Diagonalization of boundary transfer matrix

for the U_{q,p}(ŝl(3, ℂ)) ABF model

October 20, 2009

Takeo KOJIMA

Department of Mathematics, College of Science and Technology, Nihon University, Surugadai, Chiyoda-ku, Tokyo 101-0062, JAPAN

Abstract We construct a free field realization of the ground state of the boundary transfer matrix for the U_{q,p}(ŝl(3, ℂ)) ABF model. Using this ground state and type-II vertex operator, we have a diagonalization of the boundary transfer matrix.

1 Introduction

The vertex operator approach [1, 2] provides a powerful method to study solvable models. In [1] authors diagonalized the XXZ-transfer matrix on infinite spin chain by vertex operators. Sklyanin [3] begin systematic approach to boundary condition generalization in the framework of the algebraic Bethe ansatz. In periodic boundary condition, we construct a family of commuting transfer matrix from a solution of the Yang-Baxter equation. Sklyanin showed that similar construction is possible with the aid of a solution of the boundary Yang-Baxter equation. In open boundary condition, we construct a family of commuting transfer matrix from both a solution of the Yang-Baxter equation and a solution of the boundary Yang-Baxter equation. In [6] the authors diagonalized the boundary XXZ-transfer matrix on semi-infinite spin chain by vertex operators. The vertex operator approach was extended to higher-rank boundary XXZmodel in [9]

The vertex operator approach to solvable model was originally formulated for vertex type model such as the XXZ-model [1], and then extended to the face type model such as the Andrews-Baxter-Forrester (ABF) model [4, 10]. The ABF model is described by the elliptic algebra U_{q,p}(ŝl(2, ℂ)). The U_{q,p}(ŝl(3, ℂ)) ABF model we are going to study in this paper were introduced in [5], as the higher-rank generlization of the ABF model introduced in [4]. In this paper we diagonalize the boundary transfer matrix for the U_{q,p}(ŝl(3, ℂ)) ABF model by vertex operator approach. We construct a free field realization of the ground-state of the commuting boundary transfer matrix. Using this groundstate and type-II vertex operators for the elliptic algebra, we get a diagonalization of the boundary transfer matrix of the U_{q,p}(ŝl(3, ℂ)) ABF model. The result of this paper gives a higher-rank generalization of the ABF model [7].

The text is organized as follows. In section 2 we recall the boundary U_{q,p}(ŝl(3, ℂ)) ABF model, and introduce the boundary transfer matrix. In section 3 we recall the free field realization of the vertex operators. In section 4 we construct a free field realization of the ground-state of the boundary transfer matrix, and give a diagonalization of the boundary transfer matrix by using the type-II vertex opeartors.

2 Boundary U_{q,p}(ŝl(3, ℂ)) ABF model

We recall the boundary U_{q,p}(ŝl(3, ℂ)) ABF model.

2.1 Bulk Boltzmann weights

The U_{q,p}(ŝl(3, ℂ)) ABF model has two parameters x and r. We assume 0 < x < 1 and r ≧ 5 (r ∈ ℤ). We set the elliptic theta function [u] by

[u] = x^{\frac{u^{2}}{r} −u}Θ_{x^{2r}}(x^{2u}).  (2.1)

Here we have used

Θ_{q}(z) = (q, q)_{∞}(z; q)_{∞}(q/z; q)_{∞},  (2.2)
(z; q_{1}, q_{2}, ⋯, q_{m})_{∞} = ∏_{j_{1},j_{2},···,j_{m}=0}^{∞}(1 − q_{1}^{j_{1}}q_{2}^{j_{2}} ⋯ q_{m}^{j_{m}}z).  (2.3)

Let ϵ_{μ}(1 ≦ μ ≦ 3) be the orthonormal basis of ℝ^{3} with the inner product (ϵ_{μ}|ϵ_{ν}) = δ_{μ,ν}. Let us set ϵ̄_{μ} = ϵ_{μ} − ϵ where ϵ = \frac{1}{3} ∑_{ν=1}^{3} ϵ_{ν}. The type ŝl(3, ℂ) weight lattice is the linear span of ϵ̄_{μ}.

P = ∑_{μ=1}^{3} ℤϵ̄_{μ}.  (2.4)

Note ∑_{μ=1}^{3} ϵ̄_{μ} = 0. Let the simple root α_{μ} = ϵ̄_{μ} − ϵ̄_{μ+1} (μ = 1, 2). For a ∈ P we set

a_{μ,ν} = a_{μ} − a_{ν}, a_{μ} = (a + ρ|ϵ̄_{μ}),  (2.5)

where ρ = 2ϵ̄_{1} + ϵ̄_{2}. The Boltzmann weights W ( \begin{matrix}a & b \\ c & d \end{matrix} | u) are given by

W ( \begin{matrix}a + 2ϵ̄_{μ} & a + ϵ̄_{μ} \\ a + ϵ̄_{μ} & a \end{matrix} | u) = r_{1}(u),  (2.6)
W ( \begin{matrix}a + ϵ̄_{μ} + ϵ̄_{ν} & a + ϵ̄_{μ} \\ a + ϵ̄_{ν} & a \end{matrix} | u) = r_{1}(u)\frac{[u][a_{μ,ν} − 1]}{[u − 1][a_{μ,ν}]},  (2.7)
W ( \begin{matrix}a + ϵ̄_{μ} + ϵ̄_{ν} & a + ϵ̄_{ν} \\ a + ϵ̄_{ν} & a \end{matrix} | u) = r_{1}(u)\frac{[u − a_{μ,ν}][1]}{[u − 1][a_{μ,ν}]}.  (2.8)

Otherwise are zero. The function r_{1}(u) is given by

r_{1}(u) = z^{\frac{r−1}{r} \frac{2}{3}} \frac{h_{1}(z^{−1})}{h(z)}, h_{1}(z) = \frac{(x^{2}z; x^{2r}, x^{6})_{∞}(x^{2r+4}z; x^{2r}, x^{6})_{∞}}{(x^{2r}z; x^{2r}, x^{6})_{∞}(x^{6}z; x^{2r}, x^{6})_{∞}} .  (2.9)

Here we have used z = x^{2u}. The Boltzmann weights satisfy the following relations. (1)Yang-Baxter equation :

∑_{g} W ( \begin{matrix}d & e \\ c & g \end{matrix} | u_{1}) W ( \begin{matrix}c & g \\ b & a \end{matrix} | u_{2}) W ( \begin{matrix}e & f \\ g & a \end{matrix} | u_{1} − u_{2})

= ∑_{g} W ( \begin{matrix}g & f \\ b & a \end{matrix} | u_{1}) W ( \begin{matrix}d & e \\ g & f \end{matrix} | u_{2}) W ( \begin{matrix}d & g \\ c & b \end{matrix} | u_{1} − u_{2}) .  (2.10)

(2)The first inversion relation :

∑_{g} W ( \begin{matrix}c & g \\ b & a \end{matrix} | − u) W ( \begin{matrix}c & d \\ g & a \end{matrix} | u) = δ_{b,d}.  (2.11)

(3)The second inversion relation :

∑_{g} G_{g}W ( \begin{matrix}g & b \\ d & c \end{matrix} | 3 − u) W ( \begin{matrix}g & d \\ b & a \end{matrix} | u) = δ_{a,c} \frac{G_{b}G_{d}}{G_{a}} .  (2.12)

where G_{a} = [a_{1,2}][a_{1,3}][a_{2,3}]. The Boltzmann weights are related to the elliptic algebra U_{q,p}(ŝl(3, ℂ)).

2.2 Boundary Boltzmann weights

In [8] the boundary Boltzmann weights K ( \begin{matrix}& b \\ a \\ & c \end{matrix} | u) are given by

K ( \begin{matrix}& k \\ k + ϵ̄_{μ} \\ & k \end{matrix} | u) = z^{\frac{2}{r} (− \frac{r−1}{3} +(ϵ̄_{1}|k))} \frac{h(z)}{h(z^{−1})} \frac{[c − u][k_{1,μ} + c + u]}{[c + u][k_{1,μ} + c − u]}, (c ∈ ℝ; μ = 1, 2, 3). (2.13)

Otherwise are zero. The function h(z) is given in (4.18). They satisfy the boundary Yang-Baxter equation,

∑_{f,g} W ( \begin{matrix}c & f \\ b & a \end{matrix} | u − v) W ( \begin{matrix}c & d \\ f & g \end{matrix} | u + v) K ( \begin{matrix}& g \\ f \\ & a \end{matrix} | u) K ( \begin{matrix}& e \\ d \\ & g \end{matrix} | v)

= ∑_{f,g} W ( \begin{matrix}c & d \\ f & e \end{matrix} | u − v) W ( \begin{matrix}c & f \\ b & g \end{matrix} | u + v) K ( \begin{matrix}& e \\ f \\ & g \end{matrix} | u) K ( \begin{matrix}& g \\ b \\ & a \end{matrix} | v) .(2.14)

2.3 Vertex operator

Following the general scheme of algebraic approach in solvable lattice models, we give the type-I vertex operators. Let us consider the corner transfer matrices A(z), B(z), C(z), D(z) which represent NW, SW, SE, NE quadrants, respectively. The space H_{l,k} of the eigenvectors of A(z) is parametrized by l, k ∈ P. Let us introduce the type-I vertex operator Φ^{(a,b)}(z). We denote by Φ^{(k+ϵ̄_{j},k)}(z^{−1}) the half-infinite transfer matrix extending to infinity in the north. This is an operator

Φ^{(k+ϵ̄_{j},k)}(z^{−1}) : H_{l,k} → H_{l,k+ϵ̄_{j}}.

The opeartor Φ^{(a,b)}(z) = 0 for a − b ≠ ϵ̄_{j}. Similarily, we introduce dual type-I vertex operator Φ^{∗(a,b)}(z). We denote by Φ^{∗(k,k+ϵ̄_{j})}(z) the half-infinite transfer matrix extending to infinity in the west. This is an operator

Φ^{∗(k,k+ϵ̄_{j})}(z) : H_{l,k+ϵ̄_{j}} → H_{l,k}.

The opeartor Φ^{(a,b)}(z) = 0 for b − a ≠ ϵ̄_{j}. They satisfy the following relations. (1)Commutation relation

Φ^{(c,b)}(z_{1})Φ^{(b,a)}(z_{2}) = ∑_{d} W ( \begin{matrix}c & d \\ b & a \end{matrix} | u_{1} − u_{2}) Φ^{(c,d)}(z_{2})Φ^{(d,a)}(z_{1}).  (2.15)

(2)Inversion relation

∑_{g} Φ^{∗(a,g)}(z)Φ^{(g,a)}(z) = 1, Φ^{(a,b)}(z)Φ^{∗(b,c)}(z) = δ_{a,c}.  (2.16)

Later we give a free field realization of the vertex operators acting on the bosonic Fock space.

2.4 Boundary transfer matrix

We define the boundary transfer matrix

T_{B}^{(k)}(z) = ∑_{j=1,2,3} Φ^{∗(k,k+ϵ̄_{j})}(z)K ( \begin{matrix}& k \\ k + ϵ̄_{j} \\ & k \end{matrix} | u) Φ^{(k+ϵ̄_{j},k)}(z^{−1}).  (2.17)

The boundary Yang-Baxter equation implies the commutativity.

[T_{B}^{(k)}(z_{1}), T_{B}^{(k)}(z_{2})] = 0.  (2.18)

Our problem of this paper is to diagonalize the boundary transfer matrix T_{B}^{(k)}(z).

3 Free field realization

We give a free field realization of the vertex operators [11, 12].

3.1 Boson

We set the bosonic oscillators β_{m}^{i}, (i = 1, 2; m ∈ ℤ) by

[β_{m}^{j}, β_{n}^{k}] = \{ \begin{matrix}m\frac{[(r − 1)m]_{x}}{[rm]_{x}} \frac{[2m]_{x}}{[3m]_{x}} δ_{m+n,0} & (j = k) \\ −mx^{3m sgn(j−k)} \frac{[(r − 1)m]_{x}}{[rm]_{x}} \frac{[m]_{x}}{[3m]_{x}} δ_{m+n,0} & (j ≠ k). \end{matrix}  (3.1)

Here the symbol [a]_{x} = \frac{x^{a}−x^{−a}}{x−x^{−1}} . Let us set β_{m}^{3} by ∑_{j=1}^{3} x^{−2jm}β_{m}^{j} = 0. The above commutation relations are valid for all 1 ≦ j, k ≦ 3. We also introduce the zero-mode operators P_{α}, Q_{α}, (α ∈ P) by

[iP_{α}, Q_{β}] = (α|β), (α, β ∈ P).  (3.2)

In what follows we deal with the bosonic Fock space F_{l,k}(l, k ∈ P) generated by β_{−m}^{j}(m > 0) over the vacuum vector |l, k⟩ :

F_{l,k} = ℂ[{β_{−1}^{j}, β_{−2}^{j}, ⋯}_{j=1,2,3}]|l, k⟩,  (3.3)

where

β_{m}^{j}|k, l⟩ = 0, (m > 0),

P_{α}|l, k⟩ = (α |\sqrt{\frac{r}{r − 1}}l − \sqrt{\frac{r − 1}{r}}k ) |l, k⟩,

|l, k⟩ = e^{i\sqrt{\frac{r}{r−1}} Q_{l}−\sqrt{\frac{r−1}{r}} Q_{k}}|0, 0⟩.

3.2 Vertex operator

We give a free field realization of the type-I vertex operators [11], associated with the elliptic algebra U_{q,p}(ŝl(3, ℂ)) [13, 14]. Let us set P (z), Q(z), R_{−}^{j}(z), S_{−}^{j}(z)(j = 1, 2) by

P (z) = ∑_{m>0} \frac{1}{m}β_{−m}^{1}z^{m}, Q(z) = − ∑_{m>0} \frac{1}{m}β_{m}^{1}z^{−m},  (3.4)
R_{−}^{j}(z) = − ∑_{m>0} \frac{1}{m}(β_{−m}^{j} − β_{−m}^{j+1})x^{jm}z^{m}, S_{−}^{j}(z) = ∑_{m>0} \frac{1}{m}(β_{m}^{j} − β_{m}^{j+1})x^{−jm}z^{−m}. (3.5)

Let us set the basic operators U(z), F_{α_{1}}(z), F_{α_{2}}(z) on the Fock space F_{l,k}.

U(z) = z^{\frac{r−1}{3r}} e^{−i\sqrt{\frac{r−1}{r}} Qϵ̄_{1}} z^{−\sqrt{\frac{r−1}{r}} Pϵ̄_{1}} e^{P (z)}e^{Q(z)},  (3.6)
F_{α_{j}}(z) = z^{\frac{r−1}{r}} e^{i\sqrt{\frac{r−1}{r}} Q_{α_{j}}} z^{\sqrt{\frac{r−1}{r}} P_{α_{j}}} e^{R_{−}^{j}(z)}e^{S_{−}^{j}(z)}.  (3.7)

In what follows we set

π_{μ} = \sqrt{r(r − 1)}Pϵ̄_{μ}, π_{μν} = π_{μ} − π_{ν}.  (3.8)

Then π_{μν} acts on F_{l,k} as an integer (ϵ_{μ} − ϵ_{ν}|rl − (r − 1)k). We give the free field realization of the type-I vertex operators.

Φ_{1}(z) = U(z),  (3.9)
Φ_{2}(z) = ∮_{C_{1}} \frac{dw_{1}}{w_{1}} U(z)F_{α_{1}}(w_{1})\frac{[v_{1} − u + \frac{1}{2} − π_{1,2}]}{[v_{1} − u − \frac{1}{2}]},  (3.10)
Φ_{3}(z) = ∮ ∮_{C_{2}} \frac{dw_{1}}{w_{1}} \frac{dw_{2}}{w_{2}} U(z)F_{α_{1}}(w_{1})F_{α_{2}}(w_{2})\frac{[v_{1} − u + \frac{1}{2} − π_{1,3}]}{[v_{1} − u − \frac{1}{2}]} \frac{[v_{2} − v_{1} + \frac{1}{2} − π_{2,3}]}{[v_{2} − v_{1} − \frac{1}{2}]}.

(3.11)

Here we set z = x^{2u}, w_{j} = x^{2v_{j}}(j = 1, 2). We take the integration contours to be simple closed curves around the origin satisfying

C_{1} : x|z| < |w_{1}| < x^{−1}|z|,
C_{2} : x|z| < |w_{1}| < x^{−1}|z|, x|w_{1}| < |w_{2}| < x^{−1}|w_{1}|.

We identify Φ^{(k+ϵ̄_{j},k)}(z) = Φ_{j}(z).

Let us introduce the type-II vertex operators [12], associated with the elliptic algebra U_{q,p}(ŝl(3, ℂ)). The type-II vertex operators represents the excitations. Let us set r^{∗} = r − 1. Let us set the elliptic theta function [u]^{∗} by

[u]^{∗} = x^{\frac{u^{2}}{r^{∗}} −u}Θ_{x^{2r^{∗}}} (x^{2u}).  (3.12)

Let us set P^{∗}(z), Q^{∗}(z), R_{+}^{j}(z), S_{+}^{j}(z)(j = 1, 2) by

P^{∗}(z) = − ∑_{m>0} \frac{[rm]_{x}}{m[r^{∗}m]_{x}} β_{−m}^{1}z^{m}, Q^{∗}(z) = ∑_{m>0} \frac{[rm]_{x}}{m[r^{∗}m]_{x}} β_{m}^{1}z^{−m},  (3.13)

R_{+}^{j}(z) = ∑_{m>0} \frac{[rm]_{x}}{m[r^{∗}m]_{x}} (β_{−m}^{j} − β_{−m}^{j+1})x^{jm}z^{m}, S_{+}^{j}(z) = − ∑_{m>0} \frac{[rm]_{x}}{m[r^{∗}m]_{x}} (β_{m}^{j} − β_{m}^{j+1})x^{−jm}z^{−m}.

(3.14)

Let us set the basic operators V (z), E_{α_{1}}(z), E_{α_{2}}(z) on the Fock space F_{l,k}.

V (z) = z^{\frac{r}{3r^{∗}}} e^{i\sqrt{\frac{r}{r^{∗}}} Qϵ̄_{1}} z^{\sqrt{\frac{r}{r^{∗}}} Pϵ̄_{1}} e^{P^{∗}(z)}e^{Q^{∗}(z)},  (3.15)
F_{α_{j}}(z) = z^{\frac{r}{r^{∗}}} e^{−i\sqrt{\frac{r}{r^{∗}}} Q_{α_{j}}} z^{−\sqrt{\frac{r}{r^{∗}}} P_{α_{j}}} e^{R_{+}^{j}(z)}e^{S_{+}^{j}(z)}.  (3.16)

We give a free field realization of the type-II vertex operators.

Ψ_{1}^{∗}(z) = V (z),  (3.17)
Ψ_{2}^{∗}(z) = ∮C̃_{1} \frac{dw_{1}}{w_{1}} V (z)E_{α_{1}}(w_{1})\frac{[v_{1} − u − \frac{1}{2} + π_{1,2}]^{∗}}{[v_{1} − u + \frac{1}{2}]^{∗}} ,  (3.18)
Ψ_{3}^{∗}(z) = ∮ ∮C̃_{2} \frac{dw_{1}}{w_{1}} \frac{dw_{2}}{w_{2}} V (z)E_{α_{1}}(w_{1})E_{α_{2}}(w_{2})\frac{[v_{1} − u − \frac{1}{2} + π_{1,3}]^{∗}}{[v_{1} − u + \frac{1}{2}]^{∗}} \frac{[v_{2} − v_{1} − \frac{1}{2} + π_{2,3}]^{∗}}{[v_{2} − v_{1} + \frac{1}{2}]^{∗}} .

(3.19)

The integration contours C̃_{1} encloses the poles at w_{1} = x^{−1+2sr^{∗}}z, (s = 0, 1, 2, ⋯), but not the poles at w_{1} = x^{1−2sr^{∗}}z, (s = 0, 1, 2, ⋯). The integration of Ψ_{3}^{∗}(z) is carried out in the order w_{2}, w_{1} along the contours C̃_{2} which encloses the poles the poles at w_{j+1} = x^{−1+2sr^{∗}}w_{j}, (s = 0, 1, 2, ⋯; j = 1, 2), but not the poles at w_{j+1} = x^{1−2sr^{∗}}w_{j}, (s = 0, 1, 2, ⋯; j = 1, 2). Here we

set w_{0} = z. Let us set the type-II Boltzmann weights W^{∗} ( \begin{matrix}a & b \\ c & d \end{matrix} | u) are given by

W^{∗} ( \begin{matrix}a + 2ϵ̄_{μ} & a + ϵ̄_{μ} \\ a + ϵ̄_{μ} & a \end{matrix} | u) = r_{1}^{∗}(u),  (3.20)
W^{∗} ( \begin{matrix}a + ϵ̄_{μ} + ϵ̄_{ν} & a + ϵ̄_{μ} \\ a + ϵ̄_{ν} & a \end{matrix} | u) = r_{1}^{∗}(u)\frac{[u]^{∗}[a_{μ,ν} − 1]^{∗}}{[u − 1]^{∗}[a_{μ,ν}]^{∗}} ,  (3.21)

W^{∗} ( \begin{matrix}a + ϵ̄_{μ} + ϵ̄_{ν} & a + ϵ̄_{ν} \\ a + ϵ̄_{ν} & a \end{matrix} | u) = r_{1}^{∗}(u)\frac{[u − a_{μ,ν}]^{∗}[1]^{∗}}{[u − 1]^{∗}[a_{μ,ν}]^{∗}} .  (3.22)

Otherwise are zero. The function r_{1}^{∗}(u) is given by

r_{1}^{∗}(u) = z^{\frac{r}{r^{∗}} \frac{2}{3}} \frac{h_{1}^{∗}(z^{−1})}{h^{∗}(z)}, h_{1}^{∗}(z) = \frac{(z; x^{2r^{∗}}, x^{6})_{∞}(x^{4}z; x^{2r^{∗}}, x^{6})_{∞}}{(x^{2r^{∗}}z; x^{2r^{∗}}, x^{6})_{∞}(x^{6}z; x^{2r^{∗}}, x^{6})_{∞}} .  (3.23)

The Boltzmann weights satisfy the following relations. (1)Yang-Baxter equation :

∑_{g} W^{∗} ( \begin{matrix}d & e \\ c & g \end{matrix} | u_{1}) W^{∗} ( \begin{matrix}c & g \\ b & a \end{matrix} | u_{2}) W^{∗} ( \begin{matrix}e & f \\ g & a \end{matrix} | u_{1} − u_{2})

= ∑_{g} W^{∗} ( \begin{matrix}g & f \\ b & a \end{matrix} | u_{1}) W^{∗} ( \begin{matrix}d & e \\ g & f \end{matrix} | u_{2}) W^{∗} ( \begin{matrix}d & g \\ c & b \end{matrix} | u_{1} − u_{2}) .  (3.24)

(2)The first inversion relation :

∑_{g} W^{∗} ( \begin{matrix}c & g \\ b & a \end{matrix} | − u) W^{∗} ( \begin{matrix}c & d \\ g & a \end{matrix} | u) = δ_{b,d}.  (3.25)

(3)The second inversion relation :

∑_{g} G_{g}^{∗}W^{∗} ( \begin{matrix}g & b \\ d & c \end{matrix} | 3 − u) W^{∗} ( \begin{matrix}g & d \\ b & a \end{matrix} | u) = δ_{a,c} \frac{G_{b}^{∗}G_{d}^{∗}}{G_{a}^{∗}} .  (3.26)

where G_{a}^{∗} = [a_{1,2}]^{∗}[a_{1,3}]^{∗}[a_{2,3}]^{∗}. The type-II vertex operators satisfy the following relations.

Φ_{μ_{1}}^{∗}(z_{1})Φ_{μ_{2}}^{∗}(z_{2}) = ∑_{ϵ_{μ′_{1}}+ϵ_{μ′_{2}}=ϵ_{μ_{1}}+ϵ_{μ_{2}}} W^{∗} ( \begin{matrix}k + ϵ̄_{μ_{1}} + ϵ̄_{μ_{2}} & k + ϵ̄_{μ′_{1}} \\ k + ϵ̄_{μ_{2}} & k \end{matrix} | u_{2} − u_{1}) Φ_{μ′_{2}}^{∗}(z_{2})Φ_{μ′_{1}}^{∗}(z_{1}).

(3.27)

The type-I and type-II vertex operators commute modulo the ”energy function” χ(z).

Φ_{μ_{1}}(z_{1})Ψ_{μ_{2}}^{∗}(z_{2}) = χ(z_{1}/z_{2})Ψ_{μ_{2}}^{∗}(z_{2})Φ_{μ_{1}}(z_{1}).  (3.28)

Here we have set

χ(z) = z^{\frac{2}{3}} \frac{(x^{5}z^{−1}; x^{6})_{∞}(xz; x^{6})_{∞}}{(xz^{−1}; x^{6})_{∞}(x^{5}z; x^{6})_{∞}} .  (3.29)

4 Boundary state

In this section we construct the free field realization of the ground state of the boundary transfer matrix of the U_{q,p}(ŝl(3, ℂ)) ABF model, which give higher-rank generalization of [7]. We construct the free field realization of the ground state on the Fock space F_{l,k}. Precisely the Fock space F_{l,k} is different from the space of the state H_{l,k}. The space of state H_{l,k} is obtained by the cohomological argument from the Fock space F_{l,k} by using the so-called screening operators. In this paper we omit the detailed of the screening operators and consider the operators acting on the Fock space F_{l,k}.

4.1 Bogoliubov transformation

The commutation relations of bosons β_{m}^{j} are not symmetric. Hence it is convenient to introduce new generators of bosons α_{m}^{1}, α_{m}^{2} whose commutation relations are symmetric.

α_{m}^{1} = x^{−m}(β_{m}^{1} − β_{m}^{2}), α_{m}^{2} = x^{−2m}(β_{m}^{2} − β_{m}^{3}).  (4.1)

They satisfy the following commutation relations.

[α_{m}^{j}, α_{n}^{k}] = \{ \begin{matrix}m\frac{[(r − 1)m]_{x}}{[rm]_{x}} \frac{[2m]_{x}}{[m]_{x}} δ_{m+n,0} & (j = k) \\ −m\frac{[(r − 1)m]_{x}}{[rm]_{x}} δ_{m+n,0} & (j ≠ k). \end{matrix}  (4.2)

Let us set

F_{0} = − ∑_{m>0} \frac{1}{m} \frac{[rm]_{x}}{[(r − 1)m]_{x}[3m]_{x}} ([2m]_{x}α_{−m}^{1}α_{−m}^{1} + 2[m]_{x}α_{−m}^{1}α_{−m}^{2} + [2m]_{x}α_{−m}^{2}α_{−m}^{2}). (4.3)

The adjioint action of e^{F_{0}} has the effect of a Bogoliubov transformation,

e^{−F_{0}}α_{m}^{j}e^{F_{0}} = α_{m}^{j} − α_{−m}^{j} (m > 0, j = 1, 2),  (4.4)
e^{−F_{0}}α_{−m}^{j}e^{F_{0}} = α_{−m}^{j} (m > 0, j = 1, 2),  (4.5)

and

e^{−F_{0}}β_{m}^{1}e^{F_{0}} = −β_{−m}^{1} (m > 0),  (4.6)
e^{−F_{0}}β_{m}^{2}e^{F_{0}} = (x^{2m} − 1)β_{−m}^{1} − x^{2m}β_{−m}^{2} (m > 0),  (4.7)
e^{−F_{0}}β_{−m}^{j}e^{F_{0}} = β_{−m}^{j} (m > 0, j = 1, 2).  (4.8)

Proposition 4.1

e^{−F_{0}}e^{Q(z)}e^{F_{0}} = \sqrt{\frac{(x^{2r+4}z^{−2}; x^{2r}, x^{6})_{∞}(x^{2}z^{−2}; x^{2r}, x^{6})_{∞}}{(x^{2r}z^{−2}; x^{2r}, x^{6})_{∞}(x^{6}z^{−2}; x^{2r}, x^{6})_{∞}}} e^{P (1/z)}e^{Q(z)},  (4.9)
e^{−F_{0}}e^{S_{−}^{j}(z)}e^{F_{0}} = \sqrt{\frac{(1 − z^{−2})(x^{2r−2}z^{−2}; x^{2r})_{∞}}{(x^{2}z^{−2}; x^{2r})_{∞}}} e^{R_{−}^{j}(1/z)}e^{S_{−}^{j}(z)} (j = 1, 2). (4.10)

4.2 Boundary state

Let us set the bosonic operator F by

F = F_{0} + F_{1},  (4.11)

where F_{0} is given in (4.3) and F_{1} is given by

F_{1} = ∑_{m>0}(D_{1}(m)β_{−m}^{1} + D_{2}(m)β_{−m}^{2}).  (4.12)

Here we set

D_{1}(m) = \frac{x^{−m}[(r − 2c + 2 − 2π_{1,3})m]_{x} − [(r − 2c − 1)m]_{x} + x^{(r−2c−2π_{1,2})m}[m]_{x}}{m[(r − 1)m]_{x}}
− θ_{m} (\frac{x^{− \frac{m}{2}} [m]_{x}[rm/2]_{x}^{+}}{m[(r − 1)m/2]_{x}} ) ,  (4.13)
D_{2}(m) = \frac{x^{m}([(r − 2c + 2 − 2π_{1,3})m]_{x} − [(r − 2c − 2π_{1,2})m]_{x})}{m[(r − 1)m]_{x}} − θ_{m} (\frac{x^{m}[rm/2]_{x}^{+}[m/2]_{x}}{m[(r − 1)m/2]_{x}} ) ,

(4.14)

where

[a]_{x}^{+} = a^{x} + a^{−x}, θ_{m}(x) = \{ \begin{matrix}x & (m : even) \\ 0 & (m : odd) \end{matrix} .

Let us set the vector

|B⟩_{l,k} = e^{F} |l, k⟩.  (4.15)

We call this vector |B⟩_{l,k} the boundary state. Proposition 4.2 The boundary state |B⟩_{l,k} has the following properties.

e^{Q(z)}|B⟩_{l.k} = h(z)e^{P (1/z)}|B⟩_{l,k},  (4.16)
e^{S_{−}^{j}(w)}|B⟩_{l,k} = g_{j}(w)e^{R_{−}^{j}(1/w)}|B⟩_{l,k}, (j = 1, 2).  (4.17)

Here the fuctions h(z), g_{j}(w)(j = 1, 2) are given by

h(z) = \frac{(x^{2r+4}z^{−2}; x^{2r}, x^{12})_{∞}(x^{8}z^{−2}; x^{2r}, x^{12})_{∞}}{(x^{12}z^{−2}; x^{2r}, x^{12})_{∞}(x^{2r}z^{−2}; x^{2r}, x^{12})_{∞}}
× \frac{(x^{2r+−2c+4}z^{−1}; x^{2r}, x^{12})_{∞}(x^{2c+2}z^{−1}; x^{2r}, x^{12})_{∞}}{(x^{2r−2c}z^{−1}; x^{2r}, x^{12})_{∞}(x^{2c+6}z^{−1}; x^{2r}, x^{12})_{∞}}
× \frac{(x^{2r+6−2c+2π_{1,2}}z^{−1}; x^{2r}, x^{12})_{∞}(x^{2c+2π_{1,2}}z^{−1}; x^{2r}, x^{12})_{∞}}{(x^{2r+4−2c−2π_{1,2}}z^{−1}; x^{2r}, x^{12})_{∞}(x^{2c+2+2π_{1,2}}z^{−1}; x^{2r}, x^{12})_{∞}}

× \frac{(x^{2r+6−2c+2π_{1,3}}z^{−1}; x^{2r}, x^{12})_{∞}(x^{2c+2π_{1,3}}z^{−1}; x^{2r}, x^{12})_{∞}}{(x^{2r+4−2c−2π_{1,3}}z^{−1}; x^{2r}, x^{12})_{∞}(x^{2c+2+2π_{1,3}}z^{−1}; x^{2r}, x^{12})_{∞}} ,  (4.18)
g_{1}(w) = (1 − 1/w^{2})\frac{(x^{2c+1}/w; x^{2r})_{∞}(x^{2r−2c−2π_{1,2}+1}/w; x^{2r})_{∞}}{(x^{2c+2π_{1,2}−1}/w; x^{2r})_{∞}(x^{2r−2c−1}/w; x^{2r})_{∞}} ,  (4.19)
g_{2}(w) = (1 − 1/w^{2})\frac{(x^{2c+2π_{1,2}}/w; x^{2r})_{∞}(x^{2r−2c+2−2π_{1,3}}/w; x^{2r})_{∞}}{(x^{2c+2π_{1,3}−2}/w; x^{2r})_{∞}(x^{2r−2c−2π_{1,2}}/w; x^{2r})_{∞}} .  (4.20)

The parameters l, k ∈ P are determined by the boundary conditions. The parameter k represents the central height. The parameter l represents the asymptotic boundary height. In what follows we consider the case k = l ∈ P for simplicity. For more general l, k ∈ P, there exist similar boundary state. We constract the eigenvector of the commuting boundary transfer matrix, [T_{B}^{(k)}(z_{1}), T_{B}^{(k)}(z_{2})] = 0. The following is the main result of this paper.

Theorem 4.3 The boundary state |B⟩_{k,k} is the eigenvector of boundary transfer matrix T_{B}^{(k)}(z).

T_{B}^{(k)}(z)|B⟩_{k,k} = |B⟩_{k,k}.  (4.21)

Corollary 4.4 Using the type-II vertex operators, we get general eigenvectors of the boundary transfer matrix T_{B}^{(k)}(z).

T_{B}^{(k)}(z) · Ψ_{μ_{1}}^{∗}(ξ_{1}) ⋯ Ψ_{μ_{M}}^{∗} (ξ_{M})|B⟩_{k,k}
= ∏_{j=1}^{M} χ(1/zξ_{j})χ(ξ_{j}/z) · Ψ_{μ_{1}}^{∗}(ξ_{1}) ⋯ Ψ_{μ_{M}}^{∗} (ξ_{M})|B⟩_{k,k},  (4.22)

where χ(z) is given by (3.29).

Here we sketch proof of main theorem 4.3. Acting the vertex operator Φ^{(k+ϵ̄_{j},k)}(z) to the condition T_{B}^{(k)}|B⟩_{k,k} = |B⟩_{k,k} from the left, we get the following necessary and sufficient condition.

z^{\frac{1}{r} (− \frac{r−1}{3} +(ϵ̄_{1}|k))}h(z)[c − u][π_{1,j} + c + u]Φ_{j}(z^{−1})|B⟩_{k,k} = (z ↔ z^{−1}) (j = 1, 2, 3).  (4.23)

Here we used the inversion relation of the type-I vertex operators. In the case of j = 1, after some calculation, LHS becomes the following

h(z)h(z^{−1})[c − u][c + u]e^{P (z)+P (1/z)}|B⟩_{k,k},

which is invariant z ↔ z^{−1}. Hence the relation (4.23) for j = 1 holds. In the case of j = 2, 3, after some calculation as similar as [9], the relation (4.23) are reduced to the following theta identity.

\frac{[v + k + c − \frac{1}{2}][v − c − \frac{1}{2}]}{[−v + k + c − \frac{1}{2}][−v − c − \frac{1}{2}]}  (4.24)
= \frac{[c − u][k + c + u][u − v + \frac{1}{2} − k][−v − u + \frac{1}{2}] − [c + u][k + c − u][−u − v + \frac{1}{2} − k][−v + u + \frac{1}{2}]}{[c − u][k + c + u][u + v + \frac{1}{2} − k][v − u + \frac{1}{2}] − [c + u][k + c − u][−u + v + \frac{1}{2} − k][v + u + \frac{1}{2}]}.

Later the author will write complete proof of this theorem, U_{q,p}(ŝl(N, ℂ)) version and generalization of asymptotic boundary condition l ∈ P in the another place [15]. Our proof is different from those given in [7] even for U_{q,p}(ŝl(2, ℂ)) case.

Acknowledgement

The author would like to thank the organizing committee of VIII-th International Workshop Lie theory and its applications in Physics, held in Varna, Bulgaria 2009. This work is partly supported by the Grant-in Aid for Scientific Research C(21540228) from Japan Society for the Promotion of Science.

A Some formule

h(z) =

exp (− ∑_{m>0} \frac{1}{2m} \frac{[(r − 1)m]_{x}[2m]_{x}}{[rm]_{x}[3m]_{x}} z^{−2m} − ∑_{m>0} \frac{[(r − 1)m]_{x}}{[rm]_{x}} (\frac{[2m]_{x}}{[3m]_{x}} D_{1}(m) − \frac{x^{−3m}[m]_{x}}{[3m]_{x}} D_{2}(m)) z^{−m}) ,

g_{1}(z) = exp (− ∑_{m>0} \frac{1}{2m} \frac{[(r − 1)m]_{x}(x^{m} + x^{−m})}{[rm]_{x}}z^{−2m} + ∑_{m>0} \frac{[(r − 1)m]_{x}}{[rm]_{x}} (D_{1}(m) − x^{−2m}D_{2}(m))z^{−m}) ,

g_{2}(z) = exp (− ∑_{m>0} \frac{1}{2m} \frac{[(r − 1)m]_{x}(x^{m} + x^{−m})}{[rm]_{x}}z^{−2m} + ∑_{m>0} \frac{[(r − 1)m]_{x}}{[rm]_{x}} D_{2}(m)x^{−m}z^{−m}) .

References

[1] B.Davis, O.Foda, M.Jimbo, T.Miwa and A.Nakayashiki, Diagonalization of the XXZ Hamiltonian by vertex operators, Commun.Math.Phys. 151, 89- (1993).

[2] M.Jimbo and T.Miwa, Algebraic Analysis of Solvable Lattice Models, CBMS Regional Conference Series in Mathematics, Vol 85, Providence, RI:AMS, (1994).

[3] E.K.Sklyanin, Boundary condition for integrable quantum system, J.Phys.A21, 2375- (1988).

[4] G.E.Andrews, R.J.Baxter and P.J.Forerester, Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J.Stat.Phys.35, 193- (1984).

[5] M.Jimbo, T.Miwa and M.Okado, Local state probabilities of solvable lattice models : an A_{n−1}^{(1)} family, Nucl.Phys.B300 [FS22], 74- (1988).

[6] M.Jimbo,R.Kedem,T.Kojima,H.Konno and T.Miwa, XXZ chain with a boundary, Nucl.Phys.B441[FS], 437- (1995).

[7] T.Miwa and R.Weston, Boundary ABF models, Nucl.Phys.B486[PM], 517- (1997).

[8] M.T.Batchelor, V.Fridkin, A.Kuniba, Y.K.Zhou, Solutions of the reflection equation for face and vertex models associated with A_{n}^{(1)}, B_{n}^{(1)}, C_{n}^{(1)} and A_{n}^{(2)}, Phys.Lett.B376 no.4, 266- (1996).

[9] H.Furutsu and T.Kojima, The U_{q}(ŝl_{n}) analogue of the XXZ chain with a boundary, J.Math.Phys. 41, no.7, 4413- (2000).

[10] S.Lukyanov and Ya.Pugai, Multi-point local height probabilities in the integrable RSOS model,Nucl.Phys.B473[FS], 631- (1996).

[11] Y.Asai, M.Jimbo, T.Miwa, Ya.Pugai, Bosonization of vertex operators for the A_{n−1}^{(1)} face model, J.Phys.A29, 6595- (1996).

[12] H.Furutsu, T.Kojima and Y.Quano, Type-II vertex operators for the A_{n−1}^{(1)} face model, Int.J.Mod.Phys.A15,no.10, 1533- (2000).

[13] M.Jimbo, H.Konno, S.Odake and J.Shiraishi : Elliptic Algebra U_{q,p}(ŝl_{2}) : Drinfeld currents and vertex operators, Commun.Math.Phys.199, 605- (1999).

[14] T.Kojima and H.Konno : The Elliptic Algebra U_{q,p}(ŝl_{N}) and the Drinfeld Realization of the Elliptic Quantum Group B_{q,λ}(ŝl_{N}), Commun.Math.Phys.237, 405- (2003).

[15] T.Kojima , in preparation.