An integrable vertex model and the quantum affine algebra Uq(sl(2)) at the critical level

Nuclear Physics B 638 [PM] (2002) 291–327 www.elsevier.com/locate/npe

An integrable vertex model and the quantum sl(2)) at the critical level affine algebra Uq (
R.M. Gade Lerchenfeldstraße 12, 80538 Munich, Germany Received 4 February 2002; received in revised form 13 May 2002; accepted 30 May 2002

Abstract Based on infinite-dimensional modules of Uq (sl(2)), an integrable model with an inhomogeneity in the spectral parameters is set up. An appropriate generalization of Baxter’s method allows to evaluate the eigenvalues of the corner transfer matrices. The spectrum of the latter is related to sl(2)) at its critical level. Two-point functions Wakimoto modules of the quantum affine algebra Uq (
of the model are expressed by traces over composite vertex operators introduced as lattice objects. sl(2))−2 -modules is An interpretation of these operators in terms of suitable intertwiners of Uq (
conjectured.  2002 Elsevier Science B.V. All rights reserved. Keywords: Integrable models; Quantum affine algebras

1. Introduction Starting with the work of Bethe [1] on the Heisenberg model, integrable models associated to finite-dimensional modules of the quantum algebra Uq (sl(2)) have been studied extensively (for references, see [2–4], for example). In particular, a method suited for the antiferromagnetic regime has been developed founded on the affine quantum group symmetry Uq (
sl(2)) present in the infinitely extended system [5,6]. The corner transfer matrix [7] provides the key to the formulation of the connection between the lattice model and the representation theory of the algebra. As revealed in [8], the multiplicities of the corner transfer matrix (CTM) eigenvalues of the six-vertex model and its spin- n2 generalizations coincide with the weight multiplicities of irreducible highest weight representations of the quantum affine algebra at level n, n = 1, 2, 3, . . . . The configurations employed to characterize the CTM eigenvectors also E-mail address: [email protected] (R.M. Gade). 0550-3213/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 ( 0 2 ) 0 0 4 4 1 - 8

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R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

label the base vectors of irreducible highest weight representations of Uq (
sl(2)) at q = 0 [9]. Moreover, for the six-vertex model, the interpretation of CTM-eigenvectors as weight sl(2))-representation at level 1 is supported by the results of an expansion vectors of a Uq (
in small q [10]. The concept of vertex operators [11] allows to extend the mathematical description to physical objects such as space of states, transfer matrix, N -point functions etc. [5,12]. Generally, N -point functions are expressed in terms of traces over products of vertex operators. A realization of the algebra including its highest weight representations and vertex operators provides a means to evaluate vacuum to vacuum expectation values as well as form factors [13–15]. In this paper, a first step towards an analogous analysis of integrable vertex models involving infinite-dimensional modules of Uq (sl(2)) is presented. Simplicity suggests to consider the first highest weight module of the discrete series. In order to attribute a well defined weight for each state, the dual module is incorporated in addition. Both in horizontal and vertical direction, the highest weight module V and the lowest weight module V ∗ are attributed in an alternating sequence to the lines of lattice links. It turns out that the basic lattice objects such as the transfer matrix or the vertex operators should be composed using 2 × 2 plaquettes of vertices as building blocks. Furthermore, an inhomogeneity in the spectral parameters is allowed for. Here, the maximal inhomogeneity of spectral parameters per plaquette permitted by integrability is specialized such that all vertices involving both the modules V and V ∗ share the same spectral parameter. From this, a second model with all other vertices carrying the same spectral parameter follows by a symmetry map. This spectral inhomogeneity proves crucial to the arguments relating the lattice objects to the various mathematical terms [19]. Inhomogeneous structures of this type have been studied by means of algebraic ([16,17] and references therein) or functional [18] Bethe ansatz in models constructed from the 3-dimensional vector representation of the quantum affine algebra Uq (sl(2|1)) as well as its dual. In the second case, the structure is introduced for the purpose of handling the thermodynamic behavior of the system. The Boltzmann weights for the integrable model based on V and V ∗ are obtained by solving the intertwining condition for the R-matrix on the tensor product of the infinitedimensional Uq (sl(2))-modules. With the special choice of the spectral inhomogeneity, the north-west (or south-east) corner transfer matrix is diagonizable. Given this property, Baxter’s argument to determine the eigenvalues and the form of the corner transfer matrices can be generalized. The procedure involves evaluation at particular values of the spectral parameters and of the limit q → 0. While the latter is generally not welldefined for the R-matrices of single vertices, it turns out to be well-behaved for the Boltzmann weights of 2 × 2-plaquettes. Depending on the choice of the 2 × 2-plaquettes, the degeneracy of the CTM-eigenvalues is finite or infinite. In the latter case, a finite degeneracy remains after adding a statistical weight provided by the action of the generator h1 of the Cartan subalgebra of Uq (sl(2)) to links of a border column of the corner transfer matrix such that integrability is conserved. With the statistical weight, the eigenvalues can be compared to the weight space structures of Uq (
sl(2))-modules. The comparison reveals that the appropriate level takes the critical value k = −2. For the description of local properties, vertex operators are introduced as half-infinite columns of plaquettes. Their basic properties imply a formulation of the two-point functions as traces over two

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(composite) vertex operators. Generalization to N -point functions is immediate. Guided by the approach to the six-vertex model and its higher spin analogues, an interpretation of sl(2))-modules is conjectured on the the vertex operators in terms of intertwiners of Uq (
basis of their properties. A proof of existence of the relevant intertwiners is beyond the scope of the present paper. At the critical level, the representation theory of the quantum affine algebra differs drastically from the noncritical case due to the occurrence of a large sl(2))−2 -modules suited to the present context is given in center. Here a definition of Uq (
terms of Wakimoto modules. Exploiting the intertwining property, the eigenvalues of the generators of the center on the two modules attributed to a vertex operator are related. Further details and arguments concerning the existence of intertwiners will be given in a separate publication [19]. The paper is organized as follows. Section 2 gives various definitions of the affine sl(2)) used in the following and summarizes the Hopf algebra quantum algebra Uq (
structure. The integrable vertex model is described in detail in Section 3 where the Boltzmann weights for the single vertices are given. Section 4 outlines the evaluation of the limit q → 0 and Section 5 deals with the generalization of Baxter’s procedure for the corner transfer matrices built from the plaquettes. Section 6 is devoted to the interpretation sl(2)) at its critical level. The vertex of the space of CTM-states in terms of modules of Uq (
operators are introduced on the lattice model in Section 7. Collecting their properties, sl(2))−2 -modules is a relation between the lattice objects and certain intertwiners of Uq (
conjectured. Section 8 adds a summary and a short outlook. For the convenience of the sl(2)) and the reader, the universal R-matrix underlying the Hopf algebra structure of Uq (
explicit relation between Drinfeld generators and L-operators are recalled in Appendix A. sl(2))−2 -modules are given. In Appendix B, some characters of irreducible Uq (
2. Definitions sl(2)) is the C-algebra generated by {ei , fi , q hi ; i = 0, 1} subject to the defining Uq (
relations [20] q hi q hj = q hj q hi , [ei , fj ] = δi,j

q hi ej = q aij ej q hi ,

q hi − q −hi , q − q −1

q hi fj = q −aij fj q hi , (1)

ej3 ei − [3]ej2ei ej + [3]ej ei ej2 − ei ej3 = 0, fj3 fi − [3]fj2 fi fj + [3]fj fi fj2 − fi fj3 = 0,

(2)

with a00 = a11 = −a01 = −a10 = 2. The quantum affine algebra Uq (
sl(2)) is obtained by including a generator d with the property [d, ei ] = δi,0 ei ,

[d, fi ] = −δi,0 fi .

With the coproduct given by ∆(ei ) = ei ⊗ 1 + q hi ⊗ ei ,

(3)

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∆(fi ) = fi ⊗ q −hi + 1 ⊗ fi , ∆(q hi ) = q hi ⊗ q hi ,

(4)

the antipode reads S(ei ) = −q −hi ei ,

S(q hi ) = q −hi .

S(fi ) = −fi q hi ,

(5)

According to the context, it proves to be useful to work with a second realization of sl(2)) due to Drinfeld [21]. A central element γ with the property (γ 1/2 )2 = q h0 +h1 Uq (
is introduced. The Drinfeld generators {Hm , El± , γ ± 2 , K ±1 , m ∈ Z\{0}, l ∈ Z} satisfy the relations 1

[Hm , Hm ] = δm+m ,0

[2m] γ m − γ −m , m q − q −1

KHk K −1 = Hk , KEk± = q ±2 Ek± K,   [2m] ∓ 1 |m| ± γ 2 Em+l , Hm , El± = ± m ± ± El+1 El± − q ±2 El± El+1 = q ±2 El± El± +1 − El± +1 El± , 

 El+ , El− =

where [m] = ∞ 

γ

l−l  2

l  −l

ψl+l  − γ 2 φl+l  , q − q −1

q m −q −m q−q −1

ψl z−l

and   ∞    = K exp q − q −1 Hl z−l ,

l=0 ∞ 

(6)

 φ−l z = K l

−1

l=1



exp − q − q

−1

l=0

∞ 

 l

H−l z .

(7)

l=1

The Drinfeld generators are related to the Chevalley basis by q h1 = K,

E0+ = e1 ,

E0− = f1 ,

q h0 = γ K −1 ,

E1− = e0 q h1 ,

+ E−1 = q −h1 f0 .

(8)

A third description of the algebra particularly suited for the analysis of the vertex model introduced below is provided by the realization in terms of L-operators [11,22]. These sl(2)) [23]. R(z) is the unique generators are related to the universal R-matrix R(z) of Uq (
ˆ q (
sl(2))⊗U sl(2)) ⊗ C[[z, z−1 ]] with the properties element in Uq (
sl(2)), R(z)∆z (a) = ∆z (a)R(z) ∀a ∈ Uq (
(∆z ⊗ id)R(w) = R13 (zw)R23 (w), (id ⊗ ∆w )R(zw) = R13 (z)R12 (zw).

(9)

Eqs. (9) involve the maps ∆z (a) = (Dz ⊗ id)∆(a) and ∆z (a) = (Dz ⊗ id)∆ (a)

(10)

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where z is a formal variable, ∆ (a) = σ ◦ ∆(a) with σ (a ⊗ b) = b ⊗ a and Dz (fi ) = z−δi,0 fi ,

Dz (ei ) = zδi,0 ei ,

Dz (q ±hi ) = (q ±hi ),

D(z) = z.

(11)

Besides, in the second and third lines of (9) the notations R12 (z) = R(z) ⊗ id, R23 (z) = id ⊗ R(z) and R13 (z) = (σ ⊗ id)R23 (z) are used. The universal R-matrix and its inverse are related by   (S ⊗ id)R(z) = id ⊗ S −1 R(z) = R−1 (z). (12) The entire dependence of R(z) on the grading operator d is specified by R(z) = R(z)q −c⊗d−d⊗c

(13)

ˆ q (
where R(z) ∈ Uq (
sl(2))⊗U sl(2)) ⊗ C[[z, z−1 ]]. The universal R-matrix encodes a number of important results on the structure of the algebra. A consequence of the properties (9) is the universal Yang–Baxter equation R12 (z)R13 (zw)R23 (w) = R23 (w)R13 (zw)R12 (z).

(14)

Current-type operators can be constructed for any finite-dimensional representation of Uq (
sl(2)). A representation πW : Uq (
sl(2)) → End(W ) associated to the two-dimensional module W with basis {w1 , w2 } is given by f1 w1 = w2 ,

e0 w1 = w2 ,

h1 w1 = −h0 w1 = w1 ,

e1 w2 = w1 ,

f0 w2 = w1 ,

h1 w2 = −h0 w2 = −w2 .

(15)

Here πW (a)wi has been written as awi for brevity. [22] or [11], a set of 2 × 2 ∞ Following ± ∓n is obtained from R(z) by matrix generators L± (z) with entries L± (z) = (L ) z n=0 ij ±n ij   L+ (z) = (πW ⊗ id) σ ◦ R(z−1 ) ,   L− (z) = (πW ⊗ id) R −1 (z) . (16) Rewriting the Yang–Baxter equation (14) in terms of various combinations of (Rij (z))±1 and applying (16) leads to the RLL-equations   i j  ± i2 j2 ± ± R ± (z/w) i3 j3 L± L± i2 i1 (z)Lj2 j1 (w) = j3 j2 (w)Li3 i2 (z) R (z/w) i j 2 2

1 1

i2 ,j2

i2 ,j2

 i j ∓ R ± (γ ±1 z/w) i3 j3 L± i2 i1 (z)Lj2 j1 (w) 2 2

i2 ,j2

=



 ± ∓1 i j ± L∓ z/w) i2 j2 j3 j2 (w)Li3 i2 (z) R (γ

(17)

1 1

i2 ,j2

with the matrix elements R

±

(z)11 11

=R

±

(z)22 22



=q

∓ 21

 ∞   1 [n]2 ∓n −1 z exp ∓ q − q , n [2n] n=1

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q(1 − z) ± 11 R (z)11 , q2 − z z(q 2 − 1) ± 11 ± 12 R ± (z)21 R (z)11 . 12 = zR (z)21 = q2 − z

± 21 R ± (z)12 12 = R (z)21 =

(18)

Due to the property (9), the coproduct takes the form   + αc  α (α−1)c1 Lik (q 2 z) ⊗ L+ z) ∆ L+ ij (γ z) = kj (q k=1,2

≡

∞  

    −n−m −αn + γ (1−α)m L+ , Lkj m z ik n ⊗ γ

k=1,2 n,m=0

   − (α−1)c α 2 z) ⊗ L− (q αc1 z) ∆ L− Lik (q ij (γ z) = kj k=1,2

≡

∞  

    (α−1)n − Lkj −m zn+m . γ αm L− ik −n ⊗ γ

(19)

k=1,2 n,m=0

Finally, the Chevalley generators are related to the L-operators by  −  + 1 1 L21 −1 = (q − q −1 )q − 2 h1 f0 , L21 0 = −(q − q −1 )e1 q − 2 h1 ,  +  − 1 1 L12 1 = −(q − q −1 )e0 q 2 h1 , L12 0 = (q − q −1 )q 2 h1 f1 ,  ±  ± 1 1 L11 0 = q ∓ 2 h1 , L22 0 = q ± 2 h1 .

(20)

In Appendix A, explicit expressions for R(z) and for the L-operators in terms of the Drinfeld-operators are given. The vertex model investigated in the subsequent sections involves the infinite-dimensional Uq (sl(2))-modules V and V ∗ . V is an irreducible highest-weight representation with basis {vp , p = 0, 1, 2, . . .} and πV ∗ the associated lowest-weight representation. In terms of the basis elements {vp }, {vp∗ } the action of the Uq (sl(2))-generators is defined by f1 vp = q p [p + 1]vp+1 ,

∗ f1 vp∗ = −q −(p+1)[p]vp−1 ,

e1 vp = −q −(p−1)[p]vp−1 ,

∗ e1 vp∗ = q p+2 [p + 1]vp+1 ,

q h1 vp = q −2p−1vp ,

q h1 vp∗ = q 2p+1 vp∗ .

(21)

sl(2))-structure on V and V ∗ is obtained by setting A Uq (
e0 wp = −q 2 f1 wp ,

f0 wp = −q −2 e1 wp ,

q h0 wp = q −h1 wp ,

(22)

with wp = vp or wp = vp∗ . V and V ∗ are dual to each other with respect to the antiˇ automorphism S(a) defined by ˇ S(a) = (−q)2d+ 2 h1 S(a)(−q)−2d− 2 h1 1

1

∀a ∈ Uq (
sl(2)).

(23)

ˇ V ∗ may then be introduced as the dual space of V by av ∗ |v = v ∗ |S(a)v and vice versa. Using (21) and (22) in (8) and the defining relations (6), (7) yields the action of the Drinfeld

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generators Hn , n ∈ Z\0 and En± , n ∈ Z on V and V ∗ :   1 Hn vp = (−1)n+1 q −n(p−2) q n [np] + [n(p + 1)] vp , n En− vp = (−1)n q −(2n−1)p+n [p + 1]vp+1 , En+ vp = (−1)n+1 q −(2n+1)p+3n+1[p]vp−1 ,

(24)

 1 n q [n(p + 1)] + [np] vp∗ , n ∗ En+ vp∗ = (−1)n q (2n+1)p+5n+2[p + 1]vp+1 , Hn vp∗ = (−1)n q n(p+3)

∗ En− vp∗ = (−1)n+1 q (2n−1)p+3n−1[p]vp−1 .

(25)

With Eq. (A.4), the action of the L-operators is easily obtained from (24) and (25).

3. The vertex model Both quantum and auxiliary space of the vertex model considered in Section 4 are composed of alternating sequences of V and V ∗ . In Fig. 1, arrows pointing to the right or upwards (left or downwards) indicate links attributed to the module V (V ∗ ). Furthermore, the vertex model is characterized by an inhomogeneity in the spectral parameters. Depending on the configuration of V and V ∗ on the adjacent links, complex numbers are associated to a vertex as specified in Fig. 2. For even numbers of horizontal and vertical lines, the lattice model outlined above may be thought as built from elementary plaquettes of either type A or B as shown in Fig. 3. A single plaquette with general x, y, w disposes of the maximal inhomogeneity of spectral parameters allowed by integrability. Two closely related specializations y = w−1 and y = w will be considered in the sections below. To each vertex of the model, an Rmatrix depending on the spectral parameter is associated. The elements of the R-matrix RW W  (z) : W ⊗ W  → W ⊗ W  can be determined explicitly from the intertwining property ∆z (a)R(z) = R(z)∆z (a)

∀a ∈ Uq (
sl(2)).

(26)

The assignment of indices is shown in Fig. 4 for the four types of vertices. Due to the property (26), the R-matrix elements satisfy the Yang–Baxter equation RW1 W2 (z1 )RW1 W3 (z1 z2 )RW2 W3 (z2 ) = RW2 W3 (z2 )RW1 W3 (z1 z2 )RW1 W2 (z1 ), on any tensor product of three modules W1 , W2 , W3 chosen among V and condition (26) yields the R-matrix elements s,t s.t Rp,r (z) = µ(z)rp,z (z)

V ∗.

(27) Solving (28)

with the scalar factor µ(z) =

∞ 1 − q 4l+2 z 1 − q −4l−4 z l=0

1 − q 4l+4 z 1 − q −4l−2 z

(29)

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Fig. 1. The vertex model.

Fig. 2. The spectral parameters.

Fig. 3. The elementary plaquettes.

Fig. 4. The R-matrices.

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299

and 1 [p + r]! p+r,0 rp,r (z) = q (p+ 2 r)(r+1)zr

[p]!

p+r−s,s

rp,r

1

× z−s

(30)

k=0

p+r,0

(z) = q p(r+s)− 2 s(s−1)rp,r



r 1 − z q − q −1 , q − q −1 q 2(p+k) − z

(z)

s−1 [p + r − s]! q −2(p−k) − z [p + r]![s]! q − q −1 k=0

[r]! + [r − s]!

s 

q −m zm−s

m=1

×

m−1 n=0

[r + m]![s]! [r − s + m]![s − m]![m]!

q − q −1 q − q −1 [p − n]2 −2(p−n) q − z q 2(p−n−1) − z

 (31)

for p
s > 0. The remaining matrix elements of RV V (z) are related to (30) and (31) by s,t t,s Rp,r (z) = zs−p Rr,p (z)

(32)

and r,p

s,t Rp,r (z) = Rt,s (z).

(33)

s,t R-matrix elements Rp,r (z) with p + r = s + t vanish. An initial condition is satisfied at spectral parameter z = 1: s,t (1) = δp,t δs,r . Rp,r

(34)

The matrix elements of RV ∗ V ∗ (z), RV ∗ V (z) and RV V ∗ (z) are obtained by ∗ ∗

p,r

Rps ∗,t,r ∗ (z) = Rs,t (z), ∗

Rps ∗,t,r (z) = q s−p Rr,s (q −2 z−1 ), t,p

∗

s,t p−s r,s Rp,r Rt,p (q −2 z−1 ). ∗ (z) = q

(35)

4. The limit Q → 0 Applying periodic boundary conditions in horizontal direction, a transfer matrix built from the R-matrices of two horizontal lines may be defined:

T (x, y, w) = trV ⊗V ∗ Ri 2N (xy −1w−1 )Ri 2N−1 (xy −1 w) × Ri 2N−2 (xy −1w−1 ) · · · Ri 2 (xy −1 w−1 )Ri 1 (xy −1w) × Ri+1 2N (xyw−1 )Ri+1 2N−1 (xyw)

 × Ri+1 2N−2 (xyw−1 ) · · · Ri+1 2 (xyw−1 )Ri+1 1 (xyw) . (36)

Here the trace refers to the horizontal lines labeled by i and i + 1. A transfer matrix T ∗ (x, y, w) involving the trace over V ∗ ⊗ V may be introduced analogously. Two transfer

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matrices with different x, y but fixed w commute: [T (x, y, w), T (x  , y  , w)] = 0.

(37)

Apart from a scalar factor, the transfer matrix coincides with a shift operator in right (left) direction at x = 1, y = w (x = q −2 , y = w−1 ):   T (1, w, w) vr1 ⊗ vp∗ 1 ⊗ vr2 ⊗ vp∗ 2 ⊗ · · · ⊗ vrN ⊗ vp∗ N   = vrN ⊗ vp∗ N ⊗ vr1 ⊗ vp∗ 1 ⊗ · · · ⊗ vrN−1 ⊗ vp∗ N−1 ,   T (q −2 , w−1 , w) vr1 ⊗ vp∗ 1 ⊗ vr2 ⊗ vp∗ 2 ⊗ · · · ⊗ vrN ⊗ vp∗ N   = vr2 ⊗ vp∗ 2 ⊗ · · · ⊗ vrN ⊗ vp∗ N ⊗ vr1 ⊗ vp∗ 1 . (38) This can be seen taking into account the initial condition (34) for RV V (xy −1w) and for RV ∗ V ∗ (xyw−1 ). Due to this property, for x = 1, y = w the unitarity condition −1 RW
W
(z )RW
W
 (z) = 1

(39)

r  ,p ∗ ,s  ,t ∗

governs the matrix elements Pr,p∗ ,s,t ∗ (1, y, y) of the elementary plaquette of type A. In
W
 stands for V V , V ∗ V ∗ , V V ∗ or V ∗ V . Then, with the assignment of indices (39), W shown in Fig. 3, the matrix elements are given by p  ,r ∗ ,s  ,t ∗

Pp,r ∗ ,s,t ∗ (1, w, w) = δp,p δs,s  δr,r  δt,t  ,

(40)

which implies (38). With (35), the corresponding result for the choice y = w−1 reads: p  ,r ∗ ,s  ,t ∗





Pp,r ∗ ,s,t ∗ (q −2 , w−1 , w) = q s−r +t −p δp,r δp ,r  δs,t δs  ,t  .

(41)

The explicit verification of (39) using (35) implies the summation of infinitely many products of matrix elements. The structure of the universal R-matrix suggests to expand each term in powers of z ≡ (yw)2 . Each power of z occurs for only finitely many terms and is easily seen to satisfy property (39). The evaluation of P (x, y, w) : V ⊗ V ∗ ⊗ V ⊗ V ∗ → V ⊗ V ∗ ⊗ V ⊗ V ∗ is facilitated making use of the unitarity condition (Fig. 5): r  ,p ∗ ,s  ,t ∗

Pr,p∗ ,s,t ∗

(x, y, w) =

∞ 

p ∗ ,r  ,t ∗



∗

s ,n ,m −1 Xn∗ ,m,t ∗ (xyw−1 , y 2 )Ys,p w, y −2 ) ∗ ,r (xy

(42)

n,m=0

with Y : V ⊗ V ∗ ⊗ V → V ⊗ V ∗ ⊗ V and X : V ∗ ⊗ V ⊗ V ∗ → V ∗ ⊗ V ⊗ V ∗ given by Y123 (xy −1 w, y −2 ) = P13 R12 (y −2 )R13 (xy −1w)R23 (xyw)

(43)

X234 (xyw−1 , y 2 ) = P24 R34 (xy −1w−1 )R24 (xyw−1 )R23 (y 2 ).

(44)

and

Due to (35), the matrix elements of X are related to those of Y by s  ,p ∗ ,r 

r ∗ ,p,s ∗

Ys,p∗ ,r (xy −1 w, y −2 ) = Xr ∗ ,p ,s ∗ (xy −1w, y −2 ), where the indices are attached as indicated in Fig. 6.

(45)

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301

Fig. 5. Decomposition of P (x, y, w).

The operator Pˆ (x, y, w) : V ∗ ⊗ V ⊗ V ∗ ⊗ V → V ∗ ⊗ V ⊗ V ∗ ⊗ V corresponding to an elementary plaquette of type B may be decomposed analogously:  p ,r ∗ ,t  ∗ ,n,m∗ s ∗ ,p  ,r ∗ ,t  −1 2 X¯ n,m∗ ,t (xy −1w, y −2 )Y¯ss∗ ,p,r , y ). Pˆs ∗ ,p,r ∗ ,t (x, y, w) = (46) ∗ (xyw n,m=0

Eq. (32) allows to express the matrix elements of X¯ : V ⊗ V ∗ ⊗ V → V ⊗ V ∗ ⊗ V and Y¯ : V ∗ ⊗ V ⊗ V ∗ → V ∗ ⊗ V ⊗ V ∗ in terms of Y as illustrated in Fig. 6: 

∗



∗

∗

r ,p ,s s ,p,r X¯ r,p∗ ,s (xy −1 w, xyw) = Y¯s ∗ ,p ,r ∗ (xy −1w, xyw) 

s  ,p ∗ ,r 



= (xy −1 w)s −r y 2(p −p) Ys,p∗ ,r (xy −1 w, y −2 ).

(47)

Symmetry relations for Y (xy −1w, y −2 ) follow from (32) and from the Yang–Baxter equation combined with (47) (see Fig. 7): s  ,p ∗ ,r 

s,p ∗ ,r

Ys,p∗ ,r (xy −1 w, y −2 ) = Ys  ,p∗ ,r  (xy −1 w, xyw) 



r  ,p ∗ ,s 

= (xy −1w)r −s (xyw)p−p Yr,p∗ ,s (xy −1w, xyw).

(48)

Recalling the intertwining condition (26) for the R-matrix, an analogous equation can be derived for Y (xy −1w, y 2 ): ∆xw,y (a)Y = Y ∆xw,y (a) ∀a ∈ Uq (
sl(2)),

(49)

where   ∆xw,y (a) = Dxy −1 w ⊗ Dxyw ⊗ id ∆(2) (a),   ∆xw,y (a) = id ⊗ Dxyw ⊗ Dxy −1 w ∆(2) (a),

(50)

302

R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

Fig. 6. Symmetry relations.

with Dz as defined in (11) and ∆(2) (a) = (∆ ⊗ id) ◦ ∆(a)

(51)

0,0∗ ,r 0,0∗ ,r

The matrix elements Y (xy −1w, y −2 ) may be obtained by insertion of (35), (30) and (33) in (43). Guided again by the structure of the universal R-matrix, the resulting expression is expanded in powers of y −2 . The remaining matrix elements of Y (xy −1w, y −2 ) may be evaluated making use of the intertwining conditions (49). Rather complicated formulae are obtained. However, in analogy to the case of homogeneous vertex models based on finite-dimensional representations of quantum algebras [24], a remarkable simplification occurs in the limit q → 0. The limit is performed keeping x, y, w each at a fixed finite s  ,p ∗ ,r  value. Only diagonal elements Ys,p∗ ,r (xy −1w, y −2 ) do not vanish for q → 0: s  ,p ∗ ,r 

lim Ys,p∗ ,r (xy −1w, y −2 ) = δs,s  δp,p δr,r 

q→0



xw y

−p−s .

(52)

R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

303

Fig. 7. Symmetries of Y .

For the matrix elements of X(xyw−1 , y 2 ) and P (x, y, w), (45) and (42) lead to  −r−t xy p ∗ ,r  ,t ∗ lim Xp∗ ,r,t ∗ (xyw−1 , y 2 ) = δp,p δr,r  δt,t  , q→0 w  p+s−r−t y s  ,p ∗ ,r  ,t ∗ −(p+r+s+t ) . lim P ∗ ∗ (x, y, w) = δs,s  δp,p δr,r  δt,t  x q→0 s,p ,r,t w

(53)

Writing Pˆ (x, y, w) for the plaquette operator of type B, its matrix elements can be related to those of type A by means of (32) and (35):  p−t     xy yw s−r ˆ t ∗ ,r,p∗ ,s  s  ,p ∗ ,r  ,t ∗ Pt ∗ ,r  ,p∗ ,s  (x, y, w). Ps,p∗ ,r,t ∗ (x, y, w) = w2(r−s ) (54) w x Thus, in the limit q → 0, the plaquette operator Pˆ (x, y, w) approaches a diagonal matrix with elements inferred from (53): ∗



∗ 

s ,p ,r ,t lim Pˆs ∗ ,p,r ∗ ,t (x, y, w) = δs,s  δp,p δr,r  δt,t  x −2(p+r).

q→0

(55)

Several symmetry properties of P (x, y, w) and Pˆ (x, y, w) used in the following section are implied by Eqs. (33) and (35).

5. The corner transfer matrices Following the construction of the corner transfer matrix (CTM) for the eight-vertex model [25], the analogous object of the inhomogeneous model is introduced on the upper

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left quadrant of the lattice. The vertical (horizontal) boundaries of a triangular subsection AN are composed of 2N + 3 neighboring horizontal (vertical) links on the boundaries of the quadrant. The configuration of statistical variables on the diagonal boundary is kept fixed. Thus the statistical weights of configurations within the triangle provide a map  N+1  N+1 → V ∗ ⊗V ⊗ V ∗ . A(N) (x, y, w) : V ∗ ⊗V ⊗ V ∗ (56) The corresponding objects occupying the lower left, lower right and upper right quadrant of the lattice are denoted by  N  N B (N) (x, y, z) : V ⊗V ∗ ⊗ V → V ∗ ⊗V ⊗ V ∗ ,  N  N C (N) (x, y, z) : V ⊗V ∗ ⊗ V → V ⊗V ∗ ⊗ V ,  N  N D (N) (x, y, z) : V ∗ ⊗V ⊗ V ∗ → V ⊗V ∗ ⊗ V . (57) For each object, the values of all statistical variables on the links of the diagonal boundary are set to zero. In the case y = w−1 and a suitable set of values on the diagonal boundaries, the matrices A(N) (x, w−1 , w) and C (N) (x, w−1 , w) are diagonalizable due to the property s  ,p ∗ ,r  ,t ∗

Ps,p∗ ,r,t ∗

∗

∗

,r,t −1 −1 (x, y, w) = Pss,t , y ).  ,p ∗ ,r  ,t ∗ (x, w

(58)

Similarly, for y = w the property s  ,p ∗ ,r  ,t ∗

Ps,p∗ ,r,t ∗

t  ,r ∗ ,p  ,s ∗

(x, y, w) = Pt,r ∗ ,p,s ∗

(x, w, y)

(59)

ensures the diagonalizability of B (N) (x, w, w) and D (N) (x, w, w). The corner transfer matrices are introduced as the infinite lattice limits A˜ n (x, w−1 , w) = lim A(N) (x, w−1 , w), N→∞

C˜ n (x, w−1 , w) = lim C (N) (x, w−1 , w), N→∞

Bn (x, w

−1

, w) = lim B (N) (x, w−1 , w), N→∞

Dn (x, w−1 , w) = lim D (N) (x, w−1 , w). N→∞

(60)

For convenience, the normalized matrix An (x, w−1 , w) (Cn (x, w−1 , w)) may be introduced by dividing A˜ n (x, w−1 , w) (C˜ n (x, w−1 , w)) by its maximum eigenvalue. Applying the arguments of [7] to the inhomogeneous vertex model with y = w−1 allows to express the corner transfer matrices by     −1 An (x, w−1 , w) τ  ,τ = a(x, w) P (w)A(A) d (x, w)Q (w) τ  ,τ ,     (B) Bn (x, w−1 , w) τ  ,τ = b(x, w) Q(w)Ad (q −2 x −1 , w)R −1 (w) τ  ,τ ,     (C) Cn (x, w−1 , w) τ  ,τ = c(x, w) R(w)Ad (x, w)T −1 (w) τ  ,τ ,     −2 −1 −1 (w) τ  ,τ , Dn (x, w−1 , w) τ  ,τ = d(x, w) T (w)A(D) (61) d (q x , w)P where the matrices P (w), Q(w), R(w) and T (w) are independent of x, a(x, w), b(x, w), ) c(x, w) and d(x, w) denote scalar functions and A(F d (x, w) are diagonal matrices with

R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

entries 

(F )

Ad (x, w)

 τ,τ

) ατ = m(F τ (w)x ,

F = A, B, C, D.

305

(62)

Generally, the exponents ατ will depend on w, too. In (61) and (62), the subscript τ abbreviates the set of arguments {p0 , r0 , p1 , r1 , · · · , pN , rN , pN+1 } specifying the vector vp∗ 0 ⊗ vr0 ⊗ vp∗ 1 ⊗ vr1 ⊗ · · · ⊗ vp∗ N ⊗ vrN ⊗ vp∗ N+1 or vp0 ⊗ vr∗0 ⊗ vp1 ⊗ vr∗1 ⊗ . . .⊗ vpN ⊗ vr∗N ⊗ vpN+1 . In each case, p0 is assigned to a link of the plaquette where all four corner transfer (F ) matrices meet. Information on mτ (w) and ατ can be gained by consideration of special values for x and q. As stated in (41), the matrices B (N) (x, w−1 , w) and D (N) (x, w−1 , w) are diagonal for x = q −2 :  (N)    B (1, w−1 , w) τ  ,τ = D (N) (1, w−1 , w) τ  ,τ =

N

q ri δri ,r 

N+1

i

i=0

q −pi δpi ,p ≡ δτ  τ Ωτ,τ . i

(63)

i=1

The symmetry property (58) implies D = B T . Using this relation as well as (63) in (61) leads to     Bn (x, w−1 , w) τ  ,τ = DnT (x, w−1 , w) τ  ,τ  b(x, w)  Q(w)(q 2 x)−ατ Q−1 (w)Ω τ  ,τ . = (64) −2 b(q , w) Obviously, the matrices Bn (x, w−1 , w) and Dn (x, w−1 , w) are well-defined only on the subset of vectors {vp0 ⊗ vr∗0 ⊗ vp1 ⊗ vr∗1 ⊗ · · ·} with finite values of ∞ (p i=0 i + ri ). This requirement leads to the above choice of boundary conditions. The corresponding corner transfer matrices An (x, w−1 , w) and Cn (x  , w−1 , w) may be related by investigating their product at x = w2 and x  = w−2 . Use of the Yang–Baxter equation, (35) and (39) allows to derive the decomposition  (N) −2 −1  C (w , w , w)A(N) (w2 , w−1 , w) τ  ,τ     ˆ −4 )Ω A(w ˆ 4 ) · W (w2 )  = W  (w−2 ) · 1N+1 ⊗ C(w (65) τ ,τ ˆ 4 ) : V (⊗V )N−1 → V (⊗V )N−1 (C(w ˆ −4 ) : V ∗ (⊗V ∗ )N−1 → In the last equation, A(w V ∗ (⊗V ∗ )N−1 ) denotes the size-N approximation of the upper left (lower right) corner transfer matrix of a homogeneous model with the representation spaces V (V ∗ ) and the spectral parameter w4 (w−4 ) assigned to any of its horizontal or vertical lines of links and the corresponding Boltzmann weights given by the R-matrix elements ˆ −4 )). Due to equation (30)–(33) (supplemented by the first equation of (35) for C(w 4 4 ˆ ˆ (35), (A(w ))σ  ,σ = (C(w ))σ  ,σ , where σ denotes the index set {p0 , p1 , · · · , pN } characterizing the vector vp0 ⊗ vp1 ⊗ · · · ⊗ vpN or vp∗ 0 ⊗ vp∗ 1 ⊗ · · · ⊗ vp∗ N . Both vectors are mapped onto each other by ι according to ι(vp0 ⊗ vp1 ⊗ · · · ⊗ vpN ) = vp∗ 0 ⊗ vp∗ 1 ⊗ · · · ⊗ vp∗ N . ˆ 4 ) (C(w ˆ −4 )) in the limit N → ∞ may be The corner transfer matrix obtained from A(w normalized by its largest eigenvalue. In the following, the resulting matrix is called Aˆ n (w4 ) (Cˆ n (w−4 )). Further, the r.h.s. of (65) contains maps W (w2 ) : V ∗ (⊗V ⊗ V ∗ )N+1 →

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V ∗ (⊗V ∗ )N+1 (⊗V )N+1 and W  (w−2 ) : V (⊗V )N+1 (⊗V ∗ )N+1 → V (⊗V ∗ ⊗ V )N+1 . By means of the unitarity condition it is easily seen that W  (w−2 )W (w2 ) = 12N+3 .

(66)

ˆ −4 )A(w ˆ 4 ) share the Thus, the products C (N) (w−2 , w−1 , w)A(N) (w2 , w−1 , w) and C(w −2 −1 same eigenvalues. The same holds true for the products Cn (w , w , w)An (w2 , w−1 , w) and Cˆ n (w−4 )Aˆ n (w4 ). The procedure outlined in [7] yields   Aˆ n (w4 ) = Cˆ n (w4 ) = a(w) ˆ a(1) ˆ −1 Pˆ Aˆ d (w)Pˆ −1 with Aˆ d (w) σ,σ  = δσ,σ  wαˆ σ . (67) ˆ −4 ) = Furthermore, the normalization of the corner transfer matrices implies a(w ˆ 4 )a(w 2 −2 −1 2 −1 (a(1)) ˆ . Hence, all eigenvalues of Cn (w , w , w)An (w , w , w) equal one. Due to Eq. (64), this may be written as a constraint on the scalar functions and diagonal matrices in Eq. (61): a(w2 , w)b(q −2, w)c(w−2 , w)d(q −2 , w) (B) (C) (D) 2 −2 2 × A(A) d (w , w)Ad (1, w)Ad (w , w)Ad (1, w) = Ω

Eq. (68) then leads to the expressions     An (x, w−1 , w) τ  ,τ = a(x, w) (Qˇ −1 )T (w)x ατ Qˇ −1 (w) τ  ,τ     c(x, w) ατ ˇ T ˇ Q(w)x Cn (x, w−1 , w) τ  ,τ = Q (w) τ  ,τ 2 −2 a(w , w)c(w , w)

(68)

(69)

1/2 Q(w) ˇ where (m(A) = Q(w). The formula (64) for Bn (x, w−1 , w) and Dn (x, τ (w)) −1 ˇ w , w) may be rewritten by replacing Q(w) by Q(w) without further change. Following [7], the assumption of appropriate analyticity conditions of the diagonal matrix A(A) d (x, w) permits to calculate ατ from limiting cases. Due to the invariance of the Boltzmann weights under ln x → ln x + 2πi the exponents ατ are expected to be integers. As q goes to zero, An (x, w−1 , w) and Cn (x, w−1 , w) tend to diagonal matrices whose entries can be read from (53). In the limit of infinitely large N , finite eigenvalues emerge only if the values of almost all statistical variables on the lattice are zero. Thus, to each set {p0 , r0 , p1 , r1 , . . .} with only finitely many pi , ri = 0, two eigenvalues ∞  (A)    − ∞ i=0 (2i+1)·(ri +pi ) w 2 i=0 (ri +pi ) , a (x, w) τ,τ ≡ lim A(A) d (x, w) τ,τ = x



a (C)(x, w)

q→0

 τ,τ

∞ ∞  (C)  ≡ lim Ad (x, w) τ,τ = x − i=0 (2i+1)·(ri +pi ) w−2 i=0 (ri +pi ) q→0

(70)

are associated. Comparison with (62) yields the exponents ατ = −

∞ 

(2i + 1)(pi + ri )

(71)

i=0 ) of the diagonal matrices A(F d (x, w) and ∞  −1 (C) lim m(A) = w2 i=0 (pi +ri ) . τ (w) = lim mτ (w) q→0

q→0

(72)

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307

Exchanging V and V ∗ in the definitions at the beginning of this section, the lattice can be divided into corner transfer matrices A∗ (x, w−1 , w), B ∗ (x, w−1 , w), C ∗ (x, w−1 , w) and D ∗ (x, w−1 , w) built from plaquettes of type B. By means of Eq. (54), both types of corner transfer matrices are related by  ∗     A (x, w−1 , w) τ  ,τ = C(x, w−1 , w) τ  ,τ x ντ w2 i=0 (pi +ri ) ,  ∗     C (x, w−1 , w) τ  ,τ = A(x, w−1 , w) τ  ,τ x ντ w−2 i=0 (pi +ri ) ,  ∗    B (x, w−1 , w) τ  ,τ = D ∗ (x, w−1 , w) τ,τ     (73) = D(x, w−1 , w) τ  ,τ x −ντ w2 i=0 (pi −pi ) , ∞ with ντ = i=0 (pi − ri ). Insertion of (64) and (69) in (73) yields A∗ (x, w−1 , w) =

c(x, w) a(w2 , w)c(w−2 , w)

βτ ¯ T ¯ Q(w)x Q (w),

 −1 C ∗ (x, w−1 , w) = a(x, w) Q¯ T (w) x βτ Q¯ T (w),  T B ∗ (x, w−1 , w) = D ∗ (x, w−1 , w)  T  b(x, w) ¯ (w) −1 (q 2 x)−βτ Q¯ T (w), Ω Q = b(q −2 , w) ¯ where (Q(w)) τ  ,τ = w

∞

  i=0 (pi +ri )

(74)

ˇ (Q(w)) τ  ,τ and the exponents βτ are given by

βτ = ατ + ντ

(75)

as may be read from Eq. (55). Finally, the second specialization of the spectral inhomogeneity characterized by the choice y = w is related to the first one with y = w−1 by the symmetry property s  ,p ∗ ,r  ,t ∗

Ps,p∗ ,r,t ∗





p,s ∗ ,s  ,p ∗

(x, y, w) = q s+t −p−r Pr,t ∗ ,t ∗ ,r ∗ (q −2 x −1 , w, y −1 ).

(76)

The corner transfer matrices involved in the second model follow from those of the first model by ∞   (Bn (x, y, w))τ  ,τ = An (q −2 x −1 , w, y −1 ) τ  ,τ · q i=0 (ri −pi ) , ∞   (Dn (x, y, w))τ  ,τ = Cn (q −2 x −1 , w, y −1 ) τ  ,τ · q i=0 (ri −pi ) . (77) In the remaining sections, the case y = w−1 will be considered.

6. The space of CTM-states For a large class of homogeneous integrable models, the spectra of the corner transfer matrices can be identified with characters of the associated affine algebras (see references [21–23] in [5]). In the infinite lattice limit, the set of all states satisfying an appropriate boundary condition is interpreted as a module of the affine algebra with level and highest (lowest) weights implied by the characteristics of the model at hand. Relying

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on this identification, relevant operators introduced on the lattice model such as the transfer matrix, the translation operator or the quasi particle excitations have been formulated in terms of the corresponding representation theory. The concept of vertex operators [11] has been employed to develop formulae for the eigenvalues of the transfer matrix in the infinite lattice limit as well as for correlators. Bosonization techniques serve as tools for their further evaluation [5]. As a first step towards an analogous interpretation of the inhomogeneous vertex model introduced above, the spectrum of the north-east corner transfer matrix (D ∗ (x, w−1 , w))τ  ,τ given by (74) and (75) may be considered. Restricting to states τ ≡ (. . . , r1 , p1 , r0 , p0 ) with almost all pi = 0 and ri = 0, the exponents in (74) read βτ = 2 i=0 (i + 1)(ri + pi+1 ). To proceed, it is useful to lift part of the degeneracy of the eigenvalues of D ∗ (x, w−1 , w). If the eigenvalue of ∆(∞) (h1 ) on (. . . , 0, 0) is set to −1, then

∞   (∞) (ri − pi ) − 1 (. . . , r1 , p1 , r0 , p0 ) ∆ (h1 )(. . . , r1 , p1 , r0 , p0 ) = 2 i=0

= −(1 + 2ντ )(. . . , r1 , p1 , r0 , p0 ).

(78)

The lattice model may be equipped by an additional statistical weight of the form r(wk ) = uh1 wk with wk = vk or wk = vk∗ on the horizontal links on the boundaries between the corner transfer matrices A∗ (x, w−1 , w) and D ∗ (x, w−1 , w) as well as between B ∗ (x, w−1 , w) and D ∗ (x, w−1 , w). The transfer matrices of the modified vertex model still commute as a consequence of the property uh1 ⊗ uh1 R(z) = R(z)uh1 ⊗ uh1 . Incorporating the upper half of the insertion to the corner transfer matrix D ∗ (x, w−1 , w) and dividing through the scalar factor in the last equation of (74), its eigenvalues read (q 2 x)−βτ u−2ντ −1 .

(79)

βτ vanishes on all states τ = (. . . , 0, 0, p0 ). Given (78), the set of states {(. . . , 0, 0, p0 ), p0
0} may be associated to the Uq (sl(2))-representation with highest weights −1. This observation suggests to consider modules of Uq (
sl(2)) with highest weight λ−1 = (k + 1)Λ0 − Λ1 , where Λ0 and Λ1 are the fundamental weights of
sl(2) and k denotes the level [26]. For any l, k ∈ C, a level-k Verma module Ml,k with highest weight lΛ1 + (k − l)Λ0 is spanned by the action of Uq (
sl(2)) on a vector |l, k satisfying the highest weight property  + Lij n |l, k = 0 ∀n > 0, i, j = 1, 2, (80) and 

 ∓ 21 l L+ |l, k, 11 0 |l, k = q

 + 1 L22 0 |l, k = q ± 2 l |l, k.

Thus Ml,k consists of the vectors    −   −  Li1 j1 −n Li2 j2 −n · · · L− im jm −n |l, k, 1

2

m

(81)

(82)

with m
0, ir , jr = 1, 2 and nr > 0 for 1  r  m. A Z-grading on Ml,k can be introduced based on the grading operator of the algebra. In accordance with (3), the grading is given

R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

309

by the degree m   −      −  · · · L |l, k = − ns . deg L− L i1 j1 −n i2 j2 −n im jm −n 1

m

2

(83)

s=1

In general, Ml,k may have proper submodules generated from vectors other than |l, k possessing the highest weight property (80). Then the quotient of Ml,k by the maximal proper submodule yields an irreducible module Ll,k . If the state (. . . , 0, 0) is associated with the highest weight vector | − 1, k of the module L−1,k , the set {(. . . , 0, 0, p0 ), p0
0} corresponds to the set of vectors with degree zero. Examination of the configurations corresponding to a few low values of βτ taking into account (1), (2) suggests the value of the level. The states with β = 2 are given by the sets {(. . . , 0, 0, 1, p0), p0
0} and {(. . . , 0, 0, 1, 0, p0), p0
0} which may be interpreted as infinite-dimensional representation spaces of Uq (sl(2)) with highest weights 1 and −3. At general level, the vectors in M−1,k with degree 1 span three Uq (sl(2))-representation spaces with highest weights 1, −1 and −3. For k = −2, however, the second representation space is part of a proper submodule of the Verma module. Similarly, the sets of states with βτ = 4 can be related to the vectors in L−1,−2 with degree −2. The latter form infinitedimensional Uq (sl(2))-representation spaces with highest weights 3, 1, −1, −3 and −5. Comparing the character of L−1,−2 to the sum over the eigenvalues (79) of all states satisfying the boundary condition shows that such a correspondence is found at any degree. With (71) and (75) the sum can be written ∞ ∞   (q 2 x)−βτ u−2ντ −1 = (q 2 x)−2 i=0 (ri +pi+1 ) u2 i=0 (ri −pi )−1 τ

τ

=

∞ 1 1 u−1 . 1 − u−2 1 − u2 (q 2 x)−2L 1 − u−2 (q 2 x)−2L

(84)

L=1

The character of the Uq (
sl(2))-module Vl,k is introduced by χV (ρ, σ ) ≡ trVl,k (ρ d σ h1 ). l,k

(85)

For the irreducible module L−1,−2 , the character is obtained as (see Appendix B) ∞ 1 1 σ −1 χL (ρ, σ ) = . −1,−2 1 − σ −2 1 − σ 2 ρ M 1 − σ −2 ρ M

(86)

M=1

For ρ = q −4 x −2 , σ = u this coincides with the rhs of (84). Hence the set of eigenvalues sl(2)){ 12 βτ } coincides with the eigenvalues of the grading operator on the irreducible Uq (
module L−1,−2 . Due to (84)–(86), the eigenvalues of βτ − ντ − 12 coincide with the eigenvalues of the principal gradation operator dp = 2d + 12 h1 (see [5]). According to (64) and (75), ατ = βτ − ντ yields the spectrum of the corner transfer matrix D(x, w−1 , w). Alternatively, the eigenvalue of ∆(∞) (h1 ) on the state (. . . , 0, 0) may be set to zero. Then the sum over the eigenstates of the corner transfer matrix D ∗ (x, w−1 , w) is multiplied with u. Repeating the analysis above relates the set of states {τ } to the vectors of the irreducible modules L0,−2 and L−2,−2 . Indeed, the sum of the eigenvalues of

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D ∗ (x, w−1 , w) may be decomposed in terms of the corresponding characters (see (B.4)): ∞ 1 1 1 1 − u−2 1 − u2 (q 2 x)−2L 1 − u−2 (q 2 x)−2L L=1

=

1 1 − q −2 x −1 +

∞

1

1

L=1

1 − u2 (q 2 x)−2L

1 − u−2 (q 2 x)−2L

∞ 1 − u2 q −2 x −1 1 1 −2 −2 −1 2 2 −2L −2 1−u 1−q x 1 − u (q x) 1 − u (q 2 x)−2L

u−2

L=1

= χL

0,−2

(q

−4 −2

x

, u) + χL

−2,−2

(q −4 x −2 , u).

(87)

The following always adapts the choice (78) for the eigenvalues of ∆(∞) (h1 ). The value k = −2 is referred to as the critical level. In the remainder, a subscript will specify the level sl(2))−2 is distinguished by the occurrence of a large center. In contrast, of the algebra. Uq (
for k = −2, the center is one-dimensional [23]. The center of the affine algebra
sl(2)−2 is generated by the Sugawara operators [27]. q-analogues of the Sugawara operators have been constructed at general level in [22]. In terms of the L-operators introduced in Section 2 they are given by    −1 −k/2 ln z−n = q 3−2i L− z) L+ (q k/2 z) j i . l(z) = (88) ij (q n∈Z

i,j =1,2

As shown in [28], at the critical level l(z) commutes with any element of the algebra: [l(z), a] = 0

∀a ∈ Uq (
sl(2))−2 .

(89)

Eq. (89) is easily verified by means of the RLL-equations (17). l(z) gives rise to singular vectors in the Verma modules Ml,−2 . Due to (89), any vector l−n |l, −2, n
0, satisfies (80). The irreducible module L−1,−2 is obtained as the quotient of the Verma module M−1,−2 by the submodules generated from the highest weight vectors of the form l−n | − 1, −2, n = 1, 2, 3, . . . . Additional singular vectors need to be taken into account for Lm,−2 , m = −1 (see Appendix B). At the critical level, the representation theory of the algebra differs considerably from the one known for the noncritical case. As a consequence of the large center (88), the sl(2))-representations other than the characters given in (86) and (87) can be related to Uq (
highest-weight modules Lm,−2 . Another type of representations of
sl(2)−2 has been termed Wakimoto-modules in [27]. The corresponding q-deformed objects are considered in sl(2))k in [32]. Applying a slight modification context with a free boson realization of Uq (
of the construction presented in [29], the currents of Uq (
sl(2))k may be realized in terms of bosonic oscillators {bn , cn , n ∈ Z, n = 0} and zero modes qb , pb , qc , pc with commutation relations 1 [bn , bm ] = − [n]2 δn+m,0 , n 1 [cn , cm ] = [n]2 δn+m,0 , n

[qb , pb ] = −1, [qc , pc ] = 1,

(90)

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311

and a set {λn , n ∈ Z} satisfying the commutation relations [λn , λm ] =

1 [(k + 2)n][n]2 δn+m,0 . n [2n]

(91)

All other commutators vanish. The remainder of this section deals with the critical level.
At k = −2, a commutative subalgebra Hq (sl(2)) is generated by {λn }. As shown in [32], the generators l(z) = n∈Z ln z−n of the center of the algebra are realized by  −1  −1 l(z) = q −1 Λ− (qz) Λ+ (qz) (92) + qΛ+ (q −1 z) Λ− (q −1 z) , where

 ∞ λ0  ∓n . Λ± (z) = exp ± ± λ±n z 2

(93)

n=1

According to [29,32], the Drinfeld generators h1 , Hn are expressed by h1 = λ0 + 2qb  [2n]  bn + (q − q −1 )−1 q −|n| λn , n = 0. Hn = (94) [n] The currents E(z) = n∈Z En+ z−n , F (z) = n∈Z En− z−n are conveniently written in terms of the generating functions b± (z) = ±(q − q −1 )

∞ 

b±n z∓n ± ln qqb ,

n=1

 1 b(z) = − bn z−n + pb + ln zqb , [n] n=0

 1 cn z−n + pc + ln zqc . c(z) = − [n]

(95)

n=0

Given the commutation relations (90), (91) and (90), the currents E(z) = −:eb+ (z)−(b+c)(qz): + :eb− (z)−(b+c)(q F (z) = Λ+ (qz)Λ+ (q

−1

z):e

− Λ− (qz)Λ− (q

−1

−1 z)

b+ (z)+(b+c)(q −1 z)

z):e

:,

:

b− (z)+(b+c)(qz)

:

(96)

satisfy the relations implied by the defining equations (6), (7) for the Drinfeld basis (see [29], for example). The boson Fock vacuum |0 is characterized by qb |0 = qc |0 = 0 and bn |0 = cn |0 = 0 ∀n > 0.

(97)

The Fock space Fb,c is the collection of all states generated on |0 by b−n , c−n with n > 0. Specifying the action of the generators λn , Wakimoto modules associated to the vacuum representation generated from |0 may be introduced. Since each generator λn commutes with all λm and the generators in (90), a Uq (
sl(2))−2 -representation can be realized on sl(2)) generated by qb , qc , pb , pc and the tensor product of a module of the subalgebra Fq (


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bn , cn with n ∈ Z and a one-dimensional representation space κ{λ¯ n } of Hq (
sl(2)). The latter is defined by λn κ{λ¯ n } = λ¯ n κ{λ¯ n } ,

(98)

with an arbitrary set of complex numbers λ¯ n , n ∈ Z. In the following, a particular vacuum characterized by (97) and (98) will be denoted by |0{λ¯ n } ≡ |0 ⊗ κ{λ¯ n } . Eqs. (97) and (98) imply h1 |0{λ¯ n } = λ¯ 0 |0{λ¯ n } , Hn |0{λ¯ n } = (q − q −1 )q −n

[2n] ¯ λn |0{λ¯ n } [n]

∀n > 0.

(99)

The special choice λ¯ n = −δn,0 yields a realization of the highest weight vector of L−1,−2 . The highest weight module L−1,−2 is isomorphic to the tensor product of κ{−δn,0 } and the restricted Fock space F˜ , where     F˜ = (100) Kerηˆ 0 Kerpb +pc (Ft,t  ) = Kerηˆ 0 (Ft,t ). t,t  ∈Z

t ∈Z

In (100), Ft,t  denotes the subspace of Fb,c with the eigenvalue of qb (qc ) given by t (t  ) and ηˆ 0 = dw η(w) ˆ is the zero mode of the screening operator η(w) ˆ = ec(w) . Furthermore, the action of the grading operator (3) on Fb,c is realized by dA =

 n2  1 (b−n bn − c−n cn ) + qb2 − qc2 . 2 2 [n]

(101)

n=0

Taking into account the first of equations (94), the character (85) of L−1,−2 given in (86) can be expressed by the trace over the restricted Fock space:   (ρ, σ ) = σ −1 χF˜ (ρ, σ ) ≡ trF˜ ρ dA σ 2qb −1 . χL (102) −1,−2

Here the action of dA on the vacuum is fixed by dA |0{λ¯ n } = 0 for any choice of {λ¯ n }. For sets {λ¯ n } with λ¯ n = 0∀n > 0, the vector |0{λ¯ n } satisfies the highest weight property (80). As the analysis outlined in the next section reveals, these sets are not general enough to account for the present problem. Though the property (80) is not valid for general {λ¯ n }, the + + action of (L+ is js )ks · · · (Li2 j2 )k2 (Li1 j1 )k1 on |0{λ¯ n } and descendants constructed from the ˜ latter by (L− ij )−m does only create states contained in the F ⊗ Cκ{λ¯ n } . Hence, a character can be associated to the module by identifying all states which differ only by a complex number. Including a factor σ −1 , this character is given by the trace over the restricted Fock space in (102). Both the restriction to the value λ¯ 0 = −1 and to Eqs. (99) relating the eigenvalues of h1 and Hn to the subset {λ¯ n , n
0} can be released. A subclass L(N) of l,{λ¯ n } Wakimoto modules relevant to the present context is parameterized by complex parameters  ) with zi , zi , 1  i  N . Each module is characterized by an element |ul (z1 , z1 , . . . , zN , zN the properties       ) = l ul (z1 , z1 , . . . , zN , zN ) h1 ul (z1 , z1 , . . . , zN , zN (103)

R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

and

313

  + ul (z1 , z , . . . , zN , z ) Em+N−1 1 N = q −m−N

N 

  +  (j,N)  2    ), q z1 , q 2 z2 , . . . , q 2 zN Ej −1 ul (z1 , z1 . . . , zN , zN

q j Am

j =1   −   Em+N ul (z1 , z1 , . . . , zN , zN ) N    (j,N) −m−N  =q q j Am (z1 , z2 , . . . , zN )Ej− ul (z1 , z1 , . . . , zN , zN ), j =1

(104)

where m = 1, 2, . . . , and A(N−s,N) (z1 , . . . , zN ) = m

s 

(N,N)

as(N) Am+s−t (z1 , . . . , zN )

t =0

with

N 

ar(N) w−r

r=0

 N  2 zr , 1+q = w r=1

A(N,N) (z1 , . . . , zN ) = (−q 2)m m

N N  zm+1 − zm+1 k

1

k=2

z1 − zk

k  =2 k  =k

zk zk − zk 

for N = 2, 3, . . . ,

(1,1) Am (z1 ) = (−q 2 z1 )m .

(105)

 ) is not distinguished from a set obtained from it by exchanging A set (z1 , z1 , . . . , zN , zN   ) may be called a generating element in the pairs zk , zk and zk  , zk  . |ul (z1 , z1 , . . . , zN , zN

the sense that the whole Wakimoto module is contained in the set of descendants    ) (E σs )−ms . . . (E σ2 )−m2 (E σ1 )−m1 ul (z1 , z1 , . . . , zN , zN

(106)

with σi = ±, mi
1 − N if σi = +, mi
−N if σi = − and m1
m2
m2
· · ·
ms . For a given N , the values λ¯ n corresponding to (103)–(105) are λ¯ 0 = −2N − 1, N 2  

1 [n] λ¯ n = −(q − q −1 )(−q 2 )n n [2n]

zin + (zi )n



∀n = 0.

(107)

i=1

The full grading operator d satisfies [d, λn ] = nλn ∀n = 0. On the generating element of a Wakimoto module L(N) , the action of ρ d gives rise to a change in the complex l,{λ¯ n }  parameters zi , zi :       ) = ul (ρ −1 z1 , ρ −1 z1 . . . , ρ −1 zN , ρ −1 zN ). ρ d ul (z1 , z1 , . . . , zN , zN (108) (N)

Examination of the free field realization of Ll,{λ¯ } shows that the action of ρ d on any vector n

(N)  ) with H ∈ U  (
 ≡ Hl,i |ul (z1 , z1 , . . . , zN , zN |Hl,i l,i q sl(2)) can be written as  (N)    1  ). = ρ 2 (h1 −l) ρ dA Hl,i ρ −dA ul (ρ −1 z1 , ρ −1 z1 , . . . , ρ −1 zN , ρ −1 zN ρ d Hl,i

(109)

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 ) is not an eigenvector In contrast to |0{λ¯ n } in the above example, |ul (z1 , z1 , . . . , zN , zN (N)  ) ≡ of dA . However, all |Hl,i  are linear combinations of vectors |ωl,j (z1 , z1 , . . . , zN , zN  )|u (z , z , . . . , z , z ) with the property ωl,j (z1 , z1 , . . . , zN , zN l 1 1 N N     ρ dA ωl,j (z1 , z1 , . . . , zN , zN )ρ −dA ul (z1 , z1 , . . . , zN , zN )  ¯   ) d¯l,j ∈ Z. = ρ dl,j ωl,j (z1 , z1 , . . . , zN , zN (110)

 )} labelled by the index set {j }(N) A minimal collection of vectors {|ωl,j (z1 , z1 , . . . , zN , zN l (N) (N)  )|ω (z , z , . . . , allows to express all vectors |Hl,i  by |Hl,i  = αl;ij (z1 , z1 ; . . . , zN , zN l,j 1 1  ). Replacing the highest weight vector in the free field realizations of all vectors in zN , zN  ) yields a set of vectors isomorphic to the minimal set of Ll,−2 by |ul (z1 , z1 , . . . , zN , zN  ). This correspondence is reflected in the identity |ωl,j (z1 , z1 , . . . , zN , zN  1     d− 12 h1 h1 ωl,j (ρ −1 z1 , ρ −1 z1 , . . . , ρ −1 zN , ρ −1 zN χ˜ L(N) (ρ, σ ) ≡ ρ 2 l )ρ σ   l,{λ¯ n } (N)    {j }l × ωl,j (z1 , z1 , . . . , zN , zN )  ¯ ¯ ρ dl,j σ hl,j = χL (ρ, σ ), = (111) l,−2

(N)

{j }l

 ) = h ¯ l,j |ωl,j (z1 , z , . . . , zN , z ). where h1 |ωl,j (z1 , z1 , . . . , zN , zN 1 N  )} can be written as a tensor The minimal collection of vectors {|ωl,j (z1 , z1 , . . . , zN , zN  )⊗κ (N) (z , . . . , z ) denotes an F (
product F (N) (z1 , . . . , zN q sl(2))-module {λ¯ n } , where F 1 N (N) ¯ and {λn } is given by (107). In general, two Wakimoto modules L and L(M) with l,{λ¯ n }

l,{µ¯ n }

 ) and (y , y  , . . . , y , y  ) of complex parameters are different sets (z1 , z1 , . . . , zN , zN 1 1 M M orthogonal. Exceptional cases are given by M = N , yi = q 4r zi and yi = q 4r zi for   1  i  N, r ∈ Z and by M = N + 2 with yN+2 = yN+1 , yN+2 = yN+1 and yi = zi ,   4r yi = zi for 1  i  N . Thus, for ρ = q , the expression in (111) can be rewritten as a trace. A more detailed account of relations between the Wakimoto modules L(N) will be l,{λ¯ n } given in [19]. According to (84), (86) and (111), the sums of the eigenvalues of the corner transfer matrices D ∗ (x, w−1 , w) and D(x, w−1 , w) in the presence of the statistical weight uh1 co1 1 incide with the characters χL (q −4 x −2 , u) and (qx 2 )−1 χL (q −4 x −2 , (qx 2 u)−1 ), −1,−2 −1,−2 respectively. Due to the large center of the algebra, these characters can be related to a large class of Uq (
sl(2))−2 -modules. The set of eigenvalues suggests a relation between sl(2))-components of these modules. The the CTM-states of the vertex model and the Fq (
correspondence between the space of states and the representation theory of Uq (
sl(2)) at the critical level will be analyzed further in the next section.

7. Vertex operators N -point functions of integrable vertex models can be expressed as traces over suitable products of vertex operators [5,11]. Vertex operators (of type I in the nomenclature of [5])

R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

315

Fig. 8. The vertex operators.

may be interpreted in terms of objects defined on the lattice model. For the inhomogeneous vertex model, half-infinite columns of P -operators are associated to the components of two vertex operators Φ up (x, y, w) and Φ low (x, y, w) as illustrated in Fig. 8: up

up(N)

Φs,t ∗ = lim Φs,t ∗ (x, y, w), N→∞

low(N) low Φs,t (x, y, w), ∗ = lim Φs,t ∗

s, t = 0, 1, 2, . . .,

N→∞

(112)

with the maps Φs,t ∗ (x, y, w) : V ∗ (⊗V ⊗ V ∗ )N+2 → V ∗ (⊗V ⊗ V ∗ )N+2 and up(N)

low(N) (x, y, w) : V (⊗V ∗ ⊗ V )N+2 → V (⊗V ∗ ⊗ V )N+2 defined by the matrix elements Φs,t ∗



up(N)

Φs,t ∗ (x, y, w)

 τ  ,τ

 ∗  ,p ∗ 0,0∗ ,rN+1 ,pN+1 sN ,tN∗ ,rN N (x, y, w) ∗ ∗ (x, y, w)Pr ,p ∗ ,s ∗ N N N−1 ,tN−1 N+1 ,pN+1 ,sN ,tN

= Pr

∗  ∗ sN−1 ,tN−1 ,rN−1 ,pN−1 s0 ,t ∗ ,r0 ,p0∗ (x, y, w) · · · Pr ,p0∗ ,s,t ∗ ∗ ∗ (x, y, w), N−1 ,pN−1 ,sN−2 ,tN−2 0 0

× Pr 

low(N) (x, y, w) Φs,t ∗

 τ  ,τ

s,t ∗ ,p0 ,r0∗ s0 ,t0∗ ,p1 ,r1∗ (x, y, w)P (x, y, w) ∗ p1 ,r1∗ ,s1 ,t1∗ 0 0 0 ,t0

= Pp ,r ∗ ,s

s ,t ∗ ,p ,r ∗

∗ sN ,tN∗ ,pN+1 ,rN+1 ∗ ∗ (x, y, w). N+1 ,rN+1 ,0,0

× Pp1 ,r1∗ ,s2 ,t2∗ (x, y, w) · · · Pp 2

2

2 2

(113)

up(N)

The boundary conditions chosen at the uppermost vertical links of Φs,t ∗ (x, y, w) and the low(N)

lowest vertical links of Φs,t ∗

(x, y, w) imply

Φs,t ∗ (x, y, w)Ω = q t −s ΩΦs,t ∗ (x, y, w), up

up

low s−t low Φs,t ΩΦs,t ∗ (x, y, w)Ω = q ∗ (x, y, w).

(114)

Due to the symmetries (58) and (59), both vertex operators are related by low −1 Φs,t , w−1 ). ∗ (x, y, w) = Φt,s ∗ (x, y up

(115)

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R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

Eqs. (35) and (39) imply  low −1 low −2 q s−t Φs,t )Φt,s x, w, w) = 1, ∗ (x, w, w ∗ (q s,t



low low −2 q s−t Φs,t x, w, w−1 ) = 1. ∗ (x, w, w)Φt,s ∗ (q

(116)

s,t

As a consequence of the Yang–Baxter equation, the vertex operators satisfy the commutation relations low  −1 low Φs,t )Φr,p ∗ (x , w, w ∗ (x, w, w)  s,t ∗ ,r,p ∗ low  −1 = Ps  ,t ∗ ,r  ,p∗ (x  /x, w, w−1 )Φslow ),  ,t ∗ (x, w, w)Φr  ,p ∗ (x , w, w r  ,p  ,s  ,t 

low low  −1 ) Φs,t ∗ (x, w, w)Φr,p ∗ (x , w, w  ∗ ∗ s,t ,r,p −1  = Ps  ,t ∗ ,r  ,p∗ (x  /x, w−1 , w)Φslow )Φrlow  ,t ∗ (x, w, w  ,p ∗ (x , w, w).

(117)

r  ,p  ,s  ,t 

Insertion of (116) in the equations (117) for x  = q −2 x yields low −2 low s−t Φs,t x, w, w−1 )Φr,p , ∗ (q ∗ (x, w, w) = δr,t δp,s q low −2 low −1 x, w, w)Φr,p ) = δr,t δp,s q s−t . Φs,t ∗ (q ∗ (x, w, w

(118)

Generally, the corner transfer matrices of integrable vertex models act as boost operators on the vertex operators (see references in [5]). Specializing to corner transfer matrices with y = w−1 , the argument given in chapter 4 of [5] may be applied straightforwardly to the inhomogeneous model to obtain 

Q−1 (w)

T

y −ατ QT (w)Φs,t ∗ (x, w−1 , w)  T up = Φs,t ∗ (x/y, w−1 , w) Q−1 (w) y −ατ QT (w),  −1 T up Q (w) y −ατ QT (w)Φs,t ∗ (x, w−1 , w−1 )  T up = Φs,t ∗ (x/y, w−1 , w−1 ) Q−1 (w) y −ατ QT (w). up

(119)

low −1 , w) follow from (119) by means of the The corresponding equations for Φs,t ∗ (x, w symmetry (59): low −1 low −1 Q(w)y −ατ Q−1 (w)Φs,t , w) = Φs,t , w)Q(w)y −ατ Q−1 (w), ∗ (x, w ∗ (x/y, w low −1 , w−1 ) Q(w)y −ατ Q−1 (w)Φs,t ∗ (x, w low −1 , w−1 )Q(w)y −ατ Q−1 (w). = Φs,t ∗ (x/y, w

(120)

The probability Ps,t (y, w) that the statistical variables on two neighboring vertical links associated to V and V ∗ take the values s and t ∗ can be expressed in terms of vertex

R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

317

operators and corner transfer matrices by Ps,t ∗ (y, w) =

1  low tr A(x, y, w)B(x, y, w)Φs,t ∗ (x, y, w) Z  up × C(x, y, w)D(x, y, w)Φs,t ∗ (x, y, w)

(121)

with the normalization factor Z given by   low Z= tr A(x, y, w)B(x, y, w)Φs,t ∗ (x, y, w) s,t

 up × C(x, y, w)D(x, y, w)Φs,t ∗ (x, y, w) .

(122)

The remainder of this section deals again with y = w−1 . With (64), (69) and (120), the probability Ps,t ∗ (w−1 , w) may be rewritten as Ps,t ∗ (w−1 , w) =

=

=

1  low tr A(x, w−1 , w)B(x, w−1 , w)Φs,t (x, w−1 , w) Z  up × C(x, w−1 , w)D(x, w−1 , w)Φs,t ∗ (x, w−1 , w) 1 a(w−2 , w) b(x, w) Z a(x, w) b(q −2, w)  low −1 × tr A(x, w−1 , w)B(x, w−1 , w)Φs,t , w) ∗ (x, w  −2ατ q × Q(w) Q−1 (w)C(w−2 , w−1 , w) w  up −2 −1 −1 × D(q , w , w)Φs,t ∗ (x, w , w) 1 a(w−2 , w) b(x, w) Z a(x, w) b(q −2, w)   −2ατ q −1 −1 × tr A(x, w , w)B(x, w , w)Q(w) Q−1 (w) w low 2 −2 × Φs,t x, w−1 , w)C(w−2 , w−1 , w) ∗ (q w  up −2 −1 −1 × D(q , w , w)Φs,t ∗ (x, w , w) .

(123)

Applying the Yang–Baxter equation and the unitarity property (39) to the size-N approxup(N) low(N) imations Φs,t ∗ (x, w−1 , w), Φs,t (x, w−1 , w) and C (N) (w−2 , w−1 , w), D (N) (q −2 , ∗ w−1 , w) of the vertex operators and corner transfer matrices it is shown that low −2 low Φs,t x, w−1 , w)C(w−2 , w−1 , w) = C(w−2 , w−1 , w)Φs,t ∗ (w ∗ (x, w, w).

(124)

With (64) and (69), Eq. (124) leads to Ps,t ∗ (w−1 , w) =

1 b(x, w)2 s−t  −1 q tr (Q (w))T q −4ατ Q(w)T Ω 2 Z b(q −2 , w)2  up up × Φt,s ∗ (q 2 x, w−1 , w−1 )Φs,t ∗ (x, w−1 , w) .

(125)

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R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

Vertex operators Φs ∗ ,t (x, y, w) composed of plaquettes of type B are obtained by exchanging V and V ∗ in the definitions (112) and (113). Specializing to y = w−1 and using (54), they can be expressed in terms of vertex operators of type A by  up    −1 Φs ∗ ,t (x, w−1 , w) τ  ,τ = Φtlow ) τ  ,τ ∗ ,s (x, w, w ∞  up    = x t −s w−s−t + i=0 (pi +ri −pi −ri ) Φs,t ∗ (x, w, w−1 ) τ  ,τ ,  up    −1 Φs ∗ ,t (x, w, w) τ  ,τ = Φtlow , w−1 ) τ  ,τ ∗ ,s (x, w ∞   up   = x t −s ws+t + i=0 (pi +ri −pi −ri ) Φs,t ∗ (x, w−1 , w−1 ) τ  ,τ . (126) Substituting s ↔ t ∗ , t ↔ s ∗ , p ↔ r ∗ etc. in the indices of the vertex operators in Eqs. (114)–(118) yields the analogous properties of the vertex operators of type B. The boost properties (119) and (120) hold with ατ and Q(w) replaced by βτ and (Q¯ T (w))−1 and s → s ∗ , t ∗ → t. Eqs. (58), (59) and (74), (75) combined with (126) lead to the symmetry relations Ps,t ∗ (w−1 , w) = Pt,s ∗ (w, w−1 ) = Pt ∗ ,s (w−1 , w) = Ps ∗ ,t (w, w−1 ). Φ up (x, y, w)

(127)

Φ low (x, y, w)

Hence, the composite vertex operators and with suitable specifications of y and w have inversion and boost properties analogous to those of the vertex operators of the six vertex model. Repeating the steps from Eq. (121) to (125) allows to formulate N -point functions of the model as traces over the composite vertex operators. In the mathematical description of the six-vertex model and its higher spin generalizations developed in [6,12], half-infinite columns of vertices are interpreted as intertwinsl(2))k -modules with tensor products of evaluation modules and ers of highest-weight Uq (
highest-weight modules [11] with k = 1, 2, 3, . . . . The interpretation of the corner transfer matrices as characters indicates the highest-weight modules involved. According to the analysis of the previous section, the analogous object relevant to the inhomogeneous sl(2))−2 -representation and two evaluation modvertex model is a tensor product of a Uq (
ules associated to V and V ∗ , respectively. In view of the properties (116), (117) and (118) satisfied by the lattice objects introduced above, this suggest the existence of intertwiners {µ¯ }V Φ˜ {λ¯ n} n

∗V

(z, z ) : L−1,{λ¯ n } −→ L−1,{µ¯ n } ⊗ Vz∗ ⊗ Vz

(128)

In (128), the spectral parameters z and z are complex variables. The evaluation modules Vz∗ = V ∗ ⊗ C[z, z−1 ] and Vz = V ⊗ C[z, z−1 ] are equipped with the Uq (
sl(2))-structure e0 (up ⊗ zn ) = e0 up ⊗ zn+1 ,

e1 (up ⊗ zn ) = e1 up ⊗ zn ,

f0 (up ⊗ zn ) = f0 up ⊗ zn−1 ,

f1 (up ⊗ zn ) = f1 up ⊗ zn ,

h0 (up ⊗ zn ) = h0 up ⊗ zn , h1 (up ⊗ zn ) = h1 up ⊗ zn , d d = z , with up = vp∗ or up = vp . (129) dz L−1,{λ¯ n } and L−1,{µ¯ n } denote Wakimoto modules given as tensor products of suitable sl(2))-modules and the one-dimensional Hq (
sl(2))-modules κ{λ¯ n } and κ{µ¯ n } . The linear Fq (
sl(2)) and multiplication operators (128) are required to commute with the action of Uq (


R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

319

by z, z . Writing a formal series {µ¯ }V Φ˜ {λ¯ n} n

∗V

∞   {µ¯ }V ∗ V (z, z ) Φ˜ n

(z, z ) =

 {µ¯ }V ∗ V (z, z ) Φ˜ {λ¯ n}

p,r=0

r ∗ ,p

n

{λ¯ n }



=

{µ¯ }V Φ˜ {λ¯ n}

∗V

n

l,m∈Z

r ∗ ,p

⊗ vr∗ ⊗ vp ,

 r ∗ ,p;l,m

z−l (z )−m ,

(130)

 {µ¯ }V ∗ V  with linear maps Φ˜ {λ¯ n} : L−1,{λ¯ n } −→ L−1,{µ¯ n } for an operator (128), the r ∗ ,p;l,m n intertwining property reads ∞ 

{µ¯ }V Φ˜ {λ¯ n}

∗V

r ∗ ,p;l,m

n

p,r=0 l,m∈Z



    av{λ¯ n } ⊗ vr∗ ⊗ z−l ⊗ vp ⊗ (z )−m



∞   {µ¯ }V ∗ V = ∆ (a) Φ˜ {λ¯ n} (2)

p,r l,m∈Z

r ∗ ,p;l,m

n

v{λ¯ n } ⊗



vr∗

⊗z

−l





 −m

⊗ vp ⊗ (z )

a ∈ Uq (
sl(2)),



 , (131)

where v{λ¯ n } ∈ L−1,{λ¯ n } and In order to relate the sets {λ¯ n } and {µ¯ n }, Eq. (131) may be applied inserting the q-Sugawara operators l(w) = n∈Z ln w−n for a. The coproduct of l(w) at general level follows from the expression (88) for l(w) and the formulae (19) for the coproduct of the L-operators:   − k − 3 k  +  k + 1 k −1 2 2 2w L qL− ∆(l(w)) = q 2 2 2w j1 1i q i,j =1,2  − k − 3 k  +  k + 1 k −1  2 2 2w L q 2 2 2w j2 + q −1 L− 2i q

 k 1   k 1 −1 − 2 − 2 k1 ⊗ L− w L+ q 2 − 2 k1 w 1j i1 q  − k − 1 k  +  k − 1 k −1  2 2 1w L q 2 2 1 w 2j . + L− i2 q (132) Due to (8) and (22), the levels of Vz and Vz∗ are zero. Combining the relation between Drinfeld generators and L-operators given in Appendix A with Eqs. (24), (25) yields the action of the L-operators on V and V ∗ . In particular, the action of the second factor of the rhs of (132) takes the simple form  −1 + L− il (w)(L (w))lj vp ∆(2) (a) = (∆ ⊗ id) ◦ ∆(a).

l=1,2



= δi,j q 

−1

exp −(q − q

−1

 2  z n  2 n 1 [n] ) (−q ) vp , n [2n] w n=0

−1 ∗ + L− il (w)(L (w))lj vr

l=1,2

= δi,j q

−1

exp −(q − q

−1

 2  z n  2 n 1 [n] vr∗ . ) (−q ) n [2n] w n=0

(133)

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R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

Use of (133) and (88) in Eq. (132) leads to   ∆(2) (l(w)) v{µ¯ n } ⊗ vp ⊗ vr∗ 

 n  2  z n  z −2 −1 n 1 [n] (−q) + = q exp −(q − q ) n [2n] w w n=0

× l(w)v{µ¯ n } ⊗ vp ⊗ vr∗ .

(134)

Therefore, Eq. (128) needs to be supplemented by the relation  −1    ¯ ¯ q exp λ(q w) + q −1 exp −λ(qw) 

 n  2  z n  z −2 −1 n 1 [n] = q exp −(q − q ) (−q) + n [2n] w w n=0      −1 × q exp µ(q ¯ w) + q −1 exp −µ(qw) ¯ .

(135)

This relation is consistent with the inversion properties (116) and (118). As argued in [19], there exist intertwiners {µ¯ }V Φ˜ {λ¯ n}

∗V

n

(z, z ) : L−1,{λ¯ } −→ L−1,{µ¯ n } ⊗ Vz∗ ⊗ Vz (N)

(N+1)

(136)

n

with µ¯ 0 = λ¯ 0 − 2, µ¯ n = λ¯ n − (q − q −1 )(−q 2)n

 1 [n]2  n z + (z )n n [2n]

∀n = 0.

(137)

In Eq. (136), L(0) referes to the highest weight module L−1,−2 . Identifying the module −1,{λ¯ } n

(−N) n}

Ll,{λ¯

 ) with the Wakimoto associated to the set of complex numbers (z1 , z1 , . . . , zN , zN

 , q 2 z ), negative N can be parametrized by the set (q 2 z1 , q 2 z1 , . . . , q 2 zN module L(N) N l,{λ¯ n } considered as well. Similarly as in the case of the higher studied in [6], inversion spin vertex models  properties can be derived using that s,t q s−t (vt∗ ⊗ (q 2 z)l ) ⊗ (vs ⊗ (q 2 z )l ) ⊗ (vs∗ ⊗  (z )k ) ⊗ (vt ⊗ zk ) is an identity representation of Uq (
sl(2)). Writing the intertwining conditions (131) explicitly for the Chevalley generators (1) it is easily verified that

   {¯ν }V ∗ V  {µ¯ }V ∗ V 2 (138) q r−p Φ˜ {µ¯nn } (z , z) ∗ Φ˜ {λ¯ n} (q z, q 2 z ) ∗ p ,r

p,r

r ,p

n


commutes with any element of Uq (gl(2)). The intertwining property (131) for the grading operator reads

 

{y −n µ¯ }V ∗ V −1 {µ¯ }V ∗ V Φ˜ {y −n λ¯ n} (139) (y z, y −1 z ) ∗ y d = y d Φ˜ {λ¯ n} (z, z ) ∗ . n

r ,p

n

r ,p

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321

This is consistent with Eq. (108). The above considerations and the results of the preceding section suggest the identification

 up {µ¯ }V ∗ V −1 q s−t Φ˜ {λ¯ n} (w−1 x, wx) ∗ ∼ Φtlow , w−1 , w−1 ) = Φs ∗ ,t (x −1 , w, w), ∗ ,s (x n t ,s

 up {µ¯ }V V ∗ −1 q s−t Φ˜ {λ¯ n} (wx, w−1 x) ∗ ∼ Φtlow , w−1 , w) = Φs ∗ ,t (x −1 , w, w−1 ), ∗ ,s (x t ,s

n



¯ −1

y ∼ Q d

(w)

T

¯ y Q(w) , βτ

T

(140)

for a fixed x0 and {λ¯ n } and {µ¯ n } related by (137). Eq. (140) may be modified by a common factor for both spectral parameters on the left-hand sides. However, due to the boost property, traces over products of vertex operators such as (125) depend on the spectral parameters only through their ratios. Hence, varying this factor does not affect the N point functions. According to the results of Section 6, in particular, the remarks following (111), and Eq. (140), the trace in the two-point function (125) can be identified with the trace performed over a Fock space properly restricted with respect to the action of ηˆ 0 and to the eigenvalues of qb and qc . An obvious interpretation of the trace in (125) is provided by the trace over the restricted Fock space F˜ associated to the highest weight module L−1,−2 # F˜ ⊗ κ−δn,0 . In view of the translational invariance of the vertex model, it should also be possible to attribute a Wakimoto module L(N) to the position on the −1,{λ¯ n } lattice where the trace is performed. Then taking the trace amounts to summing over a minimal set of vectors characterized by the property (110). Here, for simplicity, the trace in Pt ∗ ,s (w−1 , w) is associated with the trace over F˜ . Eqs. (114) are satisfied provided that 1 Ω ∼ q − 2 h1 . Then Eqs. (140) allow to express the two-point function Pt ∗ ,s (w−1 , w) by a trace of two intertwiners of the type (136): Pt ∗ ,s (w−1 , w) =

 1 t −s  −4d−h1  ˜ −2 w−1 x −1 , q −2 wx −1 ) ∗ q trF˜ q Φ(q s ,t ZB    −1 ˜ × Φ(wx , w−1 x −1 ) t ∗ ,s .

(141)

So fare, only half-infinite tensor products have been related to modules of Uq (
sl(2)). For the six-vertex model and its higher spin generalizations, tensor products of a Uq (
sl(2))module and a dual module with respect to the antipode S or the anti-automorphism Sˇ defined by (23) are attributed to the complete quantum space. As shown in [19], for the case at hand the appropriate anti-automorphism ς is given by ς (e1 ) = −q 1−h0 e0 , ς (e0 ) = −q

1−h1

e1 ,

ς (f1 ) = −f0 q h0 −1 , ς (f0 ) = −f1 q

h1 −1

ς (h1 ) = −h0 , ς (h0 ) = −h1 .

,

(142)

The space of states is identified with

∗ς  (−N) (N+1) (−N−1) ∗ς ⊗ V ⊗ L # L ⊗ V ⊗ L , L(N) −1 −1 (xw) (xw) −1,{λ¯ } −1,{λ¯ } −1,{λ¯ } −1,{λ¯ } n

n

n

n

(143)

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R.M. Gade / Nuclear Physics B 638 [PM] (2002) 291–327

 ) with z = (q 2 xw)−1 where L−1,{λ¯ } has the generating element |u−1 (z1 , z1 , . . . , zN , zN i n ∗ς  (N)  2 −1
and zi = (q x) w∀i and the Uq (sl(2))-module structure on L−1,{λ¯ } is given by (N)

n

∀a ∈ Uq (
sl(2)),

a u| ˆ u ˇ = u|ς ˆ (a)u ˇ

(N)

uˇ ∈ L−1,{λ¯ } , n

∗ς

(N) uˆ ∈ L−1,{λ¯ } . n

(144)

The isomorphism in (143) ensures the translational invariance of the model. In fact, the class of intertwiners involving V ∗ ⊗ V is not exhausted by the intertwiners in (136). For any Hq (
sl(2))-module κ{¯νm } with ν¯ m = 0∀m > 0, an intertwiner {µ¯ n }{¯νm }V ∗ V (N) (N+1)  ˜ (z, z ) : L−1,{λ¯ } ⊗ κ{¯νm } −→ L−1,{µ¯ n } ⊗ κ{¯νm } ⊗ Vz∗ ⊗ Vz with {λ¯ n } and Φ{λ¯ }{¯ν } n m n {µ¯ n } related by (137) and properties analogous to (138) and (139) is found. Formally, the (N) (N) modules L−1,{λ¯ } in the tensor products (143) may be replaced by L−1,{λ¯ } ⊗ κ{¯νm } with n n ν¯ m = 0, ∀m > 0. The two-point function, however, depends on the Hq (
sl(2))-modules only through the differences (137) in the arguments of the intertwiners and is thus not affected by this change. For the vertex operators of type A, the corresponding relations to the intertwiners (136) read 

{µ¯ }V ∗ V low −1 (w−1 x, wx) ∗ ∼ Φt,s , w, w) q t −s w−s−t x t −s Φ˜ {λ¯ m} ∗ (x m

t ,s

up = Φs,t ∗ (x −1 , w−1 , w−1 ),

 {µ¯ }V ∗ V low −1 (wx, w−1 x) ∗ ∼ Φt,s , w, w−1 ) q t −s ws+t x t −s Φ˜ {λ¯ m} ∗ (x m

t ,s

up = Φs,t ∗ (x −1 , w−1 , w),

 T 1 y d+ 2 h1 ∼ Q−1 (w) y ατ Q(w)T ,

1

Ω ∼ q 2 h1 .

(145)

With Eqs. (140) or (145), general N -point functions may be expressed in terms of traces of 2N intertwiners.

8. Summary and outlook An integrable vertex model involving infinite-dimensional modules of Uq (sl(2)) is investigated in the spirit of the algebraic approach to integrable models with finitedimensional modules developed in [5,12]. Relying on the spectrum of the corner transfer matrices, a relation between lattice objects and objects of the representation theory of the quantum affine algebra Uq (
sl(2)) at the critical level k = −2 is suggested. For the six-vertex model and its higher spin versions at q = 0, the identification of the space of states with U -modules has been established referring to the theory of the crystal bases of quantum groups [33]. An analogous support applying to the present model would be of considerable interest. An immediate generalization of the model consists in replacing the highest weight module V by another infinite-dimensional module of Uq (sl(2)). The method employed here carries over to a vast choice of highest or lowest weight modules in the place

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of V , V ∗ provided that the combination allows for well-defined weights of the CTMstates. While the limit q → 0 is easily accessible for vertex models composed of highest weight Uq (sl(2))-modules only (or only of lowest weight modules), attempts to give an interpretation of the CTM-spectrum in terms of Uq (
sl(2))-modules at any level were not successful so fare. A possible criterion for the applicability of the method may be formulated by means of the automorphism ν of Uq (
sl(2)) exchanging e1 ↔ e0 , f1 ↔ f0 , h1 ↔ h0 . It seems that Uq (
sl(2))-modules can be attributed to the CTM-eigenvalues if the quantum space Q given as an infinite tensor product of Uq (sl(2))-modules satisfies Q # T m ν(Q), where m ∈ Z and T denotes the translation operator. A rearrangement of a finite number of rows and columns of the vertex model may be performed to obtain this property. In context with high-energy QCD, a homogeneous XXX-Heisenberg magnet involving the principal series is considered for the description of the N -gluon compound in the multi-color limit. In [34], this problem is addressed to by means of the Q-Baxter operator formalism. This method can be expected to offer an independent approach also to the present model. Using a bosonization of this algebra as well as its Wakimoto modules and vertex operators, the spectrum of the transfer matrix can be obtained. In principle, the bosonization procedure allows to proceed towards the evaluation of N -point functions and form factors. This method has been applied to homogeneous models based on finitedimensional Uq (sl(2))-modules in [14] and [15]. While rather complicated expressions result in general, important information encoded in various selection rules can be extracted from such an analysis which will be reported on elsewhere.

Appendix A For practical purposes, it often proves useful to express the universal R-matrix as well as the L-operators by means of the Drinfeld generators. An explicit formula for the universal R-matrix of quantum affine algebras has been obtained in [30]. Specializing to ˆ q (
Uq (
sl(2)), the element R(z) ∈ Uq (
sl(2))⊗U sl(2)) ⊗ C[[z]] with the properties (9) is, up to a constant, given by − 2 h1 ⊗h1 ˘ R(z) = R(z)q = R < R (δ) R > q − 2 h1 ⊗h1 1

1

(A.1)

with R< = R> =

→ n=0 ← n=1

R

(δ)

 − n n expq 2 −(q − q −1 )γ −n En+ ⊗ E−n γ z ,  + h1 n  expq 2 −(q − q −1 )q −h1 En− ⊗ E−n q z ,

= exp −(q − q

−1

 ∞  n n −n n γ 2 Hn ⊗ H−n γ 2 z . ) [2n] n=1

(A.2)

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The relation between Drinfeld generators and L-operators L+ (z) = (πW ⊗ id)(σ R(z−1 )) and L− (z) = (πW ⊗ id)R−1 (z) is most conveniently stated making use of the triangular decomposition  ±    k (z) 1 B ± (z) 0 1 0 . L± (z) = (A.3) 0 (k ± (q 2 z))−1 0 1 A± (z) 1 The form of Eq. (A.3) refers to the two-dimensional representation πW defined by (15). Using the explicit formulae (A.2) for R(z) in the definition of the L-operators yields expressions of k ± (z), A± (z) and B ± (z) in terms of Drinfeld generators: 

∞  [n] ∓ n ± ± 12 h1 −1 ∓n k (z) = q , exp ∓(q − q ) γ 2 H±n z [2n] n=1  1 1 + ∓n A± (u) = ∓(q − q −1 ) γ −( 2 ± 2 )n q ∓n E±n z , n= 12 ∓ 12 ∞ 

B ± (z) = ∓(q − q −1 )

− ∓n γ ( 2 ∓ 2 )n q ∓n E±n z . 1

1

(A.4)

n= 12 ± 12 − As apparent from (A.4), the Fourier components (L+ 12 )0 and (L21 )0 vanish.

Appendix B In this appendix, the characters of the irreducible Uq (
sl(2))−2 -modules Ll,−2 with l ∈ Z are evaluated. As pointed out in Section 6, at the critical level the q-Sugawara operator l(z) gives rise to singular vectors l−n1 l−n2 · · · l−nm |l, −2

(B.1) (sing)

in the Verma modules Ml,−2 . It has been shown in [28] that all singular vectors vl,−2 ∈ (sing)

(sing)

Ml,−2 with h1 vl,−2 = lvl,−2 are generated by the action of Fourier coefficients of l(z) on |l, −2. (B.1). Other singular vectors occur for l ∈ Z [26]. From Ml,−2 , the module Mˆ l,−2 is obtained as the quotient of Ml,−2 with the submodule generated by (L+ 21 )0 and − + + − l+1 ((L12 )0 ) for l = 0, 1, 2, . . ., by (L21 )0 for l = −1 and by (L21 )0 and ((L21 )−1 )−l−1 for l = −2, −3, . . .. For Mˆ l,−2 the characters (85) are well defined: ∞ 1 1 1 b l+1 − b−l−1 χM˜ (a, b) = , l,−2 1 − a M 1 − b2 a M 1 − b−2 a M b − b−1

l = 0, 1, 2, . . . ,

M=1

χM˜

−1,−2

χM˜

l,−2

(a, b) =

(a, b) =

∞ 1 1 1 1 , 1 − a M 1 − b2 a M 1 − b−2 a M b − b−1

M=1 ∞ l+1 b (1 − (ab2)−l−1 )

l = −2, −3, . . ..

b − b−1

M=1

1 1 1 , M 2 M 1 − a 1 − b a 1 − b−2 a M (B.2)

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To obtain an irreducible module from M˜ l,−2 , the structure of the submodules generated by (B.1) needs to be investigated. At a given degree, some of the vectors (B.1) may be linearly dependent. Explicit analysis yields

  m−1 (−1)m+1 1 l−n + −1 2(m−1) [n + · · · + n ][n − n1 − · · · − ns ] (q − q ) 1 s nt
1 m>1 s=1  n +···+n =n 1

m

× l−n1 l−n2 . . . l−nm |l, −2 = 0 for l = n − 1 and l = −n − 1,

n = 1, 2, . . . .

(B.3)

Hence, for a fixed l
0 , l−l−1 |l, −2 (or l−l−1 | − l − 2, −2) is vanishing (for l = 0) or is contained in the submodules generated by highest weight vectors of lower degrees (l = 0) and does not give rise to a different submodule. Subtracting the characters of the independent submodules from (B.2) leads to the character of an irreducible module χL : l,−2

χL

l,−2

(a, b) =

∞ 1 1 1 b l+1 − b−l−1 , b − b−1 1 − a l+1 1 − b2 a M 1 − b−2 a M

l = 0, 1, 2, . . .,

M=1

χL χL

−1,−2

l,−2

(a, b) =

(a, b) =

∞ 1 1 1 , −1 2 M b−b 1 − b a 1 − b−2 a M M=1

∞ 1 − (ab2)−l−1 1 1 , b − b−1 1 − a −l−1 1 − b2 a M 1 − b−2 a M

b l+1

M=1

l = −2, −3, . . ..

(B.4)

Upon a simple rescaling, Eq. (B.3) is recognized as a q-analogue of the Benoit–Saint– Aubin formula for the singular vectors at level r in the Verma module Vr,1 of the Virasoro algebra (see, for example, Chapter 8 of [31]). This is in agreement with the results in sl(2))−2 is isomorphic to a q[32]. There it has been shown that the center of Uq (
deformation of the classical Virasoro algebra. In the classical limit of infinite central charge c, only the Verma modules Vs,r with s = 1 have a well defined Virasoro dimension (or, alternatively, only for r = 1). Only one singular vector remains in V1,r at level r for c → ∞.

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