Boundary Lax pairs for the $A_{n}^{(1)}$ Toda field theories

Boundary Lax pairs for the A (1) n T o da field theori e s Jean Av an 1 a and Anastasia Doik ou 2 b a LPTM, Univ ersite de Cergy-Po n toise (CNRS UMR 8089), Sain t-Martin 2 2 a v en ue Adolphe Ch auvin, F-9530 2 Cergy-P on toise Ced ex, F rance b Univ ersit y of Pa tras, Department of Engineering Sciences, GR-2650 0 P atras, Greece Abstract Based on the recen t f orm ulation of a general s c heme to construct b ound ary Lax pairs, we dev elop th is systematic constru ction for the A (1) n affine T od a field theories (A TFT). W e work out explicitly the fi r st t w o mo dels of the hierarc h y , i.e. the sine- Gordon ( A (1) 1 ) and the A (1) 2 mo dels. The A (1) 2 T o da theory is the first non-trivial example of th e hierarc h y that exhibits t w o d istinct t yp es of b ou n dary conditions. W e pro vide here no v el expressions of b oun dary Lax pairs associated to b oth t yp es of b ound- ary conditions. 1 av an@u-cer g y .fr 2 adoikou@upatras.gr 1 In tro du ction The Lax represen t a tion o f a classical dynamical system consists in the formulation of tw o endomorphism-v alued ob jects (in most cases matrix- or op erator- v alued) L and M , dep end- ing on the dynamical v ariables, suc h that the equations of motion ar e con tained in the iso-sp ectral ev olution equation: ∂ t L = h L, M i (1.1) The Lax matrix L therefore liv es in some Lie algebra, finite-dimensional, lo op alg ebra, or differen tial algebra dep ending on t he sp ecific system . It is collectiv ely called “auxiliary algebra”. The sp ectrum of L , or its a sso ciated mono dromy matrix if L is a differen tial op erator, pro vides therefore a generating set of natural t ime-in v ariant candidates to b e iden tified as in tegrable Hamiltonians. Liouv ille inte grability of an y dynamical syste m asso ciated to a giv en function on t his set follows from the P oisson-comm utation of the elemen ts of the set, guaran teed b y the necessary and sufficien t condition [2, 3] of the existenc e of a generically linear P o isson structure c haracterized by a so-called classical r - matrix, for t he sp ectrum- generating op erator (Lax matrix or mono drom y matrix). A natural construction f or M , giv en an y sp ecific Hamiltonian built as a function f o n sp( L ), is then a v ailable in terms of d f , L and the r -mat r ix [1, 2 ]. In our previous pap er [4] w e hav e examined the situation when t he P oisson structure a v ailable fo r L is expressed in terms of a non-dynamical classical r - matrix plus a set of non- dynamical parameters encapsulated in a “b oundary” matrix K ob eying some purely algebraic quadratic equation in terms of t his r -matrix. This more complicated equation (actually tw o distinct f orms thereof ) represen ts a classical vers ion o f the quan tum Cherednik-Skly anin reflection algebra [5, 6, 7] . The operato r generating the Poiss on comm uting Hamiltonians then combines L and K . W e ha v e then defined a systematic construction of the M op erat o r in terms of r , L and K Note here that the denomination of K as “b o undary” matrix reflects the fact that suc h parameters enco de indeed b o undary effects in the Hamiltonians when the matr ix L is a copro duct of lo cal l i matrices on a finite lattice i = 1 , . . . , N or a mono drom y of lo cal differen tial op erators o n a finite line. When by con trast L is a purely lo cal Lie-algebra v alued matrix, suc h par a meters may b e b etter c haracterized as coupling constan ts in a folding pro cedure (one is then considering a system on a one-site space lattice f o r whic h the notion of a “b oundary” has no sense). Tw o fundamen tal situations w ere described in [4]: linear Poisson structure and quadratic P oisson s tructure for the generating operato r . The second o ne is relev ant to describe systems 1 on a lattice or a con tin uous line. W e shall here restrict ourselv es to this latter case, partic- ularizing it even more to degree-one differen tial op erators L = d/dx + l ( x ). The generating op erator is here the mo no dromy matrix of this differen tial op erator computed b etw een the t w o ends of the finite line for x . W e consider here suc h op erators for whic h the r -matrix is the one asso ciat ed to the generic affine A (1) n T o da field theories. In this wa y w e construct the asso ciated La x represen tation of A TFT with non- trivial in tegrable b oundary conditions parametrized b y the K matrix. Tw o “b oundary” reflection equations m ust b e considered, resp. c haracterized as “soliton- preserving” (SP) and “solito n non-preserving” (SNP) (see relev ant studies at quantum and classical lev el [8]–[26 ]). More general reflection equation ma y b e considered giv en a classical r -matrix, t hr o ugh the c hoice of some auxiliary-space an ti-automorphism, but this extension shall b e p o stp oned for further studies. It will b e noted that this sc heme automatically yields b oundary conditions compatible with in tegrabilit y . Careful ev aluation of the con tributions to the Hamilto nians and the M matrix at the edges o f the x line are r equired to get consisten t results. Our ess en tial purp ose here is t w o-fold. W e shall first v alidat e our general deriv at ion b y a comparison of the A (1) 1 and Soliton-non-preserving A (1) 2 cases with kno wn results on lo w- dimension A TFT obtained by case-b y-case analysis, basic ally for SNP b oundary conditions [9]. W e shall indeed sh o w that b oth deriv ations yield exactly iden tical formulations for the b oundary conditions for the dynamical fields and related Lax fo r m ulation (with a proviso, related to analyticit y conditions, to b e sp ecified later). Similar consistency c hec ks we re also ac hiev ed in [4] within the ve ctor non-linear Schrodinger con text. W e shall then provide no v el expressions for b oundary Hamiltonians and the asso ciated Lax pairs in the y et un treated case of A (1) 2 with SP boundary conditions, and w e shall address some in triguing tech nical p oin ts that arise ev en in the simplest case, i.e. the sine-Gordon mo del. 2 The general sc heme Our analysis of clas sical in tegr a ble field theories with in tegrable bo undary conditions relies on the study of an asso ciated auxiliary linear problem (see [27] and references therein). Let us first recall the bulk case (no extra “ b oundary” para meter) to fix the not a tions and recall the basic structures. The Lax pa ir [28] for m ulation [29] of classical integrable Hamiltonian systems consists in defining a n auxiliary linear differen tial pr o blem, reading in the simplest case (or der-1 2 differen tial op erato rs):  ∂ x − U ( x, t, λ )  Ψ = 0 (2.1)  ∂ t − V ( x, t, λ )  Ψ = 0 (2.2) U , V a re in general n × n matrices with en tries functions of the dynamical fields, their space deriv ativ es, and p ossibly the complex sp ectral parameter λ . Compatibilit y conditions of the t w o differen tial equations (2.1), (2.2 ) lead to the zero curv ature condition [29, 30, 31] ∂ t U − ∂ x V + h U , V i = 0 . (2.3) The latter equations g iv e rise to the corresp onding classical equations o f motio n of the system under consideration. Natural conserv ed quan tities are obtained from the mono dromy matrix T ( x, y , λ ) = P exp { Z y x U ( x ′ , t, λ ) dx ′ } (2.4) once it is assumed that U ob eys the classical linear Poiss on alg ebraic relations: n U a ( x, λ ) , U b ( y , µ ) o = h r ab ( λ − µ ) , U a ( x, λ ) + U b ( y , µ ) i δ ( x − y ) , (2.5) and consequen t ly T ( x, y , λ ) satisfies ( see [27]): n T a ( x, y , t, λ ) , T b ( x, y , t, µ ) o = h r ab ( λ − µ ) , T a ( x, y , t, λ ) T b ( x, y , t, µ ) i . (2.6) One then immediately gets: n tr T ( λ ) , tr T ( µ ) o = 0 where t ( λ ) = tr T ( λ ) (2.7) and th us tr T ( λ ) yields the relev an t conserv ed quan tities. W e are no w in terested in implem en ting non- trivial in t egr a ble b oundary conditions. W e fo cus here o n tw o distinct types of integrable b oundary conditions: the so-called soliton pre- serving (SP) and the soliton non- pr eserving (SNP). F orm ulation of the tw o distinct t yp es of b oundary conditions is ac hiev ed by defining tw o t yp es of P oisson structures for t he mo dified mono dromy matrices T . These will in fa ct repre sen t the classical v ersions of the reflection algebra R , a nd t he t wisted Y angian T written in the followin g fo r ms (see e.g. [6, 32 , 33, 9]): (I) SP (r eflection algebra) [5, 6] 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 ) r 21 ( λ 1 − λ 2 ) + T 1 ( λ 1 ) r 21 ( λ 1 + λ 2 ) T 2 ( λ 2 ) − T 2 ( λ 2 ) r 12 ( λ 1 + λ 2 ) T 1 ( λ 1 ) (2.8) 3 (I I) SNP (( q )-twisted Y angia n) [34, 35] 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 ) r t 1 t 2 21 ( λ 1 − λ 2 ) + T 1 ( λ 1 ) r t 1 12 ( λ 1 + λ 2 ) T 2 ( λ 2 ) − T 2 ( λ 2 ) r t 2 21 ( λ 1 + λ 2 ) T 1 ( λ 1 ) (2.9) In most we ll known phys ical cases, suc h as t he A (1) N − 1 r -matrices r t 1 t 2 12 = r 21 , hence all the expressions ab ov e ma y b e written in a simpler form. In order to construct represen tations of (2.8), (2.9) yielding the generating function o f P oisson-comm uting Hamiltonians realizing the in tegrals o f motio n for a new classical in te- grable sys tem, one no w intro duces non-dynamical r epresen tations ( K ± ) of the algebra R ( T ). The non- dynamical condition: n K ± 1 ( λ 1 ) , K ± 2 ( λ 2 ) o = 0 (2.10) transforms (2.8), (2.9) in to algebraic equations for K ± . W e consider then any bulk mon- o dromy matrix T with P oisson structure (2.6) and we define in addition: ˆ T ( λ ) = T − 1 ( − λ ) for SP , ˆ T ( λ ) = T t ( − λ ) for SNP . (2.11) One exp ects tha t a more self-contained f orm ulation may in v olv e a more general anti-automorphism of the auxiliary algebra where T lives . A c orresp onding reform ulation of the relev an t Poisson structures should also b e prop osed in tha t framew ork. W e shall leav e this generalizatio n for later studies. Generalized “ mono dromy ” matrices, realizing the corresp onding algebras R , T , are finally giv en b y the follow ing expressions [6, 9]: T ( x , y , t, λ ) = T ( x, y , t, λ ) K − ( λ ) ˆ T ( x, y , t, λ ) . (2.12) 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) Indeed one sho ws: n t ( x, y , t, λ 1 ) , t ( x, y , t, λ 2 ) o = 0 , λ 1 , λ 2 ∈ C . (2.14) The systematic Lax formulation in the case of op en b oundary conditions is describ ed in [4]. More precisely it w as shown in [4] that n ln t ( λ ) , U ( x, µ ) o = ∂ V ( x, λ, µ ) ∂ x + h V ( x, λ, µ ) , U ( x, µ ) i (2.15) 4 where w e define: (I) SP V ( x, λ, µ ) = t − 1 ( λ ) tr a  K + a ( λ ) T a (0 , x, λ ) r ab ( λ − µ ) T a ( x, − L, λ ) K − a ( λ ) T − 1 a (0 , − L, − λ ) + K + a ( λ ) T a (0 , − L, λ ) K − a ( λ ) T − 1 a ( x, − L, − λ ) r ba ( λ + µ ) T − 1 a (0 , x, λ )  , (2.16) (I I) SNP V ( x, λ, µ ) = t − 1 ( λ ) tr a  K + a ( λ ) T a (0 , x, λ ) r ab ( λ − µ ) T a ( x, − L, λ ) K − a ( λ ) T t a a (0 , − L, − λ ) + K + a ( λ ) T a (0 , − L, λ ) K − a ( λ ) T t a a ( x, − L, − λ ) r t a ab ( − λ − µ ) T t a a (0 , x, − λ )  , (2.17) P articular atten tion should b e paid to the b o undary p oints x = 0 , − L . Indeed, for these t w o p oin ts one has to tak e into account that T ( x, x, λ ) = ˆ T ( x, x, λ ) = I . (I) SP V (0 , t, λ, µ ) = t − 1 ( λ ) tr a  K + a ( λ ) r ab ( λ − µ ) T a (0 , − L, λ ) K − a ( λ ) T − 1 a (0 , − L, − λ ) + K + a ( λ ) T a (0 , − L, λ ) K − a ( λ ) T − 1 a (0 , − L, − λ ) r ba ( λ + µ )  V ( − L, t, λ, µ ) = t − 1 ( λ ) tr a  K + a ( λ ) T a (0 , − L, λ ) r ab ( λ − µ ) K − ( λ ) T − 1 a (0 , − L, − λ ) + K + a ( λ ) T a (0 , − L, λ ) K − a ( λ ) r ba ( λ + µ ) T − 1 a (0 , − L, − λ )  (2.18) (I I) SNP V (0 , t, λ, µ ) = t − 1 ( λ ) tr a  K + a ( λ ) r ab ( λ − µ ) T a (0 , − L, λ ) K − a ( λ ) T t a a (0 , − L, − λ ) + K + a ( λ ) T a (0 , − L, λ ) K − a ( λ ) T t a a (0 , − L, − λ ) r t a ab ( − λ − µ )  V ( − L, t, λ, µ ) = t − 1 ( λ ) tr a  K + a ( λ ) T a (0 , − L, λ ) r ab ( λ − µ ) K − ( λ ) T t a a (0 , − L, − λ ) + K + a ( λ ) T a (0 , − L, λ ) K − a ( λ ) r t a ab ( − λ − µ ) T t a a (0 , − L, − λ )  (2.19) Notice that all the b oundary info rmation, incorp orated in K ± , app ears only a t the b oundary p oin ts x = 0 , − L . W e shall presen tly see that the bulk expression has in fact no dep endence whatso ev er on the reflection K matrix, since it is canceled b y the t − 1 ( λ ) facto r . Note finally that the expressions deriv ed in (2.16)–(2.18) a re univers al, that is indep enden t of the choice of mo del. 5 3 The A (1) n A TFT: brief re view W e ar e now in a p osition to systematically construct the Lax represen tation f o r an y extension of the A (1) n A TFT following the sc heme defined a b ov e, giv en any solution K o f the algebraic b oundary equations defined b y t he A (1) n T o da classical r -matrix. N ote that the asso ciated b oundary Hamiltonians hav e been extracted thro ugh the asymptotic expansion of the op en transfer matrix in [36] for sine-Gordon and in [26] for the A (1) 2 A TFT. W e shall consider in the fo llo wing sections t w o particular examples, that is the prototype mo del of the hierarc h y , i.e. the sine-Gordon model, as w ell as the A (1) 2 case. The A (1) 2 mo del is indeed the first non-trivial example of this set t ha t may exhibit both t yp es of b oundary conditions. It is w orth noting that in sine-Gordon the tw o b oundary conditions coincide due to t he fact that the mo del is self dual. Recall first the classic al r - matrix associated to the generic A (1) n affine T o da field theory in particular is giv en b y 3 [37] 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 . (3.1) with ( e ij ) k l ≡ δ ik δ j l . Note that the classical r -matrix (3.1) is written in the so- called principal gradation as in [9, 13] (see details on the g a uge transformation c hanging t he principal to the homogeneous gradatio n in [26]). W e r ecall t he L a x pair f or a g eneric A (1) n theory [38]: 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  (3.2) Φ , Π are conjugated n -ve ctor fields, with comp onen ts φ i , π i , i ∈ { 1 , . . . , n } , u = e 2 λ n +1 is the m ultiplicativ e sp ectral parameter. T o compare with the notation used in [9] w e set m 2 16 = ˜ m 2 8 ( ˜ m denotes the mass in [9]). Note that ev en tua lly in [9] b ot h β , ˜ m are set equal to unit. W e also define: E + = n +1 X i =1 E α i , E − = n +1 X i =1 E − α i (3.3) α i are the simple ro ots plus the extended (affine) ro ot, H ( n - vector) a nd E ± α i are the algebra generators in the Cartan-W eyl basis, and they satisfy the Lie algebra relat io ns: 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 (3.4) 3 Notice that the r - matrix employ ed here is in fact r t 1 t 2 12 with r 12 being the matrix used e.g. in [11, 19] 6 Explicit expressions on the simple ro ots and the Cartan generators are presen ted b elo w. Notice that the Lax pair has the following b ehavior: V t ( x, t, − u − 1 ) = V ( x, t, u ) , U t ( x, t, u − 1 ) = U ( x, t, u ) (3.5) where t denotes usual transp osition. W e prov ide b elow explicit expressions of the simple ro ots a nd the Cartan generators for A (1) n [39]. The v ectors α i = ( α 1 i , . . . , α n i ) are the simple ro ots of the Lie algebra of rank n normalized to unit y 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 } (3.6) The fundamen tal w eigh ts µ k = ( µ 1 k , . . . , µ n k ) , k = 1 , . . . , n a re defined as ( see, e.g., [39]). α j · µ k = 1 2 δ j,k . (3.7) The extended (affine) ro ot a n +1 is pro vided b y the relation n +1 X i =1 a i = 0 . (3.8) The Cartan-W eyl generator s in the defining repre sen tation are: 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. (3.9) 4 The b oundary A (1) 1 case: sine-Go r d on mo del Let us rewrite the Lax op erator for the bulk sine Go rdon mo del 4 , U ( x, t, u ) = β 4 i π ( x ) σ 3 + mu 4 i e iβ 4 φσ 3 σ 2 e − iβ 4 φσ 3 − mu − 1 4 i e − iβ 4 φσ 3 σ 2 e iβ 4 φσ 3 (4.2) σ i are the 2-dimensional P auli matrices. 4 T o recov er the generic fo r m (3.2) from (4 .2 ) we co nsider the following identifications β i → β , φ → − φ, u → − u (4.1) Here (4.2) we cle arly consider the sine- Gordon, how ever after implementing ide ntifications (4.1) we obtain the sinh-Gordon mo de l. 7 Bearing in mind the expression for T (2.12) it is clear that w e need to consider the formal series expansion of T a nd T − 1 ( u − 1 ). But fro m the fo llowing symmetry of the Lax op erator: U ( u − 1 , φ, π ) = U ( − u. − φ, π ) (4.3) w e see tha t T ( u − 1 , φ, π ) = T ( − u, − φ, π ) . (4.4 ) W e aim at expressing the term of order u in U indep enden tly of the fields, after applying a suitable ga ug e transformation [27]. More precisely , consider the follo wing gauge transforma- tion suc h that T ( x, y , u ) = Ω( x ) ˜ T ( x, y , u ) Ω − 1 ( y ) , Ω( x ) = dia g  Ω 1 ( x ) , Ω 2 ( x )  = e i 4 β φ ( x ) σ 3 (4.5) then the gauge transformed op erator ˜ U is expressed as: ˜ U ( x, t, u ) = β 4 i f ( x ) σ 3 + mu 4 i σ 2 − mu − 1 4 i e − iβ 2 φσ 3 σ 2 e iβ 2 φσ 3 (4.6) where w e define f ( x, t ) = π ( x, t ) + φ ′ ( x, t ) . (4.7 ) Let T ′ ( u ) = T ( u − 1 ) then w e introduce the follow ing decomp osition for ˜ T , ˜ T ′ as | u | → ∞ [27] ˜ T ( x, y , u ) = ( I + W ( x, u ) ) exp[ 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 , (4.8) where the hat simply denotes that u → − u, φ → − φ . W, ˆ W are off diago nal matrices and Z , ˆ Z are purely diag onal. Also Z ( u ) = ∞ X k = − 1 Z ( k ) u k , W ( u ) = ∞ X k =0 W ( k ) u k . (4.9) Inserting the la tter exp ressions (4 .9) in (2.1) one may iden tify the matrices W ( k ) and Z ( k ) . Indeed, from equation (2.1) we conclude that the ga ug e transformed op erators satisfy: d Z dx = ˜ U ( D ) + ˜ U ( O ) W dW dx + W ˜ U ( D ) − ˜ U ( D ) W + W ˜ U ( O ) W − ˜ U ( O ) = 0 (4.10) 8 where ˜ U ( D ) , ˜ U ( O ) are the diag o nal and off diagona l parts of ˜ U respectiv ely . By solving the latter equations w e ma y identify the matrices Z , W . It is sufficien t fo r our purp oses here to iden tify o nly the first couple of terms of the expansions. Indeed based on equation (4 .10) w e conclude (see also [27]): W (0) = iσ 1 , W (1) = − iβ m f ( x ) σ 1 (4.11) Note that the leading contribution as iu → ∞ comes from the e Z 22 term. W e assumed here for simplicit y , but without losing generalit y , Sc h w a r tz b oundar y conditions at the end p oin t x = − L , that is π ( − L ) = φ ( − L ) = 0 a nd K − ∝ I . W e ma y rewrite the expression for the b oundary op erat o r V (let ˆ r ab = r ba ) 5 as: V ( x, t, u, v ) = t − 1 ( u ) e Z 22 − ˆ Z 22  (1 + ˆ W (0)) − 1 Ω(0) K + ( u )Ω(0)(1 + W (0))  22 nh (1 + W ( x )) − 1 Ω − 1 ( x ) r ( uv − 1 )Ω( x )(1 + W ( x )) i 22 + h (1 + ˆ W ( x )) − 1 Ω( x ) ˆ r ( uv )Ω − 1 ( x )(1 + ˆ W ( x )) i 22 o (4.13) but it is easy to sho w for the transfer matrix (2.13) as | u | → ∞ : t ( u ) = e Z 22 − ˆ Z 22  (1 + ˆ W (0)) − 1 Ω(0) K + ( u )Ω(0)(1 + W (0))  22 (4.14) and finally V ( x, t, u, v ) = h (1 + W ( x )) − 1 Ω − 1 ( x ) r ( uv − 1 )Ω( x )(1 + W ( x )) i 22 + h (1 + ˆ W ( x )) − 1 Ω( x ) ˆ r ( uv )Ω − 1 ( x )(1 + ˆ W ( x )) i 22 . (4.15) Again using the ansatz fo r the mono dr o m y matrix w e obtain from (2.18) for the end p oin t x = 0: V (0 , t, u, v ) = h (1 + ˆ W (0)) − 1 Ω(0) K + ( u )Ω(0)(1 + W (0)) i − 1 nh (1 + ˆ W (0)) − 1 Ω(0) K + ( u ) r ( u v − 1 )Ω(0)(1 + W (0)) i 22 + h (1 + ˆ W (0)) − 1 Ω(0) ˆ r ( uv ) K + ( u )Ω(0)(1 + W (0)) i 22 o . (4.16) The r -matrix is giv en in (3.1) and w e consider b elow t w o cases with non- diagonal and diagonal K -matrix resp ectiv ely . 5 Note that in this particular case the SP and SNP bo undary conditions coincide b ecause: r 12 ( λ ) = V 1 r t 1 12 ( − λ ) V 1 , V = antidiag (1 , 1) (4.12) 9 4.1 Non-diagonal K -matrix W e shall first examine the case with the g eneric non-diagonal K -matr ix [40, 41] K + ( λ ) = 1 κ sinh( λ + iξ ) e 11 + 1 κ sinh( − λ + iξ ) e 22 + x + sinh(2 λ ) e 12 + x − sinh(2 λ ) e 21 (4.17) ξ , x ± are a priori free indep enden t b oundar y parameters. The next step is to expand expressions (4.15) , (4.16) in p ow ers of u − 1 , and identify the first order term of the expansion. T aking in to account the expansion of W as well as ( | u | → ∞ ) K + ( u ) ∼ K +(0) + u − 1 K +(1) + O ( u − 2 ) , r ( uv − 1 ) ∼ r (0) + u − 1 r (1) + O ( u − 2 ) , ˆ r ( uv ) ∼ r (0) + u − 1 ˆ r (1) + O ( u − 2 ) (4.18) where w e define: K +(0) = x + e 12 + x − e 21 , K +(1) = e iξ κ e 11 − e − iξ κ e 22 , r (0) = 2 X i =1 e ii ⊗ e ii , r (1) = 2 v ( e 12 ⊗ e 21 + e 21 ⊗ e 12 ) , ˆ r (1) = 2 v − 1 ( e 12 ⊗ e 21 + e 21 ⊗ e 12 ) (4.19) w e ma y expand V ( u, v ) in p o w ers o f u − 1 . Multiplying the resulting expression b y a factor m 4 i w e obtain at first order: V ( x, t, v ) = β 4 i φ ′ ( x, t ) σ 3 + v m 4 i Ω( x, t ) σ 2 Ω − 1 ( x, t ) + v − 1 m 4 i Ω − 1 ( x, t ) σ 2 Ω( x, t ) (4.20) W e see that the op erat or V ( x, t, v ) at an y p oin t x 6 = 0 coincides with the bulk op erator (3.2), consisten t ly with the fa ct that H coincides with the bulk sine-Gordon b oundar y Hamilto nian [40, 36] except for x = 0 : H = Z 0 − L dx h 1 2 ( π 2 + φ ′ 2 ) + m 2 β 2 (1 − cos β φ ) i + 4 P m β 2 cos β φ (0 ) 2 − 4 Qm β 2 sin β φ (0 ) 2 (4.21) Recall that in [36] the la t ter Hamiltonian w as obtained as the first order term from the ex- pansion o f the generating function t ( u ) as | u | → ∞ , a ssuming Sc h w a rtz b oundary conditions at x = − L . The b oundary parameters P , Q are related t o the parameters ξ , κ of the K matrix as: P = e iξ − e − iξ 4 κ , Q = e iξ + e − iξ 4 κi . (4.22) W e should stress that the constrain t x + = − x − here arises by requiring that the expansions of t he tr a nsfer ma t r ix a s iu → ∞ and iu → −∞ provid e the same Hamiltonians (aga in, 10 an analyticity condition at infinit y). Suc h a requireme n t leads also to the cancelation o f b oundary terms prop ort ional to φ ′ (0). It is clear that the bulk V -op erat or is indep enden t of the c hoice of K -matrix. Exp anding carefully the b oundary expression (2.19) and multiply ing the result with a factor m 4 i w e obtain at the b oundary p oint: V ( b ) (0 , t, v ) = V (0 , t, v ) + ∆ V (0 , t, v ) , where ∆ V (0 , t, v ) = − β 4 i φ ′ (0) σ 3 − m 8  e iξ 2 κ e iβ 2 φ (0) + e − iξ 2 κ e − iβ 2 φ (0)  σ 3 (4.23) V (0 , t, v ) is provide d by the bulk expression (4.2 0 ). Note that all the b o undar y information is incorp orated at the b oundary p oint x = 0. The equations of motion and the corresp onding b oundary conditions emerge in this Lax form ulation fro m the zero curv ature condition. The zero curv ature condition for the ‘bulk’ Lax pa ir yields the f a miliar equations of motion for the sine-Gordon mo del. W e should note tha t analyticit y requiremen ts on the b oundary La x pair leads to extra constraints among the b oundary parameters, i.e. x + = − x − (see also [36]). The b o undary op erator found here is asso ciated to the b oundary Hamiltonian (4.21) . Note that a particular c hoice o f diagonal K + matrix leads to discrepancies b et w een the t w o descriptions (Hamiltonian vs Lax pair). This suggests that one has to consider as a starting p oin t a generic solution o f the reflection equation with sev eral bo undary parameters, whic h ma y then satisfy further constrain ts dictated by certain consistency req uiremen ts. The relev ant b oundary conditio ns are obta ined by considering the zero curv atur e condi- tion (2.3) at the p oint x = 0: ˙ U (0 , t, v ) − d dx V ( b ) ( x, t, v ) | x =0 + h U (0 , t, v ) , V ( b ) (0 , t, v ) i = 0 ⇒ (4.24) ˙ U (0 , t, v ) − lim δ → 0 V ( δ, t, v ) − V (0 , t, v ) − ∆ V (0 , t, v ) δ + h U (0 , t, v ) , V (0 , t, v ) + ∆ V (0 , t, v ) i = 0 . (4.25) Explicit expression of the deriv ative o f V at x = 0 in (4.25) indicates ∆ V (0 , t, v ) = 0 , (4.26) in order to eliminate a p ot ential uncomp ensated div ergence due to ∆ V . Finally fro m the ‘bulk’ zero curv ature condition a nd from the later expression the fo l- lo wing equations of motion and mixed b o undar y conditions are en tailed: ¨ φ ( x, t ) − φ ′′ ( x, t ) = − m 2 β sin( β φ ( x, t )) β φ ′ (0) = m 2 iκ cos( ξ + β 2 φ (0)) , (4.27) 11 whic h of course coincide with the equations of motion found in [4 0, 36] (recall also t he iden tification of b oundary para meters (4.22)). In a consisten t w a y the b o undary conditions are obtained exactly from the Hamiltonian thr o ugh ∂ φ ( x, t ) ∂ t = {H , φ ( x, t ) } , ∂ π ( x, t ) ∂ t = {H , π ( x, t ) } , x ∈ [ − L, 0] (4.28) b y no t icing that the contribution containing the term φ (0) in the Hamiltonian yields a δ (0) term in the equations of motio n fo r π ( x ) since { φ ( x ) , π ( y ) } = δ ( x − y ). Elimination of this term yields exactly the b oundary conditions (4.27). Requiring cancelation of the ∆ V (0) term is equiv alen t to requiring that the formal series expansion in u − 1 coincides for V (0 , t, u, v ) and V ( x, t, u, v ) at x → 0. Indeed the techn ical origin of ∆ V ( x = 0) is the no n-comm utation of limits x → 0 and u → ∞ , in particular in e Z . If these limits are required to commute then V ( x, t, u ) has its analytic b ehavior in x, u con tin ued to the limit x = 0 , whic h ma y su ggest that it can b e analytically con tin ued “b e- y ond” the b oundary . This ma y in turn b e a relev a n t consistency condition in implemen ting the notion of “gluing” differen t boundary systems. W e hav e thus established a straigh tfor- w ard a nd elegant w a y to extract the asso ciated b oundary conditions fr o m the zero curv ature condition. 4.2 Diagonal K -matrix W e shall now consider the diagonal K - ma t r ix [4 0, 4 1] K + ( λ ) = sinh( λ + iξ ) e 11 + sinh( − λ + iξ ) e 22 . (4.29) In particular w e shall b e mostly interes ted in the degenerate limit where iξ → ∞ . The relev an t b oundar y Hamiltonian, obtained from the first order term of the expansion of ln t ( λ ), is giv en b y: H = Z 0 − L dx h 1 2 ( π 2 + φ ′ 2 ) + m 2 β 2 (1 − cos β φ ) i + 2 β φ ′ (0) cos( ξ + β 2 φ (0)) sin( ξ + β 2 φ (0)) . (4.30) The b oundary con tribution in (4 .30) is not iden t ical with the κ = 0 limit of the b oundary conditions in (4.21). One needs to normalize t he K -matrix as uK ( κ → 0) to get a consisten t u -expansion, hence H in (4.30) pic ks b oundary contribution fro m higher orders in (4.18). By requiring the b oundary term, prop ortio nal to φ ′ (0) to disapp ear w e obtain the follow ing constrain t cos( ξ + β 2 φ (0)) = 0 , (4.31) 12 whic h of course may b e seen as the b oundary condition to the a sso ciated equations of motion, as w e shall see b elow . The next step as in the previous case is to expand expressions (4.1 5), (4.16 ) in p o w ers o f u − 1 , and identify the asso ciated V - op erator f r o m the first order term of the expansion. As w e ha v e seen the bulk V ( u, v )-op erator is indep enden t of the choice of b oundary conditions, i.e. the K - matrix and is giv en at an y p oin t x 6 = 0 b y (3.2). Expanding carefully the bo undar y expression (2 .1 9) and multiply ing the result with a factor − m 2 i w e obtain at the b oundary p oin t: V ( b ) (0 , t, v ) = β i ∆ 2 y + y − Ω 2 1 (0)Ω 2 2 (0) φ ′ (0) σ 3 + m 2∆ Ω 1 (0)Ω 2 (0)  v ( y + e 21 − y − e 12 ) + v − 1 ( y − e 21 − y + e 12 )  (4.32) where ∆ = y + Ω 2 1 (0) + y − Ω 2 2 (0), and y ± = ± e ± iξ . By requiring V (0) = V ( b ) (0) w e obtain the corresp onding b oundary conditio ns. Finally from the ‘bulk’ zero curv ature condition and from the later expression the follow ing equations of motion and mixed b oundary conditions are en tailed: ¨ φ ( x, t ) − φ ′′ ( x, t ) = − m 2 β sin( β φ ( x, t )) cos( ξ + β 2 φ (0)) = 0 , (4.33) whic h of course coincide with the equations of motion found in [4 0, 36] (recall also t he iden tification of b oundary parameters (4 .22) and the b oundary conditions f o und earlier. Notice that t he obtained b oundary conditions are easily obtained f r o m t he generic situation described in (4.2 7) b y simply setting the non dia gonal con tributions to zero. W e are how ev er mostly in terested in the case where the K -matrix is degenerate (in the homogeneous gradation). Consider for instance the situation w here K ( λ ) = diag ( e λ , e − λ ). The Hamiltonian and b oundary V - op erator in this case are give n resp ectiv ely b y: H = Z 0 − L dx h 1 2 ( π 2 + φ ′ 2 ) + m 2 β 2 (1 − cos β φ ) i + 2 β φ ′ (0) (4.34) and V ( b ) (0 , v ) = m 4 Ω 1 (0)Ω − 1 2 (0)  v e 21 − v − 1 e 12  . (4.35) The b oundary conditions emerging from the Hamiltonian are: φ ′ (0) = 0 whereas requiring V ( b ) (0 , v ) = V (0 , v ) in a ddition to the field space de riv at iv e b eing ze ro one more cons train t is obtained: e i β 2 φ (0) = 0 (4.36) 13 whic h of course is also automatically obtained from the b oundary conditions found previously in the f ull diag o nal case a t e − iξ = 0. Note that there is no w a y to trace the extra constraint (4.36) from Hamiltonian p oint of view although w e ha v e to no te that suc h a constrain t is not incompatible with the Hamiltonian. T o conclude w e note that in the degenerate case some imp ortan t information is automat- ically lo st when consid ering the Hamiltonian description. More precisely , in the degenerate case there is no ξ dep endence so constrain ts of the t yp e (4.36 ) disapp ear when examining the b oundary conditions from the Hamiltonian viewpo int. Whenev er we pass f r o m the most gen- eral situation to some sp ecial situation some information is lost and inconsistencies b et w een the tw o descriptions arise. This o f course happ ens only when the K - ma t rix p ossesses sev eral b oundary parameter and some of t hem a r e set t o zero or t o infinity . W e shall examine in the follow ing section a similar situation for the next mo del of the hierarc h y , the A (1) 2 theory , and w e shall see that the arising inconsistencies may b e explained in the same spirit. 5 The b oundary A (1) 2 case W e come now to the second mem ber o f the hierarc h y and the first mo del of this class exhibiting bo t h t yp es of distinct b oundary conditions SP and SNP , i.e. the A (1) 2 mo del. In this case w e ha v e: α 1 = (1 , 0) , α 2 = ( − 1 2 , √ 3 2 ) , α 3 = ( − 1 2 , − √ 3 2 ) (5.1) define also the follo wing 3 × 3 generators E 1 = E t − 1 = e 12 , E 2 = E t − 2 = e 23 , E 3 = E t − 3 = − e 31 . (5.2) The diagonal Cartan generators H 1 , 2 are then: H 1 = 1 2 ( e 11 − e 22 ) , H 2 = 1 2 √ 3 ( e 11 + e 22 − 2 e 33 ) (5.3) Let T ′ ( x, y , u ) = T ( x, y , u − 1 ) and U ′ ( x, u ) = U ( x, u − 1 ). F ollo wing the logic des crib ed previously ( see also [27 ]) for the sine-Gordon mo del, we aim at expressing the part asso ciated to E + , E − in U , U ′ resp ectiv ely independently o f the fields, th us w e consider the follo wing gauge transformation: 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 ) (5.4) 14 where w e define Ω( x ) = dia g  Ω 1 ( x ) , Ω 2 ( x ) , Ω 3 ( x )  = e β 2 Φ( x ) · H . (5.5) F rom equation (2.1) the gauge t r ansformed op erators ˜ U , ˜ U ′ are expressed 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 . (5.6) After implemen ting the gauge transforma t ions ˜ U , ˜ U ′ tak e the follo wing simple forms: ˜ U ( x, t, u ) = β 2 F · H + m 4  uE + + 1 u X −  , ˜ U ′ ( x, t, u ) = β 2 ˆ F · H + m 4  uE − + 1 u X +  (5.7) where w e define: F = Π − ∂ x Φ , ˆ F = Π + ∂ x Φ , X − = e − β Φ · H E − e β Φ · H , X + = e β Φ · H E + e − β Φ · H (5.8) ˜ T , ˜ U also satisfy (2 .1), and F , ˆ F a re v ectors with tw o compo nen ts f i , ˆ f i , i ∈ { 1 , 2 } resp ectiv ely . Consider aga in the ansatz (4.8) for ˜ T , ˜ T ′ as | u | → ∞ . As in the previous section inserting expressions (4.9) in (2.1) one then iden tifies the co efficien ts W ( k ) ij and Z ( k ) ii . Indeed from (2.1) w e obtain the follow ing fundamen tal r elat io ns: 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 (5.9) where the sup erscripts O , D denote off-diagonal and dia gonal part respectiv ely . Similar relations ma y b e obt a ined for ˆ Z , ˆ W , in this case ˜ U → ˜ U ′ . W e omit writing these equations here for brevit y . It will b e useful in what follows to introduce some compac t notatio n: β 2 F · H = dia g ( a, b, c ) , β 2 ˆ F · H = dia g (ˆ a, ˆ b, ˆ c ) , e β α i · Φ = γ i . (5.10) Explicit expressions of a, b, c and γ i are giv en b y: a = β 2 ( f 1 2 + f 2 2 √ 3 ) , b = β 2 ( − f 1 2 + f 2 2 √ 3 ) , c = − β 2 f 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 ) . (5.11) apparen tly ˆ a, ˆ b, ˆ c a re defined in the same w a y as a, b, c but with f i → ˆ f i . 15 The computation of W, ˆ W is essen tial for what follo ws. First it is imp ortant t o discu ss the leading con tribution of the ab o v e quantities as | u | → ∞ . T o ac hiev 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    . (5.12) F rom the form ulas (5 .9) the matrices W ( k ) , ˆ W ( k ) , Z ( k ) , ˆ Z ( k ) ma y b e determined. In partic- ular, we write b elow explicit expressions of these matrices for the first orders, whic h will b e necessary in the subsequen t sections (see also [26]): 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    . (5.1 3 ) F or computing the b oundary conserv ed quantities , energy and momen tum, we shall in ad- dition need the follow ing expressions: d Z (1) 11 dx = e − iπ 3 3 m 4 ( γ 1 + γ 2 + γ 3 ) + 4 e − iπ 3 3 m ( a ′ − c ′ ) + 4 e − iπ 3 6 m ( a 2 + b 2 + c 2 ) d Z (1) 22 dx = e iπ 3 3 m 4 ( γ 1 + γ 2 + γ 3 ) + 4 e iπ 3 3 m ( b ′ − a ′ ) + 4 e iπ 3 6 m ( a 2 + b 2 + c 2 ) d ˆ Z (1) 11 dx = e iπ 3 3 m 4 ( γ 1 + γ 2 + γ 3 ) − 4 e iπ 3 3 m ( ˆ b ′ − ˆ a ′ ) + 4 e iπ 3 6 m (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 ) + 4 e − iπ 3 3 m ( ˆ b ′ − ˆ c ′ ) + 4 e − iπ 3 6 m (ˆ a 2 + ˆ b 2 + ˆ c 2 ) . (5.14 ) 5.1 SNP b oundary conditions W e shall fo cus in this sec tion on the analysis of the SNP integrable b oundary conditio ns in A (1) 2 A TFT. Comparison with some already known results [9] will v a lida te our a ppro ac h, whic h then presen ts the adv an tage of b eing systematically implemen table once a non dynam- ical “b oundary” mat r ix is c hose n. The b oundary V -o p erator in this case is given by (2 .17), (2.19). W e assume here for simplicit y , but without lo sing generalit y , Sch wartz bo undar y conditions at x = − L and K − ∝ I (see also [26]). T aking into accoun t the ansatz fo r the 16 mono dromy matrix (4.8) as w ell as b earing in mind that a s u → ∞ the main con t r ibutio n for the diagonal terms comes from e Z 33 , e ˆ Z 33 (see also [26]), w e conclude V ( x, t, u, v ) = h (1 + W ( x, u )) − 1 Ω − 1 ( x ) r ( uv − 1 )Ω( x )(1 + W ( x )) i 33 + h (1 + ˆ W t ( x, u ))Ω − 1 ( x ) r t 1 ( u − 1 v − 1 )Ω( x )((1 + ˆ W ( x )) − 1 ) t i 33 . (5.15) W e recall t ha t the sup erscript t 1 denotes tra nsp osition in the first space. Also, in the ex- pressions with ‘hat’ we simply conside r Φ → − Φ. F or further tec hnical details w e refer the in terested reader to [2 6 ]. Note t ha t in this case the limit u → ∞ is easier t o consider due to the expressions (5.1 2). In any case, although tec hnically mor e in v olv ed, one can show that the u → −∞ limit pro vides the same conserv ed quan tities and Lax pairs. In the follow ing w e shall expand expression (5.15), so w e need expansions of all the in v olv ed quan tities: r ( uv − 1 ) ∼ r (0) + u − 1 r (1) + O ( u − 2 ) , r t 1 ( u − 1 v − 1 ) ∼ − r (0) − u − 1 ˆ r (1) + O ( u − 2 ) (5.16) where w e define: r (0) = 3 X i =1 e ii , r (1) = 2 v ( e 21 ⊗ e 12 + e 32 ⊗ e 23 + e 13 ⊗ e 31 ) ˆ r (1) = 2 v − 1 ( e 21 ⊗ e 21 + e 32 ⊗ e 32 + e 13 ⊗ e 13 ) (5.17) F rom the first order of t he expansion of the V -o p erator and after mu ltiplying with a facto r of − 3 m 8 w e obta in: V ( x, t, v ) = − β 2 Φ ′ ( x, t ) · H + mv 4 Ω( x, t ) E + Ω − 1 ( x, t ) − mv − 1 4 Ω − 1 ( x, t ) E − Ω( x, t ) (5 .18) whic h as an ticipated coincides with the bulk V -opera t or of A (1) 2 (3.2). In order to obtain the explicit form of the b oundary V -op era t or w e should also review kno wn results on the solutions of the r eflection equation for SNP b o undar y conditions. The generic solution for the A (1) n case in the principal gradatio n are g iv en b y [13, 18]: 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 , [ g 0 , g l ] = 0 , l > 1 (6.1) One ex p ects that the most generic solution f or the A (1) n − 1 case should b e ex pressed in terms of represen tations of the generator g 0 with all ξ α 6 = 0 and ξ α 6 = ξ β ∀ l 6 = k , i.e. (see also (5.27)) K ( λ ) = n − 1 X α =0 c α ( λ ) g α 0 (6.2) the g eneral s olution should be th us dictated b y the rank of the algebra. Note in particular that for t he sine-G ordon mo del n = 2 one recov ers the b oundary T emp erley-Lieb algebra (see e.g. [43]). In this con text a ll solutions that giv e rise to inconsistencies are sp ecial in the sens e that are eithe r degenerate ξ α = ξ β or correspond to a case with at least one zero eigen v alue ξ α = 0. T o summarize: sp ecial cases of K -matrices giv e rise to inconsistencie s, hence one needs to consider the most general p ossible solutio ns of the reflection equation with distinct inde- p enden t b oundary par a meters. In the SNP case no extra free b oundary parameters app ear and no extra constrain ts among the b oundary fields o ccur. In the A (1) 2 SP case w e conjecture that any generic ( no n-degenerate) non-diago nal solution with f ree b oundary parameters will b e appropriate. F o r the momen t w e ha v e no suc h a generic matrix at our disposal, but the inconsistencies arising g iv e us a strong hint that there should ex ist solutions with more b oundary pa rameters. 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