Explicit Free Parameterization of the Modified Tetrahedron Equation

G. von Gehlen^{a,}^{1}, S. Pakuliak^{b,2}, S. Sergeev^{b,c,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 cMax-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

Abstract The Modified Tetrahedron Equation (MTE) with affine Weyl quantum variables at N-th root of unity is solved by a rational mapping operator which is obtained from the solution of a linear problem. We show that the solutions can be parameterized in terms of eight free parameters and sixteen discrete phase choices, thus providing a broad starting point for the construction of 3-dim. integrable lattice models. The Fermat curve points parameterizing the representation of the mapping operator in terms of cyclic functions are expressed in terms of the independent parameters. An explicit formula for the density factor of the MTE is derived. For the example N = 2 we write the MTE in full detail. We also discuss a solution of the MTE in terms of bosonic continuum functions.

AMS subject classes: 82B23, 70H06; keywords: 3D integrable models

PACS: 05.45-a, 05.50+q

Introduction

The Zamolodchikov tetrahedron equation is the condition [1] for the existence of a commuting set of layer-to-layer transfer matrices for 3-dimensional lattice models, in much the same way as the Yang-Baxter equation is the analogous condition in the 2-dimensional case. Only very few solutions to these very restrictive equations have been found [1, 2]. So various Modified Tetrahedron Equations (MTE) have been studied to which more solutions can be obtained [3, 4, 5, 6, 7, 8], still leading to commuting transfer matrices or generating functionals for conserved quantities.

In this paper we shall concentrate on a particular MTE proposed in [9, 10]. The quantum variables are elements from an ultra-local affine Weyl algebra attached to every vertex of a 2-dim. graph. Since we consider the Weyl parameter to be a N-th root of unity, the N-th powers of the quantum variables form a classical system which determines the parameters of the quantum system, as has been considered in for the discrete sin-Gordon model and other models recently, e.g. [11, 12, 13, 14]. So the parameters of the eight R matrices appearing in the MTE are different, but related by functional mappings.

A linear problem discussed previously by one of us [9, 10], is used to determine the mapping which provides a multi-parameter solution to the MTE. The construction of a generating functional of the conserved quantities has been given in [9]. Here we concentrate on calculating the density factor of the MTE and to give a useful choice of the eight continuous parameters of the mapping.

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

The aim of studying these equations is at least two-fold: first a 2+1-dimensional integrable lattice model should emerge, and second, the MTE can be used by contraction [15] to construct new 2-dimensional lattice models with parameters living on higher Riemann surfaces.

The paper is organized as follows: In Sec.1 we introduce the rational mapping in the affine Weyl space and discuss how for Weyl parameter a root of unity this splits into a matrix- and functional mapping. A realization in terms of cyclic functions and a Gaussian realization is introduced. Sec. 2 discusses the modified tetrahedron equation and we calculate its weight function. In Sec. 3 we focus on the parameterization in terms of line ratios, finding eight continuous parameters and analyze the phase ambiguities. For the specific case N = 2 in Sec. 4 we show that the modified tetrahedron equations can be written quite explicitly and that of their 2^{12} matrix components there are 256 linearly independent equations. In Sec. 6 we give a summary and mention future applications.

1 The rational mapping R in the space of a triple affine Weyl algebra.

The central object of our considerations will be a mapping operator acting in the space of a triple Weyl affine algebra. It is this operator which will satisfy the MTE. Several interpretations are possible, e.g. as vertex Boltzmann weights (albeit not positive ones) of a three-dimensional lattice model. It will be a generalization of the Zamolodchikov-Bazhanov-Baxter [1, 2] Boltzmann weights in the Sergeev-Mangazeev-Stroganov [4] vertex formulation. The principle from which this mapping is obtained has been described in detail in [9]. It is a current conservation principle with a Baxter Z-invariance.

1.1 The linear problem.

To set the framework we assign to each vertex j of a 2d graph the elements u_{j}, w_{j} of an affine Weyl algebra at Weyl parameter q a root of unity:

u_{j} · w_{j} = q w_{j} · u_{j};  q = ω =^{def} e^{2π i/N} ,  N ∈ ℤ , N ≥ 2 .  (1)

Since q is a root of unity, the u_{j}^{N} and w_{j}^{N} are centers of the Weyl algebra and we shall often represent the canonical pair (u_{j}, w_{j}) by its action on a cyclic basis as unitary N × N matrices multiplied by complex parameters u_{j}, w_{j}, writing

u = u x ;  w = w z ;  (2)
| σ⟩ ≡ | σ mod N⟩ ; ⟨ σ | σ′⟩ = δ_{σ,σ′}; x | σ⟩ = | σ⟩ ω^{σ} ; z | σ⟩ = | σ + 1⟩ . (3)

The centers are represented by numbers:

u_{j}^{N} = u_{j}^{N} ,  w_{j}^{N} = w_{j}^{N} .  (4)

We define the ultra-local Weyl algebra 𝔚^{⊗Δ} as the tensor product of Δ copies of Weyl pairs

u_{j} = 1 ⊗ 1 ⊗ ... ⊗ \underbrace{u}_{{j−th \atop place}} ⊗ ... ; w_{j} = 1 ⊗ 1 ⊗ ... ⊗ \underbrace{w}_{{j−th \atop place}} ⊗ ... .  (5)

Denote the four faces around the vertex j (in which two oriented lines cross) clockwise by a, b, c, d as e.g. shown in the left side of Fig. 1 for the vertex j = 3. Imagine a current ⟨ ϕ | flowing out of the vertex into the four faces a, b, c, d, each face receiving a current ⟨ ϕ_{s}| (s = a, b, c, d) according to the values of the Weyl variables sitting at the vertex and a coupling constant κ_{j} (which may be different at each vertex):

⟨ ϕ | = ⟨ϕ_{a}| + ⟨ϕ_{b}| · q^{1/2}u_{j} + ⟨ϕ_{c}| · w_{j} + ⟨ϕ_{d}| · κ_{j} u_{j} w_{j}.  (6)

Demanding that the total current flowing out of an internal vertex is zero: ⟨ ϕ | = 0 , and demanding also that the currents flowing into the outer faces of various graphs are independent of the internal structure of the graphs (this is a Z-invariance assumption), we get a condition for the equivalence of linear problems. The right hand side of Fig.1 shows two such linear problems, one in the bottom plane, another in the upper plane. The equivalence condition determines the the mapping R_{1,2,3} between the lower (𝔴_{1}, 𝔴_{2}, 𝔴_{3}) and upper (𝔴′_{1}, 𝔴′_{2}, 𝔴′_{3}) triangle in Fig. 1 uniquely. The details of this calculation can be found in [9], in (7)-(9) below we present the result. For our case of interest q = ω it is convenient to choose the specific form (6) of the coefficients, which is unsymmetrical in a, b, c, d. There exists a fully symmetrical formulation of the linear problem valid at general q, still leading to a unique mapping R_{1,2,3} [9]. However, this will not be needed here.

1.2 The rational mapping R_{1,2,3}

The solution of the equivalence problem of the linear current flows is the following rational mapping R acting in the ring of rational functions of the generators of the ultra-local Weyl algebra [9]: For any rational function Φ we define

(R_{1,2,3} ∘ Φ) (u_{1}, w_{1}, u_{2}, w_{2}, u_{3}, w_{3}, ...) =^{def} Φ(u′_{1}, w′_{1}, u′_{2}, w′_{2}, u′_{3}, w′_{3}, ...)  (7)

where in the right hand side of (7) the u_{α} and w_{α} remain unchanged for all α ∉ {1, 2, 3}, and the primed elements are rational functions of u_{1}, ..., w_{3}, given by the definition

w′_{1} = w_{2} · Λ_{3} , w′_{2} = Λ_{3}^{−1} · w_{1} , w′_{3} = Λ_{2}^{−1} · u_{1}^{−1} ,
u′_{1} = Λ_{2}^{−1} · w_{3}^{−1} , u′_{2} = Λ_{1}^{−1} · u_{3} , u′_{3} = u_{2} · Λ_{1} ,

(8)

where

Λ_{1} ≡ u_{1}^{−1} · u_{3} − q^{1/2} u_{1}^{−1} · w_{1} + κ_{1} w_{1} · u_{2}^{−1} ,
Λ_{2} ≡ \frac{κ_{1}}{κ_{2}} u_{2}^{−1} · w_{3}^{−1} + \frac{κ_{3}}{κ_{2}} u_{1}^{−1} · w_{2}^{−1} − q^{−1/2} \frac{κ_{1} κ_{3}}{κ_{2}} u_{2}^{−1} · w_{2}^{−1} ,
Λ_{3} ≡ w_{1} · w_{3}^{−1} − q^{1/2} u_{3} · w_{3}^{−1} + κ_{3} w_{2}^{−1} · u_{3} .

(9)

κ_{1}, κ_{2}, κ_{3} ∈ ℂ are arbitrary extra parameters of the mapping R_{1,2,3}. In this subsection q can be in general position.

• \begin{matrix}u_{3} w_{3} \\ κ_{3} \end{matrix} ϕ_{a}  ϕ_{b} ϕ_{c}  ϕ_{d}  𝔴_{3}  𝔴_{1} 𝔴_{2} • 𝔴′_{2} 𝔴′_{1}  𝔴′_{3} R_{1,2,3}

Figure 1: The linear problem for the vertex with associated Weyl pair u_{3}, w_{3} and parameter κ_{3} and the visualization of R_{1,2,3}. The elements 𝔴_{1}, 𝔴′_{1}, etc. of the ultra-local affine Weyl algebras are assigned to the vertices of two auxiliary two dimensional lattices formed by the intersection of three straight lines with the auxiliarly planes.

Note that the order of the factors in the right hand sides of (8) does not matter. Each combination Λ_{i} (i = 1, 2, 3) contains only elements of the Weyl algebra which commute with the other factors in the product. For example, Λ_{3} does not have the Weyl operators u_{1} and u_{2}. From (8) we see that the rational mapping R_{1,2,3} has the three invariants:

w_{1}w_{2},  u_{2}u_{3},  u_{1}w_{3}^{−1}.  (10)

This means that this mapping has the property that the products u_{j}^{−1}u_{j}′ and w_{j}^{−1}w_{j}′ for j = 1, 2, 3 (no summation over j) depend only on three operators which we denote by u, v and w:

u = w_{2}^{−1}w_{3};  v = u_{1}u_{2}^{−1};  w = w_{1}u_{3}^{−1},  (11)

as one easily checks explicitly:

(w_{3}^{−1}w_{3}′)^{−1} = (u_{1}^{−1}u_{1}′)^{−1} = u_{1}Λ_{2}w_{3} = \frac{κ_{1}}{κ_{2}} v + \frac{κ_{3}}{κ_{2}} u − q^{−1/2} \frac{κ_{1}κ_{3}}{κ_{2}} vu;
(u_{2}^{−1}u_{2}′)^{−1} = u_{3}′u_{3}^{−1} = u_{3}^{−1}Λ_{1}u_{2} = v^{−1} + κ_{1}w − q^{1/2}v^{−1}w;
w_{1}^{−1}w_{1}′  = (w_{2}^{−1}w_{2}′)^{−1} = w_{2}Λ_{3}w_{1}^{−1} = u^{−1} + κ_{1}w^{−1} − q^{1/2}w^{−1}u^{−1}. (12)

Observe that the three operators (11) form a triple Weyl algebra: vu = q uv, vw = q wv, uw = q wu, and also at each vertex j we can regard u_{j}, w_{j} together with v_{j} ≡ κ_{j}u_{j}w_{j} as forming triple Weyl algebras.

The mapping R has the property (see [7, 5, 16, 9, 10] for the details):

Proposition 1 The invertible mapping R_{i,j,k} is an automorphism of 𝔚^{⊗Δ}.

Remark 1 The Proposition states that the rational mapping (8) is canonical, namely, it sends three copies of the ultra-local Weyl algebras into same Weyl algebras.

Later, in Sec. 2, Proposition 4, we shall discuss the second crucial property of the mapping R: it solves the Tetrahedron Equation (49).

1.3 Functional part at root of unity

In all the following we shall consider only the case that q is a root of unity (1) and use the unitary representation (3) of 𝔚^{⊗Δ} . In this representation each affine Weyl element u_{j} and w_{j} will contain one free parameter u_{j} resp. w_{j}, as written in (2).

The basic fact is that at Weyl parameter root of unity any rational automorphism of the ultra-local Weyl algebra implies a rational mapping in the space of the N-th powers of the parameters of the representation [12, 13, 14]. In our case (8) it is easy to check that for the mapping R_{1,2,3} implies

w_{1}^{′N} = w_{2}^{N} Λ_{3}^{N} , w_{2}^{′N} = \frac{w_{1}^{N}}{Λ_{3}^{N}} , w_{3}^{′N} = \frac{1}{Λ_{2}^{N} u_{1}^{N}} ,

u_{1}^{′N} = \frac{1}{Λ_{2}^{N} w_{3}^{N}} , u_{2}^{′N} = \frac{u_{3}^{N}}{Λ_{1}^{N}} , u_{3}^{′N} = u_{2}^{N} Λ_{1}^{N}

(13)

where the N-th powers of the polynomials Λ_{k} are also numbers:

\begin{aligned}Λ_{1}^{N} & = & u_{1}^{−N} u_{3}^{N} + u_{1}^{−N} w_{1}^{N} + κ_{1}^{N} w_{1}^{N} u_{2}^{−N} , \\ Λ_{2}^{N} & = & \frac{κ_{1}^{N}}{κ_{2}^{N}} u_{2}^{−N} w_{3}^{−N} + \frac{κ_{3}^{N}}{κ_{2}^{N}} u_{1}^{−N} w_{2}^{−N} + \frac{κ_{1}^{N} κ_{3}^{N}}{κ_{2}^{N}} u_{2}^{−N} w_{2}^{−N} , \\ Λ_{3}^{N} & = & w_{1}^{N} w_{3}^{−N} + u_{3}^{N} w_{3}^{−N} + κ_{3}^{N} w_{2}^{−N} u_{3}^{N} \end{aligned}  (14)

since for q = ω and a, b ∈ ℂ one has ( a u + b w)^{N} = (a u)^{N} + (b w)^{N}, using ∑_{j=ℤ_{N}} ω^{j} = 0.

Definition 1 The functional counterpart of the mapping R_{1,2,3} is the mapping R_{1,2,3}^{(f)}, acting on the space of functions of the parameters u_{j}, w_{j} (j = 1, 2, 3)

(R_{1,2,3}^{(f)} ∘ ϕ) (u_{1}, w_{1}, u_{2}, w_{2}, u_{3}, w_{3}) =^{def} ϕ(u′_{1}, w′_{1}, u′_{2}, w′_{2}, u′_{3}, w′_{3}) ,  (15)

where the primed variables are functions of the unprimed ones, defined via

u_{1}^{′N} = u_{1}^{′N} , w_{1}^{′N} = w_{1}^{′N} , etc.,  (16)

such that the u_{j}, w_{j}, u′_{j}, w′_{j} satisfy

w′_{1}w′_{2} = w_{1}w_{2} , u′_{2}u′_{3} = u_{2}u_{3} , \frac{u′_{1}}{w′_{3}} = \frac{u_{1}}{w_{3}} .  (17)

The three free phases of the N-th roots are extra discrete parameters of R_{1,2,3}^{(f)}.

We use the invariance of the three centers u_{2}u_{3}, u_{1}/w_{3} and w_{1}w_{2} to define three functions Γ_{1}, Γ_{2}, Γ_{3}:

Γ_{1}^{N} = \frac{u′_{3}^{N}}{u_{3}^{N}} = \frac{u_{2}^{N}}{u′_{2}^{N}} = (u_{3}^{−1}Λ_{1}u_{2})^{N} = \frac{u_{2}^{N}}{u_{1}^{N}} + \frac{w_{1}^{N}u_{2}^{N}}{u_{1}^{N}u_{3}^{N}} + κ_{1}^{N} \frac{w_{1}^{N}}{u_{3}^{N}} ,
Γ_{2}^{N} = \frac{u_{1}^{N}}{u′_{1}^{N}} = \frac{w_{3}^{N}}{w′_{3}^{N}} = (w_{3}Λ_{2}u_{1})^{N} = \frac{κ_{1}^{N}u_{1}^{N}}{κ_{2}^{N}u_{2}^{N}} + \frac{κ_{3}^{N}w_{3}^{N}}{κ_{2}^{N}w_{2}^{N}} + \frac{κ_{1}^{N}κ_{3}^{N}u_{1}^{N}w_{3}^{N}}{κ_{2}^{N}u_{2}^{N}w_{2}^{N}};

Γ_{3}^{N} = \frac{w′_{1}^{N}}{w_{1}^{N}} = \frac{w_{2}^{N}}{w′_{2}^{N}} = (w_{1}^{−1}Λ_{3}w_{2})^{N} = \frac{w_{2}^{N}}{w_{3}^{N}} + \frac{w_{2}^{N}u_{3}^{N}}{w_{1}^{N}w_{3}^{N}} + κ_{3}^{N} \frac{u_{3}^{N}}{w_{1}^{N}} .  (18)

so that, alternatively to using (13), (14), the functional mapping can be written as

R^{(f)} ∘ w_{1} = w_{1}Γ_{3}, R^{(f)} ∘ w_{2} = \frac{w_{2}}{Γ_{3}} , R^{(f)} ∘ w_{3} = \frac{w_{3}}{Γ_{2}} ,

R^{(f)} ∘ u_{1} = \frac{u_{1}}{Γ_{2}} , R^{(f)} ∘ u_{2} = \frac{u_{2}}{Γ_{1}} , R^{(f)} ∘ u_{3} = u_{3}Γ_{1}.

The Γ_{j} depend on 3 variables and the 3 constants κ_{j}. Their phases are arbitrary.

1.4 Matrix part at root of unity

Now we consider the matrix structure of R_{1,2,3} at q a root of unity. First of all, define

x′_{1} = \frac{u′_{1}}{u′_{1}} , z′_{1} = \frac{w′_{1}}{w′_{1}} , etc.  (19)

The normalization implies the conservation of the centers

x_{1}^{′N} = x_{1}^{N} = 1 , z_{1}^{′N} = z_{1}^{N} = 1 , etc.  (20)

The rational mapping (8) for the set of matrices (19) has the form

(x′_{1})^{−1} = \frac{κ_{1}u′_{1}}{κ_{2}u_{2}} x_{2}^{−1} + \frac{κ_{3}u′_{1}w_{3}}{κ_{2}u_{1}w_{2}} x_{1}^{−1}z_{2}^{−1}z_{3} − ω^{1/2} \frac{κ_{1}κ_{3}u′_{1}w_{3}}{κ_{2}u_{2}w_{2}} x_{2}^{−1}z_{2}^{−1}z_{3},
z′_{1} = \frac{w_{2}w_{1}}{w′_{1}w_{3}} z_{1}z_{2}z_{3}^{−1} − ω^{1/2} \frac{w_{2}u_{3}}{w′_{1}w_{3}} z_{2}x_{3}z_{3}^{−1} + \frac{κ_{3}}{w′_{1}} x_{3},

(x′_{2})^{−1} = \frac{u′_{2}}{u_{1}} x_{1}^{−1} − ω^{1/2} \frac{w_{1}u′_{2}}{u_{1}u_{3}} x_{1}^{−1}z_{1}x_{3}^{−1} + \frac{κ_{2}w_{1}u′_{2}}{u_{2}u_{3}} z_{1}x_{2}^{−1}x_{3}^{−1},  (21)

(z′_{2})^{−1} = \frac{w′_{2}}{w_{3}} z_{3}^{−1} − ω^{1/2} \frac{w′_{2}u_{3}}{w_{1}w_{3}} z_{1}^{−1}x_{3}z_{3}^{−1} + \frac{κ_{3}w′_{2}u_{3}}{w_{1}w_{2}} z_{1}^{−1}z_{2}^{−1}x_{3},
x′_{3} = \frac{u_{2}u_{3}}{u_{1}u′_{3}} x_{1}^{−1}x_{2}x_{3} − ω^{1/2} \frac{w_{1}u_{2}}{u_{1}u′_{3}} x_{1}^{−1}z_{1}x_{2} + \frac{κ_{2}w_{1}}{u′_{3}} z_{1},

(z′_{3})^{−1} = \frac{κ_{1}u_{1}w′_{3}}{κ_{2}u_{2}w_{3}} x_{1}x_{2}^{−1}z_{3}^{−1} + \frac{κ_{3}w′_{3}}{κ_{2}w_{2}} z_{2}^{−1} − ω^{1/2} \frac{κ_{1}κ_{3}u_{1}w′_{3}}{κ_{2}u_{2}w_{2}} x_{1}x_{2}^{−1}z_{2}^{−1}.

This mapping x_{j}, z_{j} ↦ x′_{j}, z′_{j}, j = 1, 2, 3, is the basic example of a class of the canonical rational mappings of the ultra-local Weyl algebra. The following lemma establishes the uniqueness of the matrix structure of any such mapping.

Lemma 1 Let x_{j}, z_{j}, j = 1 … Δ be a normalized finite-dimensional unitary basis (3) of the local Weyl algebra

x_{i} z_{j} = q z_{j} x_{i} δ_{i,j};  q^{N} = 1;  x_{j}^{N} = z_{j}^{N} = 1.  (22)

Let E: x_{j}, z_{j} ↦ x′_{j}, z′_{j} be an invertible canonical mapping in the space of rational functions of x_{j}, z_{j} such that it conserves the centers

x′_{j}^{N} = z′_{j}^{N} = 1 .

Then there exists a unique (up to a scalar multiplier) N^{Δ} × N^{Δ} matrix E such that for any Φ of eq.(7):

Φ( x′_{j}, z′_{j}) = E Φ(x_{j}, z_{j}) E^{−1}.  (23)

Proof : The ring of the rational functions of x_{j}, z_{j} at root of unity is the algebra of the polynomials of x_{j}, z_{j} with ℂ-valued coefficients. Evidently this enveloping algebra is the complete algebra of N^{Δ} × N^{Δ} matrices. Since E is invertible, the envelope of x′_{j}, z′_{j} is the same matrix algebra. Furthermore, since E is canonical and conserves the N-th powers of the Weyl elements, E is an automorphism of the matrix algebra. Finally, since the algebra of N^{Δ} × N^{Δ} matrices is the irreducible fundamental representation of the semi-simple algebra 𝔤𝔩(N^{Δ}), any such automorphism is an internal one and may be realized by the unique matrix E of (23). □

1.5 Matrix part of R_{1,2,3} at root of unity in terms of Fermat curve cyclic functions W_{p}(n)

Due to lemma 1 there exists a unique (up to a scalar factor) N^{3} × N^{3}-dimensional matrix R_{1,2,3}, such that

R_{1,2,3} x_{1} = x′_{1} R_{1,2,3} , R_{1,2,3} z_{1} = z′_{1} R_{1,2,3} , etc.  (24)

for (21).

The basis independent expression for R_{1,2,3} is not a useful object, and here we give the matrix elements of R_{1,2,3} in the basis (3) in terms of the Bazhanov-Baxter [2] cyclic functions W_{p}(x) which we shall mainly use in this paper. We define

\frac{W_{p}(n)}{W_{p}(n − 1)} = \frac{y}{1 − ω^{n} x};  W_{p}(0) = 1,  (25)

where n ∈ ℤ_{N} and p = (x, y) denotes a point on the Fermat curve

x^{N} + y^{N} = 1.  (26)

For n > 0 we have

W_{p}(n) = ∏_{ν=1}^{n} \frac{y}{1 − ω^{ν} x},  (27)

and generally W_{p}(n+N) = W_{p}(n), because of ∏_{ν=0}^{N−1}(1−ω^{ν}x) = y^{N}. One automorphism of the Fermat curve will be important for later calculations. Defining Op by

p = (x, y) ↦ Op = (ω^{−1}x^{−1}, ω^{−1/2}x^{−1}y),  (28)

we have

W_{p}(n) = \frac{1}{W_{Op}(−n) Φ(n)},  (29)

with

Φ(n) = (−1)^{n} ω^{n^{2}/2}.  (30)

At special points on the Fermat curve the W_{p}(n) take simple values: Defining

q_{0} = (0, 1);  q_{∞} = O q_{0};  q_{1} = (ω^{−1}, 0);  (31)

we get

W_{q_{0}}(n) = 1;  W_{q_{∞}}(n) = Φ^{−1}(n);  1/W_{q_{1}} = δ_{n,0}.  (32)

We now express our conjugation matrix in terms of the functions W_{p}(n) :

Proposition 2 In the basis (3) the matrix R_{1,2,3}, solving the relations (21,24), has the following matrix elements:

⟨i_{1}, i_{2}, i_{3}|R_{1,2,3}|j_{1}, j_{2}, j_{3} ⟩ =^{def} R_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}}
= δ_{i_{2}+i_{3},j_{2}+j_{3}} ω^{(j_{1}−i_{1}) j_{3}} \frac{W_{p_{1}}(i_{2} − i_{1}) W_{p_{2}}(j_{2} − j_{1})}{W_{p_{3}}(j_{2} − i_{1}) W_{p_{4}}(i_{2} − j_{1})} , (33)

where the x-coordinates of the four Fermat curve points are connected by

x_{1} x_{2} = ω x_{3} x_{4} .  (34)

In the terms of the variables u_{j}, w_{j}, κ_{j}, j = 1, 2, 3, these points are defined by

x_{1} = \frac{ω^{−1/2}}{κ_{1}} \frac{u_{2}}{u_{1}} , x_{2} = ω^{−1/2}κ_{2} \frac{u′_{2}}{u′_{1}} x_{3} = ω^{−1} \frac{u′_{2}}{u_{1}} , x_{4} = ω^{−1} \frac{κ_{2}}{κ_{1}} \frac{u_{2}}{u′_{1}} ,  (35)
\frac{y_{3}}{y_{1}} = κ_{1} \frac{w_{1}}{u′_{3}} , \frac{y_{4}}{y_{1}} = ω^{−1/2}κ_{3} \frac{w_{3}}{w_{2}} , \frac{y_{3}}{y_{2}} = \frac{w′_{2}}{w_{3}} , \frac{y_{4}}{y_{2}} = ω^{−1/2} \frac{κ_{3}}{κ_{1}} \frac{u′_{3}}{w′_{1}} ,  (36)

where the u′_{j}, w′_{j} and u_{j}, w_{j} are related by the functional transformation (15):

u′_{j} = R_{1,2,3}^{(f)} ∘ u_{j},  w′_{j} = R_{1,2,3}^{(f)} ∘ w_{j}.  (37)

Proof. We shall give the proof, that (33) produces the rational mapping, for the first line of (21) only, the other equations follow analogously. First observe that the matrix elements of the operator R_{1,2,3} satisfy several recurrent relations. In particular, we will need the recursion

R_{i_{1},i_{2}+1,i_{3}−1}^{j_{1},j_{2},j_{3}} = R_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}} · \frac{y_{1}}{y_{4}} · \frac{1 − ω^{i_{2}−j_{1}+1}x_{4}}{1 − ω^{i_{2}−i_{1}+1}x_{1}}

which can be rewritten in the form

R_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}} ω^{−j_{1}} = \frac{1}{ω^{i_{2}+1}x_{4}} R_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}} + \frac{x_{1}y_{4}}{ω^{i_{1}}y_{1}x_{4}} R_{i_{1},i_{2}+1,i_{3}−1}^{j_{1},j_{2},j_{3}} − \frac{y_{4}}{ω^{i_{2}+1}x_{4}y_{1}} R_{i_{1},i_{2}+1,i_{3}−1}^{j_{1},j_{2},j_{3}} . (38)

However, this recurrent relation is the matrix element ⟨i_{1}i_{2}i_{3}| · |j_{1}j_{2}j_{3}⟩ of the operator equality

R_{1,2,3} · (x′_{1})^{−1} = (\frac{1}{ω x_{4}} x_{2}^{−1} + \frac{x_{1}y_{4}}{y_{1}x_{4}} x_{1}^{−1}z_{2}^{−1}z_{3} − \frac{y_{4}}{ω x_{4}y_{1}} x_{2}^{−1}z_{2}^{−1}z_{3}) · R_{1,2,3}.

This coincides with first line in (21), provided the identification (35) and (36) is valid for the Fermat points (x_{1}, y_{1}) and (x_{4}, y_{4}). □

Remark 2 R is a matrix function of three continuous parameters x_{1}, x_{2}, x_{3} and three discrete parameters: the phases of y_{1}, y_{2}, y_{3}. Equivalently, one may use κ_{1} \frac{u_{1}}{u_{2}} , κ_{3} \frac{w_{3}}{w_{2}} and

\frac{w_{1}}{u_{3}} as the continuous parameters and the phases of u′_{1}, u′_{2}, w′_{1} as the discrete parame-

ters. Formulas (35,36) establish the correspondence between these choices. We call the parameterization of R_{1,2,3} in the terms of u_{j}, w_{j}, κ_{j} a ”free parameterization”.

Remark 3 Formulated in terms of mappings, the automorphism R_{1,2,3} of the ultra-local Weyl algebra at the root of unity is presented as the superposition of a pure functional mapping and the finite dimensional similarity transformation:

R_{1,2,3} ∘ Φ = R_{1,2,3} (R_{1,2,3}^{(f)} ∘ Φ) R_{1,2,3}^{−1} .  (39)

1.6 Free bosonic realization of R_{1,2,3}

Analogous to the continuum realization of the TE-Boltzmann weights proposed in eq.(4.1) of [17], also our (33) can be written in a bosonic realization. In this realization the cyclic weights W_{p}(n) of (25) are replaced by the following Gaussian weights:

W_{x}(σ) = exp (\frac{i}{2ℏ} \frac{x}{x − 1} σ^{2});  σ ∈ ℝ, x ∈ ℂ; ℑm \frac{x}{x − 1} > 0. (40)

At each vertex j of a graph define a pair of operators q_{j}, p_{j} satisfying [ q_{j′}, p_{j}] = iℏδ_{j,j′}, and scalar variables u_{j}, w_{j}. We choose a basis | σ_{j}⟩ with ⟨σ_{j′}| σ_{j}⟩ = δ(σ_{j′} − σ_{j}), such that for ψ(σ) ∈ L^{2} we have

ψ(σ_{j}) =^{def} ⟨σ_{j}|ψ⟩;  ⟨σ_{j}| q_{j} |ψ⟩ = σ_{j}ψ(σ_{j});  ⟨σ_{j}| p_{j} |ψ⟩ = \frac{ℏ}{i} \frac{∂ψ(σ_{j})}{∂σ_{j}}.  (41)

For each set of vertices we define the corresponding operators and scalars and a direct product space of the single vertex spaces. Consider now the following mapping R_{1,2,3} in the product space of three points j = 1, 2, 3 by

R_{123} q_{j} R_{123}^{−1} = ∑_{k=1}^{3} \frac{∂ log u′_{j}}{∂ log u_{k}} q_{k} + \frac{∂ log u′_{j}}{∂ log w_{k}} p_{k} ≡ q′_{j}

R_{123} p_{j} R_{123}^{−1} = ∑_{k=1}^{3} \frac{∂ log w′_{j}}{∂ log u_{k}} q_{k} + \frac{∂ log w′_{j}}{∂ log w_{k}} p_{k} ≡ p′_{j}.  (42)

where the relation between the primed and unprimed scalars u_{j}, w_{j} is given by (15) with N = 1 . The κ_{j} are further parameters (“coupling constants“) at the vertices j. As the following Proposition shows, equations (42) are the bosonic continuum analogs to eqs.(21) of the discrete case.

Proposition 3 In the basis (41) the operator R_{1,2,3} has the following kernel

\begin{matrix}⟨σ_{1}, σ_{2}, σ_{3}|R_{1,2,3}|σ′_{1}, σ′_{2}, σ′_{3}⟩ \\ = δ(σ_{2} + σ_{3} − σ′_{2} − σ′_{3}) e^{− \frac{i}{ℏ}(σ′_{1}−σ_{1}) σ′_{3}} \frac{W_{x_{1}}(σ_{2} − σ_{1}) W_{x_{2}}(σ′_{2} − σ′_{1})}{W_{x_{3}}(σ′_{2} − σ_{1}) W_{x_{4}}(σ_{2} − σ′_{1})} , \end{matrix}  (43)

with the constraint x_{1} x_{2} = x_{3} x_{4} . In terms of the variables u_{j}, w_{j}, κ_{j}, (j = 1, 2, 3), the x_{k} ∈ ℝ are defined by (obtained by putting formally ω^{−1/2} → −1 in (35)):

x_{1} = −\frac{1}{κ_{1}} \frac{u_{2}}{u_{1}} ; x_{2} = −κ_{2} \frac{u′_{2}}{u′_{1}} ; x_{3} = \frac{u′_{2}}{u_{1}} ; x_{4} = \frac{κ_{2}}{κ_{1}} \frac{u_{2}}{u′_{1}} ,  (44)

where u′_{1} and u′_{2} are defined as in (37) with N = 1.

Proof : We give the proof for one of the six equations (42), as the other equations follow analogously. Let us write shorthand |σ⟩ for |σ_{1}, σ_{2}, σ_{3}⟩ and d^{3}σ = dσ_{1}dσ_{2}dσ_{3} etc. We consider:

∫ d^{3}σ′ ⟨ σ | R_{1,2,3} | σ′ ⟩⟨ σ′ | q_{3} | σ′′ ⟩ = ∫ d^{3}σ′ ⟨ σ | q′_{3} | σ′ ⟩⟨ σ′ | R_{1,2,3} | σ′′⟩ (45)

which should be satisfied for all σ_{1}, σ_{2}, σ_{3}, σ′′_{1}, σ′′_{2}, σ′′_{3}. Written more explicitly, the kernel (43) is:

R_{σ_{1},σ_{2},σ_{3}}^{σ′_{1},σ′_{2},σ′_{3}} = δ(σ_{2} + σ_{3} − σ′_{2} − σ′_{3}) exp (\frac{i}{2 ℏ} Σ(σ, σ′));

where

Σ(σ, σ′) = \frac{w_{1}(σ_{2} − σ_{1})^{2} + u_{3}(σ_{1} − σ′_{2})^{2}}{u_{2}^{−1}w_{1}(κ_{1}u_{1} + u_{2})} + \frac{(κ_{1}u_{1}w_{2} + κ_{3}u_{2}w_{3} + κ_{1}κ_{3}u_{1}w_{3})(σ_{2} − σ′_{1})^{2}}{κ_{3}w_{3}(κ_{1}u_{1} + u_{2})}
+ \frac{u_{3}(κ_{1}u_{1}w_{2} + κ_{3}u_{2}w_{3} + κ_{1}κ_{3}u_{1}w_{3})(σ′_{2} − σ′_{1})^{2}}{(κ_{3}u_{3}w_{3} + w_{1}w_{2} + u_{3}w_{2})(κ_{1}u_{1} + u_{2})} − 2(σ′_{1} − σ_{1})σ′_{3}.  (46)

From (42) we get:

q′_{3} = \frac{(w_{1} + u_{3})u_{2}(q_{2} − q_{1}) + u_{2}u_{3}q_{3} + (u_{2} + κ_{1}u_{1})w_{1}p_{1}}{u_{2}u_{3} + u_{2}w_{1} + κ_{1}u_{1}w_{1}} .

and (45) becomes:

∫ d^{3}σ′ δ(σ_{2} + σ_{3} − σ′_{2} − σ′_{3}) exp (\frac{i}{2ℏ}Σ(σ, σ′)) σ′′_{3} δ^{3}(σ′ − σ′′)

= ∫ d^{3}σ′ δ^{3}(σ − σ′) \frac{(w_{1} + u_{3})u_{2}(σ′_{2} − σ′_{1}) + u_{2}u_{3}σ′_{3} + (u_{2} + κ_{1}u_{1})w_{1} \frac{ℏ}{i} \frac{∂}{∂σ′_{1}}}{u_{2}u_{3} + u_{2}w_{1} + κ_{1}u_{1}w_{1}} ×
× δ(σ′_{2} + σ′_{3} − σ′′_{2} − σ′′_{3}) exp (\frac{i}{2ℏ}Σ(σ′, σ′′)).  (47)

Since

(u_{2} + κ_{1}u_{1})\frac{w_{1}}{2} \frac{∂Σ(σ′, σ′′)}{∂σ′_{1}} = (w_{1} + u_{3})u_{2}σ′_{1} − u_{2}w_{1}σ′_{2} − u_{2}u_{3}σ′′_{2} + (w_{1}u_{2} + κ_{1}u_{1}w_{1})σ′′_{3},

eq.(47) reduces to

σ′′_{3}δ(σ_{2} + σ_{3} − σ′′_{2} − σ′′_{3}) = \frac{u_{2}u_{3}(σ_{2} + σ_{3} − σ′′_{2}) + (u_{2}w_{1} + κ_{1}u_{1}w_{1})σ′′_{3}}{u_{2}u_{3} + u_{2}w_{1} + κ_{1}u_{1}w_{1}} δ(σ_{2} + σ_{3} − σ′′_{2} − σ′′_{3}).

□

2 The modified tetrahedron equation

For three-dimensional integrable spin models the Tetrahedron Equation (TE) plays a role which the Yang-Baxter equation has for two-dimensional integrable spin models. The TE provides the commutativity of so-called layer-to-layer transfer matrices. In our case, where the dynamical variables form an affine Weyl algebra, we are able to define a more general equation: the Modified Tetrahedron Equation (MTE), which provides the commutativity of more complicated transfer matrices, see e.g [3, 6]. In Fig. 2 we show a graphical image of the mappings leading to the tetrahedron equation (4): the sequence of mappings Q_{1} → Q_{2} → Q_{3} → Q_{4} → Q_{5} gives the same Q_{5} as Q_{1} → Q_{8} → Q_{7} → Q_{6} → Q_{5}.

Following eq.(7) let us fix the following notation for the superposition of two mappings A and B:

((A · B) ∘ Φ) =^{def} (A ∘ (B ∘ Φ)) .  (48)

Due to the uniqueness of the mapping R discussed in section 1.2, p.3, we arrive at the

Proposition 4 The mapping R solves the Tetrahedron Equation:

R_{123} · R_{145} · R_{246} · R_{356} = R_{356} · R_{246} · R_{145} · R_{123} ,  (49)

acting in the space of the twelve affine Weyl elements u_{1}, w_{1}, u_{2}, w_{2}, …, w_{5}, u_{6}, w_{6}.

Since, as has been discussed in Sec. 1.3, any rational automorphism of the ultra-local Weyl algebra implies a rational mapping in the space of N-th powers of the parameters of the representation, its is a direct consequence of (49) that the R_{i,j,k}^{(f)} of (15) solve the tetrahedron equation with the variables u_{j}^{N}, w_{j}^{N}, j = 1, …, 8.

We want to get the functional tetrahedron equation not only for the N-th powers of the variables but for the variables u_{j} and w_{j} directly. However, not all phases of the u_{j}, w_{j} can be chosen independently. In Sec. 3.2 we shall show explicitly how to make an independent choice of phases.

Once an appropriate choice of phases has been made, we obtain the functional tetrahedron equation on the variables u_{j}, w_{j}. Now using (39) the functional tetrahedron equation can be canceled between the two sides of (49), and we are left with the Modified Tetrahedron Equation for the finite dimensional R-matrices:

R_{1,2,3} · (R_{1,2,3}^{(f)} ∘ R_{1,4,5})·(R_{1,2,3}^{(f)}R_{1,4,5}^{(f)} ∘ R_{2,4,6})·(R_{1,2,3}^{(f)}R_{1,4,5}^{(f)}R_{2,4,6}^{(f)} ∘ R_{3,5,6})

∼ R_{3,5,6} · (R_{3,5,6}^{(f)} ∘ R_{2,4,6})·(R_{3,5,6}^{(f)}R_{2,4,6}^{(f)} ∘ R_{1,4,5})·(R_{3,5,6}^{(f)}R_{2,4,6}^{(f)}R_{1,4,5}^{(f)} ∘ R_{1,2,3})

(50)

Observe that due to the cancellation of the functional tetrahedron equation there is no R_{3,5,6}^{(f)} on the left hand side of (50) and no R_{1,2,3}^{(f)} on the right hand side. Because of the uniqueness of the mapping shown in Lemma 1 on page 7, the left and right hand sides of (50) may differ only by a scalar factor, which arises when we pass from the equivalence of the mappings to the equality of the matrices. So, in matrix element notation the MTE reads

\begin{aligned}∑_{j_{1}...j_{6}} & (R^{(1)}) & {j_{1},j_{2},j_{3} \atop i_{1},i_{2},i_{3}} & (R^{(2)}) & {k_{1},j_{4},j_{5} \atop j_{1},i_{4},i_{5}} & (R^{(3)}) & {k_{2},k_{4},j_{6} \atop j_{2},j_{4},i_{6}} & (R^{(4)}) & {k_{3},k_{5},k_{6} \atop j_{3},j_{5},j_{6}} \\ & = ρ ∑_{j_{1}...j_{6}} (R^{(8)})_{i_{3},i_{5},i_{6}}^{j_{3},j_{5},j_{6}} (R^{(7)})_{i_{2},i_{4},j_{6}}^{j_{2},j_{4},k_{6}} (R^{(6)})_{i_{1},j_{4},j_{5}}^{j_{1},k_{4},k_{5}} (R^{(5)})_{j_{1},j_{2},j_{3}}^{k_{1},k_{2},k_{3}} , \end{aligned}  (51)

2 ^{3} ^{1}  ^{4}^{5}^{6}  ^{R_{1,4,5}}  ^{R_{2,4,6}}^{3}^{5}^{1}  ^{2} ^{6}^{4}^{3}^{2}  ^{5} ^{1}^{4}^{6} ^{R_{2,4,6}}^{6}^{4}  ^{2}  ^{5}^{3}  ^{1}  ^{R_{1,4,5}}^{5}  ^{4}^{2}^{6}^{1}^{3}  ^{6}^{2}^{3}^{4}  ^{5}^{1}^{4}  ^{3}^{6}^{5}  ^{1}^{2}  ^{2}^{1}  ^{4}^{5} ^{3} ^{6}^{R_{1,2,3}}  ^{R_{3,5,6}}^{R_{3,5,6}}  ^{R_{1,2,3}} Q_{4}  Q_{5}  Q_{6} Q_{3}  Q_{7} Q_{2}  Q_{1}  Q_{8}

Figure 2: Graphical image of the two equivalent ways of transforming the four-line-graph (”quadrilateral”) Q_{1} into graph Q_{5}, which leads to tetrahedron equation. Observe that each graph contains only two triangles which can be transformed by a mapping R: In graph Q_{1} either the line 124 can be moved downward through the point 3 (leading to graph Q_{2}), or the line 456 can be moved upward through point 3 (leading to Q_{8}). Both the left hand and right hand sequences of four transformations lead to the same graph Q_{5}.

where R^{(1)} corresponds to R_{1,2,3}, R^{(2)} to R_{1,2,3}^{(f)} ∘ R_{1,4,5}, etc. Here ρ is the scalar factor. The matrix elements of each R^{(j)} as functions of the Fermat-curve parameters x_{i}^{(j)}, y_{i}^{(j)} are given by Proposition 3 with functional mappings applied as shown in (50). The N^{3}-th power of the scalar factor in (51) is

ρ^{N^{3}} = \frac{det R^{(1)} det R^{(2)} det R^{(3)} det R^{(4)}}{det R^{(8)} det R^{(7)} det R^{(6)} det R^{(5)}} ,  (52)

and this can be obtained from the determinant of one single matrix R_{1,2,3} just by substituting the respective coordinates.

If the realization of the mapping by the free bosonic weight function (43) is used, the corresponding modified tetrahedron equation involves integrations over ℝ instead of the summations over ℤ_{N}. The bosonic MTE may be proven directly with the help of Gaussian integrations.

2.1 Calculation of the determinant of R_{1,2,3}

We use the representation (33) to find a closed expression for det R_{1,2,3}. The numerator term W_{p_{2}}(j_{2} − j_{1}) is diagonal, so it just contributes a factor (∏_{n} W_{p_{2}}(n))^{N^{2}} to the determinant. For later convenience, we treat the other numerator factor W_{p_{1}}(i_{2} − i_{1}) of (33) differently: using the Fermat curve automorphism (28) we write

W_{p_{1}}(n) = \frac{1}{W_{Op_{1}}(−n) Φ(n)} .

W_{Op_{1}}(i_{1}−i_{2}) is diagonal and its determinant is trivially calculated. We combine the factor Φ(i_{2} − i_{1}) with the two non-diagonal terms W_{p_{3}}, W_{p_{4}} and write:

det⟨i_{1}, i_{2}, i_{3}| R_{1,2,3}|j_{1}, j_{2}, j_{3}⟩ = ( ∏_{n=0}^{N−1} \frac{W_{p_{2}}(n)}{W_{Op_{1}}(n)})^{N^{2}}det \frac{δ_{i_{2}+i_{3},j_{2}+j_{3}} ω^{(j_{1}−i_{1}) j_{3}}}{Φ(i_{2} −i_{1}) W_{p_{3}}(j_{2} −i_{1}) W_{p_{4}}(i_{2} −j_{1})}.

(53)

We now calculate the determinant on the right hand side of (53) from its finite Fourier transform in the indices i_{1} and j_{1}. So we define and evaluate

⟨i_{1}, i_{2}, i_{3}| R′_{1,2,3}|j_{1}, j_{2}, j_{3}⟩

= δ_{i_{2}+i_{3},j_{2}+j_{3}} \frac{1}{N} ∑_{a,b∈ℤ_{N}} ω^{i_{1}a−j_{1}b} \frac{ω^{(b−a) j_{3}}}{Φ(i_{2} − a) W_{p_{3}}(j_{2} − a) W_{p_{4}}(i_{2} − b)}

= δ_{i_{2}+i_{3},j_{2}+j_{3}} \frac{1}{N} ∑_{a′,b′} ω^{i_{1}(j_{2}−a′)−j_{1}(i_{2}−b′)−(i_{2}−j_{2})a′} \frac{ω^{(i_{2}−j_{2}+a′−b′) j_{3}}}{Φ(i_{2} − j_{2}) Φ(a′) W_{p_{3}}(a′) W_{p_{4}}(b′)}

= δ_{i_{2}+i_{3},j_{2}+j_{3}} \frac{ω^{i_{1}j_{2}−i_{2}j_{1}+(j_{3}−i_{3})j_{3}}}{Φ(j_{3} − i_{3})} \frac{1}{N} ∑_{a′} \frac{ω^{(i_{3}−i_{1})a′}}{Φ(a′) W_{p_{3}}(a′)} ∑_{b′} \frac{ω^{−(j_{3}−j_{1})b′}}{W_{p_{4}}(b′)}  (54)

In the last line we have used the property

Φ(a + b) = Φ(a) Φ(b) ω^{ab}  (55)

which, supplying a factor ω^{(i_{2}−j_{2})a′}, decouples the a′- and b′-summations in the presence of δ_{i_{2}+i_{3},j_{2}+j_{3}}. The determinant of the first three factors of the last line of (54) will again be calculated from its Fourier transform. So we define and evaluate

⟨i_{1}, i_{2}, i_{3}| P |j_{1}, j_{2}, j_{3}⟩ = \frac{1}{N^{2}} ∑_{c,d} ω^{i_{1}c−j_{1}d}δ_{i_{2}+i_{3},j_{2}+j_{3}} \frac{ω^{j_{2}c−i_{2}d+(j_{3}−i_{3})j_{3}}}{Φ(i_{3})Φ(j_{3})} ω^{i_{3}j_{3}}

= δ_{i_{2}+i_{3},j_{2}+j_{3}} δ_{i_{1},−j_{2}} δ_{j_{1},−i_{2}} \frac{Φ(j_{3})}{Φ(i_{3})},  (56)

so that

det ⟨i_{1}, i_{2}, i_{3}| P |j_{1}, j_{2}, j_{3}⟩ = det || δ_{i_{2}+i_{3},j_{2}+j_{3}} δ_{i_{1},−j_{2}} δ_{j_{1},−i_{2}} || = (−1)^{N^{2}(N−1)/2}. (57)

Combining (53),(54),(56) and (57), our preliminary result is:

det < i_{1}, i_{2}, i_{3}|R_{1,2,3}|j_{1}, j_{2}, j_{3} >
= ((−1)^{(N−1)/2} ∏_{k} \frac{W_{p_{2}}(k)}{W_{Op_{1}}(k)} ∏_{m} ∑_{a} \frac{ω^{ma}}{Φ(a)W_{p_{3}}(a)} ∏_{n} ∑_{b} \frac{ω^{nb}}{W_{p_{4}}(b)})^{N^{2}} . (58)

We may simplify this formula by introducing the function V (x) on the Fermat curve

V (x) =^{def} ∏_{n=1}^{N−1} (1 − ω^{n+1} x)^{n}.  (59)

Writing (V (ω^{−1}))^{2} as product of N^{2} factors of the form (1 − ω^{n}) times powers of ω, we compute

V (ω^{−1}) = N^{N/2} e^{iπ(N−1)(N−2)/12}.  (60)

It is useful to note that

∏_{n} ∑_{a} \frac{ω^{na}}{Φ(a)} = V (ω^{−1});  ∏_{n} Φ(n) = e^{−πi(N^{2}−1)/6}.  (61)

Writing out the factors of ( x^{N(N−1)/2} V (ω^{−1} x^{−1}) ) V (x) and extracting several powers of ω, we obtain

V (ω^{−1}x^{−1}) = \frac{y^{N(N−1)}}{x^{N(N−1)/2}} \frac{e^{iπ(N−1)(N−2)/6}}{V (x)} .  (62)

We can express the terms involving p_{1} and p_{2} using V (x):

∏_{n} W_{p_{2}}(n) = \frac{V (x_{2})}{y_{2}^{N(N−1)/2}} ;  ∏_{n} \frac{1}{W_{Op_{1}}} = \frac{V (x_{1})}{y_{1}^{N(N−1)/2}} e^{−iπ(N^{2}−1)/6}.

In the ratio V (x)/V (ω x) many terms cancel, leading to

\frac{V (x)}{V (ω x)} = (\frac{y}{1 − ω x})^{N} .  (63)

This can be used to compute the p_{4}-term in (58): We define

F_{p=(x,y)} = ∏_{n} ∑_{b} \frac{ω^{n b}}{W_{p}(b)},

which, using W_{(ω x, y)}(b) = (1 − ω x) y^{−1} W_{(x,y)}(b + 1) , is seen to satisfy

\frac{F_{(ω x, y)}}{F_{(x, y)}} = ω^{N(N−1)/2} (\frac{y}{1 − ω x})^{N}, F_{q_{1}} = 1, F_{q_{∞}} = V^{∗}(ω^{−1}), F_{q_{0}} = 0.

where q_{0}, q_{∞} and q_{1} are the three special Fermat points introduced in (31). This is solved by

F_{(x,y)} = (ω x)^{N(N−1)/2} \frac{V (ω^{−1})}{V (x)} .

Finally, we consider the p_{3} term: Defining

G_{p=(x,y)} = ∏_{n} ∑_{a} \frac{ω^{na}}{Φ(a) W_{p}(a)},

which satisfies

\frac{G_{(ω x, y)}}{G_{(x, y)}} = (\frac{y}{1 − ω x})^{N} .

Using G_{q_{1}} = 1, G_{q_{∞}} = 0, G_{q_{0}} = V (ω^{−1}), we get G_{p} = V (ω^{−1})/V (x) . Inserting these results into (58), our final expression for det R_{1,2,3} is:

det R = N^{N^{3}} ((\frac{x_{4}}{y_{1} y_{2}} )^{N(N−1)/2} \frac{V (x_{1})V (x_{2})}{V (x_{3})V (x_{4})})^{N^{2}} .  (64)

The relation x_{1}x_{2} = ω x_{3}x_{4} has not yet been used and is still to be imposed here. For N = 2 eq.(64) gives

det R = 2^{8} (\frac{x_{4}}{y_{1} y_{2}} \frac{(1 − x_{1})(1 − x_{2})}{(1 − x_{3})(1 − x_{4})})^{4} = ( 4 x_{4} \frac{y_{1} y_{2}}{y_{3}^{2} y_{4}^{2}} \frac{(1 + x_{3})(1 + x_{4})}{(1 + x_{1})(1 + x_{2})} )^{4} ,

in agreement with (84). Observe that, despite the quite similar appearance of W_{p_{3}} and W_{p_{4}} in (33), different phases make (64) unsymmetrical between x_{3} and x_{4}, compare (84).

3 Parameterization of the Fermat points

Now the most important step follows: the parameterization of the Fermat points for each of the eight R-matrices. Recall that x_{4}^{(j)} is determined by the other three x_{i}^{(j)} due to (34).

Writing repeatedly the parameterization (35),(36) for the eight R-matrices, then applying repeatedly the functional mappings as it is written in (50), we get for the arguments appearing in (51):

x_{1}^{(j)} = \frac{1}{\sqrt{ω}} Ξ_{j1};  x_{2}^{(j)} = \frac{1}{\sqrt{ω}} Ξ_{j2};  x_{3}^{(j)} = \frac{1}{ω} Ξ_{j3};  x_{4}^{(j)} = \frac{x_{1}^{(j)}x_{2}^{(j)}}{ω x_{3}^{(j)}} ;

y_{31}^{(j)} = Υ_{j1};  y_{41}^{(j)} = \frac{1}{\sqrt{ω}} Υ_{j2};  y_{32}^{(j)} = Υ_{j3};  where y_{ik}^{(j)} ≡ \frac{y_{i}^{(j)}}{y_{k}^{(j)}} . (65)

where (not writing Ξ_{j,4} ):

Ξ_{jk} =( \begin{matrix}\frac{u_{2}^{(1)}}{κ_{1}u_{1}^{(1)}} & \frac{κ_{2}u_{2}^{(2)}}{u_{1}^{(2)}} & \frac{u_{2}^{(2)}}{u_{1}^{(1)}} \\ \frac{u_{4}^{(1)}}{κ_{1}u_{1}^{(2)}} & \frac{κ_{4}u_{4}^{(3)}}{u_{1}^{(5)}} & \frac{u_{4}^{(3)}}{u_{1}^{(2)}} \\ \frac{u_{4}^{(3)}}{κ_{2}u_{2}^{(2)}} & \frac{κ_{4}u_{4}^{(5)}}{u_{2}^{(5)}} & \frac{u_{4}^{(5)}}{u_{2}^{(2)}} \\ \frac{u_{5}^{(3)}}{κ_{3}u_{3}^{(2)}} & \frac{κ_{5}u_{5}^{(5)}}{u_{3}^{(5)}} & \frac{u_{5}^{(5)}}{u_{3}^{(2)}} \\ \frac{u_{2}^{(7)}}{κ_{1}u_{1}^{(6)}} & \frac{κ_{2}u_{2}^{(5)}}{u_{1}^{(5)}} & \frac{u_{2}^{(5)}}{u_{1}^{(6)}} \\ \frac{u_{4}^{(7)}}{κ_{1}^{(1)}u_{1}^{(1)}} & \frac{κ_{4}u_{4}^{(5)}}{u_{1}^{(6)}} & \frac{u_{4}^{(5)}}{u_{1}^{(1)}} \\ \frac{u_{4}^{(1)}}{κ_{2}u_{2}^{(1)}} & \frac{κ_{4}u_{4}^{(7)}}{u_{2}^{(7)}} & \frac{u_{4}^{(7)}}{u_{2}^{(1)}} \\ \frac{u_{5}^{(1)}}{κ_{3}u_{3}^{(1)}} & \frac{κ_{5}u_{5}^{(8)}}{u_{3}^{(8)}} & \frac{u_{5}^{(8)}}{u_{3}^{(1)}} \end{matrix} );  Υ_{jk} =( \begin{matrix}\frac{κ_{1}w_{1}^{(1)}}{u_{3}^{(2)}} & \frac{κ_{3}w_{3}^{(1)}}{w_{2}^{(1)}} & \frac{w_{2}^{(2)}}{w_{3}^{(1)}} \\ \frac{κ_{1}w_{1}^{(2)}}{u_{5}^{(3)}} & \frac{κ_{5}w_{5}^{(1)}}{w_{4}^{(1)}} & \frac{w_{4}^{(3)}}{w_{5}^{(1)}} \\ \frac{κ_{2}w_{2}^{(2)}}{u_{6}^{(4)}} & \frac{κ_{6}w_{6}^{(1)}}{w_{4}^{(3)}} & \frac{w_{4}^{(5)}}{w_{6}^{(1)}} \\ \frac{κ_{3}w_{3}^{(2)}}{u_{6}^{(5)}} & \frac{κ_{6}w_{6}^{(4)}}{w_{5}^{(3)}} & \frac{w_{5}^{(5)}}{w_{6}^{(4)}} \\ \frac{κ_{1}w_{1}^{(6)}}{u_{3}^{(5)}} & \frac{κ_{3}w_{3}^{(8)}}{w_{2}^{(7)}} & \frac{w_{2}^{(5)}}{w_{3}^{(8)}} \\ \frac{κ_{1}w_{1}^{(1)}}{u_{5}^{(5)}} & \frac{κ_{5}w_{5}^{(8)}}{w_{4}^{(7)}} & \frac{w_{4}^{(5)}}{w_{5}^{(8)}} \\ \frac{κ_{2}w_{2}^{(1)}}{u_{6}^{(5)}} & \frac{κ_{6}w_{6}^{(8)}}{w_{4}^{(1)}} & \frac{w_{4}^{(7)}}{w_{6}^{(8)}} \\ \frac{κ_{3}w_{3}^{(1)}}{u_{6}^{(8)}} & \frac{κ_{6}w_{6}^{(1)}}{w_{5}^{(1)}} & \frac{w_{5}^{(8)}}{w_{6}^{(1)}} \end{matrix} ). (66)

Here for any f it is implied that

f^{(2)} = R_{1,2,3}^{(f)} ∘ f^{(1)}; f^{(3)} = R_{1,2,3}^{(f)}R_{1,4,5}^{(f)} ∘ f^{(1)}; f^{(4)} = R_{1,2,3}^{(f)}R_{1,4,5}^{(f)}R_{2,5,6}^{(f)} ∘ f^{(1)}, (67)

and

f^{(5)} = R_{1,2,3}^{(f)}R_{1,4,5}^{(f)}R_{2,4,6}^{(f)}R_{3,5,6}^{(f)} ∘ f^{(1)} .  (68)

For the right hand side of (51) we define for any f

f^{(8)} = R_{3,5,6}^{(f)} ∘ f^{(1)}; f^{(7)} = R_{3,5,6}^{(f)}R_{2,4,6}^{(f)} ∘ f^{(1)}; f^{(6)} = R_{3,5,6}^{(f)}R_{2,4,6}^{(f)}R_{1,4,5}^{(f)} ∘ f^{(1)} . (69)

Due to the validity of the functional tetrahedron equation the four times transformed function f^{(¯5)} = R_{3,5,6}^{(f)}R_{2,4,6}^{(f)}R_{1,4,5}^{(f)}R_{1,2,3}^{(f)} ∘ f^{(1)} coincides with (68).

Observe that e.g.

u_{1}^{(3)} = u_{1}^{(4)} = u_{1}^{(5)}; u_{2}^{(2)} = u_{2}^{(3)}; u_{2}^{(4)} = u_{2}^{(5)}; u_{3}^{(2)} = u_{3}^{(3)} = u_{3}^{(4)}; u_{4}^{(8)} = u_{4}^{(1)} = u_{4}^{(2)};

u_{4}^{(4)} = u_{4}^{(5)} = u_{4}^{(6)}; u_{5}^{(1)} = u_{5}^{(2)}; u_{5}^{(3)} = u_{5}^{(4)}; u_{5}^{(5)} = u_{5}^{(6)}; u_{6}^{(1)} = u_{6}^{(2)} = u_{6}^{(3)}. (70)

The MTE’s leave the following four “centers” invariant, i.e. for j = 1, …, 8 we have:

ℭ_{1} = u_{4}^{(j)} u_{5}^{(j)} u_{6}^{(j)}; ℭ_{2} = \frac{u_{2}^{(j)} u_{3}^{(j)}}{w_{6}^{(j)}} ; ℭ_{3} = \frac{w_{3}^{(j)} w_{5}^{(j)}}{u_{1}^{(j)}} ; ℭ_{4} = w_{1}^{(j)} w_{2}^{(j)} w_{4}^{(j)}.  (71)

E.g. ℭ′_{1} = ℭ_{1} because variables with indices 4,5,6 are not transformed by R_{1,2,3}^{(f)}, then ℭ′′_{1} = ℭ_{1} since u_{4}u_{5} is center of R_{1,4,5}^{(f)} and 6 does not appear in R_{1,4,5}^{(f)}. u_{5}u_{6} is center of R_{2,5,6}^{(f)}, etc.

Eqs.(65), (66) provide a ”free parameterization of the MTE” which generalizes the free parameterization of the single R_{1,2,3} introduced in remark 2, p.9 . It corresponds to the following scenario: We start with twenty-four arbitrary complex numbers u_{j}^{(1)}, w_{j}^{(1)}, κ_{j}, j = 1..8, and apply repeatedly the functional mappings R^{(f)}, choosing appropriately the phases of u_{j}^{(k)}, w_{j}^{(k)}. So we obtain a parameterization of the eight R-matrices obeying the MTE.

The natural question arises: how many independent parameters does the MTE contain as a matrix identity ? As we see already from the existence of the centers (71), some of the twenty-four parameters will occur only in certain combinations.

In order to determine the independent variables in a systematic way, in the next subsection we shall use a simple constructive procedure: We express the u_{j} and w_{j} in terms of line section ratios, the parameterization being designed such as to automatically conserve the centers of the mapping. This is in the same spirit as the introdution of τfunctions in the theory of solitons. We shall find that as a matrix identity the MTE may be parameterized by eight independent continuous parameters and eight discrete phases common for left and right hand sides of MTE, and besides each of left and right hand sides contains four extra independent discrete parameters. In particular, the couplings κ_{j} can all be absorbed by a rescaling.

Note that one great advantage of the MTEs, which is not shared by Yang-Baxter equations, is that many different parameterizations can be found and can be chosen according to the particular calculations and applications one likes to do.

R_{1,2,3}^{a_{1}}  ^{d_{0}}^{d_{1}}^{c_{1}}  ^{b_{3}}^{b_{2}}  ^{a_{2}}^{b_{1}}  ^{A}^{B}^{C}^{D}  ^{c_{3}}^{d_{3}}  ^{d_{2}}^{c_{2}}^{b_{0}}^{a_{0}}  ^{c_{0}}  ^{a_{3}}  ^{c_{0}}^{d_{1}}^{d_{3}}^{D}  ^{C}  ^{B}  ^{A}^{a_{3}}^{d_{0}}^{a_{2}}^{a_{1}}^{a_{0}} ^{b_{0}}  ^{c_{1}}^{d′_{2}} ^{c′_{2}}^{b_{3}}^{c_{3}}^{b′_{2}}^{1}  ^{2}^{3}  ^{4}^{5}^{6}  ^{4}^{5}^{2′}  ^{3′}  ^{1′}^{6} ^{b_{1}} Q_{1}  Q_{2}

Figure 3: Parameterization of the arguments u_{i}^{(j)}, w_{i}^{(j)} of the rational mapping R_{i,j,k}^{(j)} in terms of line-section ratios.

3.1 Parameterization in terms of line section ratios

We now parameterize the u_{i}^{(j)}, w_{i}^{(j)} in terms of ratios of parameters which can be read off from the four line graphs (”quadrilaterals”) shown in Fig. 2. Fig. 3 gives an enlarged detail of Fig. 2 with labels attached, which are explained in the following: In the quadrilateral Q_{1} there are four directed lines A, B, C, D. The six vertex points cut the lines into four sections each. We denote the sections of line A by a_{0}, a_{1}, …, a_{3} (the indices increasing in direction of the line). Analogously we label the sections of lines B, C, D by b_{0}, …, d_{3}. This way for each quadrilateral Q_{j} we have defined 16 variables.

Now, to each vertex i of each quadrilateral Q_{j} we associate a pair of variables u_{i}^{(j)}, w_{i}^{(j)} (which determine the Fermat points in (65)), and we require these to be given in terms of the a_{0}, a_{1}, …, d_{2}, d_{3} of the respective quadrilateral as follows:

For a u_{i}^{(j)}, take Q_{j} and look from point i in direction of the arrows and select the rightpointing of the two lines. Now u_{i}^{(j)} is the ratio of the variable attached to the section before to the variable after the vertex. For the w_{i}^{(j)} take the left-pointing line, and divide the variable after the vertex by the variable before the vertex. So, in the expressions for the u_{i}^{(j)} the index of the numerator is one smaller than that for the denominator, the inverse is true for the w_{i}^{(j)}. Passing from one quadrilateral to the next one, corresponding to a mapping R, always three of the “internal“ lines are changed, and for distinction, we attach to these changed variables dashes and daggers. The eight “external“ variables a_{0}, a_{3}, …, d_{0}, d_{3} are never changed by our mappings. Of the eight “internal“ variables (these have the indices 1 and 2) five are unchanged in each mapping. To collect these definitions, we define

U_{j} = [u_{1}^{(j)}, u_{2}^{(j)}, …, u_{5}^{(j)}, u_{6}^{(j)}, w_{1}^{(j)}, w_{2}^{(j)}, …, w_{5}^{(j)}, w_{6}^{(j)}] .  (72)

Then from Figs. 2, 3 we read off, applying successively the anticlockwise mappings:

U_{1} = [\frac{c_{2}}{c_{3}} , \frac{b_{2}}{b_{3}} , \frac{b_{1}}{b_{2}} , \frac{a_{2}}{a_{3}} , \frac{a_{1}}{a_{2}} , \frac{a_{0}}{a_{1}} , \frac{d_{3}}{d_{2}} , \frac{d_{2}}{d_{1}} , \frac{c_{2}}{c_{1}} , \frac{d_{1}}{d_{0}} , \frac{c_{1}}{c_{0}} , \frac{b_{1}}{b_{0}} ];

U_{2} = [\frac{c_{1}}{c′_{2}} , \frac{b_{1}}{b_{2}′} , \frac{b_{2}′}{b_{3}} , \frac{a_{2}}{a_{3}} , \frac{a_{1}}{a_{2}} , \frac{a_{0}}{a_{1}} , \frac{d_{2}′}{d_{1}} , \frac{d_{3}}{d_{2}′} , \frac{c_{3}}{c′_{2}} , \frac{d_{1}}{d_{0}} , \frac{c_{1}}{c_{0}} , \frac{b_{1}}{b_{0}} ];

U_{3} = [\frac{c_{0}}{c′′_{1}} , \frac{b_{1}}{b′_{2}} , \frac{b′_{2}}{b_{3}} , \frac{a_{1}}{a′′_{2}} , \frac{a′′_{2}}{a_{3}} , \frac{a_{0}}{a_{1}} , \frac{d′′_{1}}{d_{0}} , \frac{d_{3}}{d_{2}′} , \frac{c_{3}}{c′_{2}} , \frac{d_{2}′}{d′′_{1}} , \frac{c′_{2}}{c′′_{1}} , \frac{b_{1}}{b_{0}} ];

U_{4} = [\frac{c_{0}}{c′′_{1}} , \frac{b_{0}}{b′′′_{1}} , \frac{b′_{2}}{b_{3}} , \frac{a_{0}}{a′′′_{1}} , \frac{a′′_{2}}{a_{3}} , \frac{a′′′_{1}}{a′′_{2}} , \frac{d′′_{1}}{d_{0}} , \frac{d′′′_{2}}{d_{1}′′} , \frac{c_{3}}{c′_{2}} , \frac{d_{3}}{d′′′_{2}} , \frac{c′_{2}}{c′′_{1}} , \begin{matrix}b_{2} & ′ \\ b_{1} & ′′′ \end{matrix} ];

U_{5} = [\frac{c_{0}}{c′′_{1}} , \frac{b_{0}}{b′′′_{1}} , \frac{b′′′_{1}}{b_{2}^{T}} , \frac{a_{0}}{a′′′_{1}} , \frac{a′′′_{1}}{a_{2}^{T}} , \frac{a_{2}^{T}}{a_{3}} , \frac{d′′_{1}}{d_{0}} , \frac{d′′′_{2}}{d′′_{1}} , \frac{c_{2}^{T}}{c′′_{1}} , \frac{d_{3}}{d′′′_{2}} , \frac{c_{3}}{c_{2}^{T}} , \frac{b_{3}}{b_{2}^{T}} ];  (73)

We write a_{2}^{T} instead of a′′′′_{2} and d_{1}^{t} instead of d_{1}^{††††}, etc. Transforming clockwise in Fig. 2, we get

U_{8} = [\frac{c_{2}}{c_{3}} , \frac{b_{2}}{b_{3}} , \frac{b_{0}}{b_{1}^{†}} , \frac{a_{2}}{a_{3}} , \frac{a_{0}}{a_{1}^{†}} , \frac{a_{1}^{†}}{a_{2}} , \frac{d_{3}}{d_{2}} , \frac{d_{2}}{d_{1}} , \frac{c_{1}^{†}}{c_{0}} , \frac{d_{1}}{d_{0}} , \frac{c_{2}}{c_{1}^{†}} , \frac{b_{2}}{b_{1}^{†}} ] ,

U_{7} = [\frac{c_{2}}{c_{3}} , \frac{b_{1}^{†}}{b_{2}^{††}} , \frac{b_{0}}{b_{1}^{†}} , \frac{a_{1}}{a_{2}^{††}} , \frac{a_{0}}{a_{1}^{†}} , \frac{a_{2}^{††}}{a_{3}} , \frac{d_{3}}{d_{2}} , \frac{d_{1}^{††}}{d_{0}} , \frac{c_{1}^{†}}{c_{0}} , \frac{d_{2}}{d_{1}^{††}} , \frac{c_{2}}{c_{1}^{†}} , \frac{b_{3}}{b_{2}^{††}} ];

etc. Transforming clockwise two times more, we arrive at an alternative expression for U_{5} , which for distinction we call U¯5 :

U¯5 = [\frac{c_{0}}{c_{1}^{t}} , \frac{b_{0}}{b_{1}^{t}} , \frac{b_{1}^{t}}{b_{2}^{††}} , \frac{a_{0}}{a_{1}^{†††}} , \frac{a_{1}^{†††}}{a_{2}^{††}} , \frac{a_{2}^{††}}{a_{3}} , \frac{d_{1}^{t}}{d_{0}} , \frac{d_{2}^{†††}}{d_{1}^{t}} , \frac{c_{2}^{†††}}{c_{1}^{t}} , \frac{d_{3}}{d_{2}^{†††}} , \frac{c_{3}}{c_{2}^{†††}} , \frac{b_{3}}{b_{2}^{††}} ] .  (74)

We see that this particular parameterization in terms of ratios automatically incorporates the invariance of the centers of the mappings. E.g. for R^{(1)} (in our notation (72) u_{j}^{(2)} is the u′_{j} of (17)):

u_{2}^{(2)}u_{3}^{(2)} = u_{2}^{(1)}u_{3}^{(1)};  u_{1}^{(2)}/w_{3}^{(2)} = u_{1}^{(1)}/w_{3}^{(1)};  w_{1}^{(2)}w_{2}^{(2)} = w_{1}^{(1)}w_{2}^{(1)}.

Also more complicated conditions are fulfilled automatically, e.g.

u_{1}^{(4)}w_{3}^{(1)}w_{5}^{(1)} = u_{1}^{(1)}w_{3}^{(2)}w_{5}^{(3)}.

Observe that because in the mapping from Q_{1} to Q_{2} the line A keeps the same intersection points, there is no a′_{i} in the formulae for the U_{j}, similarly, there are no b′′_{i}, c′′′_{i}, d_{i}^{T} , d_{i}^{†}, c_{i}^{††}, b_{i}^{†††}, a_{i}^{t}. So, altogether, there are 24 primed or daggered variables appearing. However, some coincide as we will see. From (13) and (14) we easily obtain the following relations, which allow to express their N-th powers recursively in terms of the N-th powers of the unprimed variables:

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};
a_{2}^{T}^{N}b_{2}^{′N}c_{2}^{′N} = b_{3}^{N}a_{1}^{′′′N}c_{2}^{′N} + b_{3}^{N}c_{3}^{N}a_{2}^{′′N} + κ_{3}^{N}a_{3}^{N}c_{3}^{N}b_{2}^{′N};
κ_{5}^{N}a_{2}^{′′N}b_{2}^{T}^{N}c_{2}^{′N} = κ_{3}^{N}a_{3}^{N}b_{1}^{′′′N}c_{2}^{′N} + κ_{6}^{N}b_{3}^{N}a_{2}^{′′N}c_{1}^{′′N} + κ_{3}^{N}κ_{6}^{N}a_{3}^{N}b_{2}^{′N}c_{1}^{′′N};
a_{2}^{′′N}b_{2}^{′N}c_{2}^{T} = c_{3}^{N}a_{2}^{′′N}b_{1}^{′′′N} + a_{1}^{′′′N}b_{1}^{′′′N}c_{2}^{′N} + κ_{6}^{N}a_{1}^{′′′N}b_{2}^{′N}c_{1}^{′′N};
a_{1}^{†}^{N}b_{1}^{N}c_{1}^{N} = a_{0}^{N}b_{2}^{N}c_{1}^{N} + a_{1}^{N}b_{2}^{N}c_{2}^{N} + κ_{3}^{N}a_{2}^{N}b_{1}^{N}c_{2}^{N};
κ_{5}^{N}a_{1}^{N}b_{1}^{†}^{N}c_{1}^{N} = κ_{3}^{N}a_{2}^{N}b_{0}^{N}c_{1}^{N} + κ_{6}^{N}a_{1}^{N}b_{2}^{N}c_{0}^{N} + κ_{3}^{N}κ_{6}^{N}a_{2}^{N}b_{1}^{N}c_{0}^{N};
a_{1}^{N}b_{1}^{N}c_{1}^{†}^{N} = a_{1}^{N}b_{0}^{N}c_{2}^{N} + a_{0}^{N}b_{0}^{N}c_{1}^{N} + κ_{6}^{N}a_{0}^{N}b_{1}^{N}c_{0}^{N};
a_{2}^{N}b_{2}^{N}d_{1}^{††}^{N} = a_{2}^{N}b_{1}^{†}^{N}d_{2}^{N} + a_{1}^{†}^{N}b_{1}^{†}^{N}d_{1}^{N} + κ_{6}^{N}a_{1}^{†}^{N}b_{2}^{N}d_{0}^{N};
a_{2}^{††}^{N}b_{2}^{N}d_{1}^{N} = b_{3}^{N}a_{1}^{†}^{N}d_{1}^{N} + b_{3}^{N}a_{2}^{N}d_{2}^{N} + κ_{2}^{N}a_{3}^{N}b_{2}^{N}d_{2}^{N};
κ_{4}^{N}a_{2}^{N}b_{2}^{††}^{N}d_{1}^{N} = κ_{2}^{N}a_{3}^{N}b_{1}^{†}^{N}d_{1}^{N} + κ_{6}^{N}b_{3}^{N}a_{2}^{N}d_{0}^{N} + κ_{2}^{N}κ_{6}^{N}a_{3}^{N}b_{2}^{N}d_{0}^{N};
a_{1}^{†††}^{N}c_{2}^{N}d_{2}^{N} = a_{0}^{N}c_{3}^{N}d_{2}^{N} + c_{3}^{N}d_{3}^{N}a_{1}^{′N} + κ_{1}^{N}d_{3}^{N}a_{2}^{′′N}c_{2}^{N};
κ_{4}^{N}a_{1}^{†}^{N}c_{2}^{†††}^{N}d_{2}^{N} = κ_{1}^{N}a_{2}^{††}c_{1}^{†}^{N}d_{2}^{N} + κ_{5}^{N}c_{3}^{N}a_{1}^{†}^{N}d_{1}^{††}^{N} + κ_{1}^{N}κ_{5}^{N}a_{2}^{††}^{N}c_{2}^{N}d_{1}^{††}^{N};
a_{1}^{†}^{N}c_{2}^{N}d_{2}^{†††}^{N} = d_{3}^{N}a_{1}^{†}^{N}c_{1}^{†}^{N} + a_{0}^{N}c_{1}^{†}^{N}d_{2}^{N} + κ_{5}^{N}a_{0}^{N}c_{2}^{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}.  (75)

Using these relations one verifies by straightforward substitution in (73) and (74) that

U_{5} = U¯5,  (76)

i.e. that the functional tetrahedron equations are satisfied. Eq.(76) tells us that there are eight direct relations between the dashed and daggered variables, e.g. c′′_{1} = c_{1}^{t}, etc. so that in fact due to the functional tetrahedron equations the last eight equations of (75) are superfluous. This will be important for the discussion of the freedom of phase choices when taking Nth roots.

Written in terms of our parameters a_{i}, b_{i}, etc. the arguments of the R^{(j)} (j = 1, …, 8), see (65), are

x_{1}^{(j)} = \frac{1}{\sqrt{ω}} X_{j1};  x_{2}^{(j)} = \frac{1}{\sqrt{ω}} X_{j2};  x_{3}^{(j)} = \frac{1}{ω} X_{j3};  x_{4}^{(j)} = \frac{x_{1}^{(j)}x_{2}^{(j)}}{ω x_{3}^{(j)}} ;
y_{31}^{(j)} = Y_{j1};  y_{41}^{(j)} = \frac{1}{\sqrt{ω}} Y_{j2};  y_{32}^{(j)} = Y_{j3},  (77)

where (again we do not write out x_{4}^{(j)} )

X_{jk} =( \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}}{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}}{a′′_{2}c_{1}} \\ \frac{a_{1}b_{2}′}{κ_{2}a′′_{2}b_{1}} & \frac{κ_{4}a_{0}b′′′_{1}}{b_{0}a′′′_{1}} & \frac{a_{0}b_{2}′}{a′′′_{1}b_{1}} \\ \frac{a′′_{2}b_{3}}{κ_{3}b_{2}′a_{3}} & \frac{κ_{5}a′′′_{1}b_{2}^{T}}{b′′′_{1}a_{2}^{T}} & \frac{a′′′_{1}b_{3}}{a_{2}^{T} b_{2}′} \\ \frac{b_{1}^{†}c_{2}^{†††}}{κ_{1}b_{2}^{††}c_{1}^{†}} & \frac{κ_{2}b_{0}c_{1}^{t}}{b_{1}^{t}c_{0}} & \frac{b_{0}c_{2}^{†††}}{b_{1}^{t}c_{1}^{†}} \\ \frac{a_{1}^{†}c_{3}}{κ_{1}a_{2}^{††}c_{2}} & \frac{κ_{4}a_{0}c_{2}^{†††}}{c_{1}^{†}a_{1}^{†††}} & \frac{a_{0}c_{3}}{a_{1}^{†††}c_{2}} \\ \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}}{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}}{a_{1}^{†}b_{1}} \end{matrix} ); Y_{jk} =(\begin{matrix}\frac{κ_{1}d_{3}b_{3}}{d_{2}b_{2}′} & \frac{κ_{3}c_{2}d_{1}}{c_{1}d_{2}} & \frac{d_{3}c_{1}}{d_{2}′c_{2}} \\ \frac{κ_{1}d_{2}′a_{3}}{d_{1}a′′_{2}} & \frac{κ_{5}c_{1}d_{0}}{c_{0}d_{1}} & \frac{d_{2}′c_{0}}{d_{1}′′c_{1}} \\ \frac{κ_{2}d_{3}a′′_{2}}{d_{2}′a′′′_{1}} & \frac{κ_{6}b_{1}d_{1}′′}{b_{0}d_{2}′} & \frac{d_{3}b_{0}}{d_{2}′′′b_{1}} \\ \frac{κ_{3}c_{3}a_{3}}{c′_{2}a_{2}^{T}} & \frac{κ_{6}b_{2}′c′′_{1}}{b_{1}′′′c′_{2}} & \frac{c_{3}b_{1}′′′}{c_{2}^{T} b_{2}′} \\ \frac{κ_{1}d_{2}^{†††}b_{2}^{††}}{d_{1}^{††}b_{1}^{t}} & \frac{κ_{3}c_{1}^{†}d_{0}}{c_{0}d_{1}^{††}} & \frac{d_{2}^{†††}c_{0}}{d_{1}^{t}c_{1}^{†}} \\ \frac{κ_{1}d_{3}a_{2}^{††}}{d_{2}a_{1}^{†††}} & \frac{κ_{5}c_{2}d_{1}^{††}}{c_{1}^{†}d_{2}} & \frac{d_{3}c_{1}^{†}}{d_{2}^{†††}c_{2}} \\ \frac{κ_{2}d_{2}a_{3}}{d_{1}a_{2}^{††}} & \frac{κ_{6}b_{2}d_{0}}{b_{1}^{†}d_{1}} & \frac{d_{2}b_{1}^{†}}{d_{1}^{††}b_{2}} \\ \frac{κ_{3}c_{2}a_{2}}{c_{1}a_{1}^{†}} & \frac{κ_{6}b_{1}c_{0}}{b_{0}c_{1}} & \frac{c_{2}b_{0}}{c_{1}^{†}b_{1}} \end{matrix} ). (78)

To check the validity of all the Fermat relations x_{i}^{(j)}^{N} + y_{i}^{(j)}^{N} = 1 requires using the transformation equations (75). The relations (75) involve the Nth powers of the variables, so to use them to get the Fermat coordinates, we have to take Nth roots, which entails discrete phase choices. The centers (71) are just ratios of the external variables:

ℭ_{1} = \frac{a_{0}}{a_{3}} ;  ℭ_{2} = \frac{b_{0}}{b_{3}} ;  ℭ_{3} = \frac{c_{3}}{c_{0}} ;  ℭ_{4} = \frac{d_{3}}{d_{0}} .  (79)

The external variables a_{0}, a_{3}, …, d_{0}, d_{3} are irrelevant and serve mainly to express all quantities in terms of ratios. We may choose them simply all to be unity. The ”coupling constants” κ_{j} may all be eliminated by re-scaling the eight relevant variables a_{1}, a_{2}, …, d_{1}, d_{2} as follows:

\begin{matrix}a_{1} = \frac{κ_{1}κ_{2}}{κ_{5}} \overline{a_{1}}; & a_{2} = \frac{κ_{1}κ_{6}}{κ_{3}} \overline{a_{2}}; & b_{1} = \frac{κ_{1}}{κ_{3}κ_{5}} \overline{b_{1}}; & b_{2} = \frac{κ_{1}κ_{6}}{κ_{2}κ_{3}} \overline{b_{2}}; \\ c_{1} = \frac{κ_{1}κ_{6}}{κ_{3}κ_{5}} \overline{c_{1}}; & c_{2} = \frac{κ_{6}}{κ_{2}κ_{3}} \overline{c_{2}}; & d_{1} = \frac{κ_{1}κ_{6}}{κ_{3}} \overline{d_{1}}; & d_{2} = \frac{κ_{5}κ_{6}}{κ_{2}} \overline{d_{2}}. \end{matrix}  (80)

This entails a corresponding re-scaling of the dashed and daggered variables, e.g.

\begin{matrix}b_{2}′ = \frac{κ_{1}κ_{2}}{κ_{5}κ_{6}} \overline{b_{1}′}; & c_{2}′ = \frac{κ_{1}}{κ_{5}} \overline{c_{2}′}; & d_{2}′ = \frac{κ_{1}κ_{2}}{κ_{5}} \overline{d_{2}′}; & a′′_{2} = \frac{κ_{1}κ_{2}κ_{3}}{κ_{5}κ_{6}} \overline{a′′_{2}}; \\ c′′_{1} = \frac{κ_{3}}{κ_{4}κ_{6}} \overline{c′′_{1}}; & d′′_{1} = \frac{κ_{2}κ_{3}}{κ_{6}} \overline{d′′_{1}}; & a′′′_{1} = \frac{κ_{2}κ_{3}}{κ_{6}} \overline{a′′′_{1}}; & b′′′_{1} = \frac{κ_{2}κ_{3}}{κ_{4}κ_{6}} \overline{b′′′_{1}}; \\ d′′′_{2} = \frac{κ_{3}κ_{5}}{κ_{1}} \overline{d′′′_{2}}; & a_{2}^{t} = \frac{κ_{3}κ_{5}}{κ_{1}} \overline{a_{2}^{t}}; & b_{2}^{t} = \frac{κ_{3}}{κ_{1}κ_{4}} \overline{b_{2}^{t}}; & c_{2}^{t} = \frac{κ_{3}κ_{5}}{κ_{1}κ_{4}} \overline{c_{2}^{t}}; \\ a_{1}^{†} = \frac{κ_{5}κ_{6}}{κ_{2}} \overline{a_{1}^{†}}; & b_{1}^{†} = \frac{κ_{6}}{κ_{2}} \overline{b_{1}^{†}}; & c_{1}^{†} = \frac{κ_{5}κ_{6}}{κ_{1}κ_{2}} \overline{c_{1}^{†}}; & … \end{matrix} (81)

So we can simplify eqs.(75) and (78) by taking a_{0} = a_{3} = b_{0} = … = d_{3} = 1 and κ_{1} = κ_{2} = … = κ_{6} = 1 and replacing the relevant variables by their overlined counterparts.

3.2 The choice of discrete phases.

Apart from the eight continuous parameters just discussed, the left hand and right hand sides of the MTE depend on phases choices arising from taking Nth roots. We investigate how many independent choices can be made and whether these affect the MTEs.

From (76) and (73), (74) we see that the left hand side (LHS) and right hand side (RHS) of the MTE have 8 arbitrary common phases of

a_{2}^{T} = a_{2}^{††} , b_{2}^{T} = b_{2}^{††} , c_{2}^{T} = c_{2}^{†††} , d′′′_{2} = d_{2}^{†††} , a′′′_{1} = a_{1}^{†††} , b′′′_{1} = b_{1}^{t} , c′′_{1} = c_{1}^{t} , d′′_{1} = d_{1}^{t}.

These phases correspond to the phases of u_{1}^{(5)} etc. Furthermore, the LHS of the MTE contains 4 internal phases of

c′_{2}, b_{2}′, d_{2}′, a′′_{2}

while the RHS contains 4 internal phases of

a_{1}^{†}, c_{1}^{†}, b_{1}^{†}, d_{1}^{††} .

In which way does the LHS depend on its internal phases? Consider e.g. the shift

c′_{2} ↦ q^{−1}c′_{2} .

According to (78) this shift changes the following Fermat coordinates:

x_{2}^{(1)} ↦ q^{−1} x_{2}^{(1)}, x_{4}^{(1)} ↦ q^{−1}x_{4}^{(1)} ,
x_{1}^{(2)} ↦ q^{−1} x_{1}^{(2)},  x_{3}^{(2)} ↦ q^{−1}x_{3}^{(2)} ,  y_{1}^{(4)} ↦ q y_{1}^{(4)} .

Now note that for q = ω

W_{(q^{−1}x,y)}(n) = \frac{y}{1 − x}W_{(x,y)}(n − 1) ; W_{(x,qy)}(n) = q^{n}W_{(x,y)}(n) .

Therefore our shift produces

⟨i_{1}, i_{2}, i_{3}|R^{(1)}|j_{1}, j_{2}, j_{3}⟩ ↦ \frac{y_{2}^{(1)}}{y_{4}^{(1)}} \frac{1 − x_{4}^{(1)}}{1 − x_{2}^{(1)}} ⟨i_{1}, i_{2}, i_{3}|R^{(1)}|j_{1} + 1, j_{2}, j_{3}⟩ q^{−j_{3}} ,

⟨j_{1}, i_{4}, i_{5}|R^{(2)}|k_{1}, j_{4}, j_{5}⟩ ↦ \frac{y_{1}^{(2)}}{y_{3}^{(2)}} \frac{1 − x_{3}^{(2)}}{1 − x_{1}^{(2)}} ⟨j_{1} + 1, i_{4}, i_{5}|R^{(2)}|k_{1}, j_{4}, j_{5}⟩ q^{j_{5}} ,

⟨j_{3}, j_{5}, j_{6}|R^{(4)}|k_{3}, k_{5}, k_{6}⟩ ↦ q^{j_{3}−j_{5}} ⟨j_{3}, j_{5}, j_{6}|R^{(4)}|k_{3}, k_{5}, k_{6}⟩ .

We see that the change considered produces a simple scalar factor and does not change the matrix structure of the LHS of the MTE:

LHS(q^{−1}c′_{2}) = \frac{y_{2}^{(1)}}{y_{4}^{(1)}} \frac{1 − x_{4}^{(1)}}{1 − x_{2}^{(1)}} \frac{y_{1}^{(2)}}{y_{3}^{(2)}} \frac{1 − x_{3}^{(2)}}{1 − x_{1}^{(2)}} LHS(c′_{2})

In this way we may convince ourselves:

• A change of the phases of any of the eight internal variables produces only extra scalar factors for the LHS or RHS of the MTE, while the external matrix structure does not change.

• A change of the phases for any of the eight external variables produces extra scalar factors both for the LHS and RHS of the MTE, and besides it produces a change of the external matrix structure: a shift of the indices i_{1}, ..., k_{6} and some multipliers q^{±i_{1}}, ..., q^{±k_{6}}. However, this change is the same for the LHS and RHS of the MTE.

4 Explicit form of the MTE for N = 2

For N = 2, using combined indices i = 1 + i_{1} + 2i_{2} + 4i_{3}; k = 1 + k_{1} + 2k_{2} + 4k_{3} : we can give R_{i1,i2,i3}^{k1,k2,k3} explicitly in a simple matrix form: We define

Y_{k} = \frac{y_{k}}{1 + x_{k}} = \sqrt{\frac{1 − x_{k}}{1 + x_{k}}}  for  k = 1, 2, 3, 4 ;
Z_{ik} = \frac{Y_{i}}{Y_{k}} for ik = 13, 14, 23, 24;  Z_{12} = Y_{1}Y_{2};  Z_{34} = \frac{1}{Y_{3}Y_{4}} . (82)

and get

R_{i}^{k} = ( \begin{matrix}1 & Z_{24} & 0 & 0 & 0 & 0 & Z_{23} & −Z_{34} \\ Z_{13} & Z_{13}Z_{24} & 0 & 0 & 0 & 0 & −Z_{12} & Z_{14} \\ 0 & 0 & Z_{13}Z_{24} & Z_{13} & Z_{14} & −Z_{12} & 0 & 0 \\ 0 & 0 & Z_{24} & 1 & −Z_{34} & Z_{23} & 0 & 0 \\ 0 & 0 & Z_{23} & Z_{34} & 1 & −Z_{24} & 0 & 0 \\ 0 & 0 & Z_{12} & Z_{14} & −Z_{13} & Z_{13}Z_{24} & 0 & 0 \\ Z_{14} & Z_{12} & 0 & 0 & 0 & 0 & Z_{13}Z_{24} & −Z_{13} \\ Z_{34} & Z_{23} & 0 & 0 & 0 & 0 & −Z_{24} & 1 \end{matrix} )_{ik}  (83)

The determinant can be calculated directly:

det R = (Y_{1} Y_{2} \frac{(Y_{3}^{2} + 1)(Y_{4}^{2} − 1)}{Y_{3}^{2} Y_{4}^{2}})^{4} .  (84)

For N = 2 we can also write the MTE quite explicitly. We shall write eq.(51) shorthand as

Θ _{i_{1},i_{2},i_{3},i_{4},i_{5},i_{6}}^{k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}} = ρ \overline{Θ} _{i_{1},i_{2},i_{3},i_{4},i_{5},i_{6}}^{k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}}.  (85)

where in obvious correspondence Θ and \overline{Θ} are defined to be the left- and right-hand sums of products of four R-matrices. We use (82) and abbreviate Y_{i}^{(j)} by Y_{ij} , where i labels the four points on the Fermat curve x_{1}, x_{2}, x_{3}, x_{4} = x_{1}x_{2}/(ωx_{3}), and j = 1, …, 8 denote the eight arguments of the R^{(j)}. The left hand side of (85) is, using (33):

Θ _{i_{1},i_{2},i_{3},i_{4},i_{5},i_{6}}^{k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}} = (−1)^{k_{3}k_{6}} \frac{Y_{11}^{i_{2}+i_{1}} Y_{23}^{k_{4}+k_{2}} Y_{24}^{k_{5}+k_{3}}}{Y_{42}^{i_{4}+k_{1}}} ×

× ∑_{j_{1},j_{2},j_{3},j_{4},j_{5},j_{6}}δ_{i_{2}+i_{3},j_{2}+j_{3}} δ_{i_{4}+i_{5},j_{4}+j_{5}} δ_{j_{4}+i_{6},k_{4}+j_{6}} δ_{j_{5}+j_{6},k_{5}+k_{6}} ×

× \frac{(−1)^{j_{1}(j_{3}−j_{5})+j_{6}(k_{2}−j_{2})−j_{3}(i_{1}+k_{6})+k_{1}j_{5}} Y_{21}^{j_{2}+j_{1}} Y_{12}^{i_{4}+j_{1}} Y_{22}^{j_{4}+k_{1}} Y_{13}^{j_{4}+j_{2}} Y_{14}^{j_{5}+j_{3}}}{Y_{31}^{j_{2}+i_{1}} Y_{41}^{i_{2}+j_{1}} Y_{32}^{j_{4}+j_{1}} Y_{33}^{k_{4}+j_{2}} Y_{43}^{j_{4}+k_{2}} Y_{34}^{k_{5}+j_{3}} Y_{44}^{j_{5}+k_{3}}} ,

where all exponents are understood modulo 2, i.e. being just 0 or 1. We can use three of the δ’s to eliminate the sums over e.g. j_{2}, j_{4}, j_{6}. Then the last δ gives a compatibility condition with the result that if i_{4} + i_{5} + i_{6} + k_{4} + k_{5} + k_{6} is odd, the component of Θ vanishes and these components of the MTE are trivial. These are half of the 2^{12} components. We also see that, if these occur at all, Y_{11}, Y_{23}, Y_{24} and Y_{42} will factorize (Y_{11} appears if i_{1} + i_{2} is odd, etc.). Each non-zero component has a sum over eight terms on each side. The result is:

Θ _{i_{1},i_{2},i_{3},i_{4},i_{5},i_{6}}^{k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}} = δ_{i_{4}+i_{5}+i_{6},k_{4}+k_{5}+k_{6}}(−1)^{γ} \frac{Y_{11}^{i_{2}+i_{1}} Y_{23}^{k_{4}+k_{2}} Y_{24}^{k_{5}+k_{3}}}{Y_{42}^{i_{4}+k_{1}}} ×

∑_{j_{1},j_{2},j_{4}} \frac{(−1)^{β} Y_{21}^{j_{2}+j_{1}} Y_{12}^{i_{4}+j_{1}} Y_{22}^{j_{4}+k_{1}} Y_{13}^{j_{4}+j_{2}} Y_{14}^{j_{4}+j_{2}+ι}}{Y_{31}^{j_{2}+i_{1}} Y_{41}^{i_{2}+j_{1}} Y_{32}^{j_{4}+j_{1}} Y_{33}^{k_{4}+j_{2}} Y_{43}^{j_{4}+k_{2}} Y_{34}^{k_{5}+i_{2}+i_{3}+j_{2}} Y_{44}^{i_{4}+i_{5}+j_{4}+k_{3}}}  (86)

where ι = i_{2} + i_{3} + i_{4} + i_{5} and

β = j_{1}j_{2} + j_{2}j_{4} + j_{4}j_{1} + j_{1} ι + j_{2}(i_{1} + i_{6} + k_{4} + k_{6}) + j_{4}(k_{1} + k_{2}),
γ = k_{1}(i_{4} + i_{5}) + k_{2}(i_{6} + k_{4}) + (i_{2} + i_{3})(i_{1} + k_{6}) + k_{3}k_{6}.

The analogous expression for the right hand side is:

\overline{Θ} _{i_{1},i_{2},i_{3},i_{4},i_{5},i_{6}}^{k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}} = δ_{i_{4}+i_{5}+i_{6},k_{4}+k_{5}+k_{6}}(−1)\overline{γ} \frac{Y_{25}^{k_{2}+k_{1}} Y_{18}^{i_{5}+i_{3}} Y_{17}^{i_{4}+i_{2}}}{Y_{36}^{k_{4}+i_{1}}} ×

∑_{j_{1},j_{2},j_{4}} \frac{(−1)\overline{β} Y_{16}^{j_{4}+i_{1}} Y_{15}^{j_{2}+j_{1}} Y_{26}^{j_{1}+k_{4}} Y_{27}^{j_{4}+j_{2}} Y_{28}^{j_{4}+j_{2}+\overline{ι}}}{Y_{38}^{j_{4}+i_{3}+k_{4}+k_{5}} Y_{48}^{i_{5}+j_{2}+k_{2}+k_{3}} Y_{37}^{i_{2}+j_{4}} Y_{47}^{i_{4}+j_{2}} Y_{46}^{j_{1}+j_{4}} Y_{35}^{j_{1}+k_{2}} Y_{45}^{k_{1}+j_{2}}} , (87)

with \overline{ι} = k_{2} + k_{3} + k_{4} + k_{5}, and

\overline{β} = j_{1}(k_{3} + k_{5}) + j_{2}(j_{4} + i_{4}) + j_{4}(k_{2} + k_{3} + i_{3}),
\overline{γ} = i_{1}k_{5} + i_{2}k_{6} + k_{1}k_{3} + (i_{4} + k_{6})(k_{2} + k_{3} + i_{3}).

and again all exponents are understood mod 2.

We give two examples of non-trivial components of the MTEs: First we consider the component

Θ _{0,0,0,0,0,0}^{0,0,0,0,0,0} = ρ \overline{Θ} _{0,0,0,0,0,0}^{0,0,0,0,0,0}

This is, written more explicitly, using the abbreviations of (82) adding the index j: Z_{ik,j} = Y_{i}^{(j)}/Y_{k}^{(j)} for ik = 13, 14, 23, 24 , Z_{12,j} = Y_{1}^{(j)}Y_{2}^{(j)} , Z_{34,j} = 1/(Y_{3}^{(j)} Y_{4}^{(j)}):

1 + Z_{24,1}Z_{13,2} + (Z_{23,1} − Z_{34,1}Z_{13,2})Z_{13,3}Z_{13,4}
+(Z_{23,2} − Z_{12,2}Z_{24,1})Z_{14,3}Z_{14,4} − (Z_{23,1}Z_{23,2} + Z_{34,1}Z_{12,2})Z_{34,3}Z_{34,4}
= ρ {1 + Z_{24,6}Z_{13,5} + (Z_{14,5} + Z_{24,6}Z_{34,5})Z_{24,8}Z_{24,7}
+(Z_{14,6} + Z_{12,6}Z_{13,5})Z_{23,8}Z_{23,7} − (Z_{14,6}Z_{14,5} + Z_{12,6}Z_{34,5})Z_{34,8}Z_{34,7}} . (88)

Another component:

Θ _{0,0,1,0,1,0}^{0,0,1,1,0,0} = ρ \overline{Θ} _{0,0,1,0,1,0}^{0,0,1,1,0,0}

reads analogously:

(1 + Z_{24,1}Z_{13,2})Z_{23,3}Z_{23,4} − (Z_{23,1} − Z_{13,2}Z_{34,1})Z_{12,3}Z_{12,4}

+(Z_{23,2} − Z_{24,1}Z_{12,2})Z_{13,3}Z_{24,3}Z_{13,4}Z_{24,4} + (Z_{23,1}Z_{23,2} + Z_{34,1}Z_{12,2})Z_{24,3}Z_{24,4}

= ρ {Z_{23,6} − Z_{34,6}Z_{13,5} + (Z_{23,6}Z_{14,5} − Z_{34,6}Z_{34,5})Z_{24,8}Z_{24,7}
+ (Z_{24,6} − Z_{13,5})Z_{13,6}Z_{23,8}Z_{23,7} + (Z_{34,5}Z_{13,6} − Z_{13,6}Z_{24,6}Z_{14,5})Z_{34,7}Z_{34,8}} . (89)

Each equation appears eight times for different components. In order to write the symmetries compactly, we introduce the following three mappings a, b, c of the upper or lower indices:

a(i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, i_{6}) = (i_{1} + 1, i_{2} + 1, i_{3}, i_{4} + 1, i_{5}, i_{6});
b(i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, i_{6}) = (i_{1}, i_{2}, i_{3} + 1, i_{4}, i_{5} + 1, i_{6});
c(i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, i_{6}) = (i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, i_{6} + 1),
and  ab(i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, i_{6}) = a(b(i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, i_{6})), etc.  (90)

the same also for the k_{j} instead of the i_{j}. Of course, the indices are always taken mod 2. Altogether, for N = 2 there are 2^{8} different nontrivial components, an independent set is (the same for the \overline{Θ})

Θ_{i_{1},i_{2},i_{3},0,0,0}^{k_{1},k_{2},k_{3},0,0,0};  Θ_{i_{1},i_{2},i_{3},0,0,0}^{k_{1},k_{2},k_{3},1,1,0};  Θ_{i_{1},i_{2},i_{3},0,0,0}^{k_{1},k_{2},k_{3},1,0,1};  Θ_{i_{1},i_{2},i_{3},0,0,0}^{k_{1},k_{2},k_{3},0,1,1}
for  i_{1}, i_{2}, i_{3}, k_{1}, k_{2}, k_{3} = 0, 1.  (91)

Proposition 5 For N = 2 the components of the left-hand side of the Modified Tetrahedron equation satisfy the following symmetry relations:

Θ_{i_{1},i_{2},i_{3},0,0,0}^{k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}} = Θ_{a(i_{1},i_{2},i_{3},0,0,0)}^{a(k_{1},k_{2},k_{3},k_{4},k_{5},k_{6})}
= (−1)^{i_{1}+k_{1}}Θ_{b(i_{1},i_{2},i_{3},0,0,0)}^{b(k_{1},k_{2},k_{3},k_{4},k_{5},k_{6})} = (−1)^{i_{1}+k_{1}}Θ_{ab(i_{1},i_{2},i_{3},0,0,0)}^{ab(k_{1},k_{2},k_{3},k_{4},k_{5},k_{6})}
= (−1)^{i_{2}+i_{3}+k_{2}+k_{3}}Θ_{c(i_{1},i_{2},i_{3},0,0,0)}^{c(k_{1},k_{2},k_{3},k_{4},k_{5},k_{6})} = (−1)^{i_{2}+i_{3}+k_{2}+k_{3}}Θ_{ac(i_{1},i_{2},i_{3},0,0,0)}^{ac(k_{1},k_{2},k_{3},k_{4},k_{5},k_{6})}
= (−1)^{i_{1}+i_{2}+i_{3}+k_{1}+k_{2}+k_{3}}Θ_{bc(i_{1},i_{2},i_{3},0,0,0)}^{bc(k_{1},k_{2},k_{3},k_{4},k_{5},k_{6})} = (−1)^{i_{1}+i_{2}+i_{3}+k_{1}+k_{2}+k_{3}}Θ_{abc(i_{1},i_{2},i_{3},0,0,0)}^{abc(k_{1},k_{2},k_{3},k_{4},k_{5},k_{6})},

(92)

where i_{1}, i_{2}, i_{3}, k_{1}, k_{2}, …, k_{5}, k_{6} = 0, 1. Here the mappings a, b, c are defined as in (90). The same equations are valid for the right-hand components, i.e. for Θ replaced by \overline{Θ}. For k_{4} + k_{5} + k_{6} odd, these equations are trivial.

Proof: Use the explicit formulae (86) and (87). For the relations involving the mapping a we shift the summation indices j_{1}, j_{2}, j_{4}. Then all exponents of the Y −factors are unchanged. No phase is appearing, since the shift in γ (\overline{γ}) is compensated by the shift in β (\overline{β}). For the relations involving the mappings b and/or c, also all exponents of the Y are unchanged. The phase factors appear from the shifts in γ or \overline{γ}. □

The terms

Z_{ik,j} = y_{ik}^{(j)} \frac{1 + x_{k}^{(j)}}{1 + x_{i}^{(j)}} , ik = 13, 14, 23, 24;

Z_{12,j} = \frac{y_{13}^{(j)} y_{23}^{(j)} (1 − x_{3}^{(j)}^{2})}{(1 + x_{1}^{(j)})(1 + x_{2}^{(j)})} ; Z_{34,j} = \frac{(1 + x_{3}^{(j)})(1 + x_{4}^{(j)})}{y_{31}^{(j)} y_{41}^{(j)} (1 − x_{1}^{(j)}^{2})} ,

which appear in (88) and (89) have the following explicit form, taking all boundary coefficients and all κ_{j} to be unity, compare (80),(81) :

Z_{13,1} = \frac{(b_{2}′c_{2} − b_{1})d_{2}}{c_{2} − ib_{2}} ;  Z_{14,1} = \frac{(b_{2}c′_{2} − c_{1})d_{2}}{(b_{2} + ic_{2})d_{1}} ;
Z_{23,1} = \frac{(b_{2}′c_{2} − b_{1})d_{2}′}{c_{1}b_{2}′ − ib_{1}c′_{2}} ;  Z_{24,1} = \frac{(b_{2}c′_{2} − c_{1})d_{2}′}{(b_{1}c′_{2} + ib_{2}′c_{1})d_{1}} ;
…
Z_{23,4} = \frac{(a_{2}^{††}b_{2}′ − a_{1}^{†††})c_{2}^{†††}}{b_{1}′′′a_{2}^{††} − ia_{1}^{†††}b_{2}^{††}};  Z_{24,4} = \frac{(a′′_{2}b_{2}^{††} − b′′′_{1})c_{2}^{†††}}{(a_{1}^{†††}b_{2}^{††} + ia_{2}^{††}b′′′_{1})c′′_{1}} ;  (93)

etc. Discrete sign choices of square roots come in when expressing the transformed variables e.g. b′_{2}, a′′′_{1}, etc. in terms of the original eight variables a_{1}, a_{2}, …, d_{2} via (75). E.g. from the first lines of (75):

b′_{2} = ±\frac{1}{c_{2}d_{2}} \sqrt{b_{1}^{2}d_{2}^{2} + b_{2}^{2} + c_{2}^{2}} ; c′_{2} = ±\frac{1}{b_{2}d_{2}} \sqrt{c_{1}^{2}d_{2}^{2} + (b_{2}^{2} + c_{2}^{2})d_{1}^{2}} ;

d_{2}′ = ±\frac{1}{b_{2}c_{2}} \sqrt{(b_{2}^{2} + b_{1}^{2}d_{2}^{2})c_{1}^{2} + b_{1}^{2}c_{2}^{2}d_{1}^{2}} ; a′′_{2} = ±\frac{1}{c_{1}d_{1}} \sqrt{(a_{1}^{2}d_{1}^{2} + a_{2}^{2}d_{2}^{′2})c′_{2}^{2} + c_{1}^{2}d_{2}^{′2}} .

All dashed or daggered variables are square roots of rational expressions.

From (84) the factor ρ is:

ρ = \sqrt{\frac{∏_{i=1}^{4} Z_{12,i} (1 + Z_{34,i}(Z_{13,i}Z_{14,i}^{−1} − Z_{14,i}Z_{13,i}^{−1}) − Z_{34,i}^{2})}{∏_{j=5}^{8} Z_{12,j} (1 + Z_{34,j}(Z_{13,j}Z_{14,j}^{−1} − Z_{14,j}Z_{13,j}^{−1}) − Z_{34,j}^{2})}} .  (94)

Using the components like (89) and inserting there (94) and (93), (75) we get the MTE in terms of our eight parameters a_{1}, …, d_{2} and sixteen choices of the signs of

b_{2}′; c′_{2}; d_{2}′; a′′_{2}; c′′_{1}; d′′_{1}; b′′′_{1}; d′′′_{2}; a_{1}^{†}; b_{1}^{†}; c_{1}^{†}; a_{2}^{††}; b_{2}^{††}; d_{1}^{††}; a_{1}^{†††}; c_{2}^{†††}.  (95)

We have confirmed the N = 2-MTE numerically for all 256 different components, choosing random complex numbers for the eight continuous variables and random signs for the sixteen square roots in the variables (95).

We conclude mentioning that for N = 3 we find that of the 3^{12} components of the MTE for (R^{(j)})_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}} given in (33), 2 × 3^{11} components are just 0 = 0, while 3^{11} (not all distinct) equations have non-trivial left- and right-hand sides.

5 Conclusions

In this paper we study the Modified Tetrahedron Equation (51) in which the Boltzmann weights R_{i_{1},i_{2},i_{3}}^{j_{1},j_{2},j_{3}} depend on Fermat-curve variables via cyclic weight functions, see (33). The conjugation by the R-matrix is a rational automorphism of the ultra-local Weyl algebra at the N-th root of unity. The representation of this automorphism as a functional mapping in the space of the parameters of the ultra-local Weyl algebra allows us to obtain the free parameterization of the MTE. We express the Fermat-curve variables (and so the Boltzmann weights) in terms of an independent set of eight continuous parameters and specify the sixteen phases which can be chosen independently. We derive a general expression for the scalar factor of the MTE. For simplest non-trivial case N = 2 the MTE is written out explicitly. In this case it contains of 256 linearly independent components.

The MTE allow to obtain a wide class of new integrable models: The R-matrices may be combined into the cubic blocks obeying globally the usual TE by due to the validity of the local MTE-s. The advantage of the free parameterization presented here is that it allows to get the appropriate parameterization for blocks of any size. This is the subject of a forthcoming paper.

New integrable 2-dimensional lattice models with parameters living on higher Riemann surfaces can be obtained from the MTE by a contraction process which has been described in [15].

A further important application of the MTE concerns the following: As the usual the TE leads to the commutativity of the layer-to-layer transfer matrices, while the MTE can be used to obtain exchange relations for the layer-to-layer transfer matrices. The exchange relations are related to isospectrality deformations and form the basis for a functional Bethe ansatz for three dimensional integrable spin models, see [18].

Note finally, that starting from the results of this paper, we can consider several limits of the parameterization, which are connected to various degenerations of the weights. The usual Tetrahedron equation of [4] follows from the MTE in the special regime when the free parameters u_{j}, w_{j}, j = 1 … 6 belong to the submanifold of ℂ^{12} defined by

u_{l}^{N} − R_{i,j,k}^{(f)} · u_{l}^{N} = w_{l}^{N} − R_{i,j,k}^{(f)} · w_{l}^{N} = 0 .  (96)

This variety may be parameterized in terms of spherical geometry data. E.g. R_{1,2,3} may be associated with the spherical triangle with the dihedral angles θ_{1}, θ_{2}, θ_{3} and

κ_{1}^{N} = tan^{2} \frac{θ_{1}}{2} , κ_{2}^{N} = cot^{2} \frac{θ_{2}}{2} , κ_{3}^{N} = tan^{2} \frac{θ_{3}}{2} .  (97)

Acknowledgements

This work was 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 01-01-00539 and the grant for support of scientific schools RFBR 00-15-9655. 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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