The Baxter's Q-operator for the W-algebra W N
aa r X i v : . [ n li n . S I] J un The Baxter’s Q -operatorfor the W -algebra W N October 29, 2018
Takeo KOJIMA
Department of Mathematics, College of Science and Technology, Nihon University,Surugadai, Chiyoda-ku, Tokyo 101-0062, JAPAN
Abstract
The q -oscillator representation for the Borel subalgebra of the affine symmetry U ′ q ( d sl N ) is presented. By means of this q -oscillator representation, we give the freefield realizations of the Baxter’s Q -operator Q j ( λ ) , Q j ( λ ) , ( j = 1 , , · · · , N ) for the W -algebra W N . We give the functional relations of the T - Q operators, includingthe higher-rank generaliztion of the Baxter’s T - Q relation. Key Words : CFT, q -oscillator, Q -operator, functional relation, free field realization1 Introduction
The Baxter’s T - Q -operator have various exceptional properties and play an importantrole in many aspect of the theory of integrable systems. Originally the Q -operatorwas introduced by R.Baxter [1], in terms of some special transfer matrix of the 8-vertex model. Over the last three decades, this method of the Q -operator has beendeveloped by many literatures. We would like to refer some of these literatures, writ-ten by R.Baxter [2, 3, 4, 5], by L.Takhtadzhan and L.Faddeev [6], by K.Fabricius andB.McCoy [7, 8, 9], by K.Fabricius [10], by V.Bazhanov and V.Mangazeev [11], by B.Feigin,T.Kojima, J.Shiraishi and H.Watanabe [12], by T.Kojima and J.Shiraishi [13]. However afull theory of the Q -operator for the 8-vertex model is not yet developed. For the simplermodels associated with the quantum group U q ( g ), there have been many papers whichextend, generalize, and comment on the T - Q relation. We would like to refer some of theseliteratures, including Sklyanin’s separation variable method, written by E.Sklyanin [14, 15,16], by V.Kuzunetsov, V.Mangazeev and E.Sklyanin [17], by V.Pasquier and M.Gaudin[18], by S.Derkachov [19] by S.Derkachov, G.Karakhanyan and A.Mansahov [20, 21] byS.Derkachov, G.Karakhanyan and R.Kirschner, [22] by S.Derkachov and A.Mansahov [23],by A.Belisty, S.Derkachov, G.Korchemesky and A.Manasahov [24], by C.Korff [25, 26],by A.Bytsko and J.Teschner [27], by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov [28,29, 30, 31], by M.Rossi and R.Weston [32], by P.Dorey and R.Tateo [33], by V.Bazhanov,A.Hibberd and S.Khoroshkin [34], by P.Kulish and Z.Zeitlin [35], by A.Antonov andB.Feigin [36], by I.Krichever, O.Lipan, P.Wiegmann and A.Zabrodin [38], by V.Bazhanovand N.Reshetikhin [39], by A.Kuniba, T.Nakanishi and J.Suzuki [40], by H.Boos, M.Jimbo,T.Miwa, F.Smirnov and Y.Takeyama [41, 42], by A.Chervov and G.Falqui [43]. Each pa-per added to our understanding of the great Baxter’s original paper [1]. Especially forexample the T - Q -operators acting on the Fock space of the Virasoro algebra V ir wereintroduced by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov [28, 29, 30]. They de-rived various functional relations of the T - Q operators and gave the asymptotic behaviorof the eigen-value of the T - Q operators. P.Dorey and R.Tateo [33] revealed the hiddenconnection between the vacuum expectation value of the Q -operator and the spectraldeterminant for Schr¨odinger equation. V.Bazhanov, A.Hibberd and S.Khoroshkin [34]achieved the W -algebraic generalization of [28, 29, 30, 31, 33]. In this paper we study2he higher-rank W N -generalization of [34]. We study the T - Q -operators acting on theFock space of the W -algebra W N . We give the free field realization of the Q -operator andfunctional relations of the T - Q -operators for the W -algebra W N , including the higher-rankgeneralization of the Baxter’s T - Q relation, Q i ( tq N ) + N − X s =1 ( − s T Λ + ··· +Λ s ( tq − ) Q i ( tq N − s ) + ( − N Q i ( tq − N ) = 0 , Q i ( tq − N ) + N − X s =1 ( − s T Λ + ··· +Λ s ( tq ) Q i ( tq − N +2 s ) + ( − N Q i ( tq N ) = 0 , where i = 1 , , · · · , N . The organization of this paper is as following. In section 2,we give basic definitions, including q -oscillator representation of the Borel subalgebraof the affine symmetry U ′ q ( c sl N ), which play an essential role in construction of the Q -operator. In section 3, we give the definition of the T -operator and the Q -operator. Insection 4 we give conjecturous funtional relations between the T -opeartor and the Q -operator, including Baxter’s T - Q relation. In appendix, we give supporting arguments onconjecturous formulae stated in section 4. In this section we give the different realizations of the Borel subalgebra of the affinequantum algebra U ′ q ( c sl N ), which will play an important role in construction of the Baxter’s T - Q operator. Let us fix the integer N ≧
3. Let us fix a complex number 1 < r < N .In this paper, upon this setting, we construct the Baxter’s T - Q operators on the space ofthe W -algebra W N with the central charge −∞ < C CF T < −
2, where C CF T = ( N − (cid:18) − N ( N + 1) r ( r − (cid:19) . Becuse C CF T → −∞ represents the classical limit, we call −∞ < C CF T < − C CF T <
1. We would like to note that the unitary minimalCFT is described by the central charge C CF T = ( N − (cid:16) − N ( N +1) r ( r − (cid:17) for N, r ∈ Z ,( N ≧ , r ≧ N + 2) [44]. We set parameters r ∗ = r − q = e πi r ∗ r . In what followswe use the q -integer [ n ] q = q n − q − n q − q − . 3 .1 The q -oscillator representation Let { ǫ j } be an orthonormal basis of R N , relative to the standard inner product ( ǫ i | ǫ j ) = δ i,j . Let us set ¯ ǫ j = ǫ j − ǫ where ǫ = N P Nj =1 ǫ j . We have (¯ ǫ i | ¯ ǫ j ) = δ i,j − N . Let us setthe simple roots α j = ¯ ǫ j − ¯ ǫ j +1 , (1 ≦ j ≦ N −
1) and α N = − P N − j =1 α j . Let us set thefundamental weights ω j as the dual vector of α j , i.e. ( α i | ω j ) = δ i,j . Explicitly we have ω j = ¯ ǫ + · · · + ¯ ǫ j . Let us set the weight lattice P = ⊕ Nj =1 Z ¯ ǫ j . We consider the quantumaffine algebra U ′ q ( c sl N ), which is generated by e , · · · , e N , f , · · · , f N , and h , · · · , h N , withthe defining relations,[ h i , h j ] = 0 , [ h i , e j ] = ( α i | α j ) e j , [ h i , f j ] = − ( α i | α j ) f j , [ e i , f j ] = δ i,j q h i − q − h i q − q − ,e i e j − [2] q e i e j e i + e j e i = 0 , f i f j − [2] q f i f j f i + f j f i = 0 , for ( α i | α j ) = − . Here (( α j | α k )) ≦ j,k ≦ N is the Cartan matrix of type c sl N . Let us introduce the Borel sub-algebra of U ′ q ( c sl N ). The Borel subalgebra U ′ q ( b b + ) is generated by e , · · · , e N , h , · · · , h N ,and U ′ q ( b b − ) by f , · · · , f N , h , · · · , h N . In this paper we consider the level c = 0 case, withthe central element c = h + · · · + h N . Let us introduce the q -oscillator representation o t of the Borel subalgebra U ′ q ( b b + ). The q -oscillator algebra Osc j , (1 ≦ j ≦ N − E j , E ∗ j , H j , with the defining relations,[ H j , E j ] = E j , [ H j , E ∗ j ] = −E ∗ j , q E j E ∗ j − q − E ∗ j E j = 1 q − q − . (2.1)Let us set Osc = Osc ⊗ C · · · ⊗ C Osc N − . We have [ E j , E k ] = 0, [ E ∗ j , E ∗ k ] = 0, [ E j , E ∗ k ] = 0,[ H j , H k ] = 0 for j = k . Let us set the auxiliarry operator H N = −H − H − · · · − H N − .We define homomorphism o t : U ′ q ( b b + ) → Osc by o t ( e ) = tq ( q − q − ) q −H E ∗ E ,o t ( e ) = q ( q − q − ) q −H E ∗ E , · · · o t ( e N − ) = q ( q − q − ) q −H N − E ∗ N − E N − ,o t ( e N − ) = E ∗ N − ,o t ( e N ) = q −H −H N E , (2.2) o t ( h ) = −H + H , o t ( h ) = −H + H , · · · , o t ( h N ) = −H N + H . q -oscillator representation o t satisfies level zero condition o t ( h + h + · · · + h N ) = 0.This q -oscillator representation give a higher-rank generalization of those in [34]. Bymeans of the Dynkin-diagram automorphism τ, σ , we construct a family of the q -oscillatorrepresentation o t,j , ¯ o t,j . Let us set the Dynkin-diagram automorphism τ of the affinealgebra U ′ q ( c sl N ). τ ( e ) = e , · · · , τ ( e j ) = e j +1 , · · · , τ ( e N ) = e ,τ ( h ) = h , · · · , τ ( h j ) = h j +1 , · · · , τ ( h N ) = h ,τ ( f ) = f , · · · , τ ( f j ) = f j +1 , · · · , τ ( f N ) = f . Let us set the Dynkin-diagram automorphism σ of the finite simple algebra U q ( sl N ),generated by e , · · · , e N , h , · · · , h N , f , · · · , f N . σ ( e ) = e N , · · · , σ ( e j ) = e N +2 − j , · · · , σ ( e N ) = e ,σ ( h ) = h N , · · · , σ ( h j ) = h N +2 − j , · · · , σ ( h N ) = h ,σ ( f ) = f N , · · · , σ ( f j ) = f N +2 − j , · · · , σ ( f N ) = f , and σ is esxtended to the affine vertex as σ ( e ) = e , σ ( h ) = h , σ ( f ) = f . We have theaction of τ j · σ · τ − , τ j · σ · τ − ( e i ) = e j − − i ,τ j · σ · τ − ( h i ) = h j − i ,τ j · σ · τ − ( f i ) = f j − − i , with s, j ∈ Z . We set homomorphism o t,j , ¯ o t,j : U ′ q ( b b + ) → Osc , (1 ≦ j ≦ N ), o t,j = o t · τ − j , ¯ o t,j = o ( − N t · τ j · σ · τ − , (2.3)These q -oscillator representations o t,j , ¯ o t,j will play an important role in construction ofthe Baxter’s Q -operator. Let us consider the quantum simple algebra U q ( gl N ), which is generated by E α , · · · , E α N − , H , · · · , H N , and F α , · · · , F α N − , with the defining relations,[ H i , H j ] = 0 , [ H i , E α j ] = ( δ i,j − δ i,j +1 ) E α j , [ H i , F α j ] = ( − δ i,j + δ i,j +1 ) F α j , E α i , F α j ] = δ i,j q H i − H i +1 − q − H i + H i +1 q − q − ,E α i E α j − [2] q E α i E α j E α i + E α j E α i = 0 , F α i F α j − [2] q F α i F α j F α i + F α j F α i = 0 . Let us set the root vectors, F α + α = [ F α , F α ] √ q = √ qF α F α − √ q F α F α , ¯ F α + α = [ F α , F α ] √ q = 1 √ q F α F α − √ qF α F α ,F α + ··· + α N − = [ F α N − , [ F α N − , · · · , [ F α , F α ] √ q · · · ] √ q ] √ q ,F α + ··· + α N − = [[ · · · , [ F α N − , F α N − ] √ q · · · , F α ] √ q , F α ] √ q . Let us set the automorhism σ by σ ( E α ) = E α N − , · · · , σ ( E α j ) = E α N − j , · · · , σ ( E α N − ) = E α ,σ ( H ) = − H N , · · · , σ ( H j ) = − H N − j +1 , · · · , σ ( H N ) = − H ,σ ( F α ) = F α N − , · · · , σ ( F α j ) = F α N − j , · · · , σ ( F α N − ) = F α . We have the evaluation representation ev t , ev t : U ′ q ( c sl N ) → U q ( gl N ), given by ev t ( e ) = E α , · · · , ev t ( e j +1 ) = E α j , · · · , ev t ( e N ) = E α N − ,ev t ( h ) = H − H , · · · , ev t ( h j +1 ) = H j − H j +1 , · · · , ev t ( h N ) = H N − − H N ,ev t ( f ) = F α , · · · , ev t ( f j +1 ) = F α j , · · · , ev t ( f N ) = F α N − ,ev t ( e ) = tF α + α + ··· + α N − q H + H N , ev t ( f ) = t − E α + α + ··· + α N − q − H − H N ,ev t ( h ) = H N − H .ev t ( e ) = E α , · · · , ev t ( e j +1 ) = E α j , · · · , ev t ( e N ) = E α N − ,ev t ( h ) = H − H , · · · , ev t ( h j +1 ) = H j − H j +1 , · · · , ev t ( h N ) = H N − − H N ,ev t ( f ) = F α , · · · , ev t ( f j +1 ) = F α j , · · · , ev t ( f N ) = F α N − ,ev t ( e ) = tF α + α + ··· + α N − q − H − H N , ev t ( f ) = t − E α + α + ··· + α N − q H + H N ,ev t ( h ) = H N − H . We have the conjugation ev t = σ · ev ( − ) N t · σ − . We set the irreducible highest represen-tation of U q ( gl N ) with the highest weight λ = m Λ + · · · + m N Λ N , the highest weightvector | λ i of U q ( sl N ). π ( λ ) ( E α j ) | λ i = 0 , π ( λ ) ( H j ) | λ i = m j | λ i , (1 ≦ j ≦ N ) .
6n what follows we consider the case m j − m j +1 ∈ N , (1 ≦ j ≦ N − π ( λ ) is finite dimension. Let us set the evaluation highest weight represen-tation π ( λ ) t of the affine symmetry U ′ q ( c sl N ), as π ( λ ) t = π ( λ ) · ev t , π ( λ ) t = π ( λ ) · ev t . These evaluation highest weight representation will play an important role in constructionof the T -operator T λ ( t ) , T λ ( t ). Let us introduce bosons B im , ( m ∈ Z =0 ; i = 1 , , · · · , N −
1) by[ B im , B jn ] = mδ m + n, ( α i | α j ) r − r , (1 ≦ i, j ≦ N − . (2.4)Let us set B Nm = − P N − j =1 B jm . We have the commutation relation [ B im , B jn ] = mδ m + n, ( α i | α j ) r − r ,for 1 ≦ i, j ≦ N . Let us set the zero-mode operators P λ and Q λ , ( λ ∈ P = ⊕ j Z ¯ ǫ j ) by[ P λ , iQ µ ] = ( λ | µ ) . (2.5)Let us set the Heisenberg algebra B generated by B m , · · · , B N − m , P λ , Q λ , ( λ ∈ P ) and itscompletion b B . Let us set the Fock space F l,k by B jm | l, k i = 0 , ( m >
0) (2.6) P α | l, k i = α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r rr − l − r r − r k ! | l, k i , (2.7) | l, k i = e i √ rr − Q l − i √ r − r Q k | , i . (2.8)Let us set the screening currents of the W -algebra W N by V α j ( u ) = exp i r r ∗ r Q α j ! exp r r ∗ r P α j iu ! × exp X m> m B j − m e imu ! exp − X m> m B jm e − imu ! , (1 ≦ j ≦ N ) . (2.9)Here we have added one operator V α N ( u ), which looks like affinization of the classical A N − . We can find the elliptic deformation of V α j ( u ) for j = N in [12, 13]. For Re( u ) > u ), we have V α j ( u ) V α j ( u ) = : V α j ( u ) V α j ( u ) : ( e iu − e iu ) r ∗ r , (1 ≦ j ≦ N ) ,V α j ( u ) V α j +1 ( u ) = : V α j ( u ) V α j +1 ( u ) : ( e iu − e iu ) − r ∗ r , (1 ≦ j ≦ N ) ,V α j +1 ( u ) V α j ( u ) = : V α j +1 ( u ) V α j ( u ) : ( e iu − e iu ) − r ∗ r , (1 ≦ j ≦ N ) . By analytic continuation, we have V α i ( u ) V α j ( u ) = q ( α i | α j ) V α j ( u ) V α i ( u ) , (1 ≦ i, j ≦ N ) . (2.10)Let us set z j = exp − πi r r ∗ r P ¯ ǫ j ! , (1 ≦ j ≦ N ) . (2.11)We have z z · · · z N = 1 and V α i ( u + 2 π ) = z − i z i +1 V α i ( u ) , z i V α j ( u ) = q δ i,j +1 − δ i,j V α j ( u ) z i . Let us set the nilpotent subalgebra U ′ q ( b n − ) generated by f , f , · · · , f N . We have homo-morphism sc : U ′ q ( b n − ) → b B given by sc ( f j ) = 1 q − q − Z π V α j ( u ) du, (1 ≦ j ≦ N ) . Q -operator In this section we define the Baxter’s T - Q operator by means of the trace of the universal R , and present conjecturous functional relations of the T - Q operator, which include thehigher-rank generalization of the Baxter’s T - Q relation. L -operator Let us set the universal L -operator L ∈ b B ⊗ U q ( b n − ) by L = exp − πi r r ∗ r N X j =1 P ω j ⊗ h j ! P exp (cid:18)Z π K ( u ) du (cid:19) . (3.1)Here we have set K ( u ) = N X j =1 V α j ( u ) ⊗ e j . P exp (cid:16)R π K ( u ) du (cid:17) represents the path ordered exponential P exp (cid:18)Z π K ( u ) du (cid:19) = ∞ X n =0 Z · · · Z π ≧ u ≧ u ≧ ··· ≧ u n ≧ K ( u ) K ( u ) · · · K n ( u n ) du du · · · du n . The above integrals converge in “quasi-classical domain” −∞ < C CF T < −
2. For thevalue of C CF T outside the quasi-classical domain, the integrals should be understood asanalytic continuation. Let us set U q ( c sl N ) the extension of U ′ q ( c sl N ) by the degree operator d . Let us set U q ( b n ± ) the extension of U ′ q ( b n ± ) by the degree operator d . There exists theunique universal R -matrix R ∈ U q ( b n + ) ⊗ U q ( b n − ) satisfying the Yang-Baxter equation. R R R = R R R . The universal- R ’s Cartan elements t is factored as R = q t R , t = N − X j =1 h j ⊗ h j + c ⊗ d + d ⊗ c, where ( h i | h j ) = δ i,j . We call the element R ∈ U ′ q ( b n + ) ⊗ U ′ q ( b n − ) the reduced universal R -matrix. The L -operator is an image of the reduced R -matrix [34], L = ( sc ⊗ id )( R ) . The L -operator will play an important role in trace construction of the T - Q operator. Let us set the T -operator T λ ( t ) and T λ ( t ) by T λ ( t ) = Tr π ( λ ) t exp − πi r r ∗ r N X j =1 P ω j ⊗ h j ! L ! , (3.2) T λ ( t ) = Tr π ( λ ) t exp − πi r r ∗ r N X j =1 P ω j ⊗ h j ! L ! . (3.3)Let us set an image of L as L λ ( t ) = ( id ⊗ π ( λ ) t ) ( L ), and the R -matrix R λ ,λ ( t /t ) = π ( λ ) t ⊗ π ( λ ) t ( R ). We have so-called RLL relation, R λ ,λ ( t /t ) L λ ( t ) L λ ( t ) = L λ ( t ) L λ ( t ) R λ ,λ ( t /t ) . R -matrix R λ ,λ ( t /t ) − from the right, and taking trace, we have thecommutation relation,[ T λ ( t ) , T λ ( t )] = [ T λ ( t ) , T λ ( t )] = [ T λ ( t ) , T λ ( t )] = 0 . The coefficients of the Taylor expansion of T λ ( t ) commute with each other. Hence wehave infinitly many commutative operators, which give quantum deformation of the con-servation laws of the N -th KdV equation. Let us set the Fock representation π ± j : Osc j → W ± with j = 1 , , · · · , N − W + = ⊕ k ≧ C | k i + , W − = ⊕ k ≧ C | k i − . The action is given by π + j ( H j ) | k i + = − k | k i + , π + j ( E j ) | k i + = 1 − q − k ( q − q − ) | k − i + , π + j ( E ∗ j ) | k i = | k + 1 i + ,π − j ( H j ) | k i − = k | k i − , π − j ( E j ) | k i − = 1 − q k ( q − q − ) | k − i − , π − j ( E j ) | k i − = | k + 1 i − . Let π j and π j be any representation of the q -oscillator Osc = Osc ⊗ C · · · ⊗ C Osc N − suchthat the partition Z j ( t ) , Z j ( t ) converge. Z j ( t ) = Tr π j o t,j exp − πi r r ∗ r N X j =1 P ω j ⊗ h j !! ,Z j ( t ) = Tr π j o t,j exp − πi r r ∗ r N X j =1 P ω j ⊗ h j !! . Let us set the operators A j ( t ) and A j ( t ) with j = 1 , , · · · , N A j ( t ) = 1 Z j ( t ) Tr π j o t,j exp − πi r r ∗ r N X j =1 P ω j ⊗ h j ! L ! , (3.4) A j ( t ) = 1 Z j ( t ) Tr π j o t,j exp − πi r r ∗ r N X j =1 P ω j ⊗ h j ! L ! . (3.5)Let us set the Baxter’s Q -operator Q j ( t ) and Q j ( t ) with j = 1 , , · · · , N, Q j ( t ) = t − √ rr ∗ P ¯ ǫj A j ( t ) , Q j ( t ) = t √ rr ∗ P ¯ ǫj A j ( t ) . (3.6)10e would like to note convenient relation, N X k =1 P ω k ⊗ o t,j ( h k ) = N − X k =1 ( P ¯ ǫ j − P ¯ ǫ j + k ) ⊗ H k , N X k =1 P ω k ⊗ o t,j ( h k ) = N − X k =1 ( P ¯ ǫ j − k − P ¯ ǫ j ) ⊗ H k . Here we should understand the surfix number as modulus N , i.e. ¯ ǫ j + N = ¯ ǫ j .From the Yang-Baxter equation, we have the commutation relations[ Q j ( t ) , Q j ( t )] = [ Q j ( t ) , Q j ( t )] = [ Q j ( t ) , Q j ( t )] = 0 , and [ Q j ( t ) , T λ ( t )] = [ Q j ( t ) , T λ ( t )] = [ Q j ( t ) , T λ ( t )] = [ Q j ( t ) , T λ ( t )] = 0 . The operators A j ( t ) can be written as power series. A j ( t ) = 1 + ∞ X n =1 X σ , ··· ,σ Nn ∈ Z N a ( j ) Nn ( σ , · · · , σ Nn ) × Z · · · Z π ≧ u ≧ u ≧ ··· ≧ u Nn ≧ V α σ ( u ) · · · V α σNn ( u Nn ) du · · · du Nn . Here we have set a ( j ) Nn ( σ , · · · , σ Nn ) = 1 Z j ( t ) Tr π j o t,j exp − πi r r ∗ r N X j =1 P ω j ⊗ h j ! e σ e σ · · · e σ Nn ! . The coefficients a ( j ) Nn vanishes unless n = |{ j | σ j = s }| for s ∈ Z N , and behaves like a ( j ) Nn ∼ O ( t n ). The coefficients a ( j ) Nn are determined by the commutation relations of theBorel subalgebra U q ( b n − ) and the cyclic property of the trace, hence the specific choiceof representation π j , π j is not significant as long as it converges. In [12, 13] we haveconstructed the elliptic version of the integral of the currents, Z · · · Z π ≧ u ≧ u ≧ ··· ≧ u Nn ≧ V α σ ( u ) V α σ ( u ) · · · V α σNn ( u Nn ) du du · · · du Nn . In the previous section, we show that the T - Q operators commute with each other. Inthis section we give conjecturous functional relations of the T - Q operators, which coincide11ith the previous work [34] upon N = 3 specialization. We have checked those functionalrelations up to the order O ( t ) in appendix. Some of similar formulae have been obtainedin the context of the solvable lattice models associated with U q ( c sl N ) [38, 39, 40]. At theend of this section we summarize conclusion. The T -operator is written by determinant of the Q -operators. Let us set the Youngdiagram µ = ( µ , µ , · · · , µ N ), ( µ j ≧ µ j +1 ; µ j ∈ N ). Using the same character as theYoung diagram µ , we represent the highest weight µ = µ Λ + · · · + µ N Λ N . We set c = Q ≦ j 3. We have to give separate definitions of the bosons, the q -oscillator and thescreening currents for N = 2, [28, 29, 30]. This Baxter’s T - Q relation (4.7), (4.8) coincideswith those of [38] for N ≧ 3. In [38], I.Krichever, O.Lipan, P.Wiegmann and A.Zabrodingave the conjecture that the standard objects of quantum integrable models are identifiedwith elements of classical nonlinear integrable difference equation. For simplest examplethey showed that the fusion rules for quantum transfer matrices coincide with the Hirota-Miwa’s bilinear difference equation [45, 46] (the discrete KP). They derived higher-rankgeneralization of Baxter’s T - Q relation by analysing the Hirota-Miwa’s bilinear differenceequation (classical nonlinear integrable difference equation), too. In this paper, we derivethe same Baxter’s T - Q relation by analysing the quantum field theory of the KP (quantumintegrable model). Hence this paper give a supporting argument of the conjecture onquantum and classical-discrete integrable models, by I.Krichever, O.Lipan, P.Wiegmannand A.Zabrodin [38]. As the consequence of (4.5) and (4.6), we have the bilinear formulaeof the T -operator (4.9) and (4.10).( − ( N − N − c T m Λ ( t ) = N X s =1 ( − s +1 c s Q s ( tq m + N − ) Q s ( tq − ) , (4.9)( − ( N − N − c T m Λ ( t ) = N X s =1 ( − s +1 c s Q s ( tq − m − N +1 ) Q s ( tq ) , (4.10)and ( − ( N − N − c T m (Λ + ··· +Λ N − ) ( t ) = N X s =1 ( − N + s c s Q s ( tq − m − ) Q s ( tq N − ) , (4.11)( − ( N − N − c T m (Λ + ··· +Λ N − ) ( t ) = N X s =1 ( − N + s c s Q s ( tq m +1 ) Q s ( tq − N +1 ) . (4.12)14s a consequence of the determinant formulae (4.1) and (4.2), we have the Jacobi-Trudiformulae of the T -operator. For the Young-diagram µ = ( µ , µ , · · · , µ N − , T µ ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ ( µ ′ ) ( t ) · · · τ ( µ ′ + j − ( tq j − ) · · · τ ( µ ′ + l ( µ ′ ) − ( tq l ( µ ′ ) − ) · · · · · · · · · · · · · · · τ ( µ ′ i − i +1) ( t ) · · · τ ( µ ′ i − i + j ) ( tq j − ) · · · τ ( µ ′ i − i + l ( µ ′ )) ( tq l ( µ ′ ) − ) · · · · · · · · · · · · · · · τ ( µ ′ l ( µ ′ ) − l ( µ ′ )+1) ( t ) · · · τ ( µ ′ l ( µ ′ ) − l ( µ ′ )+ j ) ( tq j − ) · · · τ ( µ ′ l ( µ ′ ) ) ( tq l ( µ ′ ) − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.13)Here we have set µ ′ = ( µ ′ , µ ′ , · · · , µ ′ N ) the transpose Young-diagram of µ , and l ( µ ′ ) = µ . We have set τ ( s ) ( t ) = T Λ + ··· +Λ s ( t ). We have τ (0) ( t ) = τ ( N ) ( t ) = 1. The aboveconjecturous functional relations of the T - Q operators, (4.1), (4.2), (4.3), (4.4), (4.5),(4.6), (4.7), (4.8), (4.9), (4.10), (4.11), (4.12), (4.13), coincide with the previous work [34]upon N = 3 specialization. In this paper we present q -oscillator representation of the Borel subalgebra U ′ q ( c sl N ), (2.2).By using this q -oscillator representation, we give the free field realization of the Baxter’s Q -operator Q j ( t ) , Q j ( t ) with j = 1 , , · · · , N , for the W N -algebra, (3.4), (3.5), (3.6). Thecommutativity of the Q -operator is direct consequence of the Yang-Baxter equation. Wegive conjecturous determinant formulae of the T - Q operator for the W N -algebra, (4.1),(4.2), (4.5), (4.6), which produce the higher-rank W N -generalization of the Baxter’s T - Q relation, (4.7), (4.8). We have checked these determinant formulae of the T - Q opera-tor, (4.1), (4.2), (4.5), (4.6) up to the order O ( t ) in appendix. Because the scheme offuntional relations works well, we conclude that the number of the Q -operators for the W N -algebra, is just 2 N , ( N ≧ W N -algebra. V.Bazhanov, A.Hibberd and S.Khoroshkin [34]gave proof of the determinant formulae for the W -algebra. Their proof is based on thetrace of the universal L -operator over Verma module, and the Bernstein-Gel’fand-Gel’fand(BGG) resolution. Because we have already established conjecturous determinant formu-lae, higher-rank generalization of complete proof seems calculation problem. However itis not so easy. 15 cknowledgements The author would like to thank Prof.V.Bazhanov and Prof.M.Jimbo for useful communica-tions. The author would like to thank Institute of Advanced Studies, Australian NationalUniversity for the hospitality during his visit to Canberra in March 2008. The authorwould like to thank Prof.P.Bouwknegt, Prof.A.Chervov, Prof.V.Gerdjikov, Prof.K.Hasegawaand Prof.V.Mangazeev for their interests in this work. This work is partly supported byGrant-in Aid for Young Scientist B (18740092) from JSPS. A Supporting Arguments In this appendix we give some supporting arguments on conjecturous formulae of thedeterminant formulae (4.1), (4.2), (4.3), (4.4), (4.5), (4.6). We check those determinantformulae up to the order O ( t ). At first we prepare the Taylor expansion of A j ( t ) , A j ( t ).Let us set π j = π j = π +1 ⊗ · · · ⊗ π + N − . Taking the trace for the basis {| n , n , · · · , n N − i =( H ∗ ) n ( H ∗ ) n · · · ( H ∗ N − ) n N − | i + ⊗ · · · ⊗ | i + } n ,n , ··· ,n N − ∈ N , we have Z j ( t ) = Tr π j exp − πi r r ∗ r N − X k =1 ( P ¯ ǫ j − P ¯ ǫ j + k ) ⊗ H k !! = N Y k =1 k = j (cid:18) − z k z j (cid:19) − , with j = 1 , , · · · , N . As the same manner as the above, we have Z j ( t ) = N Y k =1 k = j (cid:18) − z j z k (cid:19) − . Let us set a i , a i by A i ( t ) = 1 + a i t + O ( t ) , A i ( t ) = 1 + a i t + O ( t ) . Let us set J ( n ) k ,k , ··· ,k N = Z · · · Z π ≧ u ≧ u ≧ ··· ≧ u Nn ≧ V k ( u ) V k ( u ) · · · V k N ( u N ) × V k ( u N +1 ) V k ( u N +2 ) · · · V k N ( u N ) · · ·× V k ( u N ( n − ) V k ( u N ( n − ) · · · V k Nn ( u Nn ) du du · · · du Nn . J (1)1 , , ··· ,N in a i . We haveTr π i o t,i exp − πi r r ∗ r N − X k =1 P ω k ⊗ h k ! e e e · · · e N !! × J (1)1 , , ··· ,N = t ( q − q − ) N − q N − × N − Y k =1 Tr π + k exp − πi r r ∗ r ( P ¯ ǫ i − P ¯ ǫ i + k ) ⊗ H k ! E k E ∗ k ! × J (1)1 , , ··· ,N . Taking the trace and dividing Z i ( t ), we have a i = q N − z N − i z ( q − q − ) N Y k =1 k = i ( q z i − z k ) × J (1)1 , , ··· ,N + · · · , with i = 1 , , · · · , N . As the same manner as the above, we have a i = ( − N q N z N − i z ( q − q − ) N Y k =1 k = i ( q z k − z i ) × J (1)1 , , ··· ,N + · · · , with i = 1 , , · · · , N . Let us check the determinant relations between Q i ( t ) and Q i ( t ),(4.5) and (4.6). We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ( tq N − ) Q ( tq N − ) · · · Q ( tq − N +2 ) · · · · · · · · · · · · Q i − ( tq N − ) Q i − ( tq N − ) · · · Q i − ( tq − N +2 ) Q i +1 ( tq N − ) Q i +1 ( tq N − ) · · · Q i +1 ( tq − N +2 ) · · · · · · · · · · · · Q N ( tq N − ) Q N ( tq N − ) · · · Q N ( tq − N +2 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = t − √ rr ∗ P ¯ ǫi Y ≦ j 17e have t − √ rr ∗ P ¯ ǫi c i − N q N z N − i z ( q − q − ) N Y k =1 k = i ( z k q − z i ) × J (1)1 , , ··· ,N × t + · · · , which coincides with leading terms of Q i ( t ). As the same argument as the above, thecoefficients of J (1) k ,k , ··· ,k N coincide with each other up to the order O ( t ). Now we havechecked the determinant formulae (4.5) and (4.6) up to the order O ( t ). For the secondwe check the quantum Wronskian condition (4.3) and (4.4) up to the order O ( t ). Wehave Taylor expansion of determinant of Q j ( t ), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ( tq N − ) Q ( tq N − ) · · · Q ( tq − N +1 ) Q ( tq N − ) Q ( tq N − ) · · · Q ( tq − N +1 ) · · · · · · · · · · · · Q N ( tq N − ) Q N ( tq N − ) · · · Q N ( tq − N +1 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Y ≦ j Ann.Phys. Ann.Phys. Ann.Phys. Ann.Phys. Russian.Math.Surveys :5, 11-68,(1979).[7] K.Fabricius and B.McCoy: New Development in the eight vertex model, J.Stat.Phys. , 323-337, (2003).[8] K.Fabricius and B.McCoy: Functional equations and fusion matrices for the eightvertex model, Pub.Res.Inst.Math.Sci. , 905-932, (2004).[9] K.Fabricius and B.McCoy : An elliptic current operator for the eight vertex model, J.Phys. A39 , 14869-14886, (2006).[10] K.Fabricius : A new Q -matrix in the eight vertex model, J.Phys. A40 , 4075-4086,(2007).[11] V.Bazhanov and V.Mangazeev: Analytic theory of the eight-vertex model, Nucl.Phys B775 [FS] 225-282, (2007).[12] B.Feigin, T.Kojima, J.Shiraishi and H.Watanabe: The Integrals of Motion for theDeformed W -Algebra W q,t ( c sl N ), Proceeding for Representation Theory 2006 , Atami,Japan (2006).[13] T.Kojima and J.Shiraishi: The Integrals of Motion for the Deformed W -Algebra W q,t ( c sl N ) II: Proof of commutation relations, [arXiv:0709.2305], to appear in Com-mun.Math.Phys. [14] E.Sklyanin : The Quantum Toda Chain, Lec.Notes.Phys. , 196-223, (1985).[15] E.Sklyanin : Functional Bethe Ansatz, Integrable and Superintegrable systems, 8-33,World Sci.Publ.Teaneck,N, 1990. 2116] E.Sklyanin : Separation of Variables - New trends. Quantum field theory, integrablemodels and beyond (Kyoto), Prog.Theoret.Phys.Suppl. , 35-60, (1995).[17] V.Kuznetsov V.Mangazeev and E.Sklyanin : Q -operator and factorized separationchain for Jack polynomials, Indag.Math. , 451-482, (2003).[18] V.Pasquier and M.Gaudin : The periodic Toda chain and matrix generalization ofthe Bessel function recursion relations, J.Phys. A25 , 5243-5252, (1992).[19] S.Derkachov: Baxter’s Q -operator for the homogeneous XXX spin chain, J.Phys. A32 , 5299-5316, (1999).[20] S.Derkachov, G.Korchemsky and A.Mansahov : Noncompact Heisenberg spin mag-nets from high-energy QCD I: Nucl.Phys. B617 , 375-440, (2001). Baxter Q -operatorand Separation of Variables,[21] S.Derkachov, G.Korchemsky and A.Mansahov : Baxter Q -operator and Separation ofVariables for the open SL (2 , R ) spin chain, J.High.Energy.Phys. , no.10, paper053, 31pp (electronic), (2003).[22] S.Derkachov, D.Karakhanyan and R.Kirschner : Baxter Q -operators of the XXZchain and R -matrix factorization, Nucl.Phys. B738 , 368-390, (2006).[23] S.Derkachov, A.Manashov : R -matrix and Baxter operators for the noncompactquantum sl ( N, C ) invariant spin chain, SIGMA Symmetry Integrability Geom.MthodsAppl. , paper 084, 20pp (electronic), (2006).[24] A.Belisty, S. Derkachov, G.Korchemsky and A.Manashov: The Baxter Q -operatorfor the graded SL (2 | 1) spin chain, J.Stat.Mech.Theory Exp. , paper1005, 63pp(electronic), (2007).[25] C.Korff: A Q -OperatorIdentity for the Correlation Functions of the infinite XXZspin-chain, J.Phys. A39 , 3203-3219, (2006).[26] C.Korff: A Q -operator for the quantum transfer matrix, J.Phys. A40 , 3749-3774,(2007). 2227] A.Bytsko and J.Teschner : Quantization of models with non-compact quantum groupsymmetry, Modular XXZ magnet and lattice sinhGordon model, J.Phys. A39 , 12927-12981, (2006).[28] V.Bazhanov, S.Lukyanov and Al.Zamolodchikov: Integrable structure of conformalfield theory : Quantum KdV Theory and Thermodynamic Bethe Ansatz, Com-mun.Math.Phys. Commun.Math.Phys. Commun.Math.Phys. Q -operator of Conformal Field Theory, J.Stat.Phys. ,567-576, (2001).[32] M.Rossi and R.Weston : A generalized Q -operator for U q ( c sl )-vertex model, J.Phys. A35 , 10015-10032,(2002).[33] P.Dorey and R.Tateo: Anharmonic oscillators, the thermodynamic Bethe Ansatz andnonlinear integral equation, J.Phys. A32 L419-l425, (1999).[34] V.Bazhanov, A.Hibberd and S.Khoroshkin: Integrable structure of W ConformalField Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory, Nucl.Phys. B622 N = 1 KdV hierarchyII: The Q -operator, Nucl.Phys. B709 , 578-591, (2005).[36] A.Antonov and B.Feigin : Quantum Group Representation and Baxter Equation, Phys. Lett. B392 , 115-122, (1997).[37] Y.Asai, M.Jimbo, T.Miwa,A and Ya.Pugai, Bosonization of vertex operators for A (1) n − face model, J.Phys. A29 , 6595-6616, (1996).2338] I.Krichever, O.Lipan, P.Wiegmann and A.Zabrodin, Quantum integrable models anddiscrete classical Hirota equations, Commun.Math.Phys. J.Phys. A23 Int.J.Mod.Phys. A9 Commun.Math.Phys. , 263-281, (2007).[42] H.Boos, M.Jimbo, T.Miwa, F.Smirnov and Y.Takeyama, Hidden Grassmann Struc-ture in the XXZ model II: Creation Operators, [arXiv:0801.1176].[43] A.Chervov and G.Falqui : Manin matrices and Talalaev’s formula, [arXiv:0711.2236][44] V.Fateev and S.Lukyanov : The Models of Two-Dimensional Conformal QuantumField Theory with Z n Symmetry, Int.J.Mod.Phys. A3 , 507-520, (1988).[45] R.Hirota : Discrete analogue of a generalized Toda equation, J.Phys.Soc.Japan ,3785-3791, (1981).[46] T. Miwa : On Hirota’s difference equations, Proc. Japan Acad.58