The Baxters Q-operator for the W-algebra $W_N$

The Baxter’s Q - op erator for the W -algebra W N August 28, 2021 T ak eo K OJIMA Dep artment of Mathematics, Col le ge of Scienc e and T e chnolo gy, Nihon University, Surugadai, Chiyo da- ku, T okyo 10 1-0062, JAP AN Abstract The q -oscillator representati on for the Borel subalgebra of the affine symmetry U ′ q ( d sl N ) is p resen ted. By means of this q -oscillator r ep resen tation, we giv e the free field realizations of th e Baxter’s Q -op erator Q j ( λ ) , Q j ( λ ) , ( j = 1 , 2 , · · · , N ) for the W -alg ebra W N . W e g iv e the fu n ctional r elations of the T - Q op erators, including the higher-rank generaliztio n of the Baxter’s T - Q relation. Key W ords : CFT, q - o scillator, Q - o p erator, functional relation, fr ee field realization 1 1 In t ro duction The Baxter’s T - Q -op erato r hav e v arious e xceptional prop erties and play an import a n t role in man y asp ect of the theory of integrable systems . Originally the Q -op erator w as in tro duced by R.Baxter [1], in terms of some sp ecial transfer matrix of the 8- v ertex mo del. Ov er the last three decades, this metho d o f the Q -op erator has b een dev elop ed b y man y literatures. W e w ould lik e to refer some of the se literatures, writ- ten by R.Baxter [2, 3, 4 , 5], b y L.T akh tadzhan and L.F addeev [6 ], by K.F abricius and B.McCo y [7, 8, 9], b y K.F abricius [10], by V.Bazhano v and V.Mangazeev [11], b y B.F eigin, T.Ko jima, J.Shiraishi and H.W atanab e [12], by T.Ko jima and J.Shiraishi [13]. How ev er a full theory of the Q -op erator for the 8 -v ertex mo del is not yet dev elop ed. F o r the simpler mo dels asso ciated with the quantum group U q ( g ), there hav e b een man y pap ers whic h extend, generalize, and comme n t on the T - Q relation. W e would lik e to refer some of these literatures, including Skly a nin’s separation v ariable method, written b y E.Skly anin [14, 15, 16], by V.Kuzunetso v, V.Mangazeev and E.Skly anin [17], b y V.P asquier a nd M.G a udin [18], by S.Derk ac ho v [19] b y S.D erk ac ho v, G .Karakhan y an and A.Mansaho v [20, 21] b y S.Derk ac ho v, G.Karakhan y an and R.Kirsc hner, [22] b y S.Derk ac hov and A.Mansahov [23], b y A.Belist y , S.D erk ac ho v, G .Korc hemesky and A.Manasahov [24], by C.Korff [25, 26], b y A.Bytsk o and J.T esc hner [27], b y V.Bazhano v, S.Lukyano v and Al.Zamolo dc hik o v [2 8, 29, 30 , 31], by M.Rossi and R.W eston [32], b y P .Dorey and R.T ateo [33], b y V.Bazhano v, A.Hibb erd and S.Khoroshkin [3 4], by P .Kulish and Z.Zeitlin [35], b y A.Antono v a nd B.F eigin [36], b y I.Kric hev er, O.Lipan, P .Wiegmann and A.Z a bro din [38], by V.Bazhano v and N.Reshetikhin [39 ], b y A.Kuniba, T.Nak anishi and J.Suzuki [40], by H.Bo os, M.Jim b o, T.Miw a, F.Smirnov and Y.T akey ama [41, 42], by A.Cherv o v and G.F alqui [43]. Eac h pa- p er added to o ur understanding o f the great Baxter’s original pap er [1]. E sp ecially fo r example the T - Q - op erators acting on the F o c k space of the Virasoro alg ebra V ir were in tro duced by V.Bazhanov, S.Luky ano v and Al.Zamolo dchik o v [28, 29, 30 ]. They de- riv ed v arious functional r elat io ns of the T - Q op erators and g a v e the a symptotic behavior of the eigen-v alue of the T - Q op erators. P .Dorey and R.T ateo [33] rev ealed the hidden connection b et we en the v acuum exp ectation v alue of the Q - op erator a nd the s p ectral determinan t for Sc hr¨ odinger equation. V.Bazhano v, A .Hibb erd and S.Kho roshkin [34] ac hiev ed the W 3 -algebraic g eneralization of [28, 29, 30, 31, 33]. In this pap er w e study 2 the higher-rank W N -generalization of [34]. W e study t he T - Q -op erators acting on the F o ck space of the W -algebra W N . W e give the free field realization of the Q - op erator and functional relations of the T - Q - op erators for the W - algebra W N , including the higher-rank generalization of the Baxter’s T - Q relation, Q i ( tq N ) + N − 1 X s =1 ( − 1) s T Λ 1 + ··· +Λ s ( tq − 1 ) Q i ( tq N − 2 s ) + ( − 1) N Q i ( tq − N ) = 0 , Q i ( tq − N ) + N − 1 X s =1 ( − 1) s T Λ 1 + ··· +Λ s ( tq ) Q i ( tq − N +2 s ) + ( − 1) N Q i ( tq N ) = 0 , where i = 1 , 2 , · · · , N . The organization of this pap er is as following. In section 2, w e giv e basic definitions, including q -oscillator represen tation of the Borel subalgebra of the affine symmetry U ′ q ( c sl N ), whic h pla y an essen tial ro le in construction of the Q - op erator. In section 3, w e giv e the definition of the T -op erato r and the Q -op erator. In section 4 w e giv e conjecturous fun tional relations b et we en the T -o p eartor and the Q - op erator, including Baxter’s T - Q relation. In app endix, w e giv e supp orting a rgumen ts on conjecturous form ulae stated in section 4. 2 Basic Defini tion In this section w e give the differen t realizations of t he Borel subalgebra of the affine quan tum algebra U ′ q ( c sl N ), whic h will pla y an imp o r t an t r o le in construction of the Baxter’s T - Q op erator. Let us fix the in teger N ≧ 3. Let us fix a complex n um b er 1 < r < N . In this pap er, up on this setting, we construct the Baxter’s T - Q op erators on the space of the W -algebra W N with the central charge −∞ < C C F T < − 2, where C C F T = ( N − 1 )  1 − N ( N + 1) r ( r − 1)  . Becuse C C F T → −∞ represen ts t he classical limit, w e call −∞ < C C F T < − 2 “quasi- classical domain”. By anlytic con tin uation, it is p ossible to extend our theory to the CFT with cen t r a l charge C C F T < 1. W e w ould lik e to note that the unitary minimal CFT is describ ed b y t he cen tral c harge C C F T = ( N − 1)  1 − N ( N +1) r ( r − 1)  for N , r ∈ Z , ( N ≧ 2 , r ≧ N + 2) [44]. W e set par a meters r ∗ = r − 1 a nd q = e 2 π i r ∗ r . In what follo ws w e use the q -integer [ n ] q = q n − q − n q − q − 1 . 3 2.1 The q -oscillator represen tation Let { ǫ j } b e an orthonorma l basis of R N , relativ e to the standard inner pro duct ( ǫ i | ǫ j ) = δ i,j . Let us set ¯ ǫ j = ǫ j − ǫ where ǫ = 1 N P N j =1 ǫ j . W e hav e ( ¯ ǫ i | ¯ ǫ j ) = δ i,j − 1 N . Let us set the simple ro ots α j = ¯ ǫ j − ¯ ǫ j +1 , (1 ≦ j ≦ N − 1) and α N = − P N − 1 j =1 α j . Let us set the fundamen tal w eights ω j as the dual v ector of α j , i.e. ( α i | ω j ) = δ i,j . Explicitly we ha v e ω j = ¯ ǫ 1 + · · · + ¯ ǫ j . Let us set the we ight lattice P = ⊕ N j =1 Z ¯ ǫ j . W e consider the quan tum affine algebra U ′ q ( c sl N ), which is generated b y e 1 , · · · , e N , f 1 , · · · , f N , and h 1 , · · · , h N , with the defining r elat io ns, [ 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 − 1 , e 2 i e j − [2] q e i e j e i + e j e 2 i = 0 , f 2 i f j − [2] q f i f j f i + f j f 2 i = 0 , fo r ( α i | α j ) = − 1 . Here (( α j | α k )) 1 ≦ j,k ≦ N is the Cartan matrix o f t yp e c sl N . L et us in tr o duce the Borel sub- algebra of U ′ q ( c sl N ). The Borel subalgebra U ′ q ( b b + ) is generated b y e 1 , · · · , e N , h 1 , · · · , h N , and U ′ q ( b b − ) by f 1 , · · · , f N , h 1 , · · · , h N . In this pap er we consider the lev el c = 0 case, with the central elemen t c = h 1 + · · · + h N . Let us in tro duce the q -oscillator represen ta tion o t of the Borel subalgebra U ′ q ( b b + ). The q - oscillator algebra O sc j , (1 ≦ j ≦ N − 1), is generated b y elemen ts 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 − 1 E ∗ j E j = 1 q − q − 1 . (2.1) Let us se t O sc = O sc 1 ⊗ C · · · ⊗ C O sc N − 1 . W e hav e [ E j , E k ] = 0, [ E ∗ j , E ∗ k ] = 0, [ E j , E ∗ k ] = 0, [ H j , H k ] = 0 for j 6 = k . Let us se t the a uxiliarry op erator H N = −H 1 − H 2 − · · · − H N − 1 . W e define homomor phism o t : U ′ q ( b b + ) → O sc b y o t ( e 1 ) = tq 1 2 ( q − q − 1 ) q −H 2 E ∗ 1 E 2 , o t ( e 2 ) = q 1 2 ( q − q − 1 ) q −H 3 E ∗ 2 E 3 , · · · o t ( e N − 2 ) = q 1 2 ( q − q − 1 ) q −H N − 1 E ∗ N − 2 E N − 1 , o t ( e N − 1 ) = E ∗ N − 1 , o t ( e N ) = q −H 1 −H N E 1 , (2.2) o t ( h 1 ) = −H 1 + H 2 , o t ( h 2 ) = −H 2 + H 3 , · · · , o t ( h N ) = −H N + H 1 . 4 This q -oscillator represen ta tion o t satisfies lev el zero condition o t ( h 1 + h 2 + · · · + h N ) = 0. This q - oscillator represen tation giv e a higher-rank generalization o f those in [34]. By means of the Dynkin-diagram automorphism τ , σ , w e construct a family of the q -oscillator represen tatio n o t,j , ¯ o t,j . Let us set the Dynkin-diagram automorphism τ of the affine algebra U ′ q ( c sl N ). τ ( e 1 ) = e 2 , · · · , τ ( e j ) = e j +1 , · · · , τ ( e N ) = e 1 , τ ( h 1 ) = h 2 , · · · , τ ( h j ) = h j +1 , · · · , τ ( h N ) = h 1 , τ ( f 1 ) = f 2 , · · · , τ ( f j ) = f j +1 , · · · , τ ( f N ) = f 1 . Let us set the Dynkin-diagra m automorphism σ of the finite simple algebra U q ( sl N ), generated b y e 2 , · · · , e N , h 2 , · · · , h N , f 2 , · · · , f N . σ ( e 2 ) = e N , · · · , σ ( e j ) = e N +2 − j , · · · , σ ( e N ) = e 2 , σ ( h 2 ) = h N , · · · , σ ( h j ) = h N +2 − j , · · · , σ ( h N ) = h 2 , σ ( f 2 ) = f N , · · · , σ ( f j ) = f N +2 − j , · · · , σ ( f N ) = f 2 , and σ is esxtended to the affine verte x as σ ( e 1 ) = e 1 , σ ( h 1 ) = h 1 , σ ( f 1 ) = f 1 . W e ha v e the action of τ j · σ · τ − 1 , τ j · σ · τ − 1 ( e i ) = e j − 1 − i , τ j · σ · τ − 1 ( h i ) = h j − 1+ i , τ j · σ · τ − 1 ( f i ) = f j − 1 − i , with s, j ∈ Z . W e set ho momorphism o t,j , ¯ o t,j : U ′ q ( b b + ) → O sc , (1 ≦ j ≦ N ), o t,j = o t · τ − j , ¯ o t,j = o ( − 1) N t · τ j · σ · τ − 1 , (2.3) These q - oscillator represen tations o t,j , ¯ o t,j will pla y an imp o rtan t role in construction of the Baxter’s Q -o p erator. 2.2 Ev aluation highest w eigh t represen tation Let us consider the quantum simple algebra U q ( g l N ), whic h is g enerated by E α 1 , · · · , E α N − 1 , H 1 , · · · , H N , and F α 1 , · · · , F α N − 1 , 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 , 5 [ E α i , F α j ] = δ i,j q H i − H i +1 − q − H i + H i +1 q − q − 1 , E 2 α i E α j − [2] q E α i E α j E α i + E α j E 2 α i = 0 , F 2 α i F α j − [2] q F α i F α j F α i + F α j F 2 α i = 0 . Let us set the ro ot vec tors, F α 1 + α 2 = [ F α 2 , F α 1 ] √ q = √ q F α 2 F α 1 − 1 √ q F α 1 F α 2 , ¯ F α 1 + α 2 = [ F α 2 , F α 1 ] 1 √ q = 1 √ q F α 2 F α 1 − √ q F α 1 F α 2 , F α 1 + ··· + α N − 1 = [ F α N − 1 , [ F α N − 2 , · · · , [ F α 2 , F α 1 ] √ q · · · ] √ q ] √ q , F α 1 + ··· + α N − 1 = [[ · · · , [ F α N − 1 , F α N − 2 ] 1 √ q · · · , F α 2 ] 1 √ q , F α 1 ] 1 √ q . Let us set the a uto morhism σ by σ ( E α 1 ) = E α N − 1 , · · · , σ ( E α j ) = E α N − j , · · · , σ ( E α N − 1 ) = E α 1 , σ ( H 1 ) = − H N , · · · , σ ( H j ) = − H N − j +1 , · · · , σ ( H N ) = − H 1 , σ ( F α 1 ) = F α N − 1 , · · · , σ ( F α j ) = F α N − j , · · · , σ ( F α N − 1 ) = F α 1 . W e ha ve the ev aluation represen tatio n ev t , ev t : U ′ q ( c sl N ) → U q ( g l N ), given by ev t ( e 2 ) = E α 1 , · · · , ev t ( e j +1 ) = E α j , · · · , ev t ( e N ) = E α N − 1 , ev t ( h 2 ) = H 1 − H 2 , · · · , ev t ( h j +1 ) = H j − H j +1 , · · · , ev t ( h N ) = H N − 1 − H N , ev t ( f 2 ) = F α 1 , · · · , ev t ( f j +1 ) = F α j , · · · , ev t ( f N ) = F α N − 1 , ev t ( e 1 ) = tF α 1 + α 2 + ··· + α N − 1 q H 1 + H N , ev t ( f 1 ) = t − 1 E α 1 + α 2 + ··· + α N − 1 q − H 1 − H N , ev t ( h 1 ) = H N − H 1 . ev t ( e 2 ) = E α 1 , · · · , ev t ( e j +1 ) = E α j , · · · , ev t ( e N ) = E α N − 1 , ev t ( h 2 ) = H 1 − H 2 , · · · , ev t ( h j +1 ) = H j − H j +1 , · · · , ev t ( h N ) = H N − 1 − H N , ev t ( f 2 ) = F α 1 , · · · , ev t ( f j +1 ) = F α j , · · · , ev t ( f N ) = F α N − 1 , ev t ( e 1 ) = tF α 1 + α 2 + ··· + α N − 1 q − H 1 − H N , ev t ( f 1 ) = t − 1 E α 1 + α 2 + ··· + α N − 1 q H 1 + H N , ev t ( h 1 ) = H N − H 1 . W e ha v e the conjugat io n ev t = σ · ev ( − ) N t · σ − 1 . W e set the irreducible highest represen- tation of U q ( g l N ) with the highest w eigh t λ = m 1 Λ 1 + · · · + m N Λ N , the highest w eigh t v ector | 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 ) . 6 In what follo ws w e consider the case m j − m j +1 ∈ N , (1 ≦ j ≦ N − 2). In this case the represen tatio n π ( λ ) is finite dimension. Let us set the ev aluation highest w eight 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 ev aluatio n highest we ight represen tation will pla y an imp ortant role in construction of the T -op erat or T λ ( t ) , T λ ( t ). 2.3 Screening curr ent Let us introduce b osons B i m , ( m ∈ Z 6 =0 ; i = 1 , 2 , · · · , N − 1) b y [ B i m , B j n ] = mδ m + n, 0 ( α i | α j ) r − 1 r , (1 ≦ i, j ≦ N − 1) . (2.4) Let us set B N m = − P N − 1 j =1 B j m . W e hav e the commutation relation [ B i m , B j n ] = mδ m + n, 0 ( α i | α j ) r − 1 r , for 1 ≦ i, j ≦ N . Let us set the zero-mo de op erators P λ and Q λ , ( λ ∈ P = ⊕ j Z ¯ ǫ j ) by [ P λ , iQ µ ] = ( λ | µ ) . (2.5) Let us set the Heise n b erg alg ebra B generated b y B 1 m , · · · , B N − 1 m , P λ , Q λ , ( λ ∈ P ) and its completion b B . Let us set the F o c k space F l,k b y B j m | l , k i = 0 , ( m > 0) (2.6) P α | l , k i = α      r r r − 1 l − r r − 1 r k ! | l , k i , (2.7) | l , k i = e i √ r r − 1 Q l − i √ r − 1 r Q k | 0 , 0 i . (2.8) Let us set the screening curren ts of the W -a lgebra W N b y V α j ( u ) = exp i r r ∗ r Q α j ! exp r r ∗ r P α j iu ! × exp X m> 0 1 m B j − m e imu ! exp − X m> 0 1 m B j m e − imu ! , ( 1 ≦ j ≦ N ) . (2 .9 ) Here w e ha v e added one op erator V α N ( u ), whic h lo oks lik e affinization of the classical A N − 1 . W e can find the elliptic deformation of V α j ( u ) for j 6 = N in [1 2, 13]. F or Re( u 1 ) > 7 Re( u 2 ), w e ha v e V α j ( u 1 ) V α j ( u 2 ) = : V α j ( u 1 ) V α j ( u 2 ) : ( e iu 1 − e iu 2 ) 2 r ∗ r , ( 1 ≦ j ≦ N ) , V α j ( u 1 ) V α j +1 ( u 2 ) = : V α j ( u 1 ) V α j +1 ( u 2 ) : ( e iu 1 − e iu 2 ) − r ∗ r , ( 1 ≦ j ≦ N ) , V α j +1 ( u 1 ) V α j ( u 2 ) = : V α j +1 ( u 1 ) V α j ( u 2 ) : ( e iu 1 − e iu 2 ) − r ∗ r , ( 1 ≦ j ≦ N ) . By analytic contin uation, w e ha v e V α i ( u 1 ) V α j ( u 2 ) = q ( α i | α j ) V α j ( u 2 ) V α i ( u 1 ) , (1 ≦ i, j ≦ N ) . (2.10) Let us set z j = exp − 2 π i r r ∗ r P ¯ ǫ j ! , ( 1 ≦ j ≦ N ) . (2.11) W e ha ve z 1 z 2 · · · z N = 1 and V α i ( u + 2 π ) = z − 1 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 t he nilp oten t subalgebra U ′ q ( b n − ) generated by f 1 , f 2 , · · · , f N . W e hav e homo - morphism sc : U ′ q ( b n − ) → b B given b y sc ( f j ) = 1 q − q − 1 Z 2 π 0 V α j ( u ) du, (1 ≦ j ≦ N ) . 3 Baxter’s Q -op erator In this section w e define the Baxter’s T - Q op erat o r b y means of the tra ce of the univ ersal R , and presen t conjecturous functional relations o f the T - Q op erator, which include t he higher-rank generalization of the Baxter’s T - Q relation. 3.1 L -op erator Let us set the unive rsal L - op erator L ∈ b B ⊗ U q ( b n − ) by L = exp − π i r r ∗ r N X j =1 P ω j ⊗ h j ! P exp  Z 2 π 0 K ( u ) du  . (3.1) Here we hav e set K ( u ) = N X j =1 V α j ( u ) ⊗ e j . 8 Here P exp  R 2 π 0 K ( u ) du  represen ts t he path ordered exponen tial P exp  Z 2 π 0 K ( u ) du  = ∞ X n =0 Z · · · Z 2 π ≧ u 1 ≧ u 2 ≧ ·· · ≧ u n ≧ 0 K ( u 1 ) K ( u 2 ) · · · K n ( u n ) du 1 du 2 · · · du n . The ab o v e inte grals con v erge in “quasi-classical domain” −∞ < C C F T < − 2. F or the v alue of C C F T outside the quasi-classical domain, the in tegrals should b e understo o d as analytic con t inuation. L et us set U q ( c sl N ) the extension of U ′ q ( c sl N ) b y the degree op erat o r d . Let us set U q ( b n ± ) the extension of U ′ q ( b n ± ) b y the degree op erator d . There exists the unique univ ersal R -ma t r ix R ∈ U q ( b n + ) ⊗ U q ( b n − ) satisfying the Y ang-Baxter equation. R 12 R 13 R 23 = R 23 R 13 R 12 . The univ ersal- R ’s Cartan elemen ts t is factored as R = q t R , t = N − 1 X j =1 h j ⊗ h j + c ⊗ d + d ⊗ c, where ( h i | h j ) = δ i,j . W e call the elemen t R ∈ U ′ q ( b n + ) ⊗ U ′ q ( b n − ) the reduced univ ersal R -matrix. The L -op erato r is an image of the reduced R -matrix [34], L = ( sc ⊗ id ) ( R ) . The L -op erator will play an imp ortan t role in trace construction of the T - Q o p erator. 3.2 T-op erator Let us set the T -o p erator T λ ( t ) and T λ ( t ) b y T λ ( t ) = T r π ( λ ) t exp − π i r r ∗ r N X j =1 P ω j ⊗ h j ! L ! , (3.2) T λ ( t ) = T r π ( λ ) t exp − π i 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 1 ,λ 2 ( t 1 /t 2 ) = π ( λ 1 ) t 1 ⊗ π ( λ 2 ) t 2 ( R ). W e ha ve so- called RLL relation, R λ 1 ,λ 2 ( t 1 /t 2 ) L λ 1 ( t 1 ) L λ 2 ( t 2 ) = L λ 2 ( t 2 ) L λ 1 ( t 1 ) R λ 1 ,λ 2 ( t 1 /t 2 ) . 9 Multiplying the R -matrix R λ 1 ,λ 2 ( t 1 /t 2 ) − 1 from the right, and taking t r ace, w e ha v e the comm utation relation, [ T λ 1 ( t 1 ) , T λ 2 ( t 2 )] = [ T λ 1 ( t 1 ) , T λ 2 ( t 2 )] = [ T λ 1 ( t 1 ) , T λ 2 ( t 2 )] = 0 . The co efficien ts of the T aylor expansion of T λ ( t ) comm ute with eac h other. Hence w e ha v e infinitly man y comm utative op erators, whic h giv e quantum deformation of the con- serv ation la ws of the N - t h KdV equation. 3.3 Q-op erato r Let us set the F o c k represen tation π ± j : O sc j → W ± with j = 1 , 2 , · · · , N − 1, W + = ⊕ k ≧ 0 C | k i + , W − = ⊕ k ≧ 0 C | k i − . The action is giv en b y π + j ( H j ) | k i + = − k | k i + , π + j ( E j ) | k i + = 1 − q − 2 k ( q − q − 1 ) 2 | k − 1 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 2 k ( q − q − 1 ) 2 | k − 1 i − , π − j ( E j ) | k i − = | k + 1 i − . Let π j and π j b e an y represen tatio n o f the q -oscillator O sc = O sc 1 ⊗ C · · · ⊗ C O sc N − 1 suc h that the partition Z j ( t ) , Z j ( t ) con verge. Z j ( t ) = T r π j o t,j exp − 2 π i r r ∗ r N X j =1 P ω j ⊗ h j !! , Z j ( t ) = T r π j o t,j exp − 2 π i r r ∗ r N X j =1 P ω j ⊗ h j !! . Let us set the o p erators A j ( t ) and A j ( t ) with j = 1 , 2 , · · · , N A j ( t ) = 1 Z j ( t ) T r π j o t,j exp − π i r r ∗ r N X j =1 P ω j ⊗ h j ! L ! , (3.4) A j ( t ) = 1 Z j ( t ) T r π j o t,j exp − π i r r ∗ r N X j =1 P ω j ⊗ h j ! L ! . (3.5) Let us set the Ba xter’s Q -op erator Q j ( t ) and Q j ( t ) with j = 1 , 2 , · · · , N , Q j ( t ) = t − 1 2 √ r r ∗ P ¯ ǫ j A j ( t ) , Q j ( t ) = t 1 2 √ r r ∗ P ¯ ǫ j A j ( t ) . (3.6) 10 W e w ould lik e to note con v enien t r elat io n, N X k =1 P ω k ⊗ o t,j ( h k ) = N − 1 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 − 1 X k =1 ( P ¯ ǫ j − k − P ¯ ǫ j ) ⊗ H k . Here we should understand the surfix num b er as mo dulus N , i.e. ¯ ǫ j + N = ¯ ǫ j . F rom the Y ang -Baxter equation, we hav e the comm utat io n relations [ Q j 1 ( t 1 ) , Q j 2 ( t 2 )] = [ Q j 1 ( t 1 ) , Q j 2 ( t 2 )] = [ Q j 1 ( t 1 ) , Q j 2 ( t 2 )] = 0 , and [ Q j ( t 1 ) , T λ ( t 2 )] = [ Q j ( t 1 ) , T λ ( t 2 )] = [ Q j ( t 1 ) , T λ ( t 2 )] = [ Q j ( t 1 ) , T λ ( t 2 )] = 0 . The op erators A j ( t ) can b e written as p ow er series. A j ( t ) = 1 + ∞ X n =1 X σ 1 , ··· ,σ N n ∈ Z N a ( j ) N n ( σ 1 , · · · , σ N n ) × Z · · · Z 2 π ≧ u 1 ≧ u 2 ≧ ·· · ≧ u N n ≧ 0 V α σ 1 ( u 1 ) · · · V α σ N n ( u N n ) du 1 · · · du N n . Here we hav e set a ( j ) N n ( σ 1 , · · · , σ N n ) = 1 Z j ( t ) T r π j o t,j exp − 2 π i r r ∗ r N X j =1 P ω j ⊗ h j ! e σ 1 e σ 2 · · · e σ N n ! . The co efficien ts a ( j ) N n v anishes unless n = | { j | σ j = s }| f or s ∈ Z N , and b eha v es lik e a ( j ) N n ∼ O ( t n ). The co efficien ts a ( j ) N n are determined by the comm utation relations of the Borel subalgebra U q ( b n − ) and t he cyclic prop ert y o f the trace, hence the sp ecific c hoice of represen tation π j , π j is not significan t as long as it con v erges. In [12, 13] we ha v e constructed the elliptic v ersion of the inte gral of the currents , Z · · · Z 2 π ≧ u 1 ≧ u 2 ≧ ·· · ≧ u N n ≧ 0 V α σ 1 ( u 1 ) V α σ 2 ( u 2 ) · · · V α σ N n ( u N n ) du 1 du 2 · · · d u N n . 4 F unction al relation s In the previous section, w e sho w that the T - Q op erators comm ute with eac h other. In this sec tion w e giv e conjecturous functional relations of the T - Q o p erators, whic h coincide 11 with the previous w ork [34] up on N = 3 sp ecialization. W e ha v e chec k ed those f unctional relations up to the order O ( t 2 ) in app endix. Some of similar form ulae hav e b een obtained in the con text of the solv able latt ice mo dels associated with U q ( c sl N ) [38, 39, 40]. A t the end of this section w e summarize conclusion. 4.1 F unctional relations The T -op erator is written b y determinant of the Q -op erators. Let us set the Y oung diagram µ = ( µ 1 , µ 2 , · · · , µ N ), ( µ j ≧ µ j +1 ; µ j ∈ N ). Using the same c haracter as the Y oung dia gram µ , w e represen t the highest w eigh t µ = µ 1 Λ 1 + · · · + µ N Λ N . W e set c 0 = Q 1 ≦ j