Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations
aa r X i v : . [ m a t h . QA ] J un Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2008), 049, 13 pages Free Field Approach to Solutionsof the Quantum Knizhnik–Zamolodchikov Equations
Kazunori KUROKI † and Atsushi NAKAYASHIKI ‡† Department of Mathematics, Kyushu University, Hakozaki 6-10-1, Fukuoka 812-8581, Japan
E-mail: [email protected] ‡ Department of Mathematics, Kyushu University, Ropponmatsu 4-2-1, Fukuoka 810-8560, Japan
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Received February 18, 2008, in final form May 27, 2008; Published online June 03, 2008Original article is available at
Abstract.
Solutions of the qKZ equation associated with the quantum affine algebra U q ( b sl ) and its two dimensional evaluation representation are studied. The integral formulaederived from the free field realization of intertwining operators of q -Wakimoto modules areshown to coincide with those of Tarasov and Varchenko. Key words: free field; vertex operator; qKZ equation; q -Wakimoto module In 1992 I. Frenkel and N. Reshetikhin [7] had developed the theory of intertwining operatorsfor quantum affine algebras and had shown that the matrix elements of intertwiners satisfy thequantized Knizhnik–Zamolodchikov (qKZ) equations.The theory of intertwiners and qKZ equations was successfully applied to the study of solvablelattice models [9] (and references therein). As to the study of solutions of the qKZ equations,bases are constructed by Tarasov and Varchenko [22] in the form of multi-dimensional hyper-geometric integrals in the case of U q ( b sl ). However solutions of the qKZ equations for otherquantum affine algebras are not well studied [21].The method of free fields is effective to compute correlation functions in conformal field theo-ry (CFT) [3], in particular, solutions to the Knizhnik–Zamolodchikov (KZ) equations [18, 19].A similar role is expected for those of quantum affine algebras. Unfortunately it is difficult tosay that this expectation is well realized, as we shall explain below.Free field realizations of quantum affine algebras are constructed by Frenkel and Jing [6]for level one integrable representations of ADE type algebras and by Matsuo [15], Shiraishi [16]and Abada et al. [1] for representations with arbitrary level of U q ( b sl ). The latter results areextended to U q ( b sl N ) in [2]. Free field realizations of intertwiners are constructed based on theserepresentations in the case of U q ( b sl ) [9, 10, 12, 15, 4].The simplicity of the Frenkel–Jing realizations makes it possible not only to compute matrixelements but also traces of intertwining operators [9], which are special solutions to the qKZequations. The case of q -Wakimoto modules with an arbitrary level becomes more complex andthe detailed study of the solutions of the qKZ equations making use of it is not well developed.In [15] Matsuo derived his integral formulae [14] from the formulae obtained by the free fieldcalculation in the simplest case of one integration variable. However it is not known in generalwhether the integral formulae derived from the free field realizations recover those of [14, 23, 22] . See Note 1 in the end of the paper.
K. Kuroki and A. NakayashikiThe aim of this paper is to study this problem in the case of the qKZ equation with the valuein the tensor product of two dimensional irreducible representations of U q ( b sl ). More generalcases will be studied in a subsequent paper.There are mainly two reasons why the comparison of two formulae is difficult. One is thatthe formulae derived from the free field calculations contain more integration variables thanin Tarasov–Varchenko’s (TV) formulae. This means that one has to carry out some integralsexplicitly to compare two formulae. The second reason is that the formulae from free fieldscontains a certain sum. This stems from the fact that the current and screening operators arewritten as a sum which is absent in the non-quantum case. Since TV formulae have a similarstructure to those for the solutions of the KZ equation [18, 19], one needs to sum up certainterms explicitly for the comparison of two formulae. We carry out such calculations in the casewe mentioned.The plan of this paper is as follows. In Section 2 the construction of the hypergeometric solu-tions of the qKZ equation due to Tarasov and Varchenko is reviewed. The free field constructionof intertwining operators is reviewed in Section 3. In Section 4 the formulae for the highest tohighest matrix elements of some operators are calculated. The main theorem is also stated inthis section. The transformation of the formulae from free fields to Tarasov–Varchenko’s formu-lae is described in Section 5. In Section 6 the proof of the main theorem is given. Remainingproblems are discussed in Section 7. The appendix contains the list of the operator productexpansions which is necessary to derive the integral formula. Let V (1) = C v ⊕ C v be a two-dimensional irreducible representation of the algebra U q ( sl ),and R ( z ) ∈ End( V (1) ⊗ ) be a trigonometric quantum R -matrix given by R ( z ) ( v ǫ ⊗ v ǫ ) = v ǫ ⊗ v ǫ ,R ( z ) ( v ⊗ v ) = 1 − z − q z qv ⊗ v + 1 − q − q z v ⊗ v ,R ( z ) ( v ⊗ v ) = 1 − q − q z zv ⊗ v + 1 − z − q z qv ⊗ v , Let p be a complex number such that | p | < T j denote the multiplicative p -shift operatorof z j , T j f ( z , . . . , z n ) = f ( z , . . . , pz j , . . . , z n ) . The qKZ equation for the V (1) ⊗ n -valued function Ψ( z , . . . , z n ) is T j Ψ = R j,j − ( pz j /z j − ) · · · R j, ( pz j /z ) κ − hj R j,n ( z j /z n ) · · · R j,j +1 ( z j /z j +1 )Ψ , (1)where R ij ( z ) signifies that R ( z ) acts on the i -th and j -th components, κ is a complex parameter, κ − hj acts on the j -th component as κ − hj v ǫ = κ ǫ v ǫ . Let us briefly recall the construction of the hypergeometric solutions [22, 20] of the equation (1).In the remaining part of the paper we assume | q | <
1. We set( z ) ∞ = ( z ; p ) ∞ , ( z ; p ) ∞ = ∞ Y j =0 (1 − p j z ) , θ ( z ) = ( z ) ∞ ( pz − ) ∞ ( p ) ∞ . ree Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 3Let n and l be non-negative integers satisfying l ≤ n . For a sequence ( ǫ ) = ( ǫ , . . . , ǫ n ) ∈{ , } n satisfying ♯ { i | ǫ i = 1 } = l let w ( ǫ ) ( t, z ) = Y a . Then the Fock module F r,s is defined to be the free N − -module of rank one generated by thevector which satisfies N + | r, s i = 0 , ˜ a | r, s i = r | r, s i , ˜ b | r, s i = − s | r, s i , ˜ c | r, s i = 2 s | r, s i . We set F r = ⊕ s ∈ Z F r,s . A representation of the quantum affine algebra U q ( c sl ) is constructed on F r for any r ∈ C in [16].The right Fock module F † r,s and F † r are similarly defined using the vector h r, s | satisfying theconditions h r, s | N − = 0 , h r, s | ˜ a = r h r, s | , h r, s | ˜ b = − s h r, s | , h r, s | ˜ c = 2 s h r, s | . Remark 1.
We change the definition of | r, s i in [10]. Namely we use | r, s i = exp (cid:18) r k + 2) Q a + s Q b + Q c (cid:19) | , i . Let us introduce field operators which are relevant to our purpose. For x = a, b, c let x ( L ; M, N | z : α ) = − X n =0 [ Ln ] x n [ M n ][ N n ] z − n q | n | α + L ˜ x M N log z + LM N Q x ,x ( N | z : α ) = x ( L ; L, N | z : α ) = − X n =0 x n [ N n ] z − n q | n | α + ˜ x N log z + 1 N Q x . The normal ordering is defined by specifying N + , ˜ a , ˜ b , ˜ c as annihilation operators and N − , Q a , Q b , Q c as creation operators. With this notation let us define the operators J − ( z ) : F r,s −→ F r,s +1 , φ ( l ) m ( z ) : F r,s −→ F r + l,s + l − m , S ( z ) : F r,s −→ F r − ,s − , ree Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 5by J − ( z ) = 1( q − q − ) z (cid:0) J − + ( z ) − J −− ( z ) (cid:1) ,J − µ ( z ) =: exp (cid:18) a ( µ ) (cid:18) q − z ; − k + 22 (cid:19) + b (cid:0) | q ( µ − k +2) z ; − (cid:1) + c (cid:0) | q ( µ − k +1) − z ; 0 (cid:1)(cid:19) : ,a ( µ ) (cid:18) q − z ; − k + 22 (cid:19) = µ ( ( q − q − ) ∞ X n =1 a µn z − µn q (2 µ − k +22 ) n + ˜ a log q ) ,S ( z ) = − q − q − ) z ( S + ( z ) − S − ( z )) ,S ǫ ( z ) =: exp (cid:18) − a (cid:18) k + 2 | q − z ; − k + 22 (cid:19) − b (cid:0) | q − k − z ; − (cid:1) − c (cid:0) | q − k − ǫ z ; 0 (cid:1)(cid:19) : ,φ ( l ) l ( z ) =: exp (cid:18) a (cid:18) l ; 2 , k + 2 | q k z ; k + 22 (cid:19)(cid:19) : ,φ ( l ) l − r ( z ) = 1[ r ]! I r Y j =1 du j πi " · · · (cid:20)h φ ( l ) l ( z ) , J − ( u ) i q l , J − ( u ) (cid:21) q l − , . . . , J − ( u r ) q l − r +2 , where[ r ]! = [ r ][ r − · · · [1] , [ X, Y ] q = XY − qY X, and the integral in φ ( l ) l − r ( z ) signifies to take the coefficient of ( u · · · u r ) − .The operator J − ( z ) is a generating function of a part of generators of the Drinfeld realization U q ( c sl ) at level k . While the operators φ ( l ) m ( z ) are conjectured to determine the intertwiningoperator for U q ( c sl ) modules [10, 15] φ ( l ) ( z ) : W r −→ W r + l ⊗ V ( l ) z , φ ( l ) ( z ) = l X m =0 φ ( l ) m ( z ) ⊗ v ( l ) m , where W r is a certain submodule of F r specified as a kernel of a certain operator, called q -Wakimoto module [15, 12, 13, 11, 1], V ( l ) is the irreducible representation of U q ( sl ) with spin l/ V ( l ) z is the evaluation representation of U q ( c sl ) on V ( l ) .In this paper we exclusively consider the case l = 1 and set φ + ( z ) = φ (1)0 , φ − ( z ) = φ (1)1 , v = v (1)0 , v = v (1)1 . The operator S ( z ) commutes with U q ( c sl ) modulo total difference. Here modulo totaldifference means modulo functions of the form k +2 ∂ z f ( z ) := f ( q k +2 z ) − f ( q − ( k +2) z )( q − q − ) z . Remark 2.
The intertwining properties of φ ( l ) ( z ) for l ∈ Z are not proved in [10] as pointed outin [15]. However the fact that the matrix elements of compositions of φ ( l ) ( z )’s and S ( t )’s satisfythe qKZ equation modulo total difference can be proved in a similar way to Proposition 6.1in [15] using the result of Konno [11] (see (4)).Let | m i = | m, i ∈ F m, , h m | = h m, | ∈ F † m, . K. Kuroki and A. NakayashikiThey become left and right highest weight vectors of U q ( c sl ) with the weight m Λ + ( k − m )Λ respectively, where Λ , Λ are fundamental weights of b sl . Consider F ( t, z ) = h m + n − l | φ (1) ( z ) · · · φ (1) ( z n ) S ( t ) · · · S ( t l ) | m i (2)which is a function taking the value in V (1) ⊗ n . Let∆ j = j ( j + 2)4( k + 2) , s = 12( k + 2) . Set b F = n Y i =1 z ∆ m + n − l +1 − i − ∆ m + n − l − i i ! F = n Y i =1 z s ( m + n − l − i + ) i ! F, (3)Let the parameter p be defined from k by p = q k +2) . We assume | p | < b F satisfies the following qKZ equation modulototal difference of a function [15, 8, 7, 10, 11] T zj b F = b R j,j − ( pz j /z j − ) · · · b R j, ( pz j /z ) q − ( m + n/ − l +1) h j b R j,n ( z j /z n ) · · · b R j,j +1 ( z j /z j +1 ) b F , (4)where b R ( z ) = ρ ( z ) ˜ R ( z ) , ˜ R ( z ) = C ⊗ R ( z ) C ⊗ ,ρ ( z ) = q / ( z − ; q ) ∞ ( q z − ; q ) ∞ ( q z − ; q ) ∞ , ( z ; x ) ∞ = ∞ Y i =0 (1 − x i z ) , Cv ǫ = v − ǫ . Define the components of F ( t, z ) by F ( t, z ) = X ( ν ) ∈{ , } n F ( ν ) ( t, z ) v ( ν ) , v ( ν ) = v ν ⊗ · · · ⊗ v ν n , where ( ν ) = ( ν , . . . , ν n ). By the weight condition F ( ν ) ( t, z ) = 0 unless the condition ♯ { i | ν i = 0 } = l is satisfied. We assume this condition once for all. Notice that φ + ( z ) = 1( q − q − ) I du πiu [ φ − ( z ) , J − + ( u ) − J −− ( u )] q ,S ( t ) = − q − q − ) t ( S + ( t ) − S − ( t )) . Let { i | ν i = 0 } = { k < · · · < k l } , ree Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 7and F ( ν )( ǫ )( µ ) ( t, z | u ) = h m + n − l | φ − ( z ) · · · [ φ − ( z k ) , J − µ ( u )] q · · · [ φ − ( z k l ) , J − µ l ( u l )] q · · · φ − ( z n ) × S ǫ ( t ) · · · S ǫ l ( t l ) | m i . Then F ( ν ) ( t, z ) can be written as F ( ν ) ( t, z ) = ( − l ( q − q − ) − l l Y a =1 t − a X ǫ i ,µ j l Y i =1 ( ǫ i µ i ) Z C l l Y j =1 du j πiu j F ( ν )( ǫ )( µ ) ( t, z | u ) , where C l is a suitable deformation of the torus T l specified as follows.The contour for the integration variable u i is a simple closed curve rounding the origin in thecounterclockwise direction such that q k +3 z j (1 ≤ j ≤ n ), q − u j ( i < j ), q − µ i ( k +2) t a (1 ≤ a ≤ l )are inside and q k +1 z j (1 ≤ j ≤ n ), q u j ( j < i ) are outside.By the operator product expansions (OPE) of the products of φ − ( z ), J − µ ( u ), S ǫ ( t ) in theappendix, one can compute the function F ( ν )( ǫ )( µ ) ( t, z | u ) explicitly. In order to write down theformula we need some notation. Set ξ ( z ) = ( pz − ; p, q ) ∞ ( pq z − ; p, q ) ∞ ( pq z − ; p, q ) ∞ , ( z ; p, q ) ∞ = ∞ Y i =0 ∞ Y j =0 (1 − p i q j z ) . Then F ( ν )( ǫ )( µ ) ( t, z | u ) = f ( ν )( µ ) ( t, z | u )Φ( t, z ) G ( ν )( ǫ )( µ ) ( t, z | u ) , where f ( ν )( µ ) ( t, z | u ) = (1 − q ) l q P li =1 ( n + m − l − k i + i ) µ i n Y i =1 (cid:0) q k z i (cid:1) s ( m + n − l − i ) Y i Theorem 1. If ( µ ) = ( − l ) = ( − , . . . , − ) , G ( ν )( µ ) ( t, z ) = 0 . For ( µ ) = ( − l ) we have G ( ν )( − l ) ( t, z ) = q − l + l ( l − − P li =1 k i ( q − q − ) l w ( − ν ) ( t, z ) , where ( − ν ) = (1 − ν , . . . , − ν n ) . K. Kuroki and A. NakayashikiIt follows that F ( ν ) ( t, z ) is given by F ( ν ) ( t, z ) = ( − l q − ( n + m +2 − l ) l + ksn ( m + n − l ) − ksn ( n +1)+4 sl ( m − l +1) × n Y i =1 z s ( m + n − l − i ) i Y i Proposition 1. For any W ∈ F ell ˜Ψ W = Z ˜ T l l Y a =1 dt a πi ˜ F ( t, z ) ˜ W ( t, z ) , is a solution of the qKZ equation (1) , where ˜ F is defined by (5) with b F and F being given in (3) and (2) and ˜ W is defined by (6) . ree Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 9 Let A ± = { j | µ j = ±} . Suppose that the number of elements in A ± is r ± and write A ± = { l ± < · · · < l ± r ± } . Set A = A − , r = r − and l i = l − i for simplicity. Let I ( ν )( ǫ )( µ ) ( t, z ) = Z C l l Y j =1 du j πiu j b G ( ν )( ǫ )( µ ) ( t, z | u ) . Lemma 1. We have I ( ν )( ǫ )( µ ) ( t, z ) = ( − q − ) r ( q − q − ) r l X a ,...,ar =1 ai = aj ( i = j ) r Y i =1 δ ǫ ai , + r Y i =1 t a i z k li − qt a i r Y j =1 Y i We first integrate in the variables u j , j ∈ A + in the order u l +1 , . . . , u l + r + . Let us considerthe integration in u l +1 . We denote the integration contour in u i by C i . The only singularity ofthe integrand outside C l +1 is ∞ . Thus the integral is calculated by taking residue at ∞ . Sincethe integrand is of the form du l +1 u l +1 H ( u l +1 ) , where H ( u ) is holomorphic at ∞ . Then Z C l +1 du πiu H ( u ) = − Res u = ∞ duu H ( u ) = lim u −→∞ H ( u ) . In this way the integral in u l +1 is calculated. After this integration the integrand as a functionof u l +2 has a similar structure. Therefore the integration with respect to u l +2 is carried out ina similar way and so on. Finally we get I ( ν )( ǫ )( µ ) ( t, z ) = ( − r + Res u l + r + = ∞ · · · Res u l +1 = ∞ b G ( ν )( ǫ )( µ ) ( t, z | u )= Z C n − r + Y j ∈ A du j πiu j Y i ∈ A − q − − k u i z k i − q − − k u i Y i Recall that G ( ν )( ǫ )( µ ) ( t, z ) = I ( ν )( ǫ )( µ ) ( t, z ) Y aj t a i − q − t a j t a i − t a j r Y i =1 r + Y j =1 t a i − t b j Y aaj q ( t a i − t a j ) Y i,jbi>bj ( q ǫ bi t b i − q ǫ bj t b j ) × Y i,jai>bj ( qt a i − q ǫ bj t b j ) Y i,jbj>ai ( q ǫ bj t b j − qt a i ) , we have X ǫ b ,...,ǫ br + = ± r + Y j =1 ǫ b j r Y i =1 r + Y j =1 ( t a i − q − − ǫ bj t b j ) Y aaj ( t a i − t a j ) × X ǫ b ,...,ǫ br + = ± r + Y j =1 ǫ b j Y i,jbi>bj ( q ǫ bi t b i − q ǫ bj t b j ) . (7) Lemma 2. For N ≥ we have X ǫ ,...,ǫ N = ± N Y j =1 ǫ j Y i>j ( q ǫ i t i − q ǫ j t j ) = 0 . (8) Proof . Let a i ( ǫ ) = t (1 , q ǫ t i , ( q ǫ t i ) , . . . , ( q ǫ t i ) N − ) . Then the left hand side of (8) is equal to X ǫ ,...,ǫ N = ± N Y j =1 ǫ j det ( a ( ǫ ) , . . . , a N ( ǫ N )) = det X ǫ ǫ a ( ǫ ) , . . . , X ǫ N ǫ N a N ( ǫ N ) ! . (9)Since X ǫ ǫ a i ( ǫ ) = t (cid:16) , ( q − q − ) t i , . . . , ( q N − − q − ( N − ) t N − i (cid:17) , the right hand side of (9) is zero. (cid:4) By this lemma the right hand side of (7) becomes zero if r + > 0. Consequently G ( ν )( µ ) = 0 for r + > 0. Suppose that r + = 0. In this case r = l , l i = i (1 ≤ i ≤ l ) and( − q ) l ( q − q − ) − l G ( ν )( − l ) ( t, z )= Y aj t a i − q − t a j t a i − t a j . The theorem easily follows from this.2 K. Kuroki and A. Nakayashiki In this paper we study the solutions of the qKZ equation taking the value in the tensor productof the two dimensional evaluation representation of U q ( b sl ). The integral formulae are derivedfor the highest to highest matrix elements for certain intertwining operators by using free fieldrealizations. The integrals with respect to u variables corresponding to the operator J − ( u ) arecalculated and the sum arising from the expression of J − ( u ) and the screening operator S ( t )is calculated. The formulae thus obtained coincide with those of Tarasov and Varchenko. Thecalculations in this paper can be extended to the case where the vector space V (1) ⊗ n is replacedby a tensor product of more general representations. It is an interesting problem to performsimilar calculations for other quantum affine algebras [2] and the elliptic algebras [13].In Tarasov–Varchenko’s theory solutions of a qKZ equation are parametrized by elements ofthe elliptic hypergeometric space F ell while the matrix elements are specified by intertwiners.It is an interesting problem to establish a correspondence between intertwining operators andelements of F ell . With the results of the present paper one can begin to study this problem.Study in this direction will provide a new insight on the space of local fields and correlationfunctions of integrable field theories and solvable lattice models. The corresponding problem inCFT is studied in [5]. Appendix. List of OPE’s Here we list OPE’s which are necessary in this paper. Almost all of them are taken from thepaper [10]. Let C ( z ) = ( q − z ; p ) ∞ ( q z ; p ) ∞ , ( z ) ∞ = ( z ; p ) ∞ .S ǫ ( t ) S ǫ ( t ) = ( q − t ) s q ǫ t − q ǫ − ǫ t t − q − t C ( t /t ) : S ǫ ( t ) S ǫ ( t ) : , | q − t | < | t | ,φ − ( z ) S ǫ ( t ) = ( q k z ) − s ( qt/z ) ∞ ( q − t/z ) ∞ : φ − ( z ) S ǫ ( t ) : , | q − t | < | z | ,J − µ ( u ) S ǫ ( t ) = q − µ u − q − µ ( k +1) − ǫ tu − q − µ ( k +2) t : J − µ ( u ) S ǫ ( t ) : , | q − ( k +2) t | < | u | ,φ − ( z ) J − µ ( u ) = z − q µ − − k uz − q − − k u : φ − ( z ) J − µ ( u ) : , | u | < | q k +3 z | for µ = − ,J − µ ( u ) φ − ( z ) = q µ u − q k +2 − µ zu − q k +3 z : φ − ( z ) J − µ ( u ) : , | q k +1 z | < | u | for µ = + , [ φ − ( z ) , J − µ ( u )] q = (1 − q ) u ( z − q µ − − k u )( z − q − − k u )( u − q k +3 z ) : φ − ( z ) J − µ ( u ) : ,J − µ ( u ) J − µ ( u ) = q − µ u − q µ − µ u u − q − u : J − µ ( u ) J − µ ( u ) : , | q − u | < | u | ,φ − ( z ) φ − ( z ) = ( q k z ) s ξ ( z /z ) : φ − ( z ) φ − ( z ) : , | pz | < | z | . Note 1. 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