On SO$(N)$ spin vertex models

We describe the Boltzmann weights of the $D_k$ algebra spin vertex models. Thus, we find the $SO(N)$ spin vertex models, for any $N$, completing the $B_k$ case found earlier. We further check that the real (self-dual) SO$(N)$ models obey quantum algebras, which are the Birman-Murakami-Wenzl (BMW) algebra for three blocks, and certain generalizations, which include the BMW algebra as a sub-algebra, for four and five blocks. In the case of five blocks, the $B_4$ model is shown to satisfy additional twenty new relations, which are given. The $D_6$ model is shown to obey two additional relations.


Introduction.
Solvable lattice models in two dimensions are a fruitful ground to test phase transitions, universality, integrability [1] and conformal field theory [2]. For reviews see [3,4].
We will concentrate here on a type of solvable lattice models which are called vertex models. Well known among these are the six, eight and nineteen vertex models [3,4]. For recent works on vertex models see, e.g., [5,6,7,8,9]. Our purpose here is to introduce vertex models based on the algebra D k and the spin representation. This completes the SO(N ) spin vertex models for all N , where the B k models were described before in ref. [10].
We are also interested in the algebraic structure underlying these models.
We use the more general results of [11,12], which describe the three, four and five blocks algebras (where the number of blocks is the degree of polynomial equation obeyed by the Boltzmann weights), assuming only a certain ansatz for the Baxterization, described in [13], and the Yang-Baxter equation. We describe and check numerically, the algebras of B 4 , which is a five blocks theory, and the algebra of D 6 , which is a four blocks theory.
The algebras include a version of the Birman-Murakami-Wenzl algebra (BMW) [14,15], along with two new relations for four blocks and twenty new relations for the five blocks theory, which are given here for B 4 . We check that the BMW algebra is obeyed for D k , for any small even k, with a different skein relation.

D k spin vertex models.
We wish to describe a vertex model based on the algebra D k = SO(2k) and the spin representation. This solution is an element of End(V ⊗V ) where V is the spin representation of D k . We denote by α n = ǫ n −ǫ n+1 , for n = 1, 2, . . . , k−1 and α k = ǫ k−1 + ǫ k the simple roots of D k , where ǫ i are orthogonal unit vectors. The spin representation, denoted by S has the weights The last product is −1 for the anti-spinor representation, denoted byS. We find it useful to add 1/2 to these weights, and to represent weights of the spinor (anti-spinor) representation by the vector m, where m i = 0 or 1.
To start constructing the vertex model, we need a solution which commutes with the co-product of U q 2 (SO(2k)). We find it convenient to first describe a solution for the larger representationṼ = S ⊕S, namely the sum of the spinor and anti-spinor representations. Such a solution was described recently in [16].
It is the element C of End(Ṽ ⊗Ṽ ), given by andn j is equal to n j except at the jth coordinate where it is 1 − n j . Here m, n, b, c = 0 or 1 are weights of the spin or anti-spin representations shifted by 1/2. The eigenvalues of the matrix C were computed in ref. [16], and are The solution C has the disadvantage of mapping both the spin and anti-spin representations. We note, however, that C maps the representation S ⊗ S tō S ⊗S, and vice versa. Thus, to get a solution in End(S ⊗ S) all we need to do is to square the matrix C and to equate to zero all the C b,c m,n for weights m, n, b, c which are not in S. Thus, C 2 gives the solution we want. Of course, since C commutes with the co-product, so does C 2 .
Since the matrix C 2 commutes with the co-product, it has the same eigenvectors as our desired solution which obeys the Yang-Baxter equation, but not the same eigenvalues. Thus, we define the projection operators where the product is in End(S ⊗ S) and I is the identity map.
We note that for even k, P a = 0 for a which is odd, whereas for odd k, P a = 0 for a which is even. The jth eigenvalue corresponds to the representation where v is the vector representation, i.e., the anti-symmetric product of j vector representations [16]. The highest weight of the representation V j is ǫ 1 + ǫ 2 + . . . + ǫ j . Thus, the non-zero P a are in one to one correspondence with the representations that appear in the tensor product, as they should. Thus, the projection P a projects onto the representation V a .
We wish to make the connection between the solution C 2 and the D k WZW conformal model. For explanation of conformal field theory see the book [2], and references therein. To do this we define,

7)
Here r is the level of the representation and is the dual Coxeter number.
The dimension of the highest weight Λ in a WZW theory is given by where ρ is half the sum of positive roots and C Λ = Λ(Λ + 2ρ) is the Casimir of the representation. The Casimir of the representation V j is given by As explained in [10], the eigenvalues of the R matrix are given by where p j = ±1 is some sign which corresponds to whether the product in eq.
(2.6) is symmetric or anti-symmetric. In our case, the sign is given by Thus, since we know the eigenvalues of the R matrix and the projection operators from eq. (2.5), we may construct the R matrix as It can be verified that this R matrix satisfies the Yang-Baxter equation (YBE) which for the R matrix is the braiding relation, (2.14) We checked that this R matrix obeys the YBE, numerically for k = 2, 3, 4, 5, 6 and it holds, indeed, for various weights and for general q. Now, we wish to define a trigonometric solution for the YBE. For this purpose, we use the same general ansatz for Baxterization as in [10,13]. First, we need to decide on the order of the primary fields in eq. (2.6). The order which solves the YBE is given by for even k. For odd k the order is The parameters are given by [10,13], Thus, the D k theory is an m + 1 blocks theory. We thus define the parameters as Then the trigonometric solution to the YBE assumes the form [10,13], where a = 0, 1, . . . , m.
For example, for k = 6, which is a four blocks theory, the parameters are (ζ 0 , ζ 1 , ζ 2 ) = (10, 6, 2). The crossing parameter is λ = ζ 0 . The D k vertex models with k even are real (self-dual) as S = S * . For odd k the theories are not real (not self-dual), as S = S * .

BMW ′ algebra and SO(N ) spin vertex models.
We repeat here the definition of the BMW ′ algebra following [10]. We find it convenient to use an operator form for the R matrix. We define the matrix, following [4], where I (i) is the identity matrix at position i and (e rs ) lm = δ rl δ sm . The YBE, eq. (2.22), then assumes a more compact form, Let us denote the number of blocks by n. For the D k models, this is n = m + 1 = k/2 + 1 (k even), or n = m + 1 = (k + 1)/2 (for odd k). In this section, we will assume that k is even, so that the theory is real (self-dual). It is assumed that the number of blocks is greater or equal to three, n ≥ 3. The algebras of non-real theories are also interesting, but we shall not describe it here. We define the limit of the matrix X i (u) as We define the operators, and which is the Temperley-Lieb algebra [17], and which is the braiding algebra. We can also prove the relations, where The following is the skein relation which stems from the definition of the projection operators along with the ansatz, eqs. (2.20, 2.21), where a and b r are some coefficients, which can be expressed in terms of the parameters, ζ r . From the skein relation we prove, The above relation, eqs. (3.7-3.13), are part of the Birman-Murakami-Wenzl algebra (BMW) [14,15]. The rest of the relations of the BMW algebra are also obeyed, except of the skein relation, eq. (3.12), which is different for more than three blocks. These are We have verified that the full BMW ′ algebra (BMW with a different skein relation) is obeyed by the D k model, with k = 4 or 6. We did this numerically, using various heights and general q. We note that the BMW ′ algebra is obeyed also by the B k spin vertex models [10], while substituting the relevant parameters ζ r . For more than three blocks there are additional relations, except from the skein relation. These are described in the next two sections, for B 4 and D 6 .
The Boltzmann weights are stated in [10]. We checked the relations numerically for a general value of the parameter q and substituting various heights.

4-CB relations for D 6 .
We wish to check the 4-CB algebra for D 6 which is a four blocks model. The four blocks relations were given in [11]. The parameters for D 6 are, eq. (2.18).
We denoted by a i,j,k (r, s, t) the element of the algebra a i [r]a j [s]a k [t] where a i [r] is G r , G −1 r , E r or 1 r , if i = 1, 2, 3, 4, respectively.
Finally, we proceed to check these two relations, for the D 6 vertex model substituting the explicit Boltzmann weights, eqs. (2.20, 2.21). Indeed they hold for various values of the heights and for general value of q.