Theta function solutions of the qKZB equation for a face model
aa r X i v : . [ m a t h - ph ] N ov THETA FUNCTION SOLUTIONS OF THE QKZB EQUATION FOR A FACEMODEL
PETER E. FINCH, ROBERT WESTON, AND PAUL ZINN-JUSTIN
Abstract.
We consider the quantum Knizhnik–Zamolodchikov–Bernard equation for a face modelwith elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the so-called combinatorial line, in which the model is equivalent to the three-colour model , these solutionsare shown to be eigenvectors of the transfer matrix with periodic boundary conditions. Introduction
The Knizhnik–Zamolodchikov (KZ) equations [29] are a set of compatible differential equationssatisfied by conformal blocks in Conformal Field Theory. The original work on KZ equationswas concerned with the theory on a sphere, but Bernard generalised it to the torus [10] and tohigher genus Riemann surfaces [11], leading to the so-called Knizhnik–Zamolodchikov–Bernard(KZB) equations. These differential equations come from a flat connection over the moduli spaceof Riemann surfaces with L marked points; in the present work, we shall only be concerned withthe variation of the marked points, and not with the variation of the underlying Riemann surface,whose genus will always be one.The KZ equations are intimately related to the representation theory of affine algebras. In [23], difference equations, now customarily called qKZ equations, were introduced as analogues of theKZ equations for quantised affine algebras. These equations were formulated in terms of the R -matrix of the associated quantum integrable system, which satisfies the Yang–Baxter equation. Fora review of qKZ equations see [16] and references contained there.Generalising both the KZB and qKZ equations, Felder introduced the qKZB equations [17].An important new ingredient was the use of an elliptic solution of the dynamical Yang–Baxterequation. The qKZB equations can be viewed as related to a representation of an (extended) affineWeyl group, which will always be the affine symmetric group for us, on an appropriate functionalspace, dynamical R -matrices being “generalised R -matrices” in the sense of Cherednik [12]. In thelatter work, two types of solutions of the qKZ(B) equation were considered: ordinary solutions,corresponding to looking for eigenvectors of the commutative subgroup of the affine Weyl group,and symmetric solutions, which are invariant under the whole of the affine Weyl group. In thesecond case, we call the set of equations that these satisfy, the qKZ(B) system .In fact, the qKZ system predated the qKZ equation – see [42] where form factors of a quantumintegrable model are shown to satisfy the qKZ system, as well as [25] for a similar connection tocorrelation functions. In the context of dynamical R -matrices associated to so-called ABF models,the qKZB system was considered in [20] using the vertex operator approach [25]. This approachwas developed further for a range of elliptic face and vertex models in [34, 2, 24, 26, 31, 30]. Ourapproach in this paper is different, and the rationale for developing this alternative to the vertexoperator approach is discussed in Section 5.In an unrelated development, Stroganov [43] and Razumov and Stroganov [36, 37] noticed thatthe ground state energy of the (odd size, or twisted even size) XXZ spin chain at the value ∆ = − / Date : August 8, 2018.P. Finch and P. Zinn-Justin are supported by ERC grant 278124 “LIC”. The authors would like to acknowledgeand thank the Galileo Galilei Institute in Florence for their hospitality and support for the period of the scientificprogram on “Statistical Mechanics, Integrability and Combinatorics” during which part of this work was completed. of its anisotropy parameter was particularly simple. The corresponding ground state entries turnedout to have interesting connections to combinatorics (see also [38, 4]), and ∆ = − / spectral parameters ), showing in particular that they formed polynomials inthese spectral parameters. The equations satisfied by the inhomogeneous ground state turned outto be nothing but the qKZ system [15], and using related representation theory, an explicit formof the ground state entries was provided [40, 21] in terms of certain contour integrals. It is worthnoting that the solution of the qKZ system makes sense for any value of the anisotropy parameter∆, but only at ∆ = − / elliptic solution. An obvious attemptis to consider the XYZ and eight-vertex models, and indeed remarkable properties of the groundstate of the XYZ/eight-vertex model in odd size on a particular “combinatorial line” were observedin [7, 8, 9, 39] and analysed using the qKZ(B) approach in [44]. This approach was partiallysuccessful in the sense that some conjectures from [9] were proved, but no explicit formula wasfound for the ground state entries.Here we pursue a slightly different generalisation: we use the solution of the dynamical Yang–Baxter equation which corresponds to SOS models [1, 13]. We write the corresponding qKZB sys-tem, and then construct some solutions. These solutions are shown to be eigenvectors of the transfermatrix (with periodic boundary conditions) of the SOS model on the “combinatorial line”, whichin the SOS language corresponds to the three-colour model [5, 32]. We note that the three-colourmodel was also recently studied in [41], but with different (Domain Wall) Boundary Conditions;experience with the trigonometric limit suggests that there may be some connection between thiswork and ours.In slightly more detail, the results of the paper are as follows. In Section 2, we define thequantum integrable model (SOS model) and the associated qKZB system. We then show that acertain integral expression provides two solutions of it. In Section 3, we analyse in more detailthese solutions, proving that they are theta functions not only of the dynamical parameter, as isexpected, but also of all spectral parameters. We show that they satisfy recurrence relations anda “wheel condition”. In Section 4, we restrict ourselves to the value η = 2 π/ L = 4, while Appendix Bdescribes the trigonometric and rational limit of our results and Appendix C gives a pictorialrepresentation of the qKZB system and equation.2. The Solution of the Difference Equations
In this section we introduce elliptic face weights, define difference equations (qKZB system) andthen go on to construct particular solutions.2.1.
The SOS weights.
We consider SOS weights which depend upon three complex parameters η , τ and ζ (where the elliptic nome p is given by p = e iπτ ), spectral parameter u ∈ C , and a height HETA FUNCTION SOLUTIONS OF THE QKZB EQUATION FOR A FACE MODEL 3 parameter a ∈ Z [6, 13]. Defining f ( u ) = ϑ ( u | τ ) we have W (cid:18) a a ± a ± a ± (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) = f ( η + u ) f ( η − u ) ,W (cid:18) a a ± a ∓ a (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) = f ( u ) f ( aη ± η + ζ ) f ( η − u ) f ( aη + ζ ) , (2.1) W (cid:18) a a ± a ± a (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) = f ( η ) f ( aη ∓ u + ζ ) f ( η − u ) f ( aη + ζ ) , It is conventional to encode SOS weights into a dynamical R-matrix R ( u, a ) ∈ End( C ⊗ C ) [19].Here, we adopt the alternative approach of considering the R-matrix as acting on a path space.More specifically, we consider a periodic SOS model on a chain of L = 2 n sites given in terms ofthe path Hilbert space H L defined as the complex span of the set B L of basis vectors given by B L = {| a , a , . . . a L i | a i ∈ Z , | a i − − a i | = 1 } with indices taken mod( L ). In this case we define R-matrices R i ( u ) ∈ End( H L ) by (cid:10) a ′ (cid:12)(cid:12) R i ( u ) | a i = W (cid:18) a i − a ′ i a i a i +1 (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) Y j = i δ a ′ j a j , where | a i = | a , a , · · · , a L i . (2.2)With this definition the face-version of the Yang-Baxter equation (which (2.1) obey) is writtensimply as the standard R i ( u ) R i +1 ( u + v ) R i ( v ) = R i +1 ( v ) R i ( u + v ) R i +1 ( u ) , (2.3)The SOS transfer matrix t ( u ) ∈ End( H L ) is then defined by (cid:10) a ′ (cid:12)(cid:12) t ( u ) | a i = L Y i =1 W (cid:18) a ′ i − a ′ i a i − a i (cid:12)(cid:12)(cid:12)(cid:12) u − u i (cid:19) . (2.4)2.2. The qKZB System.
In this section we define the qKZB system associated with our ellipticface weights. First of all we define ρ : B L → B L ρ | a , · · · , a L − , a L i = | a L , a , · · · , a L − i . We then have
Definition 2.1 (Level- ℓ qKZB System) . Consider a function Ψ : C L → H L . The level- ℓ qKZBsystem is defined as the set of equations R i ( u i − u i +1 )Ψ( u , u , · · · , u i , u i +1 , · · · , u L ) = Ψ( u , u , · · · , u i +1 , u i , · · · , u L )(2.5) ρ Ψ( u , u , · · · , · · · , u L − , u L ) = κ Ψ( u L + s, u , u , · · · , u L − )(2.6)where s = ( ℓ + 2) η and κ ∈ C × is a constant. This pair of equations implies the following: S i Ψ( u , u , · · · , u i , u i +1 , · · · , u L ) = κ Ψ( u , u , · · · , u i + s, u i +1 , · · · , u L )(2.7)where S i := [ R i − ( u i − u i − + s ) · · · R ( u i − u + s )] ρ [ R L − ( u i − u L ) · · · R i ( u i − u i +1 )] . Note, that we shall refer to (2.7) itself referred as the qKZB equation, as opposed to qKZB system.Pictorial representations of the qKZB system and qKZB equation are given in Appendix C.We define components Ψ a ( u ) ∈ C of a function Ψ : C L → H L byΨ( u ) = X | a i∈B L Ψ a ( u ) | a i , where | a i = | a , a , · · · , a L i . PETER E. FINCH, ROBERT WESTON, AND PAUL ZINN-JUSTIN
The qKZB equation can be written in terms of the coefficients asΨ a ′ ( . . . , u i + s, . . . )(2.8) = κ − X a Ψ a ( . . . ) δ a i − a ′ i i − Y j =1 W (cid:18) a ′ j − a ′ j a j − a j (cid:12)(cid:12)(cid:12)(cid:12) u i − u j + s (cid:19) L Y j = i +1 W (cid:18) a ′ j − a ′ j a j − a j (cid:12)(cid:12)(cid:12)(cid:12) u i − u j (cid:19) . The Solution of the level-1 qKZB System.
The main result of this section is given byTheorem 2.10. We shall present a series of definitions and Lemmas that lead to this result. For agiven | a i ∈ B L let us define a vector α (vector α ) in terms of the n positions where height variablesdecrease (increase) to the left. Namely, we define α = ( α , α , · · · , α n ) , where α i ∈ { , , · · · , L } , α i < α i +1 , and a α i − a α i − = +1 ,α = ( α , α , · · · , α n ) , where α i ∈ { , , · · · , L } , α i < α i +1 , and a α i − a α i − = − . For example, for L = 4 and path a = (1 , , ,
2) we have α = (2 , α = (1 , a = (3 , , , α = (1 , α = (2 , ( j ) ( u ) ( j = 2 ,
3) expressed as integrals. These are the functions that will be shown to satisfythe qKZB system. We show also define two alternative integral expressions Ψ ( j ) ( u ) ( j = 2 , ( j ) ( u ) = Ψ ( j ) ( u ), but the alternative form will prove useful in the proof of thecyclicity relation (2.6).First of all, we define the relevant integrands. Definition 2.2.
With L = 2 n , we define the four functions I (2) , I (3) , I (2) , I (3) : C n × C n → H L bytheir components I ( j ) a ( u , · · · , u L | v , · · · , v n ) := f ( η ) n f ( a L η + ζ ) f ( j ) a L η + ζ − nη + 2 n X l =1 v l − L X m =1 u m ! (2.9) × n Y l =1 f ( a α l η + ζ − v l + u α l ) (cid:16)Q l The functions I ( j ) a ( u | v ), I ( j ) a ( u | v ) are meromorphic functions of each of the arguments( v , v , · · · , v n ) =: v , and are doubly periodic with I ( j ) a ( u | v ) = I ( j ) a ( u | v , v , · · · , v i + ( r + r τ ) π, · · · , v n ) , r , r ∈ Z I ( j ) a ( u | v ) = I ( j ) a ( u | v , v , · · · , v i + ( r + r τ ) π, · · · , v n ) , r , r ∈ Z HETA FUNCTION SOLUTIONS OF THE QKZB EQUATION FOR A FACE MODEL 5 Proof. Meromorphicity comes directly from the properties of the Jacobi theta functions. The doubleperiodicity comes from the relations ϑ ( u + ( r + r τ ) π | τ ) = ( − r + r p − r e − ir u ϑ ( u | τ ) ϑ ( u + ( r + r τ ) π | τ ) = ( − r p − r e − ir u ϑ ( u | τ )(2.11) ϑ ( u + ( r + r τ ) π | τ ) = p − r e − ir u ϑ ( u | τ )with quasi-periodic prefactors cancelling from the top and bottom of (2.9). (cid:3) Now we define functions that will ultimately be shown to satisfy the qKZB system: Definition 2.4. We define the functions Ψ ( j ) , Ψ ( j ) : C L → H L ( j = 2 , 3) in terms of componentsΨ ( j ) a ( u ) , Ψ ( j ) a ( u ) byΨ ( j ) a ( u ) = Φ ( u ) 1 c n Z C (1) α dv πi Z C (2) α dv πi · · · Z C ( n ) α dv n πi I ( j ) a ( v | u )(2.12) Ψ ( j ) a ( u ) = Φ ( u ) 1 c n Z C (1) α dv πi Z C (2) α dv πi · · · Z C ( n ) α dv n πi I ( j ) a ( v | u )(2.13) where Φ( u ) = Y ≤ i We consider the case of I ( j ) a ( v | u ): From Lemma 2.3 we know that Z D τ dv ℓ πi I ( j ) a ( v | u ) = 0where D τ is the boundary of the parallelogram in the complex plane with vertices π − − τ ) , π − τ ) , π − τ ) , π τ )( D τ can be translated if necessary to avoid poles on the boundary). The catalogue of poles occurringinside the domain D τ consists of those at u m ( m = 1 , , · · · , α ℓ ) and u m + η ( m = α ℓ , α ℓ + 1 , · · · , L )or double-periodically translated versions of them. Hence we have Z D τ dv ℓ πi I ( j ) a ( v | u ) = Z C ( ℓ ) dv ℓ πi I ( j ) a ( v | u ) + Z C ( ℓ ) dv ℓ πi I ( j ) a ( v | u ) = 0 . The proof of the statement of involving the integral of I ( j ) a ( v | u ) is similar. (cid:3) For convenience we define τ i,i +1 , which acts on functions of L variables, by τ i,i +1 g ( z , · · · , z i , z i +1 , · · · , z L ) = g ( z , · · · , z i − , z i +1 , z i , z i +2 , · · · , z L )Now we have a lemma regarding the solution of the first equation (2.5) of the qKZB system: PETER E. FINCH, ROBERT WESTON, AND PAUL ZINN-JUSTIN Lemma 2.6. The functions Ψ ( j ) ( u ) and Ψ ( j ) ( u ) ( j = 2 , 3) satisfy the following special cases of theexchange relation (2.5):( i ) τ i,i +1 Ψ ( j ) ...a,a − ,a − ... ( u ) = f ( η + u i − u i +1 ) f ( η − u i + u i +1 ) Ψ ( j ) ...a,a − ,a − ... ( u ) , ( ii ) τ i,i +1 Ψ ( j ) ...a,a +1 ,a +2 ... ( u ) = f ( η + u i − u i +1 ) f ( η − u i + u i +1 ) Ψ ( j ) ...a,a +1 ,a +2 ... ( u ) , ( iii ) Ψ ...a,a − ,a... ( u )= f ( aη + ζ ) f ( η − u i + u i +1 ) τ i,i +1 − f ( η ) f ( aη + ζ − u i + u i +1 ) f ( u i − u i +1 ) f ( aη + η + ζ ) Ψ ...a,a +1 ,a... ( u ) , ( iv ) Ψ ...a,a +1 ,a... ( u )= f ( aη + ζ ) f ( η − u i + u i +1 ) τ i,i +1 − f ( η ) f ( aη + ζ + u i − u i +1 ) f ( u i − u i +1 ) f ( aη − η + ζ ) Ψ ...a,a − ,a... ( u ) , for Ψ = Ψ ( j ) , Ψ ( j ) and j = 2 , Proof. The fact that the first two relations, (i) and (ii), are satisfied follows directly from thefollowing three identities: τ i,i +1 I ( j )( ...a,a − ,a − ... ) ( v | u ) = I ( j )( ...a,a − ,a − ... ) ( v | u ) τ i,i +1 ¯ I ( j )( ...a,a +1 ,a +2 ... ) ( v | u ) = ¯ I ( j )( ...a,a +1 ,a +2 ... ) ( v | u ) ,τ i,i +1 Φ( u ) = f ( η − u i +1 + u i ) f ( η − u i + u i +1 ) Φ( u )Moving on to statements (iii) and (iv) we provide the proof only for Ψ ( j ) ( u ) - the Ψ ( j ) ( u ) case isvery similar. From Eq. (2.9) that we have τ i ( i +1) I ( j ) ...a,a +1 ,a... ( v | u ) = f ( u i − v l ) f ( η ( a + 1) + ζ − v l + u i +1 ) f ( u i +1 − v l ) f ( η ( a + 1) + ζ − v l + u i ) I ( j ) ...a,a +1 ,a... ( v | u ) ,τ i ( i +1) I ( j ) ...a,a − ,a... ( v | u ) = f ( η − v l + u i +1 ) f ( aη + ζ − v l + u i ) f ( η − v l + u i ) f ( aη + ζ − v l + u i +1 ) I ( j ) ...a,a − ,a... ( v | u ) ,I ( j ) ...a,a − ,a... ( v | u ) = f ( η − v l + u i ) f ( aη + ζ − v l + u i +1 ) f ( u i +1 − v l ) f ( η ( a + 1) + ζ − v l + u i ) I ( j ) ...a,a +1 ,a... ( v | u ) , It follows from these relations that we have f ( aη + ζ ) f ( η − u i + u i +1 ) τ i ( i +1) h Φ ( u ) I ( j ) ...a,a +1 ,a... ( v | u ) i (2.14) − f ( u i − u i +1 ) f ( aη + η + ζ ) h Φ ( u I ( j ) ) ...a,a − ,a... ( v | u ) i − f ( η ) f ( aη + ζ − u i + u i +1 ) h Φ ( u ) I ( j ) ...a,a +1 ,a... ( v | u ) i = Ξ (( a + 1) η + ζ − v l + u i +1 , aη + ζ, η − u i +1 + u i , u i − v l ) f ( u i +1 − v l ) f (( a + 1) η + ζ − v l + u i ) h Φ x ( u ) I ( j ) ...a,a +1 ,a... ( v | u ) i f ( aη + ζ ) f ( η − u i + u i +1 ) τ i ( i +1) h Φ ( u ) I ( j ) ...a,a − ,a... ( v | u ) i (2.15) − f ( u i − u i +1 ) f ( aη − η + ζ ) h Φ ( u ) I ( j ) ...a,a +1 ,a... ( v | u ) i − f ( η ) f ( aη + ζ + u i − u i +1 ) h Φ ( u ) I ( j ) ...a,a − ,a... ( v | u ) i = Ξ( aη + ζ − v l + u i , aη + ζ, η − u i +1 + u i , η − v l + u i +1 ) f ( η − v l + u i ) f ( aη + ζ − v l + u i +1 ) h Φ ( u ) I ( j ) ...a,a − ,a... ( v | u ) i HETA FUNCTION SOLUTIONS OF THE QKZB EQUATION FOR A FACE MODEL 7 where the function Ξ( z , z , z , z ) is defined byΞ(2 z , z , z , z ) := f (2 z ) f (2 z ) f (2 z ) f (2 z ) − f ( − z + z + z + z ) f ( z − z + z + z ) f ( z + z − z + z ) f ( z + z + z − z ) − f ( z + z + z + z ) f ( z − z − z + z ) f ( z − z + z − z ) f ( z + z − z − z ) . However, a Riemann identity for ϑ states that Ξ( z , z , z , z ) = 0, and hence (2.14) and (2.15)imply (iii) and (iv) respectively. (cid:3) We note that the only cases of the exchange relation (2.5) not covered by the above lemmaare relations (i) and (ii) with Ψ ( j ) ( u ) and Ψ ( j ) ( u ) interchanged, which will follow once we showthe equality between the two. However, to do this it is useful to have an integrated form of theexpressions (2.12) and (2.13) for Ψ ( j ) ( u ) and Ψ ( j ) ( u ). Such integrated expressions can be expressedin terms of the following sets: Definition 2.7. We define the sets S α = { ( i , . . . i n ) | ≤ i l ≤ α l , i l = i m , ≤ l, m ≤ n } ,S α = { ( i , . . . i n ) | α l ≤ i l ≤ L i l = i m , ≤ l, m ≤ n } , where α can be replaced with α . For example, for a = (3 , , , 4) we have α = (2 , α = (1 , S α = { (1 , , (1 , , (2 , , (2 , } , S α = { (1 , , (2 , , (3 , } . The set S α contains the possible ( i , · · · , i n ) at which the multiple residue at v = ( u i , u i , · · · , u i n )is non-zero. Similarly, S α contains the possible ( i , · · · , i n ) at which the multiple residue at v =( u i + η, u i + η, · · · , u i n + η ) is non-zero. It follows that we have the integrated expressionsΨ ( j ) a ( u ) = [ f ( η )] n f ( a L η + ζ ) X i ∈ S α ψ ( j ) a , i ( u ) , Ψ ( j ) a ( u ) = [ f ( η )] n f ( a L η + ζ ) X i ∈ S α ψ ( j ) a , ˜ i ( u ) , (2.16) ψ ( j ) a , i ( u ) = f ( j ) a L η + ζ − nη + 2 n X l =1 u i l − L X m =1 u m ! ( Q ≤ l Lemma 2.8. We have Ψ ( j ) ( u ) = Ψ ( j ) ( u ) ( j = 2 , Proof. The strategy for proving the equality is to demonstrate equality for a particular choice of a ,and then to use the exchange relations following from Lemma 2.6 to show that the relation followsfor all a . Note that for the choice a = ( a + 1 , a + 2 , · · · , a + n, a + n − , · · · , a + 1 , a ) we have α = (1 , , · · · , n ) S α = { (1 , , · · · , n ) } α = ( n + 1 , n + 2 , · · · , n ) , S α = { ( n + 1 , n + 2 , · · · , n ) } . PETER E. FINCH, ROBERT WESTON, AND PAUL ZINN-JUSTIN That is, both sets contain a single element. Using (2.16) we haveΨ ( j ) a +1 ,...,a + n,...,a +1 ,a ( u ) = Ψ ( j ) a +1 ,...,a + n,...,a +1 ,a ( u )(2.17) = f ( j ) aη + ζ − nη + n X m =1 u m − n X m = n +1 u m ! n Y l =0 f ( aη + lη + ζ ) × Y ≤ i The functions Ψ ( j ) ( u ) satisfy the conditionΨ ( j ) a L a ...a L − ( u , u , . . . , u L ) = − Ψ ( j ) a ...a L − a L ( u , . . . , u L , u − η )(2.18) Proof. We need to split our calculation into two cases depending upon the value of ( a L − a L − ).We first consider the possibility a L − a L − = 1. On the left hand side of (2.18) we have α = 1 andusing the integral expression (2.12), the contour C (1) circles just one pole at v = u . Thus we cancomputeΨ ( j ) a L a ...a L − ( u , u , . . . , u L )= [ f ( η )] n − f (( a L − η + ζ ) f ( a L η + ζ ) Y ≤ i HETA FUNCTION SOLUTIONS OF THE QKZB EQUATION FOR A FACE MODEL 9 In order to treat the case a L − a L − = − ( j ) ( u ) = Ψ ( j ) ( u )given by Lemma 2.8. (cid:3) Lemmas 2.6, 2.8 and 2.9 lead to the main theorem: Theorem 2.10. The functions Ψ (2) ( u ) and Ψ (3) ( u ) are solutions of the qKZB system (2.5) and(2.6) with level ℓ = 1 and κ = − Proof. As Ψ ( j ) ( u ) = Ψ ( j ) ( u ) (Lemma 2.8) we find that relations proven in Lemma 2.6, i.e. (i)-(iv),cover all cases of the exchange relations. Hence Ψ (2) ( u ) and Ψ (3) ( u ) satisfy Eq. (2.5). On the otherhand Lemma 2.9 shows that Ψ (2) ( u ) and Ψ (3) ( u ) satisfy Eq. (2.6) with level ℓ = 1 and κ = − (cid:3) Finally, by way of example, the explicit form of the integrated form of the solution Ψ (3) ( u ) isgiven in Appendix A. 3. Analytic properties of the solutions The properties of Ψ ( j ) as functions of u , . . . , u L and of ζ are described in this section. In fact, in this section only , it will be convenient to write u := ζ to treat all variables simultaneously.3.1. Pseudo-periodicity. We first have the following general result: Lemma 3.1. Given a vector-valued function Ψ = P a Ψ a | a i which is analytic in the variables( u , u , . . . , u L ), and an element ( λ = π ( m + n τ ) , . . . , λ L = π ( m L + n L τ )) ∈ Λ, Ψ is a solutionof the q KZ equation (2.7) at level 1 ( s = 3 η ) iff Ψ ′ is, where Ψ ′ = P a Ψ ′ a | a i is defined by any ofthe following:(1) Ψ ′ a ( u , . . . , u L ) = Ψ( u , . . . , u m + π, . . . , u L ) for some m = 0 , , . . . , L .(2) Ψ ′ a ( u , . . . , u L ) = Ψ a ( u + πτ, . . . , u L ) e i ( a L η + ζ )+ P Lj =1 ( a j η + ζ )+ P Lj =1 ( a j − a j − ) u j ).(3) Ψ ′ a ( u , . . . , u L ) = Ψ a ( u , . . . , u m + πτ, . . . , u L ) e i ( ( a m − η + ζ )( a m − a m − )+( L − u m − P i = m u i ) forsome m = 1 , . . . , L .Furthermore, Ψ and Ψ ′ satisfy (2.7) with the same value of κ . Proof. The statement (1) is trivial because the weights W of (2.1) are invariant under u u + π and ζ ζ + π .We consider the next statement (2), i.e., ζ ζ + πτ . We have, writing explicitly the dependenceon ζ , the following “gauge” transformation W (cid:18) a bc d (cid:12)(cid:12)(cid:12)(cid:12) u − u ′ ; ζ + πτ (cid:19) = W (cid:18) a bc d (cid:12)(cid:12)(cid:12)(cid:12) u − u ′ ; ζ (cid:19) e i ( c − b ) h ( a, c | u ) h ( c, d | u ′ ) h ( b, d | u ) h ( a, b | u ′ )with h ( a, c | u ) = e i ( c − a ) u .After some compensations in the q KZ equation (2.8), it is not hard to see that P a | a i Ψ a ( u + πτ, . . . ) e i ( a L η + ζ )+ i P Lj =1 ( a j η + ζ ) Q Lj =1 h ( a j − , a j | u j ) satisfies the same q KZ equation at level 1 as Ψ.This coincides once again with the expression in the proposition.Finally, for statement (3), i.e., u m u m + πτ with m = 1 , . . . , L , we use the gauge transformation W (cid:18) a bc d (cid:12)(cid:12)(cid:12)(cid:12) u + πτ (cid:19) = e − iη W (cid:18) a bc d (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) g ( a, c ) g ( b, d ) = e − iη W (cid:18) a bc d (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) g ( a, b ) g ( c, d )with g ( a, c ) = e i ( aη + ζ )( c − a ) . We then consider Eq. (2.8) and apply the first form of the gaugetransformation for i = m , and its second form for i = m . In both cases we find that the vectorwith entries g ( a m − , a m )Ψ a ( . . . , u m + πτ, . . . ) satisfies Eq. (2.8) with same ℓ but modified constant κ ′ = κe − iη in the first case, κ ′ = κe i ( L − η in the second. This constant mismatch can be fixedfor ℓ = 1 by considering g ( a m − , a m ) e i (( L − u m − P i = m u i ) Ψ a ( . . . , u m + πτ, . . . ) which coincides withthe expression in the proposition. (cid:3) Considering that we have two independent solutions, it is natural to suspect that they are relatedto each other by such translations. Indeed, define the lattice Λ inside C L +1 ∋ ( u , u , . . . , u L ):Λ := ( π Z + πτ Z ) L +1 and its sublattice of index 2:Λ e = ( ( π ( m + n τ ) , . . . , π ( m L + n L τ )) ∈ Λ : L X i =0 n i = 0 (mod 2) ) By direct computation from the integral expressions 2.4, one obtains: Proposition 3.2. Ψ ( k ) a has pseudo-periodicity lattice Λ e , k = 2 , 3; Ψ (2) a and Ψ (3) a are exchanged byodd elements of Λ, namelyΨ ( k ) a ( . . . , u i + π, . . . ) = ( − k Ψ ( k ) a ( . . . )Ψ ( k ) a ( u + πτ, . . . ) = c e − i ( a L η + ζ )+ P Lj =1 ( a j η + ζ )+ P Lj =1 ( a j − a j − ) u j )Ψ ( σ ( k )) a ( . . . )Ψ ( k ) a ( . . . , u m + πτ, . . . ) = c m e − i ( ( a m − η + ζ )( a m − a m − )+( L − u m − P i = m u i )Ψ ( σ ( k )) a ( . . . ) m = 1 , . . . , L where σ ( k ) = 5 − k , and the c m are certain constants (which can be calculated explicitly but willnot be needed).3.2. Holomorphicity. Another key property of Ψ ( j ) ( u ) is given by the following: Proposition 3.3. The functions Ψ ( j ) ( u ) ( j = 2 , 3) are holomorphic in all the arguments u m ( m = 0 , , . . . , L ). Proof. From its definition 2.4, Ψ ( j ) , as the integral of an expression that is holomorphic in ζ , isholomorphic in ζ = u .We consider next m = 1 , . . . , L . The first step in the proof is to use Prop 3.2, so that toprove holomorphicity, we need to demonstrate the absence of poles only within the fundamentaldomain D τ of each of the arguments u m . Potential poles come from points where the arguments(or periodically shifted version of them) of the denominator of ψ ( j ) a , i ( u ) vanish within this domain.These possibilities occur at u m = u i l , ≤ m ≤ α ℓ , m = i ℓ ; u m = u i l + η, α ℓ ≤ m ≤ L. We see immediately, due to the zero coming from the numerator term P ≤ i Thus all the non-zero residues of ψ (0) a , i ( u ) appearing Ψ a ( u ) are cancelled out which proves theproposition. To clarify the proof, consider the example considered above with L = 4, a = (3 , , , α = (2 , S a = { (1 , , (1 , , (2 , , (2 , } . We find for exampleRes u = u ψ ( j ) a , (1 , ( u ) = − Res u = u ψ ( j ) a , (2 , ( u )Res u = u ψ ( j ) a , (1 , ( u ) = − Res u = u ψ ( j ) a , (1 , ( u ) . (cid:3) Theta functions. Define the matrix α ij depending on a sequence a = ( a , . . . , a L ): α ij ( a ) = L + 3 i = j = 0 a j − a j − i = 0 , j = 1 , . . . , La i − a i − j = 0 , i = 1 , . . . , LL − i = j = 1 , . . . , L − i = j, i, j = 1 , . . . , L Combining Props. 3.2 and 3.3, we conclude the following: Corollary 3.4. Ψ ( k ) a ( k = 2 , 3) is a theta function in the variables u = ζ and u , . . . , u L , withprescribed lattice of pseudo-periods Λ e and degree matrix given by α ij ( a ), i.e.,Ψ ( k ) a ( u + λ ) = c ( k ) λ , a e − i P Li,j =0 n i α ij ( a ) u j Ψ ( k ) a ( u ) , λ = ( π ( m + n τ ) , . . . , π ( m L + n L τ )) ∈ Λ e where the c ( k ) λ , a are some constants.In general, one expects solutions of the elliptic q KZ equation to have complicated analytic be-haviour (see the discussion in section 5). Here, the particular solutions which we exhibit have thesimplest possibly behaviour: they are holomorphic in all variables. Theta functions are the naturalanalogue of polynomial functions in the elliptic world, and in fact these solutions degenerate topolynomial solutions in the trigonometric or rational limit, see App. B.Note that the dimension of the space of such theta functions is 2 det( α ij ( a )) i,j =0 ,...,L − (the fac-tor of 2 being the index of Λ e inside Λ), which is given by d n = ( n + 1) n n − = 2 , , , . . . for L = 2 n = 2 , , , , . . . Recurrence relations. We now want to consider the relationship between solutions to theqKZB system of different sizes. We denote Ψ ( j ) ( u ) for system size L by Ψ ( L,j ) ( u ). This notationis also used for other functions. In the following lemma we show that one can extract the stateΨ ( L,j ) ( u ) from the state Ψ ( L +2 ,j ) ( u ). Proposition 3.5. For | a i ∈ B L and j = 2 , i ) Ψ ( L,j ) a ( u ) (cid:12)(cid:12)(cid:12) u L − = u L + η = δ a L − a L ( a L − a L − )( − n f ( a L − η + ζ ) L − Y k =1 f (2 η − u k + u L )Ψ ( L − ,j ) a ,...,a L − ( u , . . . , u L − ) , ( ii ) Ψ ( L,j ) a ( u )= ( − n f ( η ) n Ψ ( L +2) a ...,a L ,a L +1 ,a L ( u , . . . , u L , u, u + η ) − Ψ ( L +2) a ...,a L ,a L − ,a L ( u , . . . , u L , u, u + η ) o f (2 η ) f ( a L η + ζ ) Q Lk =1 f (2 η − u k + u ) Proof. To prove the first relation we need to split the proof into a L − = a L + 1 and a L − = a L − α n = L and i n = L for any i ∈ S ¯ α . We firstobserve that in such a case¯ ψ ( j ) a , i ( u )= f ( a L η + η + ζ ) f ( η ) Y ≤ l Wheel condition.Proposition 3.6. For Ψ( u ) = Ψ (2) ( u ) , Ψ (3) ( u ), if u j = u i − η and u k = u j − η for some 1 ≤ i We start by seeing this lemma only applies with L ≥ 4. We first consider the case u L = u L − − η = u L − − η . Using the recursion relations we findΨ ( L ) a ( u ) = δ a L − a L ( a L − a L − )( − n f ( a L − η + ζ ) L − Y k =1 f (2 η − u k + u L )Ψ ( L − a ,...,a L − ( u , . . . , u L − )= 0 , due to the appearance of the term f (2 η − u L − + u L ). Application of the exchange relation, Equation(2.5), proves the lemma. The reader should keep in mind that R i ( η ) is not defined and this leadsto the restriction on the ordering of i, j, k . (cid:3) It would be interesting to study the space of theta functions with the same lattice of pseudo-periods Λ e and same degree as Ψ ( k ) (cf Cor. 3.4) which satisfy the wheel condition (3.1). Thecorresponding problem in the trigonometric case was solved by Kasatani in [27], leading to a aconnection to nonsymmetric Macdonald polynomials [28].4. The Combinatorial Line The combinatorial line in the elliptic 8-vertex model occurs when η = π . In this section wewill see that when η = π , which will be assumed for the entirety of the section, that the solutionsof the qKZB system as defined in the previous section are eigenstates of the face transfer matrix.Depending on the value of the parameter ζ we find that the face model can be associated witheither the 2-height RSOS model or the three colour problem.4.1. An Eigenstate of the Transfer Matrix. On the combinatorial line we find that S i = t ( u i ) ,S i Ψ ( j ) ( u ) = − Ψ ( j ) ( u ) , for j = 2 , 3. This is the first hint that Ψ (2) ( u ) , Ψ (3) ( u ) are eigenstates of the transfer matrix. Toprove this indeed the case we will use the recursion relations determine in the previous section,however, we first require a useful identity.Consider a periodic chain with L + 2 sites with the operator R L ( u L +2 − u L ) · · · R ( u L +2 − u ) ρ acting on either solution to qKZB system. The action of this operator can be computed either viathe exchange relation, Eq. (2.5), or the definition of the R -matrix, Eq. (2.2). Equating the twodifferent ways as well as setting u L +1 = u + η and u L +2 = u leads to the identityΨ ( L +2 ,j ) a ′ ( u , . . . , u L , u, u + η )= − X | a i∈B L +2 δ a ′ L +1 a L δ a ′ L +2 a L +1 W (cid:18) a ′ L − a ′ L a L − a L (cid:12)(cid:12)(cid:12)(cid:12) u − u L (cid:19) · · · W (cid:18) a ′ a ′ a a (cid:12)(cid:12)(cid:12)(cid:12) u − u (cid:19) × W (cid:18) a L +1 a ′ a L +2 a (cid:12)(cid:12)(cid:12)(cid:12) u − u (cid:19) Ψ ( L +2 ,j ) a ( u , . . . , u L , u + η, u ) , for | a ′ i ∈ B L +2 and j = 2 , Theorem 4.1. On the combinatorial line t ( u )Ψ ( j ) ( u ) = − Ψ ( j ) ( u ) for j = 2 , Proof. The approach take is to show that Ψ ( L,j ) ( u ) being an eigenstate of t ( u ) follows fromΨ ( L +2 ,j ) ( u ) satisfying the qKZB system relations by utilising the recursion relations to relate thedifferent states. t ( u )Ψ ( L,j ) ( u )= X a , a ′ ∈B L W (cid:18) a ′ L − a ′ L a L − a L (cid:12)(cid:12)(cid:12)(cid:12) u − u L (cid:19) · · · W (cid:18) a ′ a ′ a a (cid:12)(cid:12)(cid:12)(cid:12) u − u (cid:19) W (cid:18) a ′ L a ′ a L a (cid:12)(cid:12)(cid:12)(cid:12) u − u (cid:19) Ψ ( L,j ) a ( u ) (cid:12)(cid:12) a ′ (cid:11) = X a , a ′ ∈B L +2 ( − n +1 ( a ′ L +1 − a ′ L ) f ( a ′ L η + ζ ) Q Lj =1 f ( η − u + u j ) δ a ′ L +2 a ′ L δ a ′ L +1 a L δ a ′ L +2 a L +1 W (cid:18) a L +1 a ′ a L +2 a (cid:12)(cid:12)(cid:12)(cid:12) u − u (cid:19) × W (cid:18) a ′ L − a ′ L a L − a L (cid:12)(cid:12)(cid:12)(cid:12) u − u L (cid:19) · · · W (cid:18) a ′ a ′ a a (cid:12)(cid:12)(cid:12)(cid:12) u − u (cid:19) Ψ ( L +2 ,j ) a ( u , . . . , u L , u + η, u ) (cid:12)(cid:12) a ′ · · · a ′ L (cid:11) = − X a ′ ∈B L +2 ( − n +1 ( a ′ L +1 − a ′ L ) f ( a ′ L η + ζ ) Q Lj =1 f ( η − u + u j ) δ a ′ L +2 a ′ L Ψ ( L +2 ,j ) a ′ ( u , . . . , u L , u, u + η ) (cid:12)(cid:12) a ′ · · · a ′ L (cid:11) = − Ψ ( L,j ) ( u )for j = 2 , (cid:3) On the combinatorial line we also find that due to the periodicity of f ( u ), the face weights areunchanged when all heights are shifted by three. This leads to a symmetry of the transfer matrix: t ( u ) T = T t ( u ) where (cid:10) a ′ (cid:12)(cid:12) T | a i = L Y i =1 δ a ′ i a i +3 , for | a i , | a ′ i ∈ B L . The operator T is also a symmetry of the R -matrix. From Defs. 2.2 and 2.4,Ψ (2) ( u ) , Ψ (3) ( u ) are eigenstates with eigenvalue one: T Ψ ( j ) ( u ) = Ψ ( j ) ( u ) , j = 2 , . These properties allow for the construction of related integrable models with finite-dimensionalHilbert spaces. The models depend upon the parameter ζ and for each the state Ψ( u ) maps to astate which satisfies the qKZB equations and is an eigenstate of the transfer matrix.4.2. The Three Colour Problem. In this section we define the three colour problem for ζ / ∈ π Z and show that Ψ (2) ( u ) , Ψ (3) ( u ) can be mapped onto solutions to the qKZB system of the newmodel.The Hilbert space of periodic three colour problem on a chain of L sites, H CL , is defined as thecomplex span of the following set of basis vectors B CL = {| ¯ a , ¯ a , . . . , ¯ a L i | ¯ a i ∈ Z / Z , ¯ a i − − ¯ a i = ± } . The Hilbert space has dimension 2 L + 2( − L , with odd length chains allowed. The face weightsof the three colour problem are defined as W C (cid:18) ¯ a ¯ b ¯ c ¯ d (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) = W (cid:18) a bc d (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) where ¯ x = x mod 3 , and a, b, c, d ∈ Z , | a − b | = | a − c | = | b − d | = | c − d | = 1. All other face weights are defined to bezero. Using these face weights one can define R -matrices, R Ci ( u ) and a transfer matrix, t C ( u ),using Equations (2.2) and (2.4), respectively, with weights W replaced by W C .Each sequence ¯ a : Z /L Z → Z / Z in B CL can be lifted to a sequence a : Z /L Z → Z with a i +1 − a i = ± 1; the latter is defined uniquely up to overall shift by 3. This allows to divide B CL HETA FUNCTION SOLUTIONS OF THE QKZB EQUATION FOR A FACE MODEL 15 into sectors: B CL = G | k |≤ Lk = L (mod 2) k =0 (mod 3) B CL ; k , B CL ; k = {| ¯ a , ¯ a , . . . , ¯ a L i : a i + L = a i + k } The corresponding decomposition of H CL = L k H CL ; k is preserved by the dynamics. In particular,when L is even, H CL ;0 naturally embeds itself inside H L : φ | ¯ a , . . . , ¯ a L i = + ∞ X i = −∞ | a + 3 i, a + 3 i, . . . , a L + 3 i i φ induces an isomorphism from H CL ;0 to its image, which is precisely { v ∈ H L | T v = v } . In whatfollows, we restrict R Ci ( u ) and t C ( u ) to the sector H CL ;0 .The map φ has the following properties: φR Ci ( u ) = R i ( u ) φφt C ( u ) = t ( u ) φ As Ψ (2) ( u ) , Ψ (3) ( u ) are invariant under the action of T it follows that the states φ − Ψ (2) ( u ) , φ − Ψ (3) ( u )must be solutions of the qKZB system and eigenstates of the transfer matrix, t C ( u ).In other models (see [14, 40, 22]), the solution to the qKZ(B) system at the special point wherethe shift s vanishes, is not only an eigenstate but also the ground state of the model. This, however,is not the case for the three colour problem. If we instead interpret the model as a quantum chainwith Hamiltonian H = ddu ln( t C ( u )) (cid:12)(cid:12)(cid:12)(cid:12) u =0 . (4.1)For small system sizes ( L = 4 , 6) we find that in the homogeneous case, u = · · · = u L = 0, φ − Ψ (2) ( u ) , φ − Ψ (3) ( u ) are the first excited states. We conjecture this to be true for larger systemsizes. If we take the trigonometric limit, τ → 0, then we find that φ − Ψ (2) ( u ) , φ − Ψ (3) ( u ) corre-spond to ground states of twisted sectors of the model (see Appendix B). We suspect that thereare analogous sectors in the elliptic case and determining these would provide a stepping stoneto finding suitable boundary conditions for obtaining solutions to the qKZB system in the vertexlanguage.4.3. The 2-Height RSOS model. In the following we show that when ζ ∈ π Z the model repro-duces the 2-height RSOS model. Furthermore, the solutions to qKZB system of the SOS modelare mapped to solutions to the qKZB system of the RSOS model and are both eigenstates of thetransfer matrix. We show this for ζ = 0, assumed for all the computations below, and discuss thegeneralisation.We consider the periodic RSOS model on a chain of L = 2 n sites with the Hilbert space H RSOSL defined as the complex span of the following set of basis vectors B RSOSL = {| a , a , . . . a L i | a i ∈ { , } , | a i − − a i | = 1 } . It is clear that the Hilbert space has dimension 2 regardless of chain length and is a subspace ofboth H L and H CL . Despite R i ( u ) and t ( u ) no longer being defined, due to the existence of poles,on all of H L both are defined H RSOSL . Furthermore, we find that H RSOSL is invariant under theaction of R i ( u ) and t ( u ). This implies one can construct an integrable model associated to thespace H RSOSL , this model is precisely the standard 2-height elliptic RSOS model [1].Due the homomorphicity of the two solutions to the qKZB equation both are still valid. However,one needs to consider the effect of setting ζ = 0 on coefficients Ψ a ( u ). We first compute that if i = 1 and α = 1 then ψ ( j ) a , i ( u ) = f ( j ) a L η − nη + u + 2 n X l =2 u i l − L X m =2 u m ! f ( a η ) n Y l =2 f ( a α l η − u i l + u α l ) × ( Q To summarise: in this paper, we have constructed solutions to the level-1 qKZB equation (2.7) asthe integral expressions (2.12) (see Theorem 2.10). These solutions are pseudo-periodic (Proposition3.2), holomorphic (Proposition 3.3), and therefore theta functions (Corollary 3.4) in both dynamicaland spectral parameters; they satisfy recurrence relations (Propositions 3.5) and the wheel condition(Proposition 3.6). Along the combinatorial line, they correspond to eigenvectors of the SOS periodictransfer matrix with an even number of sites. We have discussed the connection with eigenvectorsof the (R)SOS three-colour problem and 2-Height model when the free parameter is = π Z and= π Z respectively. Based on the evidence of size 4 and 6 chains we conjecture that the our qKZBsolutions correspond to the first excited states of the three-colour problem in the homogeneous u = u = · · · = u L case.Let us now discuss some connections with existing work. Firstly, solutions of qKZB equationsare constructed in [18]. Our solutions appear to differ from these solutions in several respects: theintegrands of our solutions (2.12) are theta functions, that is infinite products, whereas the inte-grands of those in [18] are double infinite products. Also, the solutions of [18] are “hypergeometricintegrals” in the sense that they generalise the generic hypergeometric solution of the KZ equation.In comparison, our solutions reduce to polynomials in the trigonometric limit, and a fortiori , in theKZ limit.A second way of constructing solutions of the qKZB equations is provided by the vertex operatorapproach to SOS models described in [20]. In this approach, qKZB solutions are given in termsof the trace of products of vertex operators, which can be written as multiple integral expressionsusing the free-field realisation of [33]. However, the required trace of vertex operators ceases tobe well-defined in the ‘critical-level’ limit ℓ = → − η = 2 π/ ℓ = 1. Our approach has been toconstruct new solutions, specifically defined at ℓ = 1 for generic µ and for which the combinatorial-line η = 2 π/ HETA FUNCTION SOLUTIONS OF THE QKZB EQUATION FOR A FACE MODEL 17 ℓ = 1 solutions correspond to eigenvectors of the transfer matrix; this fact is ultimately due to thesimple observation that along this line the shift parameter in the qKZB equation is s = 3 η = 2 π andhence that the R ( u + s ) which appears in the qKZB equation (2.7) is equal to R ( u ). This strategy issimilar to that used for the trigonometric case in [14], with one key difference. In [14], the level ℓ = 1qKZ solutions were themselves constructed by computing matrix elements (as opposed to traces) ofvertex operators between level-1 highest weight states [25]. In the trigonometric case, such matrixelements obey level-1 qKZ equations as was shown in [23]. In the elliptic case, analogous level-1one matrix elements can also be computed but do not obey level-1 qKZB equations. It is for thisreason that we were led to seek the independent construction of level-1 solutions presented in thispaper.Finally, let us consider the connection with the earlier work [44] on the 8-vertex on the com-binatorial line. There are some technical differences with [44]: our system has even size, whichis necessary in a height model with periodic boundary conditions where heights on neighbouringsites differ by ± 1. This prevents a direct identification of the present model with the eight-vertexmodel of [44] via the vertex-IRF transformation [6]. Furthermore, in the current work, we areable to provide explicit formulae for the entries of the solution of the qKZB system, leaving theinhomogeneities free – in [44], they were “half-specialised” to pairs of opposite values, resulting inexpressions involving only even theta functions and therefore a rational parameterisation; here nosuch parameterisation is available. Appendix A. The L = 4 Case We describe here our solutions of the q KZB system in size L = 4. We only consider here onesolution, Ψ = Ψ (3) , noting that the other one can be recovered by using Prop. 3.2. Once one heightis fixed, say a L = a , Ψ has (cid:0) (cid:1) = 6 components.We start with the trivial component (Eq. (2.17))Ψ a +1 ,a +2 ,a +1 ,a = ϑ ( aη + ζ ) ϑ (( a + 1) η + ζ ) ϑ (( a + 2) η + ζ ) ϑ ( u − u + η ) ϑ ( u − u + η ) ϑ (( a − η + ζ + u + u − u − u | τ )(the quasi-period is τ unless otherwise specified). Using the cyclicity relation (2.6), we can obtain3 more components which also have the same simple factorised form.The 2 remaining components are obtained from each other by cyclic rotation. We consider here( a + 1 , a, a + 1 , a ). According to Cor. 3.4, we have to search inside a space of theta functions ofdimension 12, but actually, by application of relation (iii) of Lem. 2.6 (which allows to compute thiscomponent out of Ψ a +1 ,a +2 ,a +1 ,a ), it has a prefactor of ϑ ( aη + ζ ) ϑ (( a + 1) η + ζ ), which reducesthe search to a space of dimension 4, spanned by: f k = ϑ ( aη + ζ ) ϑ (( a + 1) η + ζ ) ϑ k (( a + 12 ) η + ζ ) ( k = 1 , . . . , ϑ k ( u − u + 32 η ) ϑ k ( u − u + 32 η ) ϑ −⌈ k/ ⌉ ( u + u − u − u + ( a − η + ζ | τ )Then Ψ a,a − ,a,a − ,a = P k =1 α k f k , with α k = ( − ( k − k − ϑ ′ (0) Y ℓ = k ϑ ℓ ( η/ Appendix B. The Trigonometric and Rational Cases B.1. The Trigonometric Case. The trigonometric SOS model is defined on the same Hilbertspace as the elliptic SOS model with face weights again defined by Equation (2.1) while setting f ( u ) = sin( u ). The R -matrices and transfer matrix are defined by Equations (2.2) and (2.4),respectively. If we consider the limit τ → 0, i.e. setting the elliptic nome to one then we findlim τ → ϑ ( u | τ ) ∝ sin( u ) , lim τ → ϑ ( u | τ ) ∝ cos( u ) , lim τ → ϑ ( u | τ ) ∝ , where the proportionality is unimportant for the purpose of this discussion. From these relationswe are able to view the trigonometric SOS model as a limit of the elliptic SOS model and inferresults. Setting f (2) ( u ) = cos( u ) and f (3) ( u ) = 1 implies that the states Ψ (2) ( u ) and Ψ (3) ( u ) aresolutions of the qKZ(B) system.Solutions of the qKZ(B) system for the trigonometric and elliptic cases also share similar prop-erties, with the properties of the latter discussed in detail in Section 3. In the trigonometric caseΨ (2) ( u ) and Ψ (3) ( u ) both satisfy the recursion relations and wheel condition defined in Propositions3.5 and 3.6 . The coefficients of the solutions of the qKZ system are Laurent polynomials in thevariables q = e iη , z j = e iu j and Z = e iζ . Moreover, they are homogeneous with respect to thevariables z , . . . , z L with degree zero. The most negative/positive degree of Ψ (2) a ( u ) and Ψ (3) a ( u ) interms of a single variable z j is n and n − 1, respectively. The Laurent polynomial nature and degreeof the solutions is analogous to the holomorphicity and pseudo-periodicity of the elliptic case.The trigonometric limit of the elliptic SOS model also intersects the combinatorial line when η = π and is referred to as the combinatorial point. As such it follows that at this point Ψ (2) ( u ) , Ψ (3) ( u )are eigenstates of the transfer matrix. Furthermore, the maps to the three colour problem and 2-height RSOS model can still be applied, albeit to trigonometric versions.It was mentioned that in the elliptic three colour problem the solutions of the qKZB systemare the first excited state and not ground states. It was also stated that the authors suspect thatsolutions of the qKZB system are ground states of symmetry sector of the model. To explain thiswe use the quantum chain interpretation with the homogeneous Hamiltonian, given by Equation(4.1), of the trigonometric three-colour problem. The Hamiltonian can be written as H = 2 √ X k [ U k + I ]where U k = − √ ddu ln ( R k ( u )) (cid:12)(cid:12)(cid:12)(cid:12) u =0 − I,U k U k ± U k = U k ,U k U k = 2∆ U k . with ∆ = cos( η ) = − . This implies it is connected to the Temperley–Lieb model and subsequentlythe XXZ model with anisoptropy ∆ = cosh( η ) [3]. This allows us to use the Ansatz [35]Λ( u ) = ω (cid:20) sin( u + η )sin( u − η ) (cid:21) L q ( u − η ) q ( u ) + ω − (cid:20) sin( u )sin( u − η ) (cid:21) L q ( u + η ) q ( u )where Λ( u ) is an eigenvalue of the homogeneous transfer matrix, ω ∈ C is a twist and q ( u ) = Q li =1 sin( u − u i + η ). The Bethe equations are given by sin (cid:0) u j − η (cid:1) sin (cid:0) u j + η (cid:1) ! L = − ω l Y k =1 (cid:18) sin ( u j − u k − η )sin ( u j − u k + η ) (cid:19) . The energy and momentum of the eigenstates can be given in terms of the Bethe roots, however,these are not of interest to us. The quantity of interest is the twist ω , which can be expressed interms of the eigenvalue of the homogeneous transfer matrix,lim u →±∞ Λ( u ) = ωe ± iη ( L − l ) + ω − e ± iη ( l − L )HETA FUNCTION SOLUTIONS OF THE QKZB EQUATION FOR A FACE MODEL 19 For system sizes L = 2 , . . . , 10 we have found that the value lim u →±∞ Λ( u ) is either − L is even and either 1 or − L is odd. This constrains the allowed twists and implies for even L that the only twists allowed are cube roots of unity i.e. ω ∈ { , e iη , e − iη } .Consider only the case for L even, we find for small system sizes ( L = 2 , . . . , 8) that there is onestate with negative energy, two states with zero energy while every other state has positive energy.It was observed for L = 2 , . . . , l = L Bethe roots while the two solutions of the qKZ system of the three colour problem haveenergy 0, momentum π , twist e ± iη and l = L Bethe roots. In fact Theorem 4.1 implies that thesolutions of the qKZ system must have energy 0 and momentum π . Solving the Bethe equationsnumerically for larger L yields results consistent with the observations stated. Thus it appears thatthe true ground state of the system corresponds to the ground state of the spin-1 / / The Rational Case. The rational SOS model is defined on the same Hilbert space as thetrigonometric and elliptic SOS models with f ( u ) = u . Like the trigonometric case it can beconsidered a limiting case of the elliptic model, in fact it is also a limiting case of the trigonometriccase. This relationship to trigonometric and elliptic models is seen via the relationslim τ,t → ϑ ( ut | τ ) ∝ u, lim τ,t → ϑ ( ut | τ ) ∝ u / , lim τ,t → ϑ ( ut | τ ) ∝ , where again the proportionality is unimportant for the purpose of this discussion.Since solutions of the qKZ system form a vector space we can set f (2) ( u ) = u and f (3) ( u ) = 1,which ensures that Ψ (2) ( u ) and Ψ (3) ( u ) are solutions of the qKZ system that satisfy recursion rela-tions and the wheel condition. Additionally, each Ψ (2) a ( u ) and Ψ (3) a ( u ) is a homogeneous polynomialin the variables η, ζ, u , . . . , u L of degree n + 3 and n + 1, respectively.The rational limit does not intersect with the combinatorial line. Thus the model can not bemapped to either a 2-height RSOS model or the three colour problem. This can also be seen as aconsequence of f ( u ) not being periodic. Appendix C. A Pictorial Representation of the qKZB System In this appendix, we introduce a simple graphical representation of the qKZB system. Firstly,we use the conventional graphical representation of the SOS weights: W (cid:18) a bc d (cid:12)(cid:12)(cid:12)(cid:12) u (cid:19) = ua bcd The transfer matrix defined in Equation (2.4) is then given by (cid:10) a ′ (cid:12)(cid:12) t ( u ) | a i = u u u u · · · u L a ′ L a ′ a a L a ′ a ′ a a a ′ L − a ′ L a L a L − Let us now represent a solution of the qKZB system (2.5)-(2.6) as follows:Ψ( u , u , · · · , u L ) = u u u u · · · u L and components, with a = ( a , a , · · · , a L ), byΨ a ( u , u , · · · , u L ) = u u u u · · · u L a L a a a a L − a L The qKZB system (2.5,2.6) can then be represented very simply as u · · · u i u i +1 · · · u Lu i − u i +1 = u · · · u i +1 u i · · · u L and u u u u · · · u L a L a a a a L − a L = κ u L + s u u u · · · u L − a L − a L a a a L − a L − respectively. 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