An Elliptic Parameterisation of the Zamolodchikov Model

The Zamolodchikov model describes an exact relativistic factorized scattering theory of straight strings in (2+1)-dimensional space-time. It also defines an integrable 3D lattice model of statistical mechanics and quantum field theory. The three-string S-matrix satisfies the tetrahedron equation which is a 3D analog of the Yang-Baxter equation. Each S-matrix depends on three dihedral angles formed by three intersecting planes, whereas the tetrahedron equation contains five independent spectral parameters, associated with angles of an Euclidean tetrahedron. The vertex weights are given by rather complicated expressions involving square roots of trigonometric function of the spectral parameters, which is quite unusual from the point of view of 2D solvable lattice models. In this paper we consider a particular four-parameter specialization of the tetrahedron equation when one of its vertices goes to infinity and the tetrahedron itself degenerates into an infinite prism. We show that in this limit all the vertex weights in the tetrahedron equation can be represented as meromorphic functions on an elliptic curve. Moreover we show that a special reduction of the tetrahedron equation in this case leads precisely to an example of the tetrahedral Zamolodchikov algebra, previously constructed by Korepanov. This algebra plays important role for a"layered"construction of the Shastry's R-matrix and the 2D S-matrix appearing in the problem of the ADS/CFT correspondence for N=4 SUSY Yang-Mills theory in four dimensions. Possible applications of our results in this field are briefly discussed.


Introduction 2 The vertex formulation of the Zamolodchikov model
The Zamolodchikov model can be formulated on any 3D lattice formed by an arbitrary set of intersecting 2D planes in the Euclidean space R 3 , such that no four planes intersect at one point. In particular, this definition includes a regular cubic lattice as the simplest possibility. The fluctuating spin variables in the model take two values, denoted below 0 and 1. Here we will use the vertex formulation of the model found in [17]. The spins in this case are assigned to edges of the lattice while local Boltzmann weights are assigned to vertices at threeplane intersection points. Let R j 1 ,j 2 ,j 3 i 1 ,i 2 ,i 3 denote the weight corresponding to a configuration of six edge spins i 1 , i 2 , i 3 , j 1 , j 2 , j 3 = 0, 1, arranged as in Fig.1. This quantity could be conveniently associated with a linear operator R : acting in a direct product of three two-dimensional vector spaces, where the pairs of indices (i 1 , j 1 ), (i 2 , j 2 ) and (i 3 , j 3 ) serve as matrix indices in the first, second and third spaces, respectively. The edge states "0" and "1" correspond to the basis vectors v 0 and v 1 , The vertex weights depends on three spectral parameters, θ 1 , θ 2 , θ 3 , which are identified with dihedral angles between the three planes forming the vertex, as shown in Fig.1. To indicate this dependence we will write the weights as R(θ 1 , θ 2 , θ 3 ). Consider a spherical triangle with the angles θ 1 , θ 2 , θ 3 and let a 1 , a 2 , a 3 denote three sides of this triangle opposite to the angles θ 1 , θ 2 , θ 3 , which obey the spherical sine theorem K = sin θ 1 sin a 1 = sin θ 2 sin a 2 = sin θ 3 sin a 3 .
In the original face-spin formulation of ref. [2] the weights are explicitly symmetric with respect to all spatial symmetry transformations, generated by permutations of the lines 1, 2, 3 and inversions of their directions. Namely, the weights remains unchanged if the associated permutations of spins are accompanied by the corresponding transformations of the dihedral angles. Contrary, to this the operator R(θ 1 , θ 2 , θ 3 ) is not explicitly symmetric. Nevertheless it possesses the same spatial symmetry group of the 3D cube [17,29], but its realisation now involves linear similarity transformations of the weights. This 48-element group is generated just by two transformations, which we choose as where the superscripts t i , i = 1, 2, 3, denote the matrix transposition in the i-th space, P 13 denotes the operator permuting the spaces 1 and 3 and σ x , σ y , σ z denote the Pauli matrices, 1 In [17] the weights are given in terms of the variables β0, β1, β2, β3. Here they are re-expressed in terms of α's with the help of the identity where (i, j, k, ) is any permutation of the indices (0, 1, 2, 3).
In the following the symmetry properties will not play any essential role, since the elliptic parameterisation considered below explicitly breaks the spatial symmetry. The operator R satisfies the tetrahedron equation, which ensures the integrability of the model. The above equation involves operators acting in a direct product V = (C 2 ) ⊗ 6 , of six identical two-dimensional vector spaces 2 . Geometrically, it is associated with an arbitrary intersection of four planes. The indices in (13) numerate six intersection lines among these planes, which form extended edges of an Euclidean tetrahedron. Each of the operators R ijk depends on its own set of spectral parameters, determined by the dihedral angles corresponding to the lines i, j and k. The geometric arrangement of these angles is shown in Fig.2 (note that it is exactly the same as in [17], but different from that used in Eq. (2.2) in [5]). Altogether there are six dihedral angles corresponding to six edges of the Figure 2: Graphical representation of the tetrahedron equation (13) showing the arrangement of the internal dihedral angles of the tetrahedron tetrahedron. At this point it is worth mentioning that these angles are not independent; they satisfy one non-linear constraint, which follows from the vanishing of the Gram determinant of four unit normal vectors to faces of a tetrahedron in the Euclidean 3-space. Therefore, Eq.(13) contains only five independent parameters. In matrix form it reads Altogether there are 2 12 distinct equations, half of which is non-trivial (for the other half both sides vanish identically due to the parity conservation (8)). There are several proofs that the weights (10) indeed satisfy these equations. Firstly, thanks to the "cube-vertex" correspondence of ref. [18], this follows from the Baxter's proof [5] of the tetrahedron equation for the interactionround-a-cube (IRC) formulation of the model. Baxter used various symmetry relations and a computer-aided analysis of a pattern of signs in some equations to reduce all non-trivial equations to just two equations, which were then proven with the help of spherical trigonometry. Secondly, there exists a completely algebraic proof of (15) given in [17]. This proof (as well as its earlier variant [9] for the IRC formulation) is based on a repeated application of the so-called restricted star-triangle relation, invented in [8]. It is worth mentioning, that this relation was later interpreted as the five-term identity for a cyclic version [30] of the quantum dilogarithm [31]. Finally, an alternative (and, in fact, simpler) algebraic proof of (15), based on the auxiliary linear problem, was obtained in [32].
To conclude this Section, mention a simple fact that Eq. (13) is unaffected by arbitrary linear similarity transformations in any of the six vector spaces therein, where G i , i = 1, 2, . . . , 6 are arbitrary non-degenerate matrices. We will use this freedom to bring the expressions for matrix elements of R(θ 1 , θ 2 , θ 3 ) to the most convenient form.
3 Elliptic parameterisation of the weights

Linear transformations of the weights
For the following analysis it is convenient to introduce the operator where which differs from R 123 by a simple similarity transformation of the type (16). The parameter ξ will be specified later on. Evidently, the transformation breaks the symmetry between the three spaces in (17), since the first space is distinguished from the other two. The matrix elements of L 123 can be arranged into a set of 4 × 4 matrices L 0 0 , L 1 0 , L 0 1 and L 1 1 , acting in the tensor product of the second and third spaces, while the indices 0, 1, labeling these matrices, refer to the first space, where and We regard these matrices as two by two block matrices with 2-dimensional blocks. The indices (i 2 , j 2 ) numerate the blocks while the indices (i 3 , j 3 ) numerate matrix elements inside the blocks. From (7) it follows that therefore each of the above matrices can be viewed as an R-matrix of the eight-vertex freefermion model, which satisfy the condition

Elliptic parameterisation of the weights
Here we will show that the weights (10) can be parametrised in terms of Jacobi elliptic functions. First, note that the form of the matrices (19) and (21) and the relations (23) suggest to use Baxter's parameterisation of the eight-vertex model [33] (the latter contains the symmetric free-fermion model as a particular case). To do this define the elliptic modulus similarly to that in the symmetric eight-vertex model with the weights a, b, c, d. (cf. Eq.(5.7) of [33]). With an account of (20) and (22) the RHS of Eq.(25) is a function of the dihedral angles θ 1 , θ 2 , θ 3 . After elementary simplifications, one obtains Then using the relations (see §132 of [34]) where (i, j, k) is any permutation of (1, 2, 3) and K is defined in (3), one obtains sin φ = K 2 sin a 1 sin θ 1 sin a 2 sin a 3 = sin θ 2 sin a 3 (28) which shows that φ is the angle between the line 1 and the plane containing the lines 2 and 3, as illustrated in Fig.3. Note that value k = 0 corresponds to the right angle φ = π/2. Further, let sn x = sn(x, k), cn x = cn(x, k), dn x = dn(x, k) and cd x = cd(x, k) denote Jacobi elliptic functions of the argument x and modulus k (we follow the notations of [35]). Now let us parametrise the angles θ 2 and θ 3 in terms of new parameters w 1 and w 2 , Given θ 2 , θ 3 each of these equations have two solutions in the periodicity rectangle (we assume where K = K(k) and K = K( √ 1 − k 2 ) are the complete elliptic integral of the first kind. Now substitute (29) back into (26) and solve the resulting equation for θ 1 in terms k, w 1 , w 2 . There are four solutions of the form corresponding to four possible choices of w 1 , w 2 in (29). Only one of them 3 lead to the original value of θ 1 . This allows one to uniquely fix the required solution of (29). From now on we regard k, w 1 , w 2 as new independent variables instead of θ 1 , θ 2 , θ 3 . The latter are now defined by the formulae (29) and (31). The weights in Eq.(10) are expressed through the square roots of tangents of halves of spherical excesses α 0 , α 1 , α 2 , α 3 . Surprisingly enough, after lengthy calculations we found that these tangents have rather simple expressions in the new variables, The function f (w) is defined as It follows that the products appearing in (20) and (22) simplify to Then using addition theorems for elliptic functions, one obtains where (37) Next, we specify the parameter ξ in (17), which so far remained at our disposal, Finally, introduce the R-matrix of the symmetric eight-vertex model [33] specialized to the free-fermion case, Using this notation, the matrices (19), (21) can be written in a uniform way where the Pauli matrices σ (2) x , σ y , σ z act in the space 2. Taking into account (37) and (38) is easy to see that all expressions for the matrix elements in (40) are meromorphic double periodic functions of the variables w 1 and w 2 .
To summarize, we have shown that all matrix elements the operator are meromorphic functions of w 1 and w 2 , provided the two sets of variables, entering different sides of this equation, are related by (26), (29) and (31) and the parameter ξ is defined by (38).
We would like to stress the above elliptic parametrisation was obtained for a generic case without any special requirements for the values of θ 1 , θ 2 , θ 3 . By construction, this parametrisation breaks the symmetry of the weights (the directions 1 is distinguished). Moreover, the modulus k will, a priori, be different for different vertices of the tetrahedron. Therefore, in general, one cannot parametrise all weights in the tetrahedron equation by elliptic functions of the same modulus. Nonetheless, there exists an important four-parameter reduction of this equation, where such parametrisation is possible. This is the "prismatic limit" considered in the next section.

Reductions of the tetrahedron equation
In this section we consider certain reductions of the tetrahedron equation for the solution (10) and its connection to an example of the tetrahedral Zamolodchikov algebra constructed in [19].

The "static limit"
The term "static limit" stems from the original Zamolodchikov's work [1] where it was related to the case of slowly moving or "non-relativistic" straight strings in 2 + 1 dimensions. On the level of parameters this corresponds to a configuration where the sum of dihedral angles is equal to π for every vertex of the tetrahedron. Namely, for Eq.(15) this means The linear angles between edges at the vertices all become equal to 0 or π and the tetrahedron degenerates into four planes intersecting along the same line. Their relative orientation is fixed by three angles only, so the tetrahedron equation in this case contains three independent parameters (instead of five in the general case). We will use a special notation for the operator R(θ 1 , θ 2 , θ 3 ) specialized to this case Its matrix elements are determined by (10) where one sets There are only sixteen non-vanishing elements, This solution of the tetrahedron equation (with a slightly different parameterisation, see Sect.4.5 below) was first obtained in [19]. Two years later [17] it was understood as the vertex form of the original Zamolodchikov's solution in the static limit [1]. The operator S satisfies the relation It has a block-diagonal form with two 4×4 blocks, a trivial one, which coincides with the identity matrix, and a non-trivial one which has two eigenvalues +1 and two eigenvalues −1.
The operator S possesses left and right "bare vacuum" eigenvectors Note that this property does not hold for the general case (10) with t 0 = 0.

The planar limit
Geometrically the "planar limit" corresponds to the case when all four vertices of the tetrahedron lie in one plane, i.e. when the tetrahedron becomes "squashed" into a plane. An edge-spin solution of (13) corresponding to this case was first obtained by Hietarinta [20]. Subsequently it was understood as the planar limit [36] of the Zamolodchikov model. The details of this reduction are rather complicated and not immediately related to the main topic of this paper. Therefore, we refer interested readers to the original publication [36] where this limit is thoroughly studied.

The prismatic limit
Here we consider yet another limiting case of the equation (13), when one of the tetrahedron vertices goes to infinity (we choose it to be the vertex corresponding to R 123 ). Then the sum of dihedral angles at this vertex will satisfy an additional constraint, whereas the whole tetrahedron turns into an infinite prism. The number of independent angles reduces from five to four. The edges 1, 2 and 3 become parallel and therefore, have the same angle to the plane, containing the edges 4, 5 and 6, forming the base of the prism. Remembering the geometric definition (26) of the elliptic modulus k in the previous section, we conclude that all weights corresponding to the vertices (1,4,5) , (2,4,6) and (3,5,6) can be parametrised by elliptic functions of the same modulus.
Eqs. (26), (29) and (31) define a generic change of variables from three dihedral angles (θ 1 , θ 2 , θ 3 ) to new parameters (k, w 1 , w 2 ). Here we want to apply this substitution for three different sets of dihedral angles corresponding to the three vertices at the base of the prism (θ 1 , θ 4 , θ 5 ) → (k, u 1 , u 2 ); (π − θ 2 , θ 4 , θ 6 ) → (k, u 1 , u 3 ); (θ 3 , π − θ 5 , θ 6 ) → (k, u 2 , u 3 ). (49) As explained above from the geometric considerations the elliptic modulus k will automatically be the same for all three sets. For instance, let (b 1 , b 2 , b 3 ) be the sides of the spherical triangle with angles (θ 1 , θ 4 , θ 5 ), then we set Next define the variables u 1 , u 2 , u 3 by the relations e −iθ 4 = − (1 + sn(2u 1 ))(1 − k sn(2u 1 )) cn(2u 1 ) dn(2u 1 ) , and which are obtained by a simple specialisation of (29) and (31) for the three substitutions (49). Note that the above formulae (51) and (52) give a parameterisation of six angles of a triangular prism in terms of elliptic functions. In particular, it is easy to check that the three expressions in (52) are consistent with (48). Let us now rewrite the tetrahedron equation (13) for this case in the new variables. We will do this in several steps. First, guided by the formula (41) define three operators Here F and D(ξ) are defined in (18) and the parameters ξ 1 , ξ 2 , ξ 3 have been chosen in agreement with (38) in each of the three cases, where h(x, y) is defined in (38). Therefore, according to the result of Sect.3, all operators (53) are meromorphic double periodic functions of u 1 , u 2 , u 3 , which, of course, implicitly depend on the elliptic modulus k. Next, we need to express in the new variables the weights, corresponding to the infinitely distant vertex (1,2,3). It is convenient to define a new operator, which differs from (43) by a diagonal similarity transformation.
Using (52) it is not difficult to show that where g(x, y) = k 2 sd(x − y) sd(x + y) cn(x − y) cn(x + y) , Combining this with (45), one obtains the following expressions for the non-vanishing matrix elements of S(u 1 , u 2 , u 3 ) S 000 000 = S 110 110 = S 101 101 = S 011 011 = 1, It is easy to see that all above matrix elements are meromorphic functions of u 1 , u 2 , u 3 , implicitly depending on the elliptic modulus k. Note that even though the parameters ξ 1 , ξ 2 , ξ 3 , entering (55), were fixed by the analyticity requirements for the other three operators, defined in (53), the matrix elements of S(u 1 , u 2 , u 3 ) have automatically became meromorphic functions on the elliptic curve 4 . Note also that the similarity transformation in (55) introduces two additional parameters into (58) and breaks the symmetry relations between different matrix elements exhibited in (45). Remind that the original static limit weights (45) depend on only two independent angles.
To complete our analysis of the prismatic limit apply the similarity transformation with the matrix to both sides of the tetrahedron equation (13). Then using the definitions (53) and (55) one obtains As required, this equation contains exactly four independent parameters, namely, u 1 , u 2 , u 3 , which are indicated explicitly, and the elliptic modulus k which is implicitly assumed.

Elliptic parameterisation of the static limit
Consider the tetrahedron equation (13) in the static limit, i.e., when the angles satisfy the constraints (42). In the previous subsection we saw that upon the change of variables (52) and the similarity transformation (55) the vertex weights S 123 (u 1 , u 2 , u 3 ), corresponding to the vertex (1, 2, 3), become meromorphic functions on the elliptic curve. Remarkably, it is possible to do this for the other three vertices as well, using the elliptic functions of the same modulus. First, we need to parameterise the angles, satisfying (42), in term of the elliptic functions. To avoid confusions with the notations of the previous subsection we denote these angles θ 1 , θ 2 , . . . , θ 6 . Let us parameterise θ 1 , θ 2 , θ 3 by the same formulae as (52), and assume that This implies the expressions (56) with θ 1 , θ 2 , θ 3 replaced by θ 1 , θ 2 , θ 3 and tan θ 4 2 = i g(u 1 , u 4 ) , tan θ 5 2 = −ig(u 2 , u 4 ), tan where g(u, w) is defined in (57). Note, that these formulae contains five independent parameters: u 1 , u 2 , u 3 , u 4 and k, whereas there are only three independent angles, satisfying (42). So the above parameterisation contains two spurious parameters. Next, assume that ξ 1 , ξ 2 , ξ 3 are given by (54) and where h(u, w) is defined in (38). Applying now the similarity transformation with the matrix to both sides of the tetrahedron equation (13) and taking into account the above parameterisation, one obtains where the matrix elements of S ijk (u, v, w) are defined by (58). It appears, that this equation naturally complements the equation (61). As noted in Sect.4.1 there are only three independent angles in the tetrahedron in the static limit. However, Eq.(67) contains five independent parameters (four u's and the modulus k). Two additional parameters are not spurious in this case, as they cannot be removed by a re-parameterisation. They were introduced via the similarity transformation with the matrix (66).

Tetrahedral Zamolodchikov algebra
Here we show that some additional reduction of the tetrahedron equation (61) leads precisely to the example of the tetrahedral Zamolodchikov algebra constructed in [19]. It is convenient to introduce two 4 × 4 matrices, acting in the product of two spaces C 2 ⊗ C 2 as where (E j i ) ab = δ ia δ jb denotes the 2 × 2 matrix unit. According to (40), (41) these new matrices are simply related to the R-matrix of the free-fermion model (39), Now calculate matrix elements of tetrahedron equation (61) sandwiched between the fixed vectors and similarly for the other L s. For fixed values of a, b, c, both sides of (71) still remain operators acting in the spaces 4, 5, 6. At this point it is convenient to relabel these spaces as 1, 2, 3 (the former spaces 1, 2, 3 are no longer required, so there should be no confusions) and use the definition (68). In this way one obtains This relation is known as the definition the tetrahedral Zamolodchikov algebra. It was introduced in [19]. The coefficients S a b c d e f (u 1 , u 2 , u 3 ) can be considered as the structure constants of the algebra. They satisfy the tetrahedron equations (67), which play the role of the associativity condition for the defining relations (73). This relationship is analogues to that for the Zamolodchikov-Faddeev algebra [37] where the Yang-Baxter equation plays the role of the associativity condition.
Using addition theorems for elliptic and trigonometric functions it is not difficult to show that g(u j , u k ) = tanh(ϕ j − ϕ k ), j, k = 1, 2, 3.
where g(x, y) is given by (57). Then it follows from (56) that Subsequently, Shiroishi & Wadati [22] reproduced the calculations of [19]. They worked in essentially the same normalization and notations as this work and their expressions for the matrix elements of S 123 (given by their Eq.(5.4)) exactly coincide with our Eq.(58) with interchanged u 1 and u 2 . In addition, Eq.(73) was re-checked once again in [23] in the trigonometric limit (k = 0), see §12.A.1 in [23].