$A_n^{(1)}$ affine Toda field theories with integrable boundary conditions revisited

A (1) n affine T o da field theori e s with in tegrable b oundary conditi o ns revisite d Anastasia Doik ou 1 Univ ersit y of Bologna, Ph y sics D e partmen t, INFN Section Via Irnerio 46, Bologna 40126, Italy Abstract Generic classically in tegrable bou n dary conditions for th e A (1) n affine T o da field the- ories (A TFT) are in v estigate d. The present analysis rests pr i marily on the und e rlying algebra, defined by the classica l version of th e reflection equation. W e u se as a p roto - t yp e example the first non-trivial mo del of the h ierarch y i.e. the A (1) 2 A TFT, ho w ev er our results ma y b e generalized for any A (1) n ( n > 1). W e assu me here t wo distinct t yp es of b ound ary conditions called some times soliton preserving (SP), and so liton non-preserving (SNP) associated to tw o distinct a lgebras, i.e. the reflection alge bra and the ( q ) t wisted Y angian r esp ectiv ely . The b oun dary local in tegrals of motion are then systematically extracted from the asymp totic expansion of the asso ciated transf er matrix. In the case of SNP b oundary conditions we reco v er previously kno wn r esults. The other type of b ound ary conditions (SP), asso ciated to the reflection algebra, are no v el in th is con text and lead to a different set of conserved quan tities th at dep end on free b ou n dary parameters. It also tur ns out that the num b er of lo cal in tegrals of motions for SP b oundary conditions is ‘d ouble’ compared to those of the S NP case. 1 e-mail: doikou@bo.infn.it, adoikou@upatras .gr 1 In tro d uction In tegrabilit y in the bulk has a dmittedly attracted a great deal of researc h in terest in recen t y ears, how ev e r after the seminal w orks of [1, 2 , 3] particular emphasis has b een giv en on the issue o f incorp orating consisten t b oundary conditions in integrable mo dels. This shed new ligh t into the bulk theories themselv es, and also op ened the path to new mathematical concepts and ph ysical applications. In a more g eneral setting the in v estigation of b oth classi- cal and quantum in tegrable syste ms, particularly those with no n- trivial bo undar y conditions, turns out to b e quite signific ant esp ecially after the recen t adv ances within the AdS/CFT cor- resp ondence [4] uncov ering the imp ortant ro le o f in tegrabilit y [5]. A crucial question within this frame is what w o uld the ph ysical implications b e in b ot h gauge and string theories once non-trivial consisten t b oundary conditions, esp ecially the ones that ma y mo dify the bulk b eha vior, are imp osed t o the asso ciat ed lattice a nd contin uum in tegrable mo dels (for some recen t results see [6 ] and references therein). Therefore studies concerning the existence of consisten t b oundary conditio ns tha t preserv e integrabilit y are of particular significance and timeliness not only fo r the integrable systems themselv es, but for other activ e researc h fields. The cen tral purpose of the presen t article is the inv estigation of classical in tegra ble mo dels when general b oundaries t ha t preserv e in tegrabilit y are implemen ted. Among the v arious classes of integrable mo dels w e c ho ose to consider here a particular class that is the affine T o da field theories (A TFT) [7, 8]. The prototype mo del of this class is the sine-Gordon mo del, whic h has b een extensiv ely studied b oth in the bulk [9] as we ll as in the presence of non- trivial in tegrable b oundary conditions [10 ]. Generic affine T o da field theories with classical in tegrable b o undar y conditions w ere first analyzed more than a decade ago in [11]. A differen t p oin t of view, although regarding t he same class o f b oundary conditions 2 analyzed in [11], is presen ted in [1 2]. Sp ecifically , in [12] the A (1) 2 A TFT with ‘dynamical’ b oundary conditions –that is a quantum mec hanical system is attache d at the b oundary— is in ves tigated. F urther studies regarding the b oundary A TFT at b oth classical a nd quantum level ma y b e also found in v arious articles (see e.g. [13]–[18 ]) . Although the analysis in [11] seems quite exhaustiv e it turns out that in simply-laced A TFT a whole class of consisten t b oundary conditions is absen t. Our main ob jectiv e here is to systematically searc h for all p ossible b oundary conditions in A (1) n A TFT and ev en tually implemen t the missing ones. More prec isely , w e assume tw o distinct t yp es of b oundary conditions called soliton preserving (SP), and soliton non-preserving (SNP) a sso ciated to t w o distinct algebras, i.e. the reflection algebra [2] and the t wisted Y a ng ian [19, 20] resp ectiv ely 2 by ‘same cla ss of bo undary c o nditions’ we mean that in b oth studies [11, 12] a commo n underlying algebra –(classica l) q -twisted Y a ngian– is implicitly a ssumed. Note how ever that the analysis in [11] is classical while in [12] is quantum. 1 (see also relev ant studies in [18, 2 1, 22 , 23, 24, 25]). Dep ending on the c hoice of b oundary conditions certain ph ysical b ehav ior is en tailed. Sp ecifically , in the con text of imaginary A (1) n A TFT the b oundary conditions in tro duced in [11], kno wn as SNP , oblig e a soliton to r eflect to an anti-soliton. In real A (1) n A TFT on the other hand suc h b oundary conditions lead to the reflection of a fundamen tal particle to itself. Recall that f undamen tal particles in real A TFT a re equiv alen t to the lightest b ound states (breathers) of the imaginary theory pro vided that β → iβ ( β is the coupling constan t of the theory). It is how ev er clear that ano t her p ossibilit y arises, that is the implemen ta t ion of certain b oundary conditio ns that lead to the reflection of a soliton t o itself in imagina r y A TFT or to the reflection of a fundamental particle to its conjugat e in real A TFT. These b oundary conditions are known as soliton preserving and hav e b een extensiv ely analyzed in the f rame of inte grable quan tum spin c hains [21, 22], [26]– [3 0]. Not withstanding SP boundary conditions are someho w the ob vious ones in the framew ork of inte grable lattice mo dels they ha v e remained elusiv e in the con text of A (1) n A TFT f or quite a long time. Not e how ev er that in quan tum spin c hains in addition to the well studied SP b oundaries SNP b oundary conditions w ere first in tro duced in [31] a nd further analyzed and generalized in [21, 22, 24, 25]. It is thu s our primary ob jectiv e here to complete the study of in tegrable b oundary conditions in A TFT b y in tro ducing and fully analyzing the nov el (SP) b oundary conditions. The outline of t his article is as fo llo ws: in the next section w e presen t the basic prelimi- nary notions regarding the algebraic setting for classical mo dels on the full line and on the in terv al. In our analysis w e adopt the line of att ac k describ ed in e.g. [32], and in [33, 34] for b oundary systems. More precisely w e introduce the classical Y ang-Baxter equation a nd the underlying algebra f o r the system on t he full line. In the situatio n of a system on the in terv al we distinguish t w o types of b oundary conditions based on the classical vers ions of the reflection algebra (SP) and ( q ) t wisted Y angia n (SNP). Next the A (1) n A TFT on the full line is review ed and an explicit deriv ation of the lo cal in tegrals of mot io n b y solving the auxiliary linear problem [32] is presen ted. In section 3 b eing guided by the same logic a nd adopting Skly anin’s formulation [2] we rederiv e the in tegrals of motion of the A (1) n A TFT with SNP b oundary conditions. Note that analog ous strategy w as f ollo w ed in [33] and [34] for the classical b oundary sine-Gordon and v ector NLS mo dels resp ectiv ely . Our r esults a re in agr eemen t with the ones deduced in [11]. In section 4 w e in tro duce for the first time the no v el b oundary conditions (SP) within the contex t of A TFT. Explicit expressions of t he as- so ciated lo cal in tegrals of motion are deduced from the asymptotic expansion of t he classical transfer matrix. It is w orth stressing that the induced in tegrals of motion dep end on free b oundary parameters a s opp osed to the SNP case. In the la st section a discussion on the en tailed results is presen ted and sev eral directions for future in v estigations are pro p osed. 2 2 Preliminaries The analysis of the A TFT with integrable b oundary conditions will rely on the solutio n of the so called auxiliary linear problem [32]. Before we pro ceed to the study of classical in tegrable mo dels with consisten t b oundary conditions it will b e instructiv e to recall t he ba sic notions in the p erio dic case. Let Ψ b e a solution of the following set of equations ∂ Ψ ∂ x = U ( x, t, λ )Ψ (2.1) ∂ Ψ ∂ t = V ( x, t, λ )Ψ (2.2) with U , V b eing in general n × n matrices with en tries functions of complex v a lued fields, their deriv ativ es, and the sp ectral parameter λ . Compatibility conditio ns of the t w o differential equation ( 2.1), (2 .2) lead to the zero curv ature condition [3 5 , 36, 37] ˙ U − V ′ + h U , V i = 0 . (2.3) The latter equations giv e rise to the corresponding classical eq uations of motion of the system under consideration. The mono dromy matrix from (2 .1) may b e written as: T ( x, y , λ ) = P exp n Z x y U ( x ′ , t, λ ) dx ′ o (2.4) with T ( x, x, λ ) = 1 . The mono dromy matrix satisfies a pparen tly (2.1), and this w ill b e extensiv ely used in the presen t analysis. On the other hand within the Hamiltonian fo r malism the existence of the classical r -matrix, satisfying the classical Y a ng-Baxter equation [3 8, 39 ] h r 12 ( λ 1 − λ 2 ) , r 13 ( λ 1 ) + r 23 ( λ 2 ) i + h r 13 ( λ 1 ) , r 23 ( λ 2 ) i = 0 , (2.5) guaran tees the in tegrabilit y o f the classical system. Indeed, consider the op erator T ( x, y , λ ) satisfying n T 1 ( x, y , t, λ 1 ) , T 2 ( x, y , t, λ 2 ) o = h r 12 ( λ 1 − λ 2 ) , T 1 ( x, y , t, λ 1 ) T 2 ( x, y , t, λ 2 ) i . (2.6) Making use of the latter equation one may readily show for a system in full line: n ln tr { T ( x, y , λ 1 ) } , ln tr { T ( x, y , λ 2 ) } o = 0 (2.7) i.e. the system is integrable, and the c harges in inv olution –lo cal in tegrals of mot ion– ma y b e obtained b y expanding the ob ject ln tr { T ( x, y , λ ) } . The classical r - matrix asso ciated to the A (1) n affine T o da field theory in pa r ticular is given b y 3 [40] r ( λ ) = cosh( λ ) sinh( λ ) n +1 X i =1 e ii ⊗ e ii + 1 sinh( λ ) n +1 X i 6 = j =1 e [ sg n ( i − j ) − ( i − j ) 2 n +1 ] λ e ij ⊗ e j i . (2.8) 3 Notice that the r -matrix employed her e is in fact r t 1 t 2 12 with r 12 being the matrix used e.g. in [28, 23] 3 Note that the classical r -matrix (2.8) is written in the so called principal gra dation as is also in [11, 17]. T o express the r -matrix in the homogeneous gradat ion one implemen ts a simple gauge tra nsformation: r ( h ) ( λ ) = V ( λ ) r ( p ) ( λ ) V ( − λ ) where V ( λ ) = n +1 X j =1 e 2( j − 1) λ n +1 e j j . (2.9) Our main aim a s men tioned up on is to study the A (1) n A TFT on the interv al. F o r this purp ose w e shall employ Skly anin’s form ulation (see also [33, 34] f o r classical mo dels with in tegrable b oundary conditions). It will b e con v enien t for our purp oses here to in tro duce some useful notatio n: ˆ r ab ( λ ) = r ba ( λ ) for SP , ˆ r ab ( λ ) = r t a t b ba ( λ ) for SNP r ∗ ab ( λ ) = r ab ( λ ) for SP , r ∗ ab ( λ ) = r t b ba ( − λ ) for SNP ˆ r ∗ ab ( λ ) = r ba ( λ ) for SP , ˆ r ∗ ab ( λ ) = r t a ab ( − λ ) for SNP T ( λ ) = T − 1 ( − λ ) for SP , ˆ T ( λ ) = T t ( − λ ) for SNP . (2.10) In the situation where non- trivial integrable b o undary conditions are implemen t ed one deriv es t w o types of ‘mono drom y’ mat r ices T , whic h resp ectiv ely represen t the classical v ersions of the r eflection alg ebra R , and the twis ted Y a ngian T written in t he compact f orm b elo w (see e.g. [2, 41]): n T 1 ( λ 1 ) , T 2 ( λ 2 ) o = r 12 ( λ 1 − λ 2 ) T 1 ( λ 1 ) T 2 ( λ 2 ) − T 1 ( λ 1 ) T 2 ( λ 2 ) ˆ r 12 ( λ 1 − λ 2 ) + T 1 ( λ 1 ) ˆ r ∗ 12 ( λ 1 + λ 2 ) T 2 ( λ 2 ) − T 2 ( λ 2 ) r ∗ 12 ( λ 1 + λ 2 ) T 1 ( λ 1 ) . (2.11) The mo dified ‘mo no dromy ’ mat r ices, compatible with t he corresp onding algebras R , T a re giv en b y the following expressions [2, 11]: T ( x, y , t, λ ) = T ( x, y , t, λ ) K − ( λ ) ˆ T ( x, y , t, λ ) (2.12) and the generating function of the inv olutive quan tities is defined as t ( x, y , t, λ ) = tr { K + ( λ ) T ( x, y , t, λ ) } (2.13) where K ± c -n um b er r epresen tations of the algebra R ( T ) satisfying (2.11) for SP and SNP resp ectiv ely , a nd also n K ± 1 ( λ 1 ) , K ± 2 ( λ 2 ) o = 0 . (2.14) Due to (2.11) it can b e sho wn t ha t n t ( x, y , t, λ 1 ) , t ( x, y , t, λ 2 ) o = 0 , λ 1 , λ 2 ∈ C . (2.15) T ec hnical details on the pro o f of classical in tegrabilit y are pro vided e.g. in [2, 1 1, 34 ]. 4 2.1 Classical in tegrals of motion in A TFT W e shall exemplify our in v estigation using the first non-trivial mo del of the A TFT hierarc h y that exhibits b oth t yp es of b oundary conditions, that is the A (1) 2 case. Recall the Lax pair for a generic A (1) n theory [8 ]: V ( x, t, u ) = β 2 ∂ x Φ · H + m 4  u e β 2 Φ · H E + e − β 2 Φ · H − 1 u e − β 2 Φ · H E − e β 2 Φ · H  U ( x, t, u ) = β 2 Π · H + m 4  u e β 2 Φ · H E + e − β 2 Φ · H + 1 u e − β 2 Φ · H E − e β 2 Φ · H  (2.16) Φ , Π are n -v ector fields, with comp onents φ i , π i , i ∈ { 1 , . . . , n } , u = e 2 λ n +1 is the m ulti- plicativ e sp ectral parameter. T o compare with the notation used in [11] w e set m 2 16 = ˜ m 2 8 ( ˜ m denotes the mass in [11]). Note that ev en tually in [11] b oth β , ˜ m are set equal to unit. Also define: E + = n +1 X i =1 E α i , E − = n +1 X i =1 E − α i (2.17) α i are the simple r o ots, H ( n -v ector) and E ± α i are the algebra generator s in the Cartan-W eyl basis corresp o nding to simple ro o t s, and they satisfy the Lie algebra relations: h H , E ± α i i = ± α i E ± α i , h E α i , E − α i i = 2 α 2 i α i · H (2.18) Explicit ex pressions on the simple ro ot s and the Cartan generators are prese n ted in Appendix A. Notice that the Lax pair has the f o llo wing b eha vior: V t ( x, t, − u − 1 ) = V ( x, t, u ) , U t ( x, t, u − 1 ) = U ( x, t, u ) (2.19) where t denotes usual tra nsp o sition. Our ob jectiv e as men tioned is to examine the system with non-trivial b oundaries, th us w e consider represen tations of the asso ciated underlying algebras expressed b y T . T o reco ver the lo cal inte grals of mo t io n of t he considered system w e shall follow the quite standard pro cedure and expand ln t ( u ) in p ow ers of u − 1 . An alternat ive strategy w ould b e to deriv e the mo dified Lax pair, compatible with the b oundary conditions c hosen, and hence the asso ciated equations o f motion (see e.g. [11]). A systematic deriv ation of b oundary Lax pairs independen tly of the c hoice of mo del is discuss ed in [42]. T o expand the op en transfer matrix and derive the lo cal in tegrals of motion w e shall need the expansions of T ( x, y , u ), T ( x, y , u − 1 ) and K ± ( u ). In what follows in the presen t section w e basically introduce the 5 necessary preliminaries fo r suc h a deriv ation, and w e also r epro duce the know n integrals of motion for the A TFT on the full line. Let T ′ ( x, y , u ) = T ( x, y , u − 1 ) and U ′ ( x, u ) = U ( x, u − 1 ). F ollo wing the logic describ ed in [32] for t he sine-Go rdon mo del, w e aim at expressing the part asso ciated to E + , E − in U , U ′ resp ectiv ely independently o f the fields, after applying a suitable gauge transformation. More precisely , consider the follo wing gauge transforma t io n: T ( x, y , u ) = Ω( x ) ˜ T ( x, y , u ) Ω − 1 ( y ) , T ′ ( x, y , u ) = Ω − 1 ( x ) ˜ T ′ ( x, y , u ) Ω( y ) Ω( x ) = e β 2 Φ( x ) · H . (2.20) Then fr om equation (2.1) w e obtain that the gauge transformed op erators U , U ′ can b e expresse d as: ˜ U ( x, t, u ) = Ω − 1 ( x ) U ( x, t, u ) Ω( x ) − Ω − 1 ( x ) d Ω( x ) dx ˜ U ′ ( x, t, u ) = Ω( x ) U ′ ( x, t, u ) Ω − 1 ( x ) − Ω( x ) d Ω − 1 ( x ) dx . (2.21) After implemen ting the gauge transformations the op erators ˜ U , ˜ U ′ tak e the follo wing simple form: ˜ U ( x, t, u ) = β 2 Θ · H + m 4  uE + + 1 u X −  , ˜ U ′ ( x, t, u ) = β 2 ˆ Θ · H + m 4  uE − + 1 u X +  (2.22) where w e define: Θ = Π − ∂ x Φ , ˆ Θ = Π + ∂ x Φ , X − = e − β Φ · H E − e β Φ · H , X + = e β Φ · H E + e − β Φ · H (2.23) ˜ T , ˜ U also satisfy (2.1), and Θ , ˆ Θ are n v ectors with comp onen ts θ i , ˆ θ i resp ectiv ely . Consider no w the fo llo wing ansatz for ˜ T , ˜ T ′ as | u | → ∞ [32] ˜ T ( x, y , u ) = ( I + W ( x, u ) ) ex p[ Z ( x, y , u )] ( I + W ( y , u )) − 1 , ˜ T ′ ( x, y , u ) = ( I + ˆ W ( x, u )) exp[ ˆ Z ( x, y , u ) ] ( I + ˆ W ( y , u )) − 1 , (2.24) where W , ˆ W are off diagonal matrices i.e. W = P i 6 = j W ij E ij , and Z , ˆ Z are purely diagonal Z = P n +1 i =1 Z ii E ii . Also Z ( u ) = ∞ X k = − 1 Z ( k ) u k , W ij = ∞ X k =0 W ( k ) u k . (2.25) Inserting the la t t er expressions (2.25) in (2.1) one ma y iden tify the co efficien ts W ( k ) ij and Z ( k ) ii . Indeed from (2.1) w e obtain t he following fundamen tal relations: d Z dx = ˜ U ( D ) + ( ˜ U ( O ) W ) ( D ) dW dx + W ˜ U ( D ) − ˜ U ( D ) W + W ( ˜ U ( O ) W ) ( D ) − ˜ U ( O ) − ( ˜ U ( O ) W ) ( O ) = 0 (2.26) 6 where the sup erscripts O, D denote off - diagonal a nd diag o nal pa rt respective ly . Similar relations ma y b e obtained for ˆ Z , ˆ W , in this case ˜ U → ˜ U ′ . W e omit writing these equations here fo r brevit y . It will b e useful in what follows to in tro duce some notation: β 2 Θ · H = diag( a, b, c ) , β 2 ˆ Θ · H = diag( ˆ a, ˆ b, ˆ c ) , e β α i · Φ = γ i (2.27) explicit expression o f a, b, c and γ i can b e found in App endix B (B.4); notice that a + b + c = 0. F rom the first of equations (2.26) we ma y deriv e the matrices Z , ˆ Z . Indee d one ma y easily sho w that: d Z (0) dx = m 4    W (1) 21 + ζ a W (1) 32 + ζ b − W (1) 13 + ζ c    = 0 d ˆ Z (0) dx = m 4    − ˆ W (1) 31 + ζ ˆ a ˆ W (1) 12 + ζ ˆ b ˆ W (1) 23 + ζ ˆ c    = 0 (2.28) it is clear that the latter quan tities are zero b ecause of the form of W (1) ij , ˆ W (1) ij see App endix B. Also the higher order Z ( k ) , ˆ Z ( k ) are giv en by : d Z ( k ) dx = m 4    W ( k +1) 21 − γ 3 W ( k − 1) 31 W ( k +1) 32 + γ 1 W ( k − 1) 12 − W ( k +1) 13 + γ 2 W ( k − 1) 23    d ˆ Z ( k ) dx = m 4    − ˆ W ( k +1) 31 + γ 1 ˆ W ( k − 1) 21 ˆ W ( k +1) 12 + γ 2 ˆ W ( k − 1) 32 ˆ W ( k +1) 23 − γ 3 ˆ W ( k − 1) 13    k > 0 . (2.29) The computation of W, ˆ W is essen tial for defining the diagonal elemen ts. First it is imp ortant to discuss the leading contribution of the ab ov e quan tities as | u | → ∞ . T o achie v e this w e shall need the explicit form of Z ( − 1) , ˆ Z ( − 1) : Z ( − 1) ( x, y ) = m ( x − y ) 4    e iπ 3 e − iπ 3 − 1    , ˆ Z ( − 1) ( x, y ) = m ( x − y ) 4    e − iπ 3 e iπ 3 − 1    . (2.30) The information a b o v e will b e extensiv ely used in what follo ws. 7 Before we pro ceed with the analysis o f integrable b oundary conditions in A TFT let us first repro duce the know n lo cal inte grals of motion in the p erio dic case, emerging from the expansion ( | u | → ∞ ) ln [ tr T ( u )] = ln [ tr { (1 + W ( L, u )) e Z ( L, − L,u ) (1 + W ( − L, u )) − 1 } ] . (2.31) Notice that in the case of p erio dic b oundary conditio ns w e put our system in the ‘whole’ line ( x = L, y = − L ), and consider Sch wartz b oundary conditions, i.e. the fields and their deriv ativ es v a nish at the end p o ints ± L . Bearing in mind tha t a s u → −∞ the leading con tribution of e Z , ( e ˆ Z ) (see (2.30)) comes fro m the e Z 33 , ( e ˆ Z 33 ) term, the expression ab ov e b ecomes ln [ tr T ( u → −∞ )] = X k = − 1 Z ( k ) 33 u k . (2.32) T o repro duce the familiar lo cal in tegrals of mot io n we shall need b oth Z ( L, − L, u ) , ˆ Z ( L, − L, u ). Let I 1 = − 12 m β 2 Z (1) 33 ( L, − L, u ) = Z L − L dx  2 X i =1 θ 2 i + m 2 β 2 3 X i =1 e β α i · Φ  , I − 1 = − 12 m β 2 ˆ Z (1) 33 ( L, − L, u ) = Z L − L dx  2 X i =1 ˆ θ 2 i + m 2 β 2 3 X i =1 e β α i · Φ  I 2 = 3 m 2 2 β 3 Z (2) 33 ( L, − L, u ) = Z L − L dx  8 β 3 ( abc − bc ′ ) − m 2 2 β 3 ( γ 1 c + γ 2 a + γ 3 b )  I − 2 = 3 m 2 2 β 3 ˆ Z (2) 33 ( L, − L, u ) = Z L − L dx  8 β 3 (ˆ a ˆ b ˆ c + ˆ b ˆ c ′ ) − m 2 2 β 3 ( γ 1 ˆ c + γ 2 ˆ a + γ 3 ˆ b )  . . . (higher lo cal in tegrals of motion) (2.33) the momen tum and Hamiltonian (and the higher conserv ed quan t it ies) of the A TFT are given b y: P 1 = 1 2 ( I − 1 − I 1 ) = Z L − L dx 2 X i =1  π i φ ′ i − π ′ i φ i  H 1 = 1 2 ( I 1 + I − 1 ) = Z L − L dx  2 X i =1 ( π 2 i + φ ′ 2 i ) + m 2 β 2 3 X i =1 e β α i · Φ  P 2 = 1 2 ( I − 2 − I 2 ) = 1 2 Z L − L dx  8 β 3 (ˆ a ˆ b ˆ c − abc ) + 8 β 3 ( bc ′ + ˆ b ˆ c ′ ) + m 2 2 β 3 ( γ 1 ( c − ˆ c ) + γ 2 ( a − ˆ a ) + γ 3 ( b − ˆ b ))  H 2 = 1 2 ( I 2 + I − 2 ) 8 = 1 2 Z L − L dx  8 β 3 ( abc + ˆ a ˆ b ˆ c ) − 8 β 3 ( bc ′ − ˆ b ˆ c ′ ) − m 2 2 β 3 ( γ 1 ( c + ˆ c ) + γ 2 ( a + ˆ a ) + γ 3 ( b + ˆ b ))  . . . (2.34) Note that the b oundary terms are absen t in the expressions ab ov e, since w e considered Sc h w a rtz type b oundary conditions. Also, in the generic situation, for an y A (1) n , the sum in the momen tum P 1 and the kinetic term of the Hamiltonian H 1 runs fro m 1 to n , whereas the sum in the p oten tial term o f the Hamiltonian runs from 1 to n + 1. 3 SNP b oundary conditi ons W e turn now to our main concern, whic h is the study of in tegrable b oundary conditions in A TFT. W e shall first discuss the b oundary conditions that already hav e b een analyzed in [11]. Ba sed on the underlying algebra, that is the classical analogue of the q -twis ted Y angian w e shall repro duce the previously kno wn results [11], so this section serv es basically as a w arm up exercise. In the subsequen t section we shall analyze in detail the no v el b oundary conditions (SP) asso ciat ed to the classical ve rsion of the reflection a lgebra. T o obtain the relev ant lo cal integrals of motion we shall expand the following ob ject (consider no w x = 0 , y = − L ): ln t ( u ) = ln tr n K + ( u ) T ( u ) K − ( u ) T t ( u − 1 ) o = ln tr n K + ( u ) Ω(0) ˜ T ( u ) Ω − 1 ( − L ) K − ( u ) Ω( − L ) ˜ T t ( u − 1 ) Ω − 1 (0) o (3.1) F or simplicit y here, but without really lo sing generalit y w e consider Sc h wartz b oundary conditions at the b oundary p oint − L and K − ( u ) ∝ I . Also K + ( u ) = K t ( u − 1 ) where K is an y c -n um b er solution of the t wisted Y angian. T aking also in to account the ansatz (2.24) we conclude ln t ( u ) = ln tr n (1 + ˆ W t (0 , u ))Ω − 1 (0) K + ( u ) Ω(0) (1 + W (0 , u )) e Z (0 , − L,u )+ ˆ Z (0 , − L,u ) o . (3.2) Recall from the previous section that as u → −∞ the leading con tribution of e Z , e ˆ Z comes from the e Z 33 , e ˆ Z 33 terms (see (2.30)), hence ln t ( u ) = Z 33 (0 , − L, u ) + ˆ Z 33 (0 , − L, u ) + ln[(1 + ˆ W t (0 , u ))Ω − 1 (0) K + ( u ) Ω(0) (1 + W (0 , u ))] 33 = ∞ X k = − 1 Z ( k ) 33 + ˆ Z ( k ) 33 u k + ∞ X k =0 f k u k . (3.3) 9 T o obtain the explicit for m of the b oundary con tributions to the integrals of motion w e should first review know n results on the solution of the reflection equation for SNP b oundary conditions. The g eneric solution for the A (1) n case in t he principal gradation ar e giv en by [17, 2 2 ]: K ( λ ) = ( g e λ + ¯ g e − λ ) n +1 X i =1 e ii + X i>j f ij e λ − 2 λ n +1 ( i − j ) e ij + X i 1) A TFT will b e also presen ted in a separate publication. F ina lly , a similar exhaustiv e analysis regarding principal c hiral mo dels (partial results ma yb e found in [4 6]) will b e pa r ticularly relev an t esp ecially b earing in mind the ph ysical significance of a sp ecific sup er-symmetric principal chiral mo del within the AdS/CFT corresp ondence [47, 4 8 ]. Ac kno wledgmen ts: I am indebted to J. Av an for useful commen ts. I wish to thank INFN, Bologna Section, and Univers it y o f Bologna for kind ho spitalit y . This w ork was partly sup- p orted b y INF N, Bolog na section, through gran t TO12. A App endix In this app endix w e provide explicit express ions of the simple ro ots and the Cartan generators for A (1) n [49]. The v ectors α i = ( α 1 i , . . . , α n i ) are the simple ro ots o f the Lie algebra of rank 17 n normalized to unity α i · α i = 1, i.e. α i =  0 , . . . , 0 , − r i − 1 2 i , i th ↓ r i + 1 2 i , 0 , . . . , 0  , i ∈ { 1 , . . . n } (A.1) Also define the fundamen tal w eigh ts µ k = ( µ 1 k , . . . , µ n k ) , k = 1 , . . . , n as (see, e.g., [49]). α j · µ k = 1 2 δ j,k . (A.2) The extended (affine) ro o t a n +1 is provided by the relation n +1 X i =1 a i = 0 . (A.3) W e giv e b elo w the Cartan-W eyl generators in the defining represen ta tion: E α i = e i i +1 , E − α i = e i +1 i , E α n = − e n +1 1 , E − α n = − e 1 n +1 H i = n X j =1 µ i j ( e j j − e j +1 j +1 ) , i = 1 , . . . , n (A.4) F or A (1) 2 in particular w e hav e: α 1 = (1 , 0) , α 2 = ( − 1 2 , √ 3 2 ) , α 3 = ( − 1 2 , − √ 3 2 ) (A.5) define also the follow ing 3 × 3 generators E 1 = E t − 1 = e 12 , E 2 = E t − 2 = e 23 , E 3 = E t − 3 = − e 31 (A.6) where w e define the matrices e ij as ( e ij ) k l = δ ik δ j l . The diagonal Cartan generators H 1 , 2 are H 1 = 1 2 ( e 11 − e 22 ) , H 2 = 1 2 √ 3 ( e 11 + e 22 − 2 e 33 ) (A.7) B App endix F rom the form ulas (2 .2 6), (2.29) the matrices W ( k ) , ˆ W ( k ) , Z ( k ) , ˆ Z ( k ) ma y b e determined. In particular, w e write b elo w explicit expressions of these matrices fo r the first o rders. W (0) = ˆ W (0) =    0 e iπ 3 1 e iπ 3 0 − 1 e 2 iπ 3 e − iπ 3 0    , m 4 W (1) =    0 e 2 iπ 3 a c − a 0 b e iπ 3 c − b 0    , m 4 ˆ W (1) =    0 − ˆ b − ˆ a − e − iπ 3 ˆ b 0 − ˆ c ˆ a − e iπ 3 ˆ c 0    . (B.1) 18 The higher order quan tities are more complicated a nd w e g ive the mat r ix en tries b elow fo r W (2) , ˆ W (2) (define also, ζ = 4 m ): W (2) 12 = 1 3 ( − 2 γ 3 + γ 1 + γ 2 ) + ζ 2 3 (2 a ′ + b ′ ) + ζ 2 3 ( − 2 a 2 − bc ) , W (2) 21 = e − iπ 3 3 ( − 2 γ 3 + γ 1 + γ 2 ) + ζ 2 e − iπ 3 3 ( a ′ − c ′ ) + ζ 2 e − iπ 3 3 ( c 2 − ab ) W (2) 13 = 1 3 ( − 2 γ 2 + γ 1 + γ 3 ) + ζ 2 3 ( − b ′ + c ′ ) + ζ 2 3 ( b 2 − ac ) , W (2) 31 = 1 3 (2 γ 2 − γ 1 − γ 3 ) + ζ 2 3 ( − a ′ − 2 c ′ ) + ζ 2 3 (2 c 2 + ab ) , W (2) 23 = − 1 3 (2 γ 1 − γ 2 − γ 3 ) + ζ 2 3 (2 b ′ + c ′ ) + ζ 2 3 ( − 2 b 2 − ac ) W (2) 32 = − e iπ 3 3 (2 γ 1 − γ 2 − γ 3 ) + ζ 2 e iπ 3 3 ( − a ′ + b ′ ) + ζ 2 e iπ 3 3 ( a 2 − bc ) (B.2) and ˆ W (2) 12 = e − iπ 3 3 ( − 2 γ 2 + γ 1 + γ 3 ) + ζ 2 e − iπ 3 3 ( ˆ b ′ − ˆ c ′ ) + ζ 2 e − iπ 3 3 ( ˆ c 2 − ˆ a ˆ b ) , ˆ W (2) 21 = 1 3 ( − 2 γ 2 + γ 1 + γ 3 ) + ζ 2 3 (2 ˆ b ′ + ˆ a ′ ) + ζ 2 3 ( − 2 ˆ b 2 − ˆ a ˆ c ) ˆ W (2) 13 = − 1 3 ( − 2 γ 1 + γ 3 + γ 2 ) − ζ 2 3 (2ˆ a ′ + ˆ c ′ ) + ζ 2 3 (2ˆ a 2 + ˆ b ˆ c ) , W (2) 31 = e iπ 3 3 (2 γ 1 − γ 2 − γ 3 ) + ζ 2 e iπ 3 3 ( ˆ b ′ − ˆ a ′ ) + ζ 2 e iπ 3 3 ( − ˆ b 2 + ˆ a ˆ c ) , ˆ W (2) 23 = − 1 3 ( − 2 γ 3 + γ 2 + γ 1 ) + ζ 2 3 (ˆ a ′ − ˆ c ′ ) + ζ 2 3 ( − ˆ a 2 + ˆ b ˆ c ) ˆ W (2) 32 = 1 3 ( − 2 γ 3 + γ 1 + γ 2 ) + ζ 2 3 ( ˆ b ′ + 2 ˆ c ′ ) + ζ 2 3 ( − 2ˆ c 2 − ˆ a ˆ b ) (B.3) where the prime denotes deriv ativ e with resp ect to x , also a, b, c, and γ i are defined in (2.27) and ha v e the following explicit forms: a = β 2 ( θ 1 2 + θ 2 2 √ 3 ) , b = β 2 ( − θ 1 2 + θ 2 2 √ 3 ) , c = − β 2 θ 2 √ 3 , γ 1 = e β φ 1 , γ 2 = e β ( − 1 2 φ 1 + √ 3 2 φ 2 ) , γ 3 = e β ( − 1 2 φ 1 − √ 3 2 φ 2 ) . (B.4) Moreo v er using the expressions ab o v e a nd (2.29) we ha v e: d Z (1) 11 dx = e − iπ 3 3 m 4 ( γ 1 + γ 2 + γ 3 ) + ζ e − iπ 3 3 ( a ′ − c ′ ) + ζ e − iπ 3 6 ( a 2 + b 2 + c 2 ) d Z (1) 22 dx = e iπ 3 3 m 4 ( γ 1 + γ 2 + γ 3 ) + ζ e iπ 3 3 ( b ′ − a ′ ) + ζ e iπ 3 6 ( a 2 + b 2 + c 2 ) d Z (1) 33 dx = − 1 3 m 4 ( γ 1 + γ 2 + γ 3 ) − ζ 3 ( c ′ − b ′ ) − ζ 6 ( a 2 + b 2 + c 2 ) 19 d ˆ Z (1) 11 dx = e iπ 3 3 m 4 ( γ 1 + γ 2 + γ 3 ) − ζ e iπ 3 3 ( ˆ b ′ − ˆ a ′ ) + ζ e iπ 3 6 (ˆ a 2 + ˆ b 2 + ˆ c 2 ) d ˆ Z (1) 22 dx = e − iπ 3 3 m 4 ( γ 1 + γ 2 + γ 3 ) + ζ e − iπ 3 3 ( ˆ b ′ − ˆ c ′ ) + ζ e − iπ 3 6 (ˆ a 2 + ˆ b 2 + ˆ c 2 ) d ˆ Z (1) 33 dx = − 1 3 m 4 ( γ 1 + γ 2 + γ 3 ) + ζ 3 (ˆ a ′ − ˆ c ′ ) − ζ 6 (ˆ a 2 + ˆ b 2 + ˆ c 2 ) (B.5) Finally w e rep o rt Z (2) ii , ˆ Z (2) ii : d Z (2) 11 dx = e iπ 3 3  γ ′ 2 − γ ′ 3 − ζ 2 ( c ′′ − c 2 ′ ) + ζ 2 ca ′ + ( γ 1 c + γ 2 a + γ 3 b ) − ζ 2 abc  d Z (2) 22 dx = e − iπ 3 3  − γ ′ 1 + γ ′ 3 − ζ 2 ( a ′′ − a 2 ′ ) + ζ 2 ab ′ + ( γ 1 c + γ 2 a + γ 3 b ) − ζ 2 abc  d Z (2) 33 dx = 1 3  − γ ′ 1 + γ ′ 2 + ζ 2 ( b ′′ − b 2 ′ ) − ζ 2 bc ′ − ( γ 1 c + γ 2 a + γ 3 b ) + ζ 2 abc  d ˆ Z (2) 11 dx = e − iπ 3 3  − γ ′ 1 + γ ′ 2 − ζ 2 ( ˆ b ′′ − ˆ b 2 ′ ) + ζ 2 ˆ b ˆ a ′ + ( γ 1 ˆ c + γ 2 ˆ a + γ 3 ˆ b ) − ζ 2 ˆ a ˆ b ˆ c  d ˆ Z (2) 22 dx = e iπ 3 3  − γ ′ 2 + γ ′ 3 − ζ 2 ( ˆ c ′′ − ˆ c 2 ′ ) + ζ 2 ˆ c ˆ b ′ + ( γ 1 ˆ c + γ 2 ˆ a + γ 3 ˆ b ) − ζ 2 ˆ a ˆ b ˆ c  d ˆ Z (2) 33 dx = 1 3  − γ ′ 1 + γ ′ 3 + ζ 2 (ˆ a ′′ − ˆ a 2 ′ ) − ζ 2 ˆ a ˆ c ′ − ( γ 1 ˆ c + γ 2 ˆ a + γ 3 ˆ b ) + ζ 2 ˆ a ˆ b ˆ c  . (B.6) C App endix W e presen t here the b oundary contributions in the expansion of the classical op en transfer matrix for b o th types of b oundary conditions: SNP b oundary conditions : Recall that in this case the expansion o f the generating func- tion o f the lo cal in tegrals of motion is giv en in (3.3). After some tedious a lgebra we obtain for the b oundar y terms: f 0 = ln(3 ¯ g ) , f 1 = 1 3 ¯ g  e β 2 α 3 · Φ(0) f 31 − e β 2 α 2 · Φ(0) f 23 − e β 2 α 1 · Φ(0) f 12  + ζ 3  c (0) − b (0) + ˆ c (0) − ˆ a (0)  f 2 = − 1 3 ¯ g  f 21 e − β 2 α 1 · Φ(0) + f 32 e − β 2 α 2 · Φ(0) − f 13 e − β 2 α 3 · Φ(0)  − ζ 3 ¯ g  f 12 e β 2 α 1 · Φ(0) ( c (0) + ˆ c (0)) − f 23 e β 2 α 2 · Φ(0) b (0) + f 31 e β 2 α 3 · Φ(0) ˆ a (0)  − ζ 2 3  ˆ a (0) c (0) + b (0) ˆ c (0)  + 1 3  (2 γ 1 (0) − γ 2 (0) − γ 3 (0)) − ζ 2 (ˆ a ′ (0) + b ′ (0)) + ζ 2 (ˆ a 2 (0) + b 2 (0))  − 1 2  1 3 ¯ g ( e β 2 α 3 · Φ(0) f 31 − e β 2 α 2 · Φ(0) f 23 − e β 2 α 1 · Φ(0) f 12 ) + ζ 3 ( c (0) − b (0) + ˆ c (0) − ˆ a (0))  2 . (C.1) 20 SP b oundary conditions : W e shall need for our purp oses here t he asymptotics of K ± as | u | → ∞ : K + ( | u | → ∞ , ξ + ) ∼ e 33 − e − 2 iξ + u e 11 + 1 u 2 e 22 + O ( u − 3 ) K − ( | u | → ∞ , ξ − ) ∼ e 11 − e − 2 iξ − u e 33 + 1 u 2 e 22 + O ( u − 3 ) . (C.2) Then from the expansion of the b oundary terms in (4.3), (4.4 ) w e obta in the following explicit quan tities: f + 0 = h + 0 = ln[ Ω 2 33 (0) 3 ] , f + 1 = − ζ e iπ 3 ˆ b (0) + ζ e − iπ 3 c (0) − e − 2 iξ + e − β α 3 · Φ(0) , f + 2 = n Ω 2 22 (0)Ω − 2 33 (0) − e − 4 iξ + 2 Ω 4 11 (0)Ω − 4 33 (0) + ζ e − 2 iξ + 2 Ω 2 11 (0)Ω − 2 33 (0)  c (0) + ˆ c (0)  − 1 6  2 γ 2 (0) − γ 1 (0) − γ 3 (0)  − ζ 2 6  b ′ (0) − c ′ (0)  + ζ 2  − c 2 (0) 6 + a 2 (0) 12 + b 2 (0) 12 + ˆ b 2 (0) 4 o + i √ 3 n ζ e − 2 iξ + 2 Ω 2 11 (0)Ω − 2 33 (0)  ˆ c ( 0) − c (0)  − 1 6  2 γ 2 (0) − γ 1 (0) − γ 3 (0)  − ζ 2 6  b ′ (0) − c ′ (0)  + ζ 2  − c 2 (0) 6 + a 2 (0) 12 + b 2 (0) 12 − ˆ b 2 (0) 4 o h + 1 = − ζ e iπ 3 b (0) − e − 2 iξ + e − β α 3 · Φ(0) h + 2 = n Ω 2 22 (0)Ω − 2 33 (0) − e − 4 iξ + 2 Ω 4 11 (0)Ω − 4 33 (0) + ζ e − 2 iξ + 2 Ω 2 11 (0)Ω − 2 33 (0)  c (0) + ˆ c (0)  + 1 3  γ 1 (0) − γ 2 (0)  + ζ 2 6  ˆ b ′ (0) − ˆ c ′ (0) − b ′ (0) + a ′ (0)  + ζ 2  ˆ a 2 (0) 12 + ˆ b 2 (0) 12 + ˆ c 2 (0) 12 + b 2 (0) 6 − a 2 (0) 12 − c 2 (0) 12 o + i √ 3 n ζ e − 2 iξ + 2 Ω 2 11 (0)Ω − 2 33 (0)  c (0) − ˆ c (0)  + 1 6  2 γ 3 (0) − γ 1 (0) − γ 2 (0)  + ζ 2 6  ˆ b ′ (0) − ˆ c ′ (0) + b ′ (0) − a ′ (0)  + ζ 2  ˆ a 2 (0) 12 + ˆ b 2 (0) 12 + ˆ c 2 (0) 12 − b 2 (0) 6 + a 2 (0) 12 + c 2 (0) 12 o . . . (C.3) Similar express ions a re o btained for f − n , h − n : f − 0 = h − 0 = ln[ Ω − 1 11 ( − L ) 3 ] , f − 1 = ζ e − iπ 3 a ( − L ) − e − 2 iξ − e − β α 3 · Φ( − L ) f − 2 = n Ω 2 11 ( − L )Ω − 2 22 ( − L ) − e − 4 iξ − 2 Ω 4 11 ( − L )Ω − 4 33 ( − L ) + ζ e − 2 iξ − 2 Ω 2 11 ( − L )Ω − 1 33 ( − L )  a ( − L ) + ˆ a ( − L )  + 1 6  2 γ 2 ( − L ) − γ 1 ( − L ) − γ 3 ( − L )  + ζ 2  − a ′ ( − L ) 18 − 4 c ′ ( − L ) 18 + 2 b ′ ( − L ) 18  + ζ 2  2 c 2 ( − L ) 12 − a 2 ( − L ) 12 + 2 b 2 ( − L ) 12 o + i √ 3 n ζ e − 2 iξ − 2 Ω 2 11 ( − L )Ω − 2 33 ( − L )  ˆ a ( − L ) − a ( − L )  + 1 6  2 γ 2 ( − L ) − γ 1 ( − L ) − γ 3 ( − L )  21 + ζ 2  − a ′ ( − L ) 18 − 4 c ′ ( − L ) 18 + 2 b ′ ( − L ) 18  + ζ 2  2 c 2 ( − L ) 12 − a 2 ( − L ) 12 + 2 b 2 ( − L ) 12 o h − 1 = − ζ e − iπ 3 ˆ b ( − L ) − e − 2 iξ − e − β α 3 · Φ( − L ) h − 2 = n Ω 2 11 ( − L )Ω − 2 22 ( − L ) − e − 4 iξ − 2 Ω 4 11 ( − L )Ω − 4 33 ( − L ) + ζ e − 2 iξ − 2 Ω 2 11 ( − L )Ω − 1 33 ( − L )  a ( − L ) + ˆ a ( − L )  − 1 2  γ 1 ( − L ) − γ 2 ( − L )  + ζ 2  − ˆ b ′ ( − L ) 6 + ˆ c ′ ( − L ) 6 − a ′ ( − L ) 6 + b ′ ( − L ) 6  + ζ 2  − ˆ c 2 ( − L ) 12 − ˆ a 2 ( − L ) 12 + ˆ b 2 ( − L ) 6 + c 2 ( − L ) 12 + a 2 ( − L ) 12 + b 2 ( − L ) 12 o + i √ 3 n ζ e − 2 iξ − 2 Ω 2 11 ( − L )Ω − 2 33 ( − L )  − ˆ a ( − L ) + a ( − L )  + 1 6  − 2 γ 3 ( − L ) + γ 1 ( − L ) + γ 2 ( − L )  + ζ 2  − ˆ b ′ ( − L ) 6 + ˆ c ′ ( − L ) 6 + a ′ ( − L ) 6 − b ′ ( − L ) 6  + ζ 2  − ˆ c 2 ( − L ) 12 − ˆ a 2 ( − L ) 12 + ˆ b 2 ( − L ) 6 − a 2 ( − L ) 12 − b 2 ( − L ) 12 − c 2 ( − L ) 12 o . (C.4) References [1] I.V. Cherednik, Theor. Math. Ph ys. 61 (1 9 84) 977. [2] E.K. Skly anin, F unct. Anal. Appl. 21 ( 1 987) 164; E.K. Skly anin, J. Ph ys. A21 (1988) 2375. [3] J. Cardy , Nucl. Ph ys. B275 (1986) 2 0 0; J. Cardy , Nucl. Phys . B324 (1989 ) 581. [4] J.M. Maldacena, Adv. Theor. Math. Phys . 2 (1998) 231; E. Witten, Adv. Theor. Math. Ph ys. 2 (199 8) 253; S.S. Gubser, I.R. Klebanov and A.M. Poly ak ov , Ph ys. Lett. B428 (19 98) 105. [5] J.A. Minahan and K. Zarem b o, JHEP 03 (2003) 013. [6] D.M. Hofman and J. Maldacena, . [7] A.V. Mikhailo v, Sov. Ph ys. JETP L etters 30 (1979) 414; A.V. Mikhailo v, M.A. Olshanetsk y and A.M. Pe relomov , Comm un. Math. Ph ys. 79 (1981) 47 3. [8] D.I. Olive and N. T urok, Nucl. Phy s, B215 (1983 ) 470; D.I. Olive and N. T urok, Nucl. Ph ys. B25 7 (19 85) 277; D.I. Olive and N. T urok, Nucl. Ph ys. B26 5 (19 86) 469. 22 [9] A.B. Zamolo dc hik o v and Al.B. Z amolo dch ik o v, Ann. Ph ys. 120 (19 79) 2 53. [10] S. Ghoshal and A.B. Z amolo dch ik o v, Int. J. Mo d. Ph ys. A9 (19 94) 3841. [11] E. Corr ig an, P .E. Dor ey , R.H. Rietdijk, R. Sa saki, Ph ys. Lett. B333 (1994) 83; P . Bow co c k, E. Corrigan, P .E. Dorey a nd R .H. Rietdijk, Nucl. Ph ys. B445 (1995) 469; P . Bow co c k, E. Corrigan and R .H. Rietdijk, Nucl. Ph ys. B465 (19 96) 3 50. [12] V.V. Bazhanov, A.N. Hibb erd and S.M. Khor o shkin, Nucl. Ph ys. B622 (2002) 4 75. [13] A. F ring and R. K¨ ob erle, Nucl. Ph ys. B421 (199 4 ) 159 ; A. F ring and R. K¨ ob erle, Nucl. Ph ys. B 419 (19 94) 6 47. [14] R. Sasaki, I nterfac e b etwe en Physics and Masthematics , eds W. Nahm and J-M Shen, (W orld Scien t ific 1994) 201. [15] S. P enati and D. Zanon, Ph ys. Lett. B358 (1995) 63. [16] G. D elius, Phy s. Lett. B444 (1998) 2 17. [17] G.M. Ga nden b erger, Nucl. Ph ys. B542 (19 99) 659; G.M. Ga ndenberger, hep-th/9911178 . [18] G. D elius and N. Mac k a y , Comm un. Math. Ph ys. 233 ( 2003) 173. [19] G.I. Olshanski, Twisted Y angians and infinite-dimensional classical Lie algebras in ‘Quan tum Gr oups’ (P .P . Kulish, Ed.), Lecture notes in Math. 1510 , Springer (1 992) 103; A. Molev, M. Nazaro v and G .I. Olshanski, Russ. Math. Surv eys 51 (199 6) 20 6. [20] A.I. Molev, E. Ragoucy and P . Sorba, Rev. Math. Ph ys. 15 (200 3 ) 789 ; A.I. Molev, Handb o o k of A lgebr a , V ol. 3, (M. Hazewink el, Ed.), Elsevie r, (2003), pp. 907. [21] D. Arnaudon, J. Av an, N. Cramp´ e, A. Doik ou, L. F rappat and E. Rag o ucy , J. Stat. Mec h. 0408 (20 04) P00 5; D. Arnaudon, N. Cramp´ e, A. Doikou, L. F rappat and E. Ragoucy , J. Stat. Mec h. 0502 (2005) P007 . [22] D. Arnaudon, N. Cramp´ e, A. Doikou, L. F rappat and E. Ragoucy , In t. J. Mo d. Ph ys. A21 (2 0 06) 1537 ; D. Arnaudon, N. Cramp e, A. D o ik ou, L. F ra ppa t and E. Ra g oucy , Ann. H. P oincare v ol. 7 (2006 ) 121 7. [23] A. Do ikou, Nucl. Ph ys. B725 ( 2 005) 493. 23 [24] A. Do ikou, J. Math. Ph ys. 46 053 504 (200 5 ); A. Doikou, SIGMA 3 (2007) 00 9 . [25] N. Cramp´ e and A. Doik ou, J. Math. Phy s. 48 023511 (2 007). [26] H.J. de V ega and A. Go nzalez–Ruiz, Nucl. Ph ys. B417 (1994) 55 3; H.J. de V ega and A. Gonzalez–Ruiz, Phys . Lett. B332 (1994) 123; H.J. de V ega and A. Gonzalez–Ruiz, J. Ph ys. A26 (1993) L519. [27] L. Mezincescu and R.I. Nepo mec hie, Nucl. Phys . B372 (1 9 92) 597; S. Artz, L. Mezinces cu and R.I. Nep o mec hie, J.Ph ys. A28 (1995 ) 513 1. [28] A. Do ikou and R.I. Nep omec hie, Nucl. Ph ys. B521 (19 98) 5 47; A. Doikou and R.I. Nep omec hie, Nucl. Ph ys. B530 (19 98) 6 41. [29] J. Abad and M.Rios, Ph ys. Lett. B352 (199 5 ) 92. [30] W. Galleas and M.J. Martins, Phy s. Lett. A335 (2005) 1 67; R. Malara and A. Lima-San tos, J. Stat. Mec h. 0609 (2006) P013 ; W.-L. Y ang and Y.-Z. Zhang, JHEP 0412 (2004) 019; W.-L. Y ang and Y.-Z. Zhang, hep-th/0504048. [31] A. Do ikou, J. Ph ys. A33 (20 0 0) 87 97. [32] L.D. F addeev and L.A. T akh tak a ja n, Hamiltonian Metho ds in the The ory of Solitons , (1987) Springer- V erlag. [33] A. MacIn t yre, J. Ph ys. A28 ( 1 995) 1089. [34] A. Do ikou, D. Fiorav a nti and F. Rav anini, Nucl. Ph ys. B790 (2008) 465. [35] M.J. Ablow itz, D.J. Ka up, A.C. New ell and H. Segur, Stud. Appl. Math. 53 (1 974) 24 9 . [36] M.J. Ablowitz and J.F. Ladik, J. Math. Ph ys. 17 (1976) 101 1. [37] V.E. Z akharo v and A.B. Shabat, Anal. Appl. 13 (1979) 13. [38] E.G. Skly anin, Preprin t LO MI E-3 - 97, Leningrad, 1 9 79. [39] M.A. Semenov -Tian-Shansky , F unct. Anal. Appl. 17 ( 1 983) 259 . [40] M. Jimbo, Comm un. Math. Ph ys. 102 (1986) 53. [41] J.-M. Maillet, Phy s. Lett. B162 (1985) 137. [42] J. Av an and A. Doik ou, arXiv.071 0.1538 . 24 [43] P . Baseilhac a nd G .W. Delius, J. Ph ys A34 (2 001) 825 9 . [44] D. Bernard and A. Leclair, Commun . Math. Ph ys. 142 (1991) 99. [45] O. Babelon, D. Bernard and M. T alon, Intr o duction to classic al Inte gr able systems , (2003) Cambridge monog r a phs in Mathematical Ph ysics. [46] G.W. D elius, N.J. MacKa y and B.J. Short, Phy s. Lett. B522 (2 001) 335; Erratum-ibid. B524 (20 02) 401; N.J. MacKay , J. Phy s. A35 (2002) 7865. [47] I. Bena, J. P olc hinski and R. Roiban, Ph ys. Rev. D69 ( 2 004) 046002; A.M. Poly ako v, Mo d. Ph ys. Lett. A19 (2 004) 164 9 . [48] V.A. Kazako v, A. Marshak o v, J.A. Minahan and K. Za r em b o, JHEP 0405 (2004) 024; V.A. Kazako v and K. Zarem b o, JHEP 0 4 10 (2 004) 060 ; N. Beisert, V.A. Ka zak o v, K. Sak ai and K. Zarem b o, Comm un. Math. Ph ys. 263 (2006) 659. [49] H. Georg i, Lie A lgebr as in Pa rticle Physics (Benjamin/Cummings, 1982). 25