X′, X; Y′, Y) .

(41)

Proof: Each R^{(j)} is determined by its three Fermat points x_{1}^{(j)}, x_{2}^{(j)}, x_{3}^{(j)} . From [9] these points are known in terms of the line-section parameters, see (18). Inserting the theta-function expressions for the line-sections from Table 1 into (18) one finds that the x_{i}^{(j)} for j = 2, …, 8 are obtained from those for j = 1 , eqs.(31), by the substitutions seen in (41). □

Using the correspondence between the labels 1, …, 6 and the line labels U, X, Y, Z , see (36), R^{(1)} = R_{123} may also labeled as R^{XYZ} etc., and we write the MTE as

R^{XYZ}(v) R^{UYZ}(v + I_{X}^{X′}) R^{UXZ}(v) R^{UXY}(v + I_{Z}^{Z′})
= ρ R^{UXY}(v) R^{UXZ}(v + I_{Y}^{Y′}) R^{UYZ}(v) R^{XYZ}(v + I_{U}^{U′}).  (42)

This notation also indicates directly the three pairs of points on the algebraic curve which parameterize the matrices R^{(j)} in (41).

In [9] we had shown that using simple re-scalings, out of the 24 line-section parameters listed in Table 1 and the 6 parameters κ_{1}, …, κ_{6} , only eight parameters are independent. Here we have eight points on Γ_{g} which can be chosen freely. In addition, 16 phases from taking the N-th roots can be chosen freely. In terms of the line-section parameters, the choice of the independent phases is the same as the one explained in [9].

3 The fused vertex weight ℜ

3.1 Open N_{1} × N_{2} × N_{3} box

The natural graphical interpretation of the R-matrix is a three dimensional vertex, i.e. the intersection of three planes in 3-D space. The six indices σ_{j}, σ′_{j} are associated to the edges of the vertex, recall Fig. 1.

The next step is the consideration of intersection of three sets of N_{1}, N_{2} and N_{3} parallel planes. This produces a finite open cubic lattice of the size N_{1} × N_{2} × N_{3}. We call the corresponding vertex object ℜ, which is the result of the fusion of elementary R-matrices.

The lattice is defined as in (1), but now n_{j} = 0, … N_{j} − 1 . Let ℜ_{123} be the matrix associated with the open cube (more precisely, the open parallelepiped):

ℜ_{123} ≡ ⟨σ⃗_{1}, σ⃗_{2}, σ⃗_{3}| ℜ |σ′_{1}, σ′_{2}, σ′_{3}⟩ = ∑_{{σ}} ∏_{n} ⟨σ_{1,n}, σ_{2,n+e_{2}}, σ_{3,n}|R_{n}|σ_{1,n+e_{1}}, σ_{2,n}, σ_{3,n+e_{3}}⟩ .

(43)

Here the six external multi-spin variables (i.e. the indices of the matrix ℜ_{123}) are associated with the six surfaces of the cube:

σ⃗_{1} = {σ_{1:0,n_{2},n_{3}}}, σ⃗_{2} = {σ_{2:n_{1},N_{2},n_{3}}}, σ⃗_{3} = {σ_{3:n_{1},n_{2},0}}, n_{j} = 0, ..., N_{j} − 1  (44)

and

σ′_{1} = {σ_{1:N_{1},n_{2},n_{3}}}, σ′_{2} = {σ_{2:n_{1},0,n_{3}}}, σ′_{3} = {σ_{3:n_{1},n_{2},N_{3}}}, n_{j} = 0, ..., N_{j} − 1 , (45)

and the summation in (43) is taken with respect to all internal indices σ_{j,n}. Anticipating what will be needed in (62) in order to prove that the fused weights satisfy a MTE of the same form as we had in (42), we introduce a reversed numbering for σ_{2} and σ′_{2}, so that the ”initial” external indices are σ_{1} = 0, σ_{2} = N_{2}, σ_{3} = 0. This reversed numbering in the second space is dictated also by our choice of the line orientations in the lattice and, as consequence of this, in the auxiliary plane (see Fig. 4).

In our next step we want to parameterize all R_{n} in (43) such that the fused weight ℜ again satisfies a Modified Tetrahedron Equation. We shall show that using a theta-function parameterization this is possible and the ℜ_{ijk} obtained will depend on 6(N_{1} + N_{2} + N_{3}) free parameters.

We use again the generic algebraic curve Γ_{g}, and one vector v ∈ ℂ^{g}. As in (30) each R_{n} will depend on three pairs of points on Γ_{g}, and to each R_{n} we assign different three pairs:

X′_{n_{1}}, X_{n_{1}}, Y′_{n_{2}}, Y_{n_{2}}, Z′_{n_{3}}, Z_{n_{3}},  n_{j} = 0, ..., N_{j} − 1.  (46)

However, the argument v will be shifted for each R_{n} by an amount I_{n} which depends on the points assigned to ”previous” neighbors: We define

R_{n}^{(123)} = R(v + I_{n}; X′_{n_{1}}, X_{n_{1}}; Y′_{n_{2}}, Y_{n_{2}}; Z′_{n_{3}}, Z_{n_{3}}) ,  n_{j} = 0, ..., N_{j} − 1,  (47)

n_{3}=0 n_{3}=1  n_{2}=0  n_{2}=1  n_{1}=0  n_{1}=1 (1,0,1) (1,1,1) (0,0,0)  (0,1,0) (1,0,0)  (0,1,1)

Figure 4: Top: 3-dimensional view of the oriented cube N_{1} = N_{2} = N_{3} = 2 (heavy lines) which is formed by six planes (indicated by dashed lines). Bottom: the horizontal auxiliary plane with the three pairs of lines arising from the intersection of the three pairs of planes with the auxiliary plane. This is a generalization of Figs. 1 and 2: If we consider e.g. the point (0, 1, 0) to correspond to the point A of Fig. 1, then the inner triangle in the auxiliary plane corresponds to the left shaded triangle of Fig. 1 and to the left part of Fig. 2. So the initial external spin (Weyl) variables (44) can be considered to sit at the three times four intersection points of the auxiliary plane. In order to get a similar picture for the final external variables we have to place the auxiliary plane above the cube.

where

I_{n} = ∑_{m_{1}=0}^{n_{1}−1} I_{X_{m_{1}}}^{X′_{m_{1}}} + ∑_{m_{2}=0}^{n_{2}−1} I_{Y_{m_{2}}}^{Y′_{m_{2}}} + ∑_{m_{3}=0}^{n_{3}−1} I_{Z_{m_{3}}}^{Z′_{m_{3}}} .  (48)

Now eqs. (43) and (47) define the matrix function

ℜ_{123}(v) = ℜ(v; X′, X; Y′, Y; Z′, Z)  (49)

where X′, X, Y′, Y, Z′, Z stand for the ordered lists of divisors,

X = (X_{0}, X_{1}, ..., X_{N_{1}−1}), X′ = (X′_{0}, X′_{1}, ..., X′_{N_{1}−1}), Y = (Y_{0}, Y_{1}, …), etc.  (50)

As to the index structure, recall (9),

ℜ_{123} ∈ End (V^{N_{2}N_{3}} ⊗ V^{N_{1}N_{3}} ⊗ V^{N_{1}N_{2}}) ,  (51)

where in the same way as before we enumerate the number of V^{N_{j}N_{k}} in the tensor product (e.g (43) are the matrix elements of ℜ_{123}). In Fig. 5 we show the intersection lines of

[figure: n_{2} = 0; n_{2} = 2; Y′, Y; n_{3} = 0; n_{3} = 2; Z′, Z; n_{1} = 0; n_{1} = 2; X′, X; V_{1}; V_{3}; V_{2}]

Figure 5: Ordering of the indices of the matrix ℜ_{123}, shown for the case N_{1} = N_{2} = N_{3} = 3 by drawing the auxiliary triangle in the auxiliary plane, as in the bottom part of Fig. 4.

the planes of a N_{1} × N_{2} × N_{3} cube which appear in an auxiliary plane (as in the bottom part of Fig. 4), which intersects the “initial” edges corresponding to (44). On the section the N_{1} + N_{2} + N_{3} planes become lines, and the edges of the cubic lattice become the intersection points of N_{1}N_{2} + N_{2}N_{3} + N_{1}N_{3} lines in the auxiliary plane. The intersection points are gathered into three sets V_{1} = V^{N_{2}N_{3}} etc., and the index of V_{j} is the number of the corresponding V^{N_{k}N_{l}} in the tensor product in (51). Fig. 5 is helpful to arrange the numbering in (48) and the correct assignment of X′_{n_{1}}, X_{n_{1}}, etc.

3.2 The Modified Tetrahedron equation for the fused weights

For writing the MTE, apart from the three pairs of sets X′, X; Y′, Y; Z′, Z of (49), (50) we need a fourth pair

U = (U_{0}, U_{1}, ..., U_{N_{0}−1} ); U′ = (U′_{0}, U′_{1}, ..., U′_{N_{0}−1} ).

Applying the definition (43), in addition to (49) we construct the matrices

ℜ_{145}(v) = ℜ(v; U′, U; Y′, Y; Z′, Z) ; ℜ_{246}(v) = ℜ(v; U′, U; X′, X; Z′, Z);
ℜ_{356}(v) = ℜ(v; U′, U; X′, X; Y′, Y).

Their index structure is defined by

\begin{matrix}V_{1} = V^{N_{2}N_{3}} , V_{2} = V^{N_{3}N_{1}} , V_{3} = V^{N_{1}N_{2}} , \\ V_{4} = V^{N_{0}N_{3}} , V_{5} = V^{N_{0}N_{2}} , V_{6} = V^{N_{0}N_{1}} . \end{matrix}  (52)

so that e.g. ℜ_{145} is acting in a space of dimension N^{N_{2}N_{3}+N_{0}N_{3}+N_{0}N_{2}} . For ℜ_{145} in analogy to the definitions (47),(48) one uses

R_{n}^{(145)} = R(v + I_{n}; U′_{n_{0}}, U_{n_{0}}; Y′_{n_{2}}, Y_{n_{2}}; Z′_{n_{3}}, Z_{n_{3}})  (53)

with

I_{n} = ∑_{m_{0}=0}^{n_{0}−1} I_{U_{m_{0}}}^{U′_{m_{0}}} + ∑_{m_{2}=0}^{n_{2}−1} I_{Y_{m_{2}}}^{Y′_{m_{2}}} + ∑_{m_{3}=0}^{n_{3}−1} I_{Z_{m_{3}}}^{Z′_{m_{3}}} ,  (54)

similarly for ℜ_{246} and ℜ_{356} .

Theorem 2 The matrices ℜ defined in (43) obey the modified tetrahedron equation

ℜ_{123}(v) ℜ_{145}(v + I_{X}) ℜ_{246}(v) ℜ_{356}(v + I_{Z}) =
ρ ℜ_{356}(v) ℜ_{246}(v + I_{Y}) ℜ_{145}(v) ℜ_{123}(v + I_{U})

(55)

where

I_{U} = ∑_{n_{0}=0}^{N_{0}−1} I_{U_{n_{0}}}^{U′_{n_{0}}} ; I_{X} = ∑_{n_{1}=0}^{N_{1}−1} I_{X_{n_{1}}}^{X′_{n_{1}}} ;
I_{Y} = ∑_{n_{2}=0}^{N_{2}−1} I_{Y_{n_{2}}}^{Y′_{n_{2}}} ; I_{Z} = ∑_{n_{3}=0}^{N_{3}−1} I_{Z_{n_{3}}}^{Z′_{n_{3}}} .

(56)

Proof: The main content of this theorem is the appearance of the specific set of shifts (48), (56). For the proof it is convenient to introduce some compact notations. Instead of using the number labels for the ℜ we shall the labels U, X, Y, Z just as these were introduced in (36) and (42) for the single vertex matrices R. So, for the box ℜ-matrix and its sets of divisors we write,

ℜ_{123}(v) ⟹ ℜ^{XYZ}(v);  ℜ_{145}(v) ⟹ ℜ^{UYZ}(v);  etc.  (57)

In this short notation , formulas (43), (47) imply the definition

ℜ^{XYZ}(v) = ∏_{n_{1}=0↑N_{1}−1} ∏_{n_{2}=N_{2}−1↓0} ∏_{n_{3}=0↑N_{3}−1} R^{X_{n_{1}}Y_{n_{2}}Z_{n_{3}}} (v + I_{n})  (58)

where we use ordered products

∏_{n_{1}=0↑N_{1}−1} 𝔣_{n_{1}} = 𝔣_{0}𝔣_{1} ⋯ 𝔣_{N_{1}−1} , ∏_{n_{2}=N_{2}−1↓0} 𝔣_{n_{2}} = 𝔣_{N_{2}−1} ⋯ 𝔣_{1}𝔣_{0} .  (59)

I_{n} for the triple (X, Y, Z) is given by (48). Each R^{X_{n_{1}}Y_{n_{2}}Z_{n_{3}}} acts non-trivially just in three of all the spaces (52) and the order is relevant just for neighboring indices X_{n_{1}} or Y_{n_{2}} or Z_{n_{3}}. The analogous modifications required to get the other matrices ℜ_{145}, etc. should be evident.

Now we turn to the proof of the theorem which will be by mathematical induction. The theorem claims the validity of the MTE for arbitrary N_{0}, N_{1}, N_{2}, N_{3}. For the initial point N_{0} = N_{1} = N_{2} = N_{3} = 1 the MTE holds because it is just (42). Then to prove the theorem, we reduce the MTE (55) for some N_{j} to MTEs with N′_{j} ≤ N_{j}. Thus one has four similar steps of the induction. Here we consider the induction step for the X-direction, the other steps follow analogously.

We split the list X into two sublists of length N_{1}^{(1)} and N_{1}^{(2)} = N_{1} − N_{1}^{(1)}:

X^{(1)} = (X_{0}, X_{1}, ..., X_{N_{1}^{(1)}−1}) , X^{(2)} = (X_{N_{1}^{(1)}}, ..., X_{N_{1}−1}) ,  (60)

so that I_{X^{(1)}} = ∑_{n_{1}=0}^{N_{1}^{(1)}−1} I_{X_{n_{1}}} and I_{X^{(2)}} = I_{X} − I_{X^{(1)}}. According to this splitting and due to the definition (58)

ℜ^{XYZ}(v) = ℜ^{X^{(1)}YZ}(v) ℜ^{X^{(2)}YZ}(v + I_{X^{(1)}}) ,
ℜ^{UXZ}(v) = ℜ^{UX^{(2)}Z}(v + I_{X^{(1)}}) ℜ^{UX^{(1)}Z}(v) ,
ℜ^{UXY}(v) = ℜ^{UX^{(2)}Y}(v + I_{X^{(1)}}) ℜ^{UX^{(1)}Y}(v) .

(61)

The meaning of notations X^{(1)} and X^{(2)} used in (61) should be evident from (58). Observe also that because of the reverse numbering with respect to the middle superscript of ℜ in (58), the factors in the latter two equations appear in reverse order. Since ℜ^{UYZ}(v) contains no X, neither as subscript nor in the argument, it will not be split. However, we have to put ℜ^{UYZ}(v + I_{X}) = ℜ^{UYZ}(v + I_{X^{(1)}} + I_{X^{(2)}}).

Now substituting (61) to the LHS of (55) written in our new superscript notation, and abbreviating v_{1} ≡ v + I_{X^{(1)}} we get

ℜ^{XYZ}(v) ℜ^{UYZ}(v + I_{X}) ℜ^{UXZ}(v) ℜ^{UXY}(v + I_{Z})

= ℜ^{X^{(1)}YZ}(v) ℜ^{X^{(2)}YZ}(v + I_{X^{(1)}}) ℜ^{UYZ}(v + I_{X^{(1)}} + I_{X^{(2)}}) ℜ^{UX^{(2)}Z}(v + I_{X^{(1)}}) ×

× ℜ^{UX^{(1)}Z}(v) ℜ^{UX^{(2)}Y}(v + I_{X^{(1)}} + I_{Z}) ℜ^{UX^{(1)}Y}(v + I_{Z})

= ℜ^{X^{(1)}YZ}(v) [ℜ^{X^{(2)}YZ}(v_{1}) ℜ^{UYZ}(v_{1} + I_{X^{(2)}}) ℜ^{UX^{(2)}Z}(v_{1}) ℜ^{UX^{(2)}Y}(v_{1} + I_{Z}) ] ×
×ℜ^{UX^{(1)}Z}(v)ℜ^{UX^{(1)}Y}(v + I_{Z})
= ℜ^{X^{(1)}YZ}(v) [ℜ^{UX^{(2)}Y}(v_{1}) ℜ^{UX^{(2)}Z}(v_{1} + I_{Y}) ℜ^{UYZ}(v_{1}) ℜ^{X^{(2)}YZ}(v_{1} + I_{U}) ] ×
×ℜ^{UX^{(1)}Z}(v)ℜ^{UX^{(1)}Y}(v + I_{Z})
= ℜ^{UX^{(2)}Y}(v_{1}) ℜ^{UX^{(2)}Z}(v_{1} + I_{Y}) ×

× [ℜ^{X^{(1)}YZ}(v) ℜ^{UYZ}(v + X^{(1)}) ℜ^{UX^{(1)}Z}(v) ℜ^{UX^{(1)}Y}(v + I_{Z})] ℜ^{X^{(2)}YZ}(v_{1} + I_{U})

= ℜ^{UX^{(2)}Y}(v_{1}) ℜ^{UX^{(2)}Z}(v_{1} + I_{Y}) ×
× [ℜ^{UX^{(1)}Y}(v) ℜ^{UX^{(1)}Z}(v + I_{Y}) ℜ^{UYZ}(v) ℜ^{X^{(1)}YZ}(v + I_{U})] ℜ^{X^{(2)}YZ}(v_{1} + I_{U})
= ℜ^{UX^{(2)}Y}(v + I_{X^{(1)}}) ℜ^{UX^{(1)}Y}(v) ℜ^{UX^{(2)}Z}(v + I_{X^{(1)}} + I_{Y}) ℜ^{UX^{(1)}Z}(v + I_{Y}) ×
×ℜ^{UYZ}(v) ℜ^{X^{(1)}YZ}(v + I_{U}) ℜ^{X^{(2)}YZ}(v + I_{X^{(1)}} + I_{U})
= ℜ^{UXY}(v) ℜ^{UXZ}(v + I_{Y}) ℜ^{UYZ}(v) ℜ^{XYZ}(v + I_{U}) .  (62)

From the third to the fourth line of (62) within the inserted brackets we used the MTE for the smaller set (U, X^{(2)}, Y, Z). In order to be able to isolate the terms containing X^{(2)} the order of factors in the second line of (61), which was used in the first step, is crucial. Going from the fifth to the sixth line in the brackets we used the MTE for (U, X^{(1)}, Y, Z) . In the other steps of (62) we just commuted or combined various terms. The last step is made possible by the ”reverse” order of factors in the last line of (61). □

4 Special cases: Solving the tetrahedron equation

4.1 Compact algebraic curve

Let now N_{0} = N_{1} = N_{2} = N_{3} = M . Starting from the generic parameterization of the MTE (55), the usual tetrahedron equation is obtained if

ℜ(v) ≡ ℜ(v + I)  and  ρ = 1 .  (63)

This is the case when

I_{U} = I_{X} = I_{Y} = I_{Z} = 0  on  Jac(Γ_{g}) ,  (64)

and the ratios of θ-functions (25) are periodical.

According to Abel’s theorem, equation (64) means that there are four meromorphic functions u, x, y, z on Γ_{g} with the divisors

(u) = ∑_{n_{0}=0}^{M−1} U′_{n_{0}} − U_{n_{0}}, (x) = ∑_{n_{1}=0}^{M−1} X′_{n_{1}} − X_{n_{1}}, (y) = ∑_{n_{2}=0}^{M−1} Y′_{n_{2}} − Y_{n_{2}}, (z) = ∑_{n_{3}=0}^{M−1} Z′_{n_{3}} − Z_{n_{3}}.

(65)

As it is well known (see e.g. Theorem 10-23 of [20]), conditions (64) are a strong restriction for the type of Γ_{g}: Γ_{g} is the algebraic curve given by the polynomial equation

P (x, y) {def \atop =} ∑_{a,b=0}^{M} x^{a} y^{b} p_{a,b} = 0 .  (66)

The choice of any pair of x, y, z, u produces an equivalent polynomial equation. The form of the polynomial P (x, y) provides the restriction for the genus,

g ≤ (M − 1)^{2} .  (67)

Thus we come to

Theorem 3 Let Γ_{g} be the compact algebraic curve defined by polynomial equation (66). Let four sets of U′_{n_{0}}, U_{n_{0}}, X′_{n_{1}}, X_{n_{1}}, Y′_{n_{2}}, Y_{n_{2}} and Z′_{n_{3}}, Z_{n_{3}}, n_{k} = 0, ..., M − 1, are the divisors of four meromorphic functions u, x, y, z on Γ_{g}. Then the tetrahedron equation is satisfied

\begin{matrix}ℜ_{123}(x, y, z) ℜ_{145}(u, y, z) ℜ_{246}(u, x, z) ℜ_{356}(u, x, y) = \\ ℜ_{356}(u, x, y) ℜ_{246}(u, x, z) ℜ_{145}(u, y, z) ℜ_{123}(x, y, z) , \end{matrix}  (68)

where four matrices are the same matrix function of different arguments,

ℜ(x, y, z) = ℜ(v; X′, X; Y′, Y; Z′, Z) etc.  (69)

defined via (43),(47),(49) and (65).

According to the conventional terminology, one may say that u, x, y, z are the spectral parameters, the moduli of Γ_{g} are the moduli of the tetrahedron equation, and vector v is a kind of deformation parameter.

Theorem 3 may be also be proved differently, without mentioning the simple MTE. In this alternative approach one considers the auxiliary linear problem for the whole box and the corresponding mappings. See e.g. [21, 22] for the description of this method and [23] for the parameterization of the classical equations of motion. Remarkably, in this alternative way the spectral curve (66) appears naturally from the linear problem.

4.2 Simple tetrahedron equation for ZBB-case recovered

As it was mentioned, (7) is the R-matrix for the Zamolodchikov-Bazhanov-Baxter model. Our scheme contains this case. Formally ZBB’s tetrahedron equation corresponds to (68) with M = 1. It gives g = 0, i.e. the spectral curve is the sphere with

E(X, Y) = \frac{X − Y}{\sqrt{dX dY}} , Θ() ≡ 1 .  (70)

Here the formal theta-function has no argument since the Jacobian is 0-dimensional. Conditions (64) therefore are out of use, and the parameterization (30)(31) contains the N-th

roots of the cross-ratios like \frac{(X − Z)(X′ − Z′)}{(X′ − Z)(X − Z′)}. One may show, the number of indepen-

dent cross-ratios is the number of variables X, X′, ... minus three. Therefore the single R-matrix contains 6 − 3 = 3 independent complex parameters (as it should be), and the simple tetrahedron equation contains 8 − 3 = 5 independent complex parameters (again as it should be). It means that the tetrahedral condition, which appears in Zamolodchikov’s parameterization of R in the terms of spherical triangles, is taken into account automatically. Moreover, the parameterization with the help of the cross-ratios takes automatically into account the geometric structure of any set of the planes in three dimensional Euclidean space. The parameterization of the inhomogeneous Zamolodchikov-Bazhanov-Baxter model in the terms of cross-ratios corresponding to the g = 0 limit of (30),(31)(47) was already used in [24].

4.3 Chessboard model

Previously derived “chessboard models” of the lattice statistical mechanics based on the modified tetrahedron equation [10, 11] are of course related to our considerations. The term “chessboard” appeared as the visual interpretation of the cubic lattice with M = 2 to be homogeneous. It means that the cubic lattice consists of eight different types of vertices (i.e. eight different types of the Boltzmann weights) – a kind of three dimensional analogue of the chess board with eight different colors of the cells.

The models described in [10, 11] are at the first the so-called IRC-type models, but with the help of vertex-IRC correspondence [8] one may construct their vertex reformulation. Thus the model implicitly described in [11] is equivalent to M = 2, g = 1 of our scheme. For g = 1 the curve and its Jacobian are isomorphic, so that without loss of generality one may chose

I_{Y}^{X} = X − Y ∈ ℂ/ℤ + ℤτ .  (71)

Further, one may use θ_{1}

Θ(v) = θ_{1}(v, τ) ≡ ∑_{n=−∞}^{∞} e^{iπτ(n+1/2)^{2}+2iπ(v+1/2)(n+1/2)}  (72)

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

X′_{0} − X_{0} + X′_{1} − X_{1} = Y′_{0} − Y_{0} + Y′_{1} − Y_{1} = Z′_{0} − Z_{0} + Z′_{1} − Z_{1} = U′_{0} − U_{0} + U′_{1} − U_{1} = 1. (73)

Note, 1 in the right hand side of (73) is equivalent to τ (due to the Jacobi transform), while 0 instead of 1 in the right hand side of (73) gives a trivial model.

The model explicitly described before in [10] corresponds to M = 2, g = 1, X_{0} = X_{1}, X′_{0} = X′_{1}, Y_{0} = Y_{1}, Y′_{0} = Y′_{1}, Z_{0} = Z_{1}, Z′_{0} = Z′_{1} etc. with the condition (73). This choice makes the identification of the parameters in (43)

R_{n} = R_{n+e_{1}+e_{2}} = R_{n+e_{1}+e_{3}} = R_{n+e_{2}+e_{3}} ,  (74)

so the cells of this three dimensional chess board have only two “colors”.

Note that the vertex-IRC duality is not exact because it changes the boundary conditions.

5 Conclusions

We considered a large class of integrable 3-D lattice models which have Weyl variables at N-th root of unity as dynamic variables. We have shown how the Boltzmann weights can be conveniently parameterized in terms of N-th roots of theta functions on a Jacobian of a compact Riemann surface. The Fermat relations of the points determining the Boltzmann weights are simple Fay identities and the classical equations determining the scalar parameters are a consequence of a double Fay identity. In the modified tetrahedron equation we have four pairs of arbitrary points on the Riemann surface in simple permuted combinations.

This parameterization allows a compact formulation of the rules to form fused Boltzmann weights ℜ ∈ End ℂ^{3NM^{2}} which are the partition functions of an open boxes of arbitrary size. The ℜ obey the modified tetrahedron equation and are again parameterized terms on N-th roots of theta-functions on the Jacobian of a genus g = (M − 1)^{2} compact Riemann surface Γ_{g}. The spectral parameters of the vertex weight ℜ are three meromorphic functions on the spectral curve Γ_{g}. For the case that the Jacobi transforms become trivial the ℜ obey the simple tetrahedron equation. The Zamolodchikov-Baxter-Bazhanov model and the Chessboard model are obtained as special cases.

So, the scheme discussed here contains and generalizes many known 3-D integrable models. The hope is that the framework is now sufficiently general to contain physically interesting models with a non-trivial phase structure. However, to get information on partition functions, either analytically or approximately, is still a very difficult open problem. There is no way known to generalize Baxter’s special method [26] by which he obtained the partition function of the ZBB-model.

Acknowledgments

This work has been supported in part by the contract INTAS OPEN 00-00055 and by the Heisenberg-Landau program HLP-2002-11. S.P.’s work was supported in part by the grants CRDF RM1-2334-MO-02, RFBR 03-02-17373 and the grant for support of scientific schools 1999.2003.2. S.S.’s work was supported in part by the grants CRDF RM1-2334-MO-02 and RFBR 01-01-00201.

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