The integrals of motion for the elliptic deformation of the Virasoro and $W_N$ algebra

The in tegrals of motion for the el l iptic deformation of th e Virasoro and W N algebra T.KOJI MA Dep ar tment of Mathematics, Col le ge of Scienc e and T e chno lo gy, Nihon University, Surugadai, Chiyo d a-ku, T okyo 101-0062 , JA P AN J.SHIRAISHI Gr aduate scho ol of M athematic al Scienc e, University of T okyo, Komab a, M e gr o-ku, T okyo 153-89 14, JAP AN Abstract W e review the free field realization of the deformed Virasoro algebra V ir q ,t and the deformed W algebra W q ,t ( d g l N ). W e explicitly construct t w o classes of infin itly many comm utativ e op erators I m , G m , ( m ∈ N ), in terms of these algebras. Th ey can b e regarded as the elliptic deformation of the lo cal an d nonlo cal in tegrals of motion for the c onformal field th eory [1 ,2 ,3,4,5]. This review is based o n the works [15,16 ,17]. Key wor ds: Exactly Solv ed Mo del, Virasoro algebra, W-algebra, Quantum group, Conformal field theory , Elliptic qu an tum group, Deformed W -algebra 1 In tro duction The Kortew eg-de V ries (KdV) equation o ccupies a cen tral place in the mo dern theory of completely integrable systems. Because of its integrabilit y , the KdV equation has infinitly man y conserv ation la ws. The Hamiltonian aspects of the K dV theory connected it to the conformal field theory . The quan tization of the second Poiss on brac k et { , } P .B . of the KdV giv es rise to the Virasoro algebra : [ L m , L n ] = ( m − n ) L m + n + 1 12 c C F T m ( m 2 − 1) δ m + n, 0 . The quan tum field theory of the KdV theory b ecomes the conformal field theory asso ciated with the Virasoro algebra [1,2,3]. V.Bazhano v, S.Lukyano v, Al.Zamolo dc hiko v [1] constructed quan tum field theoretical a nalogue of the comm uting transfer matrix T ( z ) acting the highest w eight module o f the Vira soro algebra. The comm uting transfer matrix T ( z ) is constructed as the trace o f an image of the univ ersal R -matrix asso ciated with the quantum a ffine symmetry U q ( c sl 2 ). Preprint submitted to Elsevier 13 No vem b er 2018 Hence the commutativit y [ T ( z ) , T ( w )] = 0 is a direct conseque nce of the Y ang-Baxter relatio n. W e call t he co efficien ts of the asymptotic expansion of log T ( z ), ( z → ∞ ), the lo cal in tegr a ls of motion for the Virasoro algebra. They reco v er t he conserv a tion laws of the KdV in the classical limit c C F T → ∞ . W e call the co efficien ts of the T a ylor expansion of T ( z ) the nonlo cal in tegra ls of motion for the Virasoro algebra. See also the generalization to the W N algebra [4,5]. In this pap er we construct t he elliptic deformatio n of the in tegrals of motion for the conformal field theory [1,4,5]. In this pap er w e construct tw o classes o f infinitly man y comm utativ e op erators I m , G m , ( m ∈ N ), asso ciated with the deformed Virasoro algebra and the deformed W - a lgebra W q ,t ( d g l N ). Because it is not so easy to calculate the t r ace of the image of the univers al R - matrix of the elliptic quan tum group, w e prefer the completly differn t metho d of the construction for the in tegrals of motion in the elliptic deformation of the conformal field theory . Instead of considering the transfer matrix T ( z ), we directly giv e the explicit for mulae of the in tegrals of motion I n and G n for the deformed W - algebra W q ,t ( d g l N ). The comm utativity of the in tregrals of motion are not understo o d as direct consequence of the Y ang-Baxter relation. They ar e understo o d as consequence of the commutativ e family of the F eigin- Odesskii algebra [14]. The orga nization of this pap er is as follow s. In section 2 w e g iv e reviews on the deformed Virasoro algebra and the deformed W - algebra W q ,t ( d g l N ). In section 3 w e giv e explicit form ulae of the in tegrals of motion f o r the deformed Virasoro algebra and the deformed W - algebra, and stat e the main theorem. 2 Elliptic deformation of the Virasoro algebra and the W N -algebra In this section we review the elliptic deformation of the Virasoro algebra and the W N -algebra. W e fix three parameters x, r , s suc h that 0 < x < 1 , Re( r ) > 0 and Re( s ) > 0. Let us set r ∗ = r − 1. W e set the para meters τ b y x = exp  − π √ − 1 /r τ  W e relate t wo v a riables z and u by z = x 2 u . The sym b ol [ u ] r stands for the Jacobi theta function [ u ] r = x u 2 r − u Θ x 2 r ( z ) ( x 2 r ; x 2 r ) ∞ , Θ q ( z ) = ( z ; q ) ∞ ( q /z ; q ) ∞ ( q ; q ) ∞ , (1) where ( z ; q ) ∞ = Q ∞ j =0 (1 − q j z ). The elliptic theta function satisfies the quasi- p erio dicities, [ u + r ] r = − [ u ] r , [ u + r τ ] r = − e − π √ − 1 τ − 2 π √ − 1 u/r [ u ] r . (2) 2 The sym b ol [ a ] stands for q -in teger [ a ] = x a − x − a x − x − 1 . 2.1 Bosons F or N = 2 , 3 , 4 , · · · , w e in tro duce the b osons β j m , ( m ∈ Z 6 =0 ; 1 ≦ j ≦ N ), whic h satisfy the comm uttion relation, [ β i n , β j m ] = n [( r − 1) n ] [ r n ] δ n + m, 0 ×      [( s − 1) n ] [ sn ] (1 ≦ i = j ≦ N ) − [ n ] [ sn ] x sn sgn(i − j) (1 ≦ i 6 = j ≦ N ) (3) F or N = 2 , 3 , 4 , · · · , w e in tro duce the zero-mo de op erators P λ and Q λ . Let ǫ j , (1 ≦ j ≦ N ) be an orthonormal basis in R N relativ e to the standard inner pro duct ( | ). Let us set ¯ ǫ i = ǫ i − ǫ, ǫ = 1 N P N j =1 ǫ j . Let us set α j = ¯ ǫ j − ¯ ǫ j +1 . Let P λ , Q λ b e the zero mo de op erators defined by the comm utation relation [ iP λ , Q µ ] = ( λ | µ ) , ( λ, µ ∈ N X j =1 Z ¯ ǫ j ) . (4) The action of the Dynkin-diagram auto morphism η on the b osons is giv en b y η ( β 1 m ) = x − 2 s N m β 2 m , · · · , η ( β N − 1 m ) = x − 2 s N m β N m , η ( β N m ) = x 2 s N ( N − 1) m β 1 m . (5) The action of the Dynkin-diagram automorphism η on the zero-mo de op erator is giv en b y η ( P λ ) = P η ( λ ) , η ( Q λ ) = Q η ( λ ) , η (¯ ǫ j ) = ¯ ǫ j +1 , (1 ≦ j ≦ N ) , (6) where w e understand ¯ ǫ 1 = ¯ ǫ N +1 . Let us in tro duce the F o ck space F l,k , ( l , k ∈ P N j =1 Z ¯ ǫ j ), of the b osons, generated b y β j − m , ( m > 0) o v er the v acuum vec tor | l , k i , ( l , k ∈ P N j =1 Z ¯ ǫ j ), β j − m | l , k i = 0 , ( m > 0; j = 1 , 2 , · · · , N ) , (7) P α | l , k i =   α       l s r r − 1 − k s r − 1 r   | l , k i , (8) | l , k i = exp   s r r − 1 Q l − s r − 1 r Q k   | 0 , 0 i . (9) 3 2.2 Deforme d W -alge b r a In this section, we review the defor med Virasoro a lg ebra and t he deformed W algebra W q ,t ( d g l N ), follow ing [7,8,9,10,1 5,16,17]. Definition 1 F or N = 2 , 3 , 4 , · · · , the deforme d W -algebr a W q ,t ( d g l N ) is gener ate d by the gener ators T ( j ) m , (1 ≦ j ≦ N , m ∈ Z ) , with the definin g r elations (10) of the series T j ( z ) = P m ∈ Z T ( j ) m z − m . f i,j ( z 2 /z 1 ) T i ( z 1 ) T j ( z 2 ) − f j,i ( z 1 /z 2 ) T j ( z 2 ) T i ( z 1 ) = c i X k =1 k − 1 Y l =1 ∆( x 2 l +1 ) δ x j − i +2 k z 2 z 1 ! f i − k ,j + k ( x − j + i ) T i − k ( x − k z 1 ) T j + k ( x k z 2 ) − δ x − j + i − 2 k z 2 z 1 ! f i − k ,j + k ( x j − i ) T i − k ( x k z 1 ) T j + k ( x − k z 2 ) ! , (1 ≦ i ≦ j ≦ N ) , (10) wher e we use d the delta-function δ ( z ) = P m ∈ Z z m . Her e we have set the c on- stant c and the structur e functions ∆( z ) and f i,j ( z ) , (1 ≦ i, j ≦ N ) by c = − (1 − x 2 r )(1 − x − 2( r − 1) ) (1 − x 2 ) , ∆( z ) = (1 − x 2 r − 1 z )(1 − x 1 − 2 r z ) (1 − xz )(1 − x − 1 z ) , (11) f i,j ( z ) (12) = exp ∞ X m =1 (1 − x 2 rm )(1 − x − 2( r − 1) m )(1 − x 2 mM in ( i,j ) )(1 − x 2 m ( s − M ax ( i,j )) ) m (1 − x 2 m )(1 − x 2 sm ) ( x | i − j | z ) m ! . Example Up on the sp ecialization N = s = 2, we ha v e the deformed Virasoro algebra V ir q ,t . Up on this sp ecialization the generators T (2) m can b e regarded as T (2) m = 1. The generato r s T (1) m = T m satisfy the follo wing defining relatio n. ∞ X l =0 f l ( T n − l T m + l − T m − l T n + l ) = c ( c 2 n − x − 2 n ) δ n + m, 0 , ( 1 3) where the structure constant f l is giv en b y P ∞ l =0 f l z l = f 1 , 1 ( z ). In the CFT limit ( x → 1), w e get the Virasoro a lgebra with the central c ha r ge c C F T = 1 − 6 r ( r − 1) . [ L m , L n ] = ( m − n ) L m + n + 1 12 c C F T m ( m 2 − 1) δ m + n, 0 . (14) Prop osition 2 F or N = 2 the d e forme d W -algebr a W q ,t ( c g l 2 ) is r e alize d by the b osons (3), (4) on the F o ck sp ac e. 4 T 1 ( z ) = Λ 1 ( z ) + Λ 2 ( z ) , T 2 ( z ) =: Λ 1 ( x − 1 z )Λ 2 ( xz ) : , (15) wher e we have set Λ 1 ( z ) = x − √ r ( r − 1) P α 1 : exp   X m 6 =0 1 m ( x r m − x − r m ) β 1 m z − m   : , (16) Λ 2 ( z ) = x − √ r ( r − 1) P α 2 : exp   X m 6 =0 1 m ( x r m − x − r m ) β 2 m z − m   : . (17) Prop osition 3 F or N = 3 , 4 , · · · the def o rme d W -algebr a W q ,t ( d g l N ) is r e alize d by the b osons (3), (4) on the F o ck sp ac e. T j ( z ) = X 1 ≦ s 1 0 should b e mo v ed to the rig h t. 2.3 Scr e enin g curr en t In this section we review the screening curren ts for the deformed Virasoro algebra and the deformed W algebra, fo llowing [7 ,8,9,10,11,12,13,15,16,1 7]. Definition 4 F or N = 2 we intr o duc e the o p er ator F j ( z ) , ( j = 1 , 2 ) , c al le d the scr e ening curr ent for the deforme d W -algebr a W q ,t ( c g l 2 ) . We define F 1 ( z ) = e − i √ r ∗ r Q α 1 z − √ r ∗ r P α 1 + r ∗ r : exp   X m 6 =0 1 m ( β 1 m − β 2 m ) z − m   : , (20) F 2 ( z ) = e − i √ r ∗ r Q α 2 z − √ r ∗ r P α 2 + r ∗ r : exp   X m 6 =0 1 m ( − x sm β 1 m + x − sm β 2 m ) z − m   : . (21) 5 Definition 5 F or N = 3 , 4 , · · · we intr o duc e the op er ator F j ( z ) , (1 ≦ j ≦ N ) , c al le d the scr e en ing curr ent for the deforme d W -algebr a W q ,t ( d g l N ) . L et us set F j ( z ) = e − i √ r r − 1 Q α j ( x ( 2 s N − 1) j z ) − √ r r − 1 P α j + r r − 1 × : exp   X m 6 =0 1 m ( β j m − β j +1 m )( x 2 s N z ) − m   : , (1 ≦ j ≦ N − 1) , (22) F N ( z ) = e − i √ r r − 1 Q α N ( x 2 s − N z ) − √ r r − 1 P ¯ ǫ N + r 2( r − 1) z √ r r − 1 P ¯ ǫ 1 + r 2( r − 1) × : exp   X m 6 =0 1 m ( x − 2 sm β N m − β 1 m ) z − m   : . (23) Prop osition 6 F or N = 2 the scr e ening curr e nts F 1 ( z ) , F 2 ( z ) satisfy the fol lowing c ommutation r elations. [ u 1 − u 2 ] r [ u 1 − u 2 + 1] r F j ( z 1 ) F j ( z 2 ) = [ u 2 − u 1 ] r [ u 2 − u 1 + 1] r F j ( z 2 ) F j ( z 1 ) , ( j = 1 , 2) , (24)  u 1 − u 2 + s 2 − 1  r  u 1 − u 2 − s 2  r F 2 ( z 1 ) F 1 ( z 2 ) =  u 2 − u 1 + s 2 − 1  r  u 2 − u 1 − s 2  r F 1 ( z 2 ) F 2 ( z 1 ) . (25) Prop osition 7 F or N = 2 the c ommutation r elations b etwe en Λ j ( z ) and F j ( z ) ar e given by [Λ 1 ( z 1 ) , F 1 ( z 2 )] = ( x − r ∗ − x r ∗ ) δ ( x r z 1 /z 2 ) A ( x − r z 2 ) , (26) [Λ 2 ( z 1 ) , F 1 ( z 2 )] = ( x r ∗ − x − r ∗ ) δ ( x − r z 1 /z 2 ) A ( x r z 2 ) , (27) [Λ 1 ( z 1 ) , F 2 ( z 2 )] = ( x r ∗ − x − r ∗ ) δ ( x − r + s z 1 /z 2 ) η ( A ( x r z 2 )) , (28) [Λ 2 ( z 1 ) , F 2 ( z 2 )] = ( x − r ∗ − x r ∗ ) δ ( x r − s z 1 /z 2 ) η ( A ( x − r z 2 )) , (29) wher e we have set A ( z ) = e i √ r ∗ r Q α 1 z √ r ∗ r P α 1 + r ∗ r : exp   X m 6 =0 1 m ( x r m β 1 m − x − r m β 2 m ) z − m   : . (30) Prop osition 8 F or N = 3 , 4 , 5 , · · · , the scr e ening curr ents F j ( z ) satisfy the fol lowing c ommutation r elations. 6  u 1 − u 2 − s N  r F j ( z 1 ) F j +1 ( z 2 ) =  u 2 − u 1 + s N − 1  r F j +1 ( z 2 ) F j ( z 1 ) , (31) [ u 1 − u 2 ] r [ u 1 − u 2 + 1] r F j ( z 1 ) F j ( z 2 ) = [ u 2 − u 1 ] r [ u 2 − u 1 + 1] r F j ( z 2 ) F j ( z 1 ) , (32) for 1 ≦ j ≦ N . We understand F N +1 ( z ) = F 1 ( z ) . We have F i ( z 1 ) F j ( z 2 ) = F j ( z 2 ) F i ( z 1 ) , otherwise . (33) Prop osition 9 F or N = 3 , 4 , 5 , · · · the c ommutation r elation s b etwe en Λ j ( z ) and F j ( z ) ar e given by [Λ j ( z 1 ) , F j ( z 2 )] = ( x − r ∗ − x r ∗ ) δ ( x − 2 s N j + r z 1 /z 2 ) A j ( x 2 s N j − r z 1 ) , (1 ≦ j ≦ N − 1) , (34) [Λ j +1 ( z 1 ) , F j ( z 2 )] = ( x r ∗ − x − r ∗ ) δ ( x − 2 s N j − r z 1 /z 2 ) A j ( x 2 s N j + r z 2 ) , (1 ≦ j ≦ N − 1) , (35) [Λ N ( z 1 ) , F N ( z 2 )] = ( x − r ∗ − x r ∗ ) δ ( x r − 2 s z 1 /z 2 ) A N ( x − r z 2 ) , (36) [Λ 1 ( z 1 ) , F N ( z 2 )] = ( x r ∗ − x − r ∗ ) δ ( x − r z 1 /z 2 ) A N ( x r z 2 ) , (37) wher e we have set A j ( z ) = e i √ r ∗ r Q α j x − √ r r ∗ ( P ¯ ǫ j + P ¯ ǫ j +1 ) ( x − j z ) √ r ∗ r P α j + r ∗ r × : exp   X m 6 =0 1 m ( x r m β j m − x − r m β j +1 m ) z − m   : , ( 1 ≦ j ≦ N − 1) , (38) A N ( z ) = e i √ r ∗ r Q α N x − √ r r ∗ ( P ¯ ǫ N + P ¯ ǫ 1 ) ( x 2 s − N z ) √ r ∗ r P ¯ ǫ N + r ∗ 2 r z − √ r ∗ r P ¯ ǫ 1 + r ∗ 2 r × : exp   X m 6 =0 1 m ( x ( r − 2 s ) m β N m − x − r m β 1 m ) z − m   : . (39) 3 In tegrals of Motion In this section w e review the in tegrals of mo t io n for the deformed Virasoro algebra and the deformed W - algebra, fo llowing [1 5,16,17]. 7 3.1 L o c a l inte g r als of motion I n W e define the op erators I n , ( n = 1 , 2 , 3 , · · · ), whic h we call the lo cal in tegrals of motion for the W q ,t ( d g l N ), ( N = 2 , 3 , 4 , · · · ). Definition 10 F or the r e gime Re( s ) > 2 a nd Re( r ∗ ) < 0 , we define I n = Z · · · Z C n Y j =1 dz j z j T 1 ( z 1 ) T 1 ( z 2 ) T 1 ( z 3 ) · · · T 1 ( z n ) × Y 1 ≦ j 0 and Re( r ) > 0 should b e understo o d as analytic contin uation. 3.2 Nonlo c al inte gr als of motion G n W e define the op erators G m , ( m = 1 , 2 , 3 , · · · ), which we call the nonlo cal in tegrals of motion for the W q ,t ( d g l N ), ( N = 2 , 3 , 4 , · · · ). Definition 11 F or N = 2 and the r e gime 0 < Re( s ) < 2 a n d Re( r ) > 0 , we define G m = Z · · · Z I Y t =1 , 2 m Y j =1 dz ( t ) j z ( t ) j F 1 ( z (1) 1 ) F 1 ( z (1) 2 ) · · · F 1 ( z (1) m ) F 2 ( z (2) 1 ) F 2 ( z (2) 2 ) · · · F 2 ( z (2) m ) × Y t =1 , 2 Y 1 ≦ j 0 , we defi n e G m = Z · · · Z I N Y t =1 m Y j =1 dz ( t ) j z ( t ) j F 1 ( z (1) 1 ) F 1 ( z (1) 2 ) · · · F 1 ( z (1) m ) × F 2 ( z (2) 1 ) F 2 ( z (2) 2 ) · · · F 2 ( z (2) m ) · · · F N ( z ( N ) 1 ) F N ( z ( N ) 2 ) · · · F N ( z ( N ) m ) × N Y t =1 Y 1 ≦ j 0 and Re( r ) > 0 should b e understo o d as a na lytic con t inuation. 3.3 Main r esults In this section w e state the main results. Theorem 13 F or N = 2 , 3 , 4 , · · · , the lo c al inte gr a l s of motion I n and the nonlo c al inte gr als of motion G m c ommute with e ach other. [ I n , I m ] = [ I n , G m ] = [ G n , G m ] = 0 , ( m, n = 1 , 2 , · · · ) . (46) 9 These comm utativities are understo o d as consequen ce of comm uting family of the F eigin-Odesskii algebra [14]. Theorem 14 F or N = 2 , 3 , 4 , · · · , the lo c al inte gr a l s of motion I n and the nonlo c al inte gr al s of motion G n ar e invariant under the action of the Dynkin- diagr am automorphism η . η ( I n ) = I n , η ( G n ) = G n , ( n = 1 , 2 , · · · ) . (47) Ac kno wledgemen t W e w ould lik e to thank Prof.B.F eigin, Prof .M.Jimbo and Mr.H.W atanab e for useful comm unications. W e would lik e to thank Professors V.Bazhanov, A.Bela vin, P .Bou wknegt, A.Cherv o v, S.Duzhin, V.Gerdjiko v, K.Hasegaw a, P .Kulish, W-X.Ma, V.Mangazeev, K.T ak em ura and M.W adati for their interes ts in this w ork. This w ork is pa rtly supp orted b y Gran t-in Aid for Y oung Scien- tist B (18 740092) and Gra n t-in Aid for Scien tific Researc h C (16540183 ) from JSPS. 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