Brief lectures on duality, integrability and deformations
BBrief lectures on duality, integrability and deformations Ctirad Klimˇc´ık
Aix Marseille Universit´e, CNRS, Centrale MarseilleI2M, UMR 737313453 Marseille, France
Abstract
We provide a pedagogical introduction to some aspects of integrability, dualitiesand deformations of physical systems in 0+1 and in 1+1 dimensions. In particular,we concentrate on the T-duality of point particles and strings as well as on theRuijsenaars duality of finite many-body integrable models, we review the conceptof the integrability and, in particular, of the Lax integrability and we analyze thebasic examples of the Yang-Baxter deformations of non-linear σ -models. The cen-tral mathematical structure which we describe in detail is the E -model which is thedynamical system exhibiting all those three phenomena simultaneously. The lastpart of the paper contains original results, in particular a formulation of sufficientconditions for strong integrability of non-degenerate E -models.Keywords: integrable systems, nonlinear sigma models, T-duality, Ruijsenaarsduality A recent progress in the theory of integrable nonlinear σ -models in two space-timedimensions has brought to light the relevance of the so-called E -models for theintegrability story. This relevance is intriguing, because the original motivation[34, 36, 28] for the introduction of the E -models was rather to understand thedynamics of the T-dualizable σ -models and the integrability came only later as awelcome bonus [26].The role of this review is to describe in a succinct manner how the stories ofthe T-duality and of the integrable deformations meet together on the playgroundof the E -models. On the top of it, we explain what is the Ruijsenaars duality in Based on the lectures given in Mathematical Sciences Institute, ANU, Canberra, SimonsCenter for Geometry and Physics, Stony Brook, Tohoku Forum for Creativity, Sendai and in theSantiago de Compostela ”Integrability, dualities and deformations” webinar. a r X i v : . [ h e p - t h ] J a n he many-body integrable systems [48] and we also formulate some original resultsconcerning sufficient conditions guaranteeing that a given E -model is strongly Laxintegrable.The plan of the paper is as follows: in Section 2, we briefly explain what wemean by ”duality” and then we concentrate on the T-duality of point particlesand strings. In the case of strings, we start with the well-known exposition of theAbelian T-duality, then we provide a reformulation of the story in terms of the E -model based on an Abelian Drinfeld double and, finally, we release the conditionof the Abelianity to recover the full-fledged Poisson-Lie T-duality. We describebriefly also the degenerate version of the E -model dynamics which gives rise to theso called dressing cosets T-duality. In Section 3, we focus on the case of 0 + 1dimension, namely we give a precise definition of the integrability of a many-bodydynamical system and we explain in this context what is the so called Ruijsenaarsduality. In Section 4, we introduce a special case of the so called Lax integrability,which, unlike the general integrability, can be easily generalized from 0 + 1 to 1 + 1-dimensions. In this context, we distinguish a weak and a strong Lax integrability,in the latter case one must specify also the so called r -matrix as we shall illustrateon the example of the principal chiral model. Finally, we devote Section 5 to thedescription of the Yang-Baxter deformations of the principal chiral model. Wefirst describe a prototypical one-parameter deformation known under the name ofYang-Baxter σ -model [26] and then we deal in detail with a three-parametric caseof the so called bi-YB-WZ σ -model introduced in Ref.[19, 30, 31]. In the lattercase, we give a simplified proof of the strong Lax integrability than that givenin Ref.[31] and, as an original result, we give sufficient conditions for a general E -model to be strongly Lax integrable. Duality is a very large term relevant in a vast and very diversified amount ofexamples. Therefore, we have first to make more precise the conceptual frameworkwhich will allow us to express what we mean by the duality in the present article.Consider a set of ”structures” S and associate to a given structure a ∈ S a set M ( a ) of ”points of views”. Looking at the structure a from a point ofview m ∈ M ( a ), we may or we may not recognize some pattern b from the set of”patterns” W . If we do recognize it, we refer to the point of view m as ”meaningful”and we say that the structure a exhibits the pattern b .Given a structure a ∈ S , it may happen that there is no meaningful point ofview to look at it (i.e. a exhibits no pattern from the predefined set W ), whichis, course, an utterly uninteresting case. It also happens, that there is just onemeaningful point of view and since there is then a one-to-one correspondence be-tween the pattern and the structure one is tempted to treat those two notionsinterchangeably. However, interestingly, for some structures there may exist more2han one meaningful point of view. In particular, if there are precisely two mean-ingful points of views m and m making the structure a to exhibit two differentpatterns b and b , we say that b and b are mutually dual to each other (we speakabout the ”self-duality” of b , if m (cid:54) = m but b = b ). If there are more than twomeaningful points of view for the structure a , we speak about a ”plurality”.An amusing example : The set S of the ”structures” is the set of drawings .Given a drawing a ∈ S , we can look at it from different angles (points of view).The set of patterns W is the set of human faces . The point of view is meaningful,if we recognize in the drawing such a human face. Example of duality
A drawing from a non-meaningful point of viewLooking at the same drawing from two meaningful points of view. The pictures come from the address http://immenseworlds.blogspot.com/2007/10/reverse-pictures-same-upside-down.html .1 T-duality for point particles In this section, the set of ”structures” will be the set of 2 n -dimensional Hamil-tonian dynamical systems , this is to say the set of pairs (
P, H ), where P is a2 n -dimensional symplectic manifold and H a distinguished function on P calledthe Hamiltonian.Given the dynamical system ( P, H ), the ”point of view” is every choice of local(Darboux) coordinates x i , p i , i = 1 , . . . , n on P in which the symplectic form takesthe canonical form ω = dp i ∧ dx i . (1)The point of view is meaningful if the Hamiltonian H expressed in the Darbouxcoordinates x i , p i takes the form H = 12 g ij ( x ) ( p i + eA i ( x )) ( p j + eA j ( x )) + eφ ( x ) , (2)where g ij ( x ) is a non-degenerate symmetric tensor field. The corresponding ”pat-tern” exhibited by the dynamical system ( P, H ) is then the electro-magnetic-gravitational background ( φ ( x ) , A i ( x ) , g ij ( x )) on the Riemannian manifold definedby the requirements p i = 0, i = 1 , . . . , n , where g ij ( x ) is the inverse metric tensor. Remark 2.1:
Recall for completeness that writing down the Hamiltonian equationof motions for the Hamiltonian (2) and eliminating the momenta p i gives the geodesicequation in the presence of the Lorentz force¨ x i + Γ ijk ( x ) ˙ x j ˙ x k + eg ij ( x )( F jk ( x ) ˙ x k + ∂ j φ ( x )) = 0 , (3)Γ ijk = 12 g im ( ∂ k g mj + ∂ j g mk − ∂ m g jk ) , F jk = ∂ j A k − ∂ k A j . (4)The charged geodesics equation (3) can be obtained also from a second order action S [ x ( t )] = 12 (cid:90) dt g ij ( x ) ˙ x i ˙ x j − e (cid:90) dt ( A i ( x ) ˙ x i + φ ( x )) . (5) The question we wish to ask is the following one: If the dynamical system (
P, H )admits Darboux coordinates yielding the meaningful form (2) of its Hamiltonian H , can it exist another meaningful point of view, i.e. another system of Darbouxcoordinates ˜ x i , ˜ p i which would also yield the meaningful form (2) of the Hamilto-nian but would exhibit a different background geometry ( ˜ φ (˜ x ) , ˜ A i (˜ x ) , ˜ g ij (˜ x ))?To answer this question, we first remark that if the Darboux coordinates ˜ x i , ˜ p i are related to the Darboux coordinates x i , p i by a point-like canonical transforma-tion ˜ x j = ˜ x j ( x ) , ˜ p j = ∂x k ∂ ˜ x j p k , (6)then the meaningful Hamiltonian (2) becomes meaningful also in the coordinates˜ x i , ˜ p i where it reads H = 12 ˜ g ij (˜ x )(˜ p i + e ˜ A i (˜ x ))(˜ p j + e ˜ A j (˜ x )) + e ˜ φ (˜ x ) , (7)4 g ij (˜ x ) = ∂ ˜ x i ∂x k ∂ ˜ x j ∂x l g kl ( x ) , ˜ A i (˜ x ) = ∂x j ∂ ˜ x i A j ( x ) , ˜ φ (˜ x ) = φ ( x ) . (8)However, the duality realized by the point-like transformation (6) is not an inter-esting one because the transformation formulas (8) give geometrically the sameelectro-magnetic-gravitational background induced by the diffeomorphism ˜ x j =˜ x j ( x ).To find a nontrivial duality, the original Darboux coordinates x i , p i and thedual ones ˜ x i , ˜ p i must be therefore related by a canonical transformation which isnot the point-like one (6). An example of such nontrivial transformation was givenin [31] and we describe it here in detail.Consider a four-dimensional manifold P = R ∗ × R ∗ , where R ∗ stands for two-dimensional plane without origin. Introducing two copies of the standard polarcoordinates r, φ and ρ, f on R ∗ , we consider the following four-parametric electro-gravitational background T ( µ, γ, m, c ) on Mϕ = 12 (cid:18) γ + 1 r (cid:19) + 12 (cid:18) c + 1 ρ (cid:19) , (9a) A = 0 , (9b) ds = 11 + µ r dr + r γ r dφ + 11 + m ρ dρ + ρ c ρ df . (9c)The dynamics of the charged point particle of the positive charge e in thebackground T ( µ, γ, m, c ) is then governed by the Hamiltonian H = 12 (1 + µ r ) p r + 1 + γ r r ( p φ + e ) + 12 (1 + m ρ ) p ρ + 1 + c ρ ρ ( p f + e ) (10)and by the symplectic form ω = p r ∧ dr + p φ ∧ dφ + p ρ ∧ dρ + p f ∧ df. (11)We now express the coordinates p r , r > , p ρ , ρ > , p φ , φ, p f , f on the phasespace P in terms of new coordinates P R , R > , P R , R > , P Φ , Φ , P F , F as follows r = R (cid:112) R P R + P + e (cid:112) R P R + P F + e , p r = P R (cid:112) R P R + P F + e (cid:112) R P R + P + e , p φ = P Φ , (12a) ρ = R (cid:112) R P R + P F + e (cid:112) R P R + P + e , p ρ = P R (cid:112) R P R + P + e (cid:112) R P R + P F + e , p f = P F , (12b) f = F + P F (cid:112) P F + e arctan (cid:32) RP R (cid:112) P F + e (cid:33) − P F (cid:112) P F + e arctan (cid:32) R P R (cid:112) P F + e (cid:33) , (12c) φ = Φ − P Φ (cid:112) P + e arctan (cid:32) RP R (cid:112) P + e (cid:33) + P Φ (cid:112) P + e arctan (cid:32) R P R (cid:112) P + e (cid:33) . (12d)5he transformation (12) is the diffeomorphism of the phase space P with theinverse diffeomorphism given by R = ρ (cid:113) ρ p ρ + p φ + e (cid:113) ρ p ρ + p f + e , P R = p ρ (cid:113) ρ p ρ + p f + e (cid:113) ρ p ρ + p φ + e , P Φ = p φ , (13a) R = r (cid:113) r p r + p f + e (cid:113) r p r + p φ + e , P R = p r (cid:113) r p r + p φ + e (cid:113) r p r + p f + e , P F = p f , (13b) F = f − p f (cid:113) p f + e arctan ρp ρ (cid:113) p f + e + p f (cid:113) p f + e arctan rp r (cid:113) p f + e , (13c)Φ = φ + p φ (cid:113) p φ + e arctan ρp ρ (cid:113) p φ + e − p φ (cid:113) p φ + e arctan rp r (cid:113) p φ + e . (13d)Moreover, the transformation (12) is the symplectic diffeomorphism (or sym-plectomorphism) of the phase space P because it preserves the symplectic form ω .Indeed, inserting the formulas (12) into (11) gives ω = dP R ∧ dR + dP Φ ∧ d Φ + dP R ∧ d R + dP F ∧ dF. (14)It remains to show that the canonical transformation (12) can be interpretedas the T-duality symplectomorphism. For that, we express the Hamiltonian (10)in terms of the new Darboux coordinates P R , R > , P R , R > , P Φ , Φ , P F , F . Theresult is H = 12 (1+ m R ) P R + 1 + γ R R ( P + e )+ 12 (1+ µ R ) P R + 1 + c R R ( P F + e ) . (15)The comparison of the formula (15) with Eq.(10) shows that the role of theparameters m and µ got exchanged while the parameters c and γ remained in theirplaces. Said in other words, the Hamiltonian (15) describes the dynamics of thecharged point particle in the dual background T ( m, γ, µ, c )˜ ϕ = 12 (cid:18) γ + 1 R (cid:19) + 12 (cid:18) c + 1 R (cid:19) , (16a)˜ A = 0 , (16b)˜ ds = 11 + m R dR + R γ R dφ + 11 + µ R d R + R c R dF . (16c)To conclude the argument that this point-particle T-duality indeed does some-thing non-trivial, it is sufficient to show that the flipping of the parameters µ and m may indeed alter the Riemannian geometry of the dual background T ( m, γ, µ, c )6ith respect to that of the original one T ( µ, γ, m, c ). For that, consider for examplethe background T (0 , , ,
1) with the metric ds = dr + r dφ + dρ + ρ df ρ . (17)We see that this is the Riemannian geometry of the direct product of the Euclideanplane with the Euclidean black hole [55]. The metric corresponding to the dualbackground T (1 , , ,
1) is˜ ds = 11 + r dr + r dφ + dρ + ρ ρ df . (18)We work out easily that the respective Ricci scalars of the metrics (17) and (18)read Ric = 41 + ρ , (cid:103) Ric = − ρ ) . (19)We thus observe that the Riemannian geometries (17) and (18) are inequivalent,because Ric is strictly positive while (cid:103)
Ric acquires also negative values.
The appropriate description of T-duality in string theory is achieved by choosingas the set of ”structures” the so called loop dynamical systems, i.e. the triples(
LD, ω, H ), where LD is the space of loops (every point of the manifold LD is amap from the unit circle S to some finite-dimensional manifold D ), ω a symplecticform on LD and H is a distinguished function on LD called the Hamiltonian.The ”point of view” is a local system of loop Darboux coordinates x i ( σ ) , p i ( σ )on LD in which the symplectic form takes the form ω = (cid:73) δp i ( σ ) ∧ δx i ( σ ) . (20)Note that the symbol (cid:72) stands for the integration over the loop parameter σ and,as usual, δ denotes the de Rham exterior derivative in the infinite-dimensionalcontext.The point of view is meaningful if the Hamiltonian H expressed in the loopDarboux coordinates x i ( σ ) , p i ( σ ) takes the form H = 12 (cid:73) g ij ( x ) (cid:0) p i − b ik ( x ) x (cid:48) k (cid:1) (cid:0) p j − b jl ( x ) x (cid:48) l (cid:1) + 12 (cid:73) g ij ( x ) x (cid:48) i x (cid:48) j . (21)Finally, the ”pattern” exhibited by the dynamical system ( LD, ω, H ) is the Kalb-Ramond-gravitational background ( b ij ( x ) , g ij ( x )) on the Riemannian manifold T spanned by the coordinates x i , i = 1 , . . . , n .7liminating the coordinates p i ( σ ) from the Hamiltonian equations of motionof the system (20), (21) gives the geodesic surface equation of the string exposedsimultaneously to the gravitational and to the Kalb-Ramond Lorentz force: ∂ + ∂ − x i + Γ ijk ( x ) ∂ + x j ∂ − x k + 12 g il ( x ) h ljk ( x ) ∂ + x j ∂ − x k = 0 . Here ∂ ± = ∂ τ ± ∂ σ and the Kalb-Ramond field strength h ljk ( x ) is the exteriorderivative of the 2-form field b ik ( x ): h ljk ( x ) = ∂ l b jk ( x ) + ∂ j b kl ( x ) + ∂ k b lj ( x ) . (22)The second-order action leading to the geodesic surface equation reads S [ x ( τ, σ )] = (cid:90) dτ (cid:73) ( g ij ( x ) + b ij ( x )) ∂ + x i ∂ − x j ≡ (cid:90) dτ (cid:73) e ij ( x ) ∂ + x i ∂ − x j . (23)The crucial fact underlying the string T-duality story is as follows: Thereare examples of the loop dynamical systems ( LD, ω, H ) admitting more than onemeaningful points of view. This means that there are at least two sets of the localDarboux coordinates, say p i ( σ ) , x i ( σ ) and ˜ p i ( σ ) , ˜ x i ( σ ), such that the Hamiltonianof this system exhibits simultaneously two geometric patterns H = 12 (cid:90) dσg ij ( x ) (cid:0) p i − b ik ( x ) x (cid:48) k (cid:1) (cid:0) p j − b jl ( x ) x (cid:48) l (cid:1) + 12 (cid:90) dσg ij ( x ) x (cid:48) i x (cid:48) j == 12 (cid:90) dσ ˜ g ij (˜ x ) (cid:16) ˜ p i − ˜ b ik (˜ x )˜ x (cid:48) k (cid:17) (cid:16) ˜ p j − ˜ b jl (˜ x )˜ x (cid:48) l (cid:17) + 12 (cid:90) dσ ˜ g ij (˜ x )˜ x (cid:48) i ˜ x (cid:48) j (24)where the geometric data b ik ( x ) , g ik ( x ) need not be related by a general coordinatetransformation to the geometric data ˜ b ik (˜ x ) , ˜ g ik (˜ x ). Remark 2.2:
The flipping of the dual indices upside down is against the usual con-ventions but this notation is often very practical in the T-duality business.
Remark 2.3:
An equivalent description of the T-duality phenomenon is as follows:consider two manifolds T and ˜ T , each one equipped with the metric and with the Kalb-Ramond field. They are said to be T-dual to each other if the associated string dynamicalsystems are non-trivially equivalent. The non-trivial equivalence means the existence ofa symplectomorphism between the phase spaces of the strings moving in the geometry T and in the geometry ˜ T which is not the point canonical transformation. Moreover,this symplectomorphism must take the ˜ T -Hamiltonian into the T -Hamiltonian. The classical string equations make perfect sense also in the case where the three form h ljk ( x ) is closed but is not exact, but we stick for the moment to discussion of the cases wherethe Kalb-Ramond potential b ik ( x ) exists globally. .3 Abelian T-duality The prototypical example of the string T-duality is the so called Abelian T-duality[24, 51]. Topologically, the target manifold T is the direct product of n circles T = S × S × · · · × S , the i th circle is parametrized by an angle x i ∈ [0 , π [. Themetric g ij and the Kalb-Ramond field b ij on T are taken to be constant matrices, b antisymmetric while g symmetric and positive definite. The dynamics of classicalstrings in this background is defined by the second order Lagrangian S [ x ( τ, σ )] = (cid:90) dτ (cid:73) ( g ij + b ij ) ∂ + x i ∂ − x j , (25)which corresponds to the dynamical system with the Darboux symplectic form ω = (cid:73) δp i ( σ ) ∧ δx i ( σ ) (26)and with the Hamiltonian H = 12 (cid:90) dσg ij (cid:0) p i − b ik x (cid:48) k (cid:1) (cid:0) p j − b jl x (cid:48) l (cid:1) + 12 (cid:90) dσg ij x (cid:48) i x (cid:48) j . (27)We now introduce new variables ˜ p i , ˜ x i on the phase space by the formulae˜ p i := x (cid:48) i , ˜ x i = (cid:90) p i . (28)The new variables are also the Darboux ones because it is easy to check that itholds ω = (cid:73) δ ˜ p i ( σ ) ∧ δ ˜ x i ( σ ) . (29)Moreover, the Hamiltonian (27) can be rewritten in the new variables as H = 12 (cid:73) ˜ g ij (cid:16) ˜ p i − ˜ b ik x (cid:48) k (cid:17) (cid:16) ˜ p j − ˜ b jl x (cid:48) l (cid:17) + 12 (cid:73) ˜ g ij ˜ x (cid:48) i ˜ x (cid:48) j , (30)which means that it corresponds to the second order Lagrangian S [˜ x ( τ, σ )] = (cid:90) dτ (cid:73) (cid:16) ˜ g ij + ˜ b ij (cid:17) ∂ + ˜ x i ∂ − ˜ x j , (31)where ˜ x i are again the angle variables ranging from 0 to 2 π . Here the dual geo-metric data ˜ g , ˜ b are functions of the original data g , b in the sense of the formula(˜ g ik + ˜ b ik )( g kj + b kj ) = δ ij . (32)Said in other words, the dual geometry ˜ g , ˜ b is obtained by taking the symmetricand the antisymmetric part of the inverse matrix ( g + b ) − .The simplest set up is the one-dimensional one when the string moves on asingle circle S parametrized by the angle coordinate x ≡ x . In this case there9s no Kalb-Ramond field and there is one constant component of the metric ten-sor g = R . The T-duality then establishes the dynamical equivalence of stringsmoving on circle backgrounds with flipped radia: S [ x ] = R (cid:90) dτ (cid:73) ∂ + x∂ − x, ˜ S [˜ x ] = 1 R (cid:90) dτ (cid:73) ∂ + ˜ x∂ − ˜ x. (33) Remark 2.4:
There are some subtle points related to the canonical transformation (28)because the prime (standing for the ∂ σ derivative) kills the zero mode of the variable x and the integral of p i is defined up to a constant. We do not discuss those subtleties herein order to move on rapidly towards the non-Abelian generalizations but the interestedreader may consult Ref.[36] for a more detailed discussion in the classical and also in thequantum case. E -model formulation of the Abelian T-duality Consider a direct product Lie group D = U (1) × U (1) and its loop group LD consisting of the set of maps from the circle S into D equipped with the pointwisegroup multiplication. If we parametrize the elements l of LD by the phases l = ( e ix , e i ˜ x ) , (34)then the quantity l (cid:48) l − = i x (cid:48) ⊕ i˜ x (cid:48) (35)is the element of the Lie algebra L D . Because it holds p = ˜ x (cid:48) , we can rewrite theHamiltonian of the simplest R → /R example of the Abelian T-duality as H = 12 (cid:73) (cid:18) p R + Rx (cid:48) (cid:19) = 12 (cid:73) (cid:0) l (cid:48) l − , E l (cid:48) l − (cid:1) D , (36)where the bilinear form ( ., . ) D on the Lie algebra D and the operator E : D → D are defined as(i u ⊕ i v, i w ⊕ i z ) D = uz + vw, E (i u ⊕ i v ) = R − i v ⊕ R i u, u, v, w, z ∈ R . (37)Furthermore the Darboux property of the phase space coordinates p, x and ˜ p, ˜ x gives the following expression for the symplectic form: ω = − (cid:73) (cid:0) l − δl ∧ , ( l − δl ) (cid:48) (cid:1) D . (38)Indeed, from Eqs.(34),(37) and (28) we infer − (cid:73) (cid:0) l − δl ∧ , ( l − δl ) (cid:48) (cid:1) D == 12 (cid:73) δp i ( σ ) ∧ δx i ( σ )+ 12 (cid:73) δ ˜ p i ( σ ) ∧ δ ˜ x i ( σ ) = (cid:73) δp i ( σ ) ∧ δx i ( σ ) = (cid:73) δ ˜ p i ( σ ) ∧ δ ˜ x i ( σ ) . (39)10e can play a similar game for the higher-dimensional Abelian T-dualityrelating the models (25) and (31). Consider a direct product Lie group D = U (1) n × U (1) n and parametrize the elements l of its loop group LD by the phases l = ( e ix , . . . , e ix n , e i˜ x , . . . , e i˜ x n ) = ( e x j T j , e ˜ x j ˜ T j ) := ( e x , e ˜ x ) , (40)where T j , ˜ T j form basis of the Lie algebra D .Then the quantity l (cid:48) l − = x (cid:48) j T j ⊕ ˜ x (cid:48) j ˜ T j = x (cid:48) ⊕ ˜ x (cid:48) is the element of the Lie algebra L D . Because it holds p j = ˜ x (cid:48) j , we can rewrite theHamiltonian (27) of the higher-dimensional Abelian T-duality as H = 12 (cid:73) g ij (cid:0) p i − b ik x (cid:48) k (cid:1) (cid:0) p j − b jl x (cid:48) l (cid:1) + 12 (cid:90) dσg ij x (cid:48) i x (cid:48) j = 12 (cid:73) (cid:0) l (cid:48) l − , E l (cid:48) l − (cid:1) D , (41)where the bilinear form ( ., . ) D on the Lie algebra D and the operator E : D → D are defined as ( u ⊕ v , w ⊕ z ) D = u j z j + v j w j , (42) E ( u ⊕ v ) = g jk ( v k − b kl u l ) T j ⊕ (cid:0) ( g jk − b jm g ml b lk ) u k + b jm g mk v k (cid:1) ˜ T j . (43)Furthermore the Darboux property of the phase space coordinates p i , x i and ˜ p i , ˜ x i gives the following formula for the symplectic form (26): ω = − (cid:73) (cid:0) l − δl ∧ , ( l − δl ) (cid:48) (cid:1) D . (44) We have observed in the previous subsection that the higher-dimensional AbelianT-duality is structurally the same as the one-dimensional one. This means that1) There is an (Abelian) Lie group D of even dimension, the Lie algebra D of whichis equipped with a non-degenerate symmetric ad-invariant bilinear form ( ., . ) D ofmaximally Lorentzian signature (+ · · · + , − · · · − ).2) There are two half-dimensional connected isotropic Lie subgroups K and ˜ K of D where the term ”isotropic” means that the restriction of the bilinear form ( ., . ) D on the Lie algebras K and ˜ K vanishes ( K and ˜ K are respectively generated by thebasis T j and ˜ T j ).3) There is a linear operator E : D → D which squares to the identity on D , i.e. E = Id, E is self-adjoint with respect to the bilinear form ( ., . ) D , i.e. ( E ., . ) D =( ., E . ) D and, finally, the E -dependent symmetric bilinear form ( ., E . ) D is strictlypositive definite.The distinction between the higher-dimensional case and the one-dimensionalone is hidden just in the choice of the group D . In the higher-dimensional case wehave D = U (1) n × U (1) n and in the one-dimensional it is D = U (1) × U (1).11he moral of the story is as follows: to the data 1), 2) and 3) there is naturallyassociated the so called E -model, which is the loop dynamical system living on thephase space LD , with the symplectic form ω = − (cid:73) (cid:0) l − δl ∧ , ( l − δl ) (cid:48) (cid:1) D (45)and with the Hamiltonian H E = 12 (cid:73) (cid:0) l (cid:48) l − , E l (cid:48) l − (cid:1) D . (46)In the case of the Abelian group D , we have seen that the E -model is the dynamicalsystem describing the strings moving on the background (25) but also on thebackground (31).What is the Poisson-Lie T-duality? In the E -model picture, it is the generaliza-tion of the Abelian T-duality in which we just replace the Abelian Lie group D bya non-Abelian one. It is really as simple as this! Indeed, we are now going to showthat taking a non-Abelian Lie group D supplemented with the structures 1), 2)and 3), the corresponding E -model (45), (46) is the dynamical system describingthe strings moving on two geometrically different backgrounds. Remark 2.5 : Every Lie group D fulfilling the properties 1) and 2) is referred to asthe Drinfeld double. The Drinfeld doubles are so abundant that they were not evenclassified, the Poisson-Lie T-duality constitutes therefore a genuine factory to produceplenty of examples of the T-dual geometries. How to extract two mutually dual Riemannian-Kalb-Ramond geometries fromthe E -model data ( LD, ω, H E )? It is particularly simple to achieve this goal ina special case where the Drinfeld double is perfect , which means that D is topo-logically the direct product D = K × ˜ K and this direct product decompositionis compatible with the group multiplication. Said in other words, every element l of the perfect Drinfeld double can be written in an unique way as the groupmultiplication (in the sense of the group D ) of one element of the group K andanother of ˜ K : l = k ˜ h, k ∈ K, ˜ h ∈ ˜ K. (47)Inserting the decomposition (47) into (45) and into (46), we obtain easily ω = δ (cid:18)(cid:73) ( ˜Λ , k − δk ) D (cid:19) , (48) H E ( k, ˜Λ) = 12 (cid:73) ( ∂ σ kk − + k ˜Λ k − , E ( ∂ σ kk − + k ˜Λ k − )) D , (49)where ˜Λ = ∂ σ ˜ h ˜ h − is a ˜ K -valued field playing the role of a generalized momentum.Knowing the symplectic form and the Hamiltonian in the variables k, ˜Λ, wecan write down the first order action of the corresponding dynamical system as S E = (cid:90) dτ (cid:73) (cid:18) ( ˜Λ , k − ∂ τ k ) D −
12 ( ∂ σ kk − + k ˜Λ k − , E ( ∂ σ kk − + k ˜Λ k − )) D (cid:19) . (50)12he quadratic dependence on ˜Λ makes then easy to find the second order actiondepending solely on k ∈ K . It reads: S E ( k ) = 12 (cid:90) dτ (cid:73) (cid:0) ( E + Π( k )) − ∂ + kk − , ∂ − kk − (cid:1) D . (51)Here the linear operator E : ˜ K → K is defined by the operator E : D → D inthe way that its graph { ˜ x + E ˜ x, ˜ x ∈ ˜ K} coincides with the image of the operatorId+ E . As far as the k -dependent operator Π( k ) : ˜ K → K is concerned it is givenby the expression Π( k ) = −J Ad k ˜ J Ad k − ˜ J . (52)Here Ad k stands for the adjoint action on D of the element k ∈ K ⊂ D and J , ˜ J are projectors; J projects to K with the kernel ˜ K and ˜ J projects to ˜ K with thekernel K .Starting from the opposite decomposition of the Drinfeld double l = ˜ kh, ˜ k ∈ ˜ K, h ∈ K (53)and repeating the same derivation with the roles of K and ˜ K exchanged, we arriveat the dual second order action for the dual field ˜ k living in ˜ K :˜ S ˜ E (˜ k ) = 12 (cid:90) dτ (cid:73) (cid:18)(cid:16) ˜ E + ˜Π(˜ k ) (cid:17) − ∂ + ˜ k ˜ k − , ∂ − ˜ k ˜ k − (cid:19) D , (54)where ˜Π(˜ k ) = − ˜ J Ad ˜ k J Ad ˜ k − J . (55)Of course, the linear operators E : ˜ K → K and ˜ E : K → ˜ K are not independentobjects because their graphs coincide, being equal to the image of the operatorId+ E . It facts, it turns out that they are inverse to each other. Remark 2.6:
The action (51) as well as the dual action (54) define the Riemannian-Kalb-Ramond geometries respectively on the group targets K and ˜ K . It is not difficultto extract from (51), (54) explicit formulas for the metrics and for the Kalb-Ramondfields but we prefer to stick on the elegant coordinate invariant formulations (51), (54)of those Riemannian-Kalb-Ramond geometries. Remark 2.7:
Remarkably, the operator Π( k ) : ˜ K → K (as well as its dual ˜Π(˜ k ) : K → ˜ K ) defines a Poisson bracket on the group manifolds K (on ˜ K ). Remark 2.8:
If one of the subgroups K , ˜ K is Abelian and the other is non-Abelian,the Poisson-Lie T-duality reduces to the so called non-Abelian T-duality introduced inRefs.[10, 17, 18].In particular, if f , f are two functions on the group K , their bracket defined by theformula: { f , f } K ( k ) = ( ∇ L f , Π( k ) ∇ L f ) D (56) urns out to be the Poisson one. Here ∇ L is ˜ K -valued differential operator acting on thefunctions on K as( ∇ L f, x ) D ( k ) := ( ∇ Lx f )( k ) ≡ df ( e sx k ) ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 , x ∈ K . (57)Actually, the Poisson bracket (56) is of the so called Poisson-Lie type.An example of the perfect double D was constructed by Lu and Weinstein [40] andit is the special complex linear group SL ( n, C ) viewed as real group (i.e. it has thedimension 2( n −
1) as the real manifold). The non-degenerate symmetric ad-invariantbilinear form ( ., . ) D on the Lie algebra sl ( n, C ) is defined by taking the imaginary partof the trace ( X, Y ) D = (cid:61) tr ( XY ) , X, Y ∈ sl ( n, C ) (58)and has indeed the maximally Lorentzian signature ( n − , n − K and ˜ K are respectively the special unitary group SU ( n ) andthe upper-triangular group AN with real positive numbers on the diagonal the productof which is equal to 1.Every E -model yields a nontrivial T-duality pattern even in the case when the double D is not perfect. In this non-perfect case, the most convenient way of casting the actionsof the pair of mutually dual σ -models is in terms of the following expressions S E ( l ) = 14 (cid:90) δ − (cid:73) (cid:18) δll − , [ ∂ σ ll − , δll − ] (cid:19) D + (59a)+ 14 (cid:90) dτ (cid:73) (cid:18) ∂ + ll − , Q − l ∂ − ll − (cid:19) D − (cid:90) dτ (cid:73) (cid:18) Q + l ∂ + ll − , ∂ − ll − (cid:19) D , ˜ S E ( l ) = 14 (cid:90) δ − (cid:73) (cid:18) δll − , [ ∂ σ ll − , δll − ] (cid:19) D + 14 (cid:90) dτ (cid:73) (cid:18) ∂ + ll − , ˜ Q − l ∂ − ll − (cid:19) D − (cid:90) dτ (cid:73) (cid:18) ˜ Q + l ∂ + ll − , ∂ − ll − (cid:19) D . (59b)Here l ( τ, σ ) ∈ D is a field configuration, δ − is a (symbolic) inverse of the de Rhamdifferential and Q ± l , ˜ Q ± l : D → D are the projectors fully characterized by their respectiveimages and kernelsIm( Q ± l ) = Ad l ( ˜ K ) , Im( ˜ Q ± l ) = Ad l ( K ) , Ker( Q ± l ) = Ker( ˜ Q ± l ) = (1 ± E ) D . (60)It may seem, that both the σ -model (59a) as well as its dual (59b) live on the target D but, actually, the former lives on the space of cosets D/ ˜ K while the latter on the space D/K . Indeed, this is because the models (59a) and (59b) enjoy respectively the gaugesymmetries l ( τ, σ ) → l ( τ, σ )˜ h ( τ, σ ) , ˜ h ( τ, σ ) ∈ ˜ K, (61a) l ( τ, σ ) → l ( τ, σ ) h ( τ, σ ) , h ( τ, σ ) ∈ K. (61b)To verify that these gauge symmetries take indeed place, we need the standard Polyakov-Wiegmann identity W Z D ( gh ) = W Z D ( g ) + W Z D ( h ) + 2 (cid:73) (cid:18) ( g − δg, ∂ σ hh − ) D − ( δhh − , g − ∂ σ g ) D (cid:19) , (62) here W Z D ( g ) := δ − (cid:73) ( δgg − ∧ , [ ∂ σ gg − , δgg − ]) D In the case of the perfect double D , there is a natural global gauge fixing l = k , k ∈ K which transforms the expression (59a) into the action (51), while in the dual casethe natural gauge fixing is l = ˜ k , ˜ k ∈ K which transforms the expression (59b) into theaction (54). There exists a natural generalization of the Poisson-Lie T-duality known under the nameof the dressing coset T-duality [36]. The basic ingredients of this more general dualityis again the Drinfeld double D , its half-dimensional maximally isotropic Lie subgroups K , ˜ K and the self-adjoint operator E : D → D squaring to the identity, but newly weneed a (strictly less than half-dimensional) isotropic Lie subgroup F ∈ D such that theadjoint action of F on D commutes with E and the bilinear form ( ., E . ) D restricted tothe Lie subalgebra F ⊂ D is non-degenerate. The actions of the mutually dual σ -modelsare then again given by the expressions (59), but now the projectors Q ± l , ˜ Q ± l : D → D are determined by more general requirementsIm( Q ± l ) = Ad l ( ˜ K ) , Im( ˜ Q ± l ) = Ad l ( K ) , Ker( Q ± l ) = Ker( ˜ Q ± l ) = E ± ⊕ F , (63)where E ± ≡ (1 ± E ) D ∩ ( F ⊕ EF ) ⊥ . (64)The properties (63) of the projectors Q ± l , ˜ Q ± l guarantee that the respective gauge sym-metries (61) of the σ -models (59) get augmented by one more common gauge symmetry l ( τ, σ ) → f ( τ, σ ) l ( τ, σ ) , f ( τ, σ ) ∈ F. (65)This means that the σ -model (59a) lives on the space of double cosets F \ D/ ˜ K while thedual σ -model (59b) lives on the space F \ D/K . If the subgroup F is trivial, the dressingcoset T-duality reduces to the Poisson-Lie T-duality of the previous section.What is the common symplectic structure of the dual pair of the dressing cosetmodels? It is given by the symplectic reduction of the symplectic form (45) with respectto the left action of the loop group LF on the loop group LD . This action is generatedby the moment maps of the form ( l (cid:48) l − , x ) D , x ∈ F therefore the reduction is defined bythe following constraints imposed on the unreduced phase space LD ( l (cid:48) l − , F ) D = 0 . (66)The space of the orbits of the loop group LF on the constraint surface (66) is the reducedphase space or, equivalently, the phase space of the dressing coset σ -models.In order to express the Hamiltonian H dc of the dressing coset σ -models, we need firstto decompose the Lie algebra D as the following direct sum of vector spaces D = ( F ⊕ EF ) ⊥ ⊕ ( F ⊕ EF ) . (67)Given y ∈ D , we denote by y the first term in the decomposition (67). Then theHamiltonian is given by H dc = 12 (cid:73) (( l (cid:48) l − ) , E ( l (cid:48) l − ) ) D . (68) Integrability and the Ruijsenaars duality
The Ruijsenaars duality connects the duality story with that of integrability. It takesplace in the context of the dynamics of a finite number of degrees of freedom, where thenotion of integrability is by now perfectly understood in the mathematical literature. Onthe other hand, in the physical literature there can be sometimes found semi-rigorousformulations (even at the textbook level) which often miss the point what the integrabil-ity really is. In particular, those inexact formulations do not emphasize or even mentionthe crucial issue of completeness of the commuting evolution flows generated by theHamiltonians in involution. As the result, those imprecise definitions of integrabilityare quite empty in the sense that they suit too many dynamical systems and do notguarantee the special properties of the truly integrable systems.In what follows, we first provide clarifications of the concepts of completeness and po-larization, before giving the precised definition of the integrability and of the Ruijsenaarsduality.
Given a symplectic manifold P and a dynamical system ( P, H ) on it, the evolution flow u t generated by the Hamiltonian H is called complete if it can be smoothly prolongedto both forward and backward infinities t → ±∞ . The point u ∈ P is called the origin of the flow. The complete dynamical system (
P, H ) is such that all evolution flows on the sym-plectic manifold P , whatever are their origins, are complete.It may seem that that the notion of the complete dynamical system is quite difficultto handle in practice because the detailed knowledge of all evolution flows appears to beneeded to determine whether a given Hamiltonian does yield a complete dynamics or not.However, sometimes shortcut arguments may help to settle the issue of completeness evenwithout having at hand a detailed description of the flows. For example, if the symplecticmanifold P is compact then all flows are complete whatever is the smooth Hamiltoniangenerating the flows.As a non-compact example of the completeness of the flows, consider the manifold P n = { ( q , . . . , q n , p , . . . , p n ) ∈ R n , q > · · · > q n } (69)equipped with a symplectic form ω = dp j ∧ dq j . (70)For the Hamiltonian, we take the Calogero-Moser one H CM ( p j , q j ; g ) = 12 n (cid:88) j =1 p j + 12 n (cid:88) j (cid:54) = k g ( q j − q k ) , (71)where g is a coupling constant.Since the energy E CM ( u ) on any flow u t ∈ P n is conserved and the Calogero-MoserHamiltonian is given by the positive fonction, the momenta as well as the velocities f the Calogero-Moser particles are all bounded on the totality of the given flow u t ,therefore the particles cannot escape into infinity in a finite time. At the same time,the particles with positions q k ( t ) and q k +1 ( t ) always avoid the singular points q k = q k +1 (in a finite or in an infinite time) again because of the positivity of the Calogero-MoserHamiltonian and because of the conservation of energy. The Calogero-Moser dynamicalsystem ( P n , H CM ) is therefore complete.On the other hand, it has to be stressed that dynamical systems with non-completeevolution flows are quite frequent and easy to construct. We may e.g. consider somecomplete Calogero-Moser flow passing through a point u ∈ P n and pick some openneighbourhood O u of u which does not contain the totality of the flow. The openneighbourhood O u is itself a symplectic manifold and the same Hamiltonian H CM whichgenerates the complete flow on P n will yield a non-complete flow on O u . The dynamicalsystem ( O u , H CM ) is therefore non-complete. Denote by Fun( P ) the commutative and associative algebra with respect to the point-wise multiplication which consists of all smooth functions on a symplectic manifold P .A polarization of P is any maximal Poisson commuting subalgebra A ⊂
Fun( P ). Saidin other words, A is the commutative and associative subalgebra of Fun( P ) such thatthe Poisson bracket { f , f } vanishes whenever f , f ∈ A . Moreover, A is not strictlycontained in any subalgebra of Fun( P ) having the same property.Before advancing further, it is perhaps worth making a connection of our definition ofthe polarization with the one usually given, say, in the context of geometric quantization,where the polarization is understood as the smooth distribution of Lagrangian subspacesof the tangent spaces of the symplectic manifold. In fact, given the polarization A , thoseLagrangian subspaces are spanned by the Hamiltonian vector fields of the form { a, . } , a ∈ A , where { ., . } denotes the Poisson bracket corresponding to the symplectic form.The concept of polarization is not only important for the geometric quantizationbut it plays an essential role also in the study of purely classical integrable systems.Actually, this classical role is even two-fold. First of all, the polarization is necessary forthe physical interpretation of the flows: The dynamical system with physical interpretation is a triple ( P, A , H ), where P isa symplectic manifold, A is a polarization and H is a Hamiltonian. The polarization A bears the name ”physical position polarization”.The principal reason for the terminology ”physical interpretation” or ”physical posi-tion polarization” comes from the fact that for physical applications it is often necessaryto distinguish which coordinates on the phase space P correspond to physical positions ofparticles and which to momenta. If the phase space has the topology R n and possessesa global Darboux parametrization ( p j , q j ) ∈ R n , then the set of all functions f ( q j ) on R n , which are independent on p j , constitutes the physical position polarization A . Thispolarization subalgebra is generated by the global coordinates q j which are functionallyindependent because everywhere on R n it holds dq ∧ dq ∧ · · · ∧ dq n (cid:54) = 0 . (72) f the space of the physical positions is some topologically nontrivial manifold M (called the configuration space), the phase space is the cotangent bundle T ∗ M . In thiscase, the effective separation of the ”position” and the ”momenta” can still be donevia the concept of the polarization, although no global Darboux coordinates exist andneither do exist global position coordinates q j which would generate the subalgebra A . Remarkably, the easiest way to proceed in this topologically non-trivial case doesnot consist in covering M by some local coordinate patches but it remains global andconceptual. If we denote by π the canonical projection π : T ∗ M → M , then the physicalposition polarization A on T ∗ M consists of all functions on T ∗ M which are the pull-backs π ∗ f of functions f on M . For completeness, we mention that the symplectic form ω on T ∗ M is given by ω = dα, (73)where α is a 1-form on T ∗ M defined at a point ( x, β ) ∈ T ∗ M as α | ( x,β ) = π ∗ β, x ∈ M, β ∈ T ∗ x M. (74)In the case of the Calogero-Moser dynamical system ( P n , H CM ), the things do notrequire the level of sophistication described in the previous paragraph, since the globalposition coordinates q j exist albeit they are restricted by the inequalities q > q > · · · >q n . The physical position polarization A is therefore the subalgebra of functions on P n which do not depend on the momenta p j : A = { f ∈ Fun( P n ) , ∂ p j f = 0 , j = 1 , . . . , n } . (75)The polarization subalgebra A defined in this way stands simply for the coordinate-invariant way of introducing the physical configuration space in which the Calogero-Moser particles move. Now we describe the second role played by the concept of the polarization in the storyof the classical integrability.A polarization B is called complete if for every Hamiltonian H ∈ B the dynamicalsystem ( P, H ) is complete.
The integrable dynamical system ( P, B ) is any complete polarization B on the sym-plectic manifold P . The complete polarization B bears the name ”the action variablespolarization”.In practice, the integrable dynamical system ( P, B ) is often constructed starting froma given complete dynamical system ( P, H ) (like the Calogero-Moser one) by finding thecomplete polarization subalgebra B which contains the Hamiltonian H as its element.Since the elements of B do Poisson-commute among themselves, they all commute inparticular with H and thus constitute maximal set of conserved integrals of motion ininvolution.It is perhaps worth remarking in this respect, that many authors define the inte-grability in a similar way, that is by claiming that the maximal set of the conservedintegrals of motion must generate a polarization subalgebra. However, this claim is not ery restrictive if the completeness of this polarization is not required. In reality, it isthe requirement of the completeness which makes the integrable systems so rare andinteresting. In particular, as it was pointed out in Ref.[49], the completeness of the po-larization guarantees the validity of the Liouville-Arnold theorem [39, 1] which says thefollowing: If a dynamical system ( P, B ) is integrable then its phase space P can be de-composed into open, connected, pair-wise non-intersecting, flow-invariant parts U α ⊂ P such that P = ∪ α ¯ U α and on each part U α there can be introduced the so-called action-angle variables I j , θ j . Those special variables verify few important properties. First ofall, the action variables I α are restrictions to U α of suitable elements of the completepolarization B , they are functionally independent and the restriction of the symplecticform ω on U α takes the form ω | U α = dI j ∧ dθ j . (76)Moreover, the Hamiltonian H restricted onto U α depends exclusively on the action vari-ables I j and does not depend on the angles θ j . It follows that the dynamics of the systembecomes explicitly determined: the action variables I j are constants of motion and theevolution flows of the variables θ j are complete on U α and linear in time. Actually, ifthe phase space P is non-compact some of the thetas need not be angles and they canbe non-compact, too.It should be finally pointed out that the existence of a complete polarization B containing the given complete Hamiltonian H is by no means granted. Many dynamicalsystems ( P, H ) are not integrable and the difficult problem is to find out for whichHamiltonian H it exists the complete polarization B which contains it and for which onein turn such a polarization does not exist. An integrable dynamical system with physical interpretation is a triple ( P, A , B ), where A , B are the polarizations on the symplectic manifold P ; the physical position polarization A may be arbitrary but the action variables polarization B must be complete.Very often, however, the physical position polarization A turns out to be complete,too. In particular, this is always the case when the phase space P has the structure ofthe cotangent bundle T ∗ M and A = π ∗ Fun( M ). In any case, if the polarization A iscomplete then, mathematically speaking, it appears in the triple ( P, A , B ) on the samefooting as the complete polarization B . The Ruijsenaars duality then flips the roles ofthe polarizations A and B , so, from the dual point of view, it is not A which providesthe physical interpretation to the integrable system ( P, B ) but it is B which furnishes theintegrable system ( P, A ) with the physical interpretation. The Ruijsenaars dual of theintegrable dynamical system ( P, A , B ) with physical interpretation is thus the integrabledynamical system ( P, B , A ) with physical interpretation.In practice, the study of the Ruijsenaars duality concerns certain deformations ofthe complete dynamical systems of the Calogero-Moser type. One typically starts withthe study of such complete deformed dynamical system with physical interpretation( P, A , H ( c )), where c = ( c , . . . , c s ) is the set of the deformation parameters. The firsttask consists in looking for the complete polarization B containing H ( c ). If B can befound, then the second task consists in looking for a particular element ˜ H ( c ) ∈ A which ould have either the same or, possibly, another deformed Calogero-Moser form whenexpressed in the (original point of view) action-angle variables.We now illustrate the phenomenon of the Ruijsenaars duality on two examples forthe simplest non-trivial case n = 2. We shall study the case of arbitrary n in the nextsection.We start with the non-deformed Calogero-Moser Hamiltonian H CM ( p , p , q , q ; g ) = 12 p + 12 p + g ( q − q ) (77)defined on the phase space P = { ( q , q , p , p ) ∈ R , q > q } . (78)The manifold P is equipped with a symplectic form ω = dp ∧ dq + dp ∧ dq (79)giving rise to the Poisson bracket of functions a, b on P { a, b } = ∂ q a∂ p b − ∂ p b∂ q a + ∂ q a∂ p b − ∂ p b∂ q a. (80)Consider a map ρ : P → P defined by˜ q = 12 (cid:18) g ( q − q ) + ( p − p ) (cid:19) −
12 ( p + p ) , (81a)˜ q = − (cid:18) g ( q − q ) + ( p − p ) (cid:19) −
12 ( p + p ) , (81b)˜ p = −
12 ( p − p )( q − q ) (cid:18) g ( q − q ) + ( p − p ) (cid:19) − + 12 ( q + q ) , (81c)˜ p = 12 ( p − p )( q − q ) (cid:18) g ( q − q ) + ( p − p ) (cid:19) − + 12 ( q + q ) . (81d)The map ρ is involutive, which means that ρ ◦ ρ = Id. As such, ρ is bijective and, infact, it is the symplectomorphism of the phase space P , because it is easy to establishthat it holds ω = dp ∧ dq + dp ∧ dq = d ˜ p ∧ d ˜ q + d ˜ p ∧ d ˜ q . (82)Recall that the physical position polarization A of the n = 2 Calogero-Moser system isdefined as A = { f ∈ Fun( P ) , ∂ p f = ∂ p f = 0 } . (83)Is this polarization complete? To answer affirmatively this question, we observe thatevery element f ( q , q ) ∈ ˜ A generates the flows u t linear in time q j ( u t ) = q j ( u ) , p j ( u t ) = p j ( u ) − ( ∂ q j f )( q ( u ) , q ( u )) t, j = 1 , , (84)all flows generated by given f ( q , q ) ∈ A are therefore obviously complete. efine B = ρ ∗ A = { f ∈ Fun( P ); f = ˜ f ◦ ρ for ˜ f ∈ A } (85)or, equivalently, B = { f ∈ Fun( P ) , ∂ ˜ p f = ∂ ˜ p f = 0 } . (86)Being the pull-back of the complete polarization A by the symplectomorphism, B isalso the complete polarization. Moreover, the Calogero-Moser Hamiltonian H CM belongsto B , because it holds H CM = 12 p + 12 p + g ( q − q ) = 12 ˜ q + 12 ˜ q . (87)We thus succeeded to embed the complete n = 2 Calogero-Moser dynamical system( P , H CM ) into the integrable system ( P , B ), proving in this way its integrability. Ofcourse, we have even more than that at hand, since we have also the physical interpre-tation ( P , A , B ) of this dynamical system.What about the Ruijsenaars dual ( P , B , A )? Is there some dual Hamiltonian˜ H ∈ A which would have (possibly deformed) Calogero-Moser form in the action-anglevariables ˜ q j , ˜ p j ? Obviously there is a one because of the involutivity of the symplecto-morphism ρ ! Indeed we set˜ H = 12 ( q ) + 12 ( q ) = 12 ˜ p + 12 ˜ p + g (˜ q − ˜ q ) . (88)The existence of the dual Calogero-Moser Hamiltonian is usually interpreted by sayingthat the Ruijsenaars dual of the n = 2 Calogero-Moser dynamical system is the same n = 2 Calogero-Moser dynamical system.Our second elementary example of the Ruijsenaars duality is slightly more involved,although it still concerns the same four-dimensional phase space P (78) equipped withthe symplectic form ω (79). However, we consider now the following hyperbolic Suther-land deformation of the Calogero-Moser Hamiltonian H hypS ( c ; p , p , q , q ; g ) = 12 p + 12 p + g c sinh q − q c . (89)Note that it holdslim c →∞ H hypS ( c ; p , p , q , q ; g ) = H CM ( p , p , q , q ; g ) , (90)which means that the n = 2 hyperbolic Sutherland dynamical system ( P , H hypS ) is theone-parametric deformation of the n = 2 Calogero-Moser system ( P , H CM ).Is n = 2 hyperbolic Sutherland model ( P , H hypS ) integrable? Or, said differently,does it exist an integrable model ( P , B ) such that H hypS ∈ B ? Yes, it does. To proveit, we construct an appropriate symplectomorphism ρ c, : P → P such that the pull-back ρ ∗ c, A of the complete (physical position) polarization A (83) will be the completepolarization containing the hyperbolic Sutherland Hamiltonian H hypS .Consider a point-canonical transformation α : ( q , q , p , p ) → ( x , x , π , π ) de-fined by x = 12 ( q + q ) + c sinh q − q c , x = 12 ( q + q ) − c sinh q − q c , (91) = 12 ( p + p ) + p − p q − q c , π = 12 ( p + p ) − p − p q − q c (92)and also another point-canonical transformation ˜ α : (˜ q , ˜ q , ˜ p , ˜ p ) → (˜ x , ˜ x , ˜ π , ˜ π )defined by ˜ x = 12 (˜ q + ˜ q ) + ˜ q − ˜ q ˜ p − ˜ p c , ˜ x = 12 (˜ q + ˜ q ) − ˜ q − ˜ q ˜ p − ˜ p c , (93)˜ π = 12 (˜ p + ˜ p ) + c sinh ˜ p − ˜ p c , ˜ π = 12 (˜ p + ˜ p ) − c sinh ˜ p − ˜ p c . (94)Finally, recall the CM action-angle symplectomorphism ρ : ( x , x , π , π ) → (˜ x , ˜ x , ˜ π , ˜ π )given by Eqs.(81)˜ x = 12 (cid:18) g ( x − x ) + ( π − π ) (cid:19) −
12 ( π + π ) , (95a)˜ x = − (cid:18) g ( x − x ) + ( π − π ) (cid:19) −
12 ( π + π ) , (95b)˜ π = −
12 ( π − π )( x − x ) (cid:18) g ( x − x ) + ( π − π ) (cid:19) − + 12 ( x + x ) , (95c)˜ π = 12 ( π − π )( x − x ) (cid:18) g ( x − x ) + ( π − π ) (cid:19) − + 12 ( x + x ) . (95d)Define the symplectomorphism ρ c, : P → P by ρ c, = ˜ α − ◦ ρ ◦ α (96)which correspond to the composition of the canonical transformations ( q j , p j ) → ( x j , π j ) → (˜ x j , ˜ π j ) → (˜ q j , ˜ p j ). Then we define the complete polarization B c, par B c, = ρ ∗ c, A (97)which means that the generators of A c, are ˜ q j = ρ ∗ c, q j . The hyperbolic SutherlandHamiltonian belongs to the complete polarization B c, because it holds H hypS ( c ; p , p , q , q ; g ) = 12 p + 12 p + g c sinh q − q c = 12 ˜ q + 12 ˜ q . (98)So far we have established the integrability of the n = 2 hyperbolic Sutherland model( P , H hypS ) by constructing the integrable system ( P , B c, ) such that H hypS ∈ B c, . Ifwe include the physical interpretation in the game, what would be the Ruijsenaars dual( P , B c, , A ) of the hyperbolic Sutherland model ( P , A , B c, )? Is there some dualHamiltonian ˜ H ∈ A which would have a deformed Calogero-Moser form in the action-angle variables (˜ q j , ˜ p j ), ˜ q j ∈ B c, ? Yes, there is. It is given by the formula [50]˜ H = c (cid:18) cosh q c + cosh q c (cid:19) − c = c (cid:18) cosh ˜ p c + cosh ˜ p c (cid:19) (cid:18) g c (˜ q − ˜ q ) (cid:19) − c = ˜ H ratRS ( c ; ˜ p , ˜ p , ˜ q , ˜ q ; g ) . (99)Indeed, it is easy to verify that it holdslim c →∞ ˜ H ratRS ( c ; ˜ p , ˜ p , ˜ q , ˜ q ; g ) = 12 ˜ p + 12 ˜ p + g (˜ q − ˜ q ) = H CM (˜ p , ˜ p , ˜ q , ˜ q ; g ) . (100) Remark 3.1:
The form of the Ruijsenaars-Schneider Hamiltonian (99) in the tildedvariables may surprise some readers because of the non-separation of the kinetic and thepotential energy. Moreover, even in the case of the vanishing coupling constant g , the”free” Hamiltonian H ratRS has non-polynomial dependence on the momenta. Neverthe-less, such Hamiltonians naturally appear e.g. in the description of the dynamics of themultisolitons in the sine-Gordon theory [50].It turns out that the Ruijsenaars self-duality of the Calogero-Moser model ( P n , H CM )as well as the duality between the hyperbolic Sutherland and rational Ruijsenaars-Schneider models takes place also in the case of arbitrary n [48, 16]. The correspondingHamiltonians then read H hypS ( c ; p j , q j ; g ) = 12 n (cid:88) j =1 p j + 12 n (cid:88) j (cid:54) = k g c sinh q j − q k c , (101) H ratRS ( c ; ˜ p j , ˜ q j ; g ) = c n (cid:88) j =1 cosh (cid:18) ˜ p j c (cid:19) (cid:89) k (cid:54) = j (cid:115) g c (˜ q k − ˜ q j ) − nc . (102) Remark 3.2:
It is perhaps worth noting that further deformation of the duality pattern(101) and (102) restore the self-duality of the Calogero-Moser system [48]. Namely, re-placing in the expression (102) the rational functions of the dual coordinates ˜ q j by appro-priate hyperbolic functions gives the Hamiltonian of the self-dual hyperbolic Ruijsenaars-Schneider model.Before proving the Ruijsenaars self-duality of the Calogero-Moser system for arbi-trary n , we adapt the Ruijsenaars duality in our general schema of dualities outlined atthe beginning of Section 2. Thus the ”structure” is a symplectic manifold P equippedwith two complete polarizations A and B . The ”meaningful point of view” is any choiceof the local Darboux coordinates p j , q j such that q j are the local restrictions of the func-tions from one of the two polarizations and there is an element from the other polarizationwhich looks in the coordinates q j , p j like the deformed Calogero-Moser Hamiltonian. n In this section, we prove the Ruijsenaars self-duality of the Calogero-Moser system forarbitrary n [48]. Recall that the phase space of this system is the manifold P n definedas P n = { ( q , . . . , q n , p , . . . , p n ) ∈ R n , q > · · · > q n } (103)equipped with a symplectic form ω = dp j ∧ dq j (104) nd the Hamiltonian is given by H CM ( p j , q j ; g ) = 12 n (cid:88) j =1 p j + 12 n (cid:88) j (cid:54) = k g ( q j − q k ) . (105)The physical position polarization A n is defined as A n = { f ∈ Fun( P n ) , ∂ p j f = 0 , j = 1 , . . . , n } . (106)The completeness of A n can be easily shown by the generalization of the n = 2 argument(84).In the case n = 2, we have obtained the complete polarization B containing theCalogero-Moser Hamiltonian as the pull-back ρ ∗ A of the complete physical positionpolarization A by the symplectomorphism ρ : P → P given by Eqs.(81). In the caseof arbitrary n , the complete polarization B n containing the Calogero-Moser Hamiltonianis also obtained as the pullback ρ ∗ n A n , however the description of the symplectomorphism ρ n : P n → P n is known only in terms of implicit functions. Concretely, ρ n : ( p j , q j ) → (˜ p j , ˜ q j ) is defined by the relations q j = λ j ( ˜ L ) , ˜ q j = − λ j ( L ) , j = 1 , . . . , n (107)where λ j ( ˜ L ) and λ j ( L ) are respectively the ordered eigenvalues of the matrices ˜ L and L with the matrix elements given by L jk = p j δ jk − (1 − δ jk ) i gq j − q k , (108)˜ L jk = ˜ p j δ jk − (1 − δ jk ) i g ˜ q j − ˜ q k . (109)We infer from Eq.(107) H CM ( p j , q j ; g ) = 12 n (cid:88) j =1 p j + 12 n (cid:88) j (cid:54) = k g ( q j − q k ) = 12 tr L = 12 n (cid:88) j =1 ˜ q j ∈ A n (110)and also H CM (˜ p j , ˜ q j ; g ) = 12 n (cid:88) j =1 (˜ p j ) + 12 n (cid:88) j (cid:54) = k g (˜ q j − ˜ q k ) = 12 tr ˜ L = 12 n (cid:88) j =1 ( q j ) ∈ ˜ A n . (111)Provided that Eqs.(107) indeed defines a symplectomorphism, we conclude from Eq.(110) that the Calogero-Moser system is integrable for arbitrary n and, from Eq. (111),that the Calogero-Moser system is self-dual.We now prove the canonicity of the transformation (107) by a nice argument due toKazhdan, Kostant and Sternberg [23].Let v ∈ R n be a vector having all components v , . . . , v n equal to 1 and let us lookfor two Hermitian n × n matrices X, Y verifying the condition XY − Y X = i g ( − v ⊗ v † ) . (112) he Hermitian matrix X can be always diagonalized by an unitary matrix k , thereforewe can write X, Y as X = kDk † , Y = kLk † , where D is the diagonal matrix D = diag( q , ..., q n ) , q ≥ q ≥ ... ≥ q n . (113)Eq. (112) then tells us three things:1) It holds q > q > ... > q n .2) The unitary matrix k has v as its eigenvector.3) L is the matrix of the form L jk = p j δ jk − (1 − δ jk ) i gq j − q k . (114)As a consequence, it holds also[ D, L ] = i g ( − v ⊗ v † ) . (115)Consider the following two-form ω defined on the manifold of the solutions X, Y ofthe equation (112) ω = tr ( dY ∧ dX ) = tr (cid:16) ([ k † dk, L ] + dL ) ∧ ([ k † dk, D ] + dD ) (cid:17) == tr ( dL ∧ dD ) + tr ( k † dk ∧ d [ L, D ]) + 12 tr ( k † dk ∧ [ k † dk, [ L, D ]]) == tr ( dL ∧ dD ) = dp j ∧ dq j . (116)Note that the passage from the second to the third line of Eq.(116) has used Eq.(115)in a substantial way (notably the fact that v is the eigenvector of the unitary matrix k ).So far we have shown that the space of solutions X, Y of the equation (112) can beconveniently parametrized as P n × U ( n ) v , where P n was defined in Eq.(103) and U ( n ) v is the group of the unitary matrices which has v as their eigenvector. Alternatively,we may parametrize the same space of solutions of Eq.(112) by first diagonalizing theHermitian matrix Y , that is, by writing X, Y as X = ˜ k ˜ L ˜ k † , Y = − ˜ k ˜ D ˜ k † where˜ D = diag(˜ q , ..., ˜ q n ) , ˜ q > ˜ q > ... > ˜ q n , (117)˜ L jk = ˜ p j δ jk − (1 − δ jk ) i g ˜ q j − ˜ q k (118)and ˜ k ∈ U ( n ) v .We evaluate the form ω in the dual parametrization X = k ˜ Lk † , Y = k ˜ Dk † , whichgives ω = tr ( dY ∧ dX ) = d ˜ p j ∧ d ˜ q j . (119)The canonicity of the transformation (107) is now the consequence of Eqs.(116) and(119). Lax integrability
In the context of infinite-dimensional phase spaces, there appears no widely accepteddefinition of integrability in terms of complete polarizations, and the term ”integrabil-ity” often used in the infinite-dimensional literature means actually the so called
Laxintegrability . In fact, the concept of the Lax integrability exists also for the finite di-mensional integrable systems ( P, A , B ) where it plays an important technical role in theconstruction of the action-angle symplectomorphism ρ : P → P such that B = ρ ∗ A .In the infinite number of dimensions, the Lax integrability gets promoted to a higherdegree of importance and, de facto, it plays a prominent conceptual role there. Let (
P, H ) be a finite-dimensional dynamical system, V a vector space and L ( V ) thespace of linear operators on V . A Hamiltonian dynamical system ( P, H ) is referred toas (strongly) Lax integrable if there exist L ( V )-valued function L on P (called the Laxmatrix) such that its spectral invariants generate a polarization B ⊂
Fun( P ) containing H . Recall in this respect that the spectral invariant of the operator L ∈ L ( V ) is anyfunction f : L ( V ) → R such that f ( S − LS ) = f ( L ) for every invertible operator S acting on V .In practice, it is useful to give sufficient conditions guaranteeing the Lax integrability.The most frequently used sufficient conditions amount to the existence of two morematrix-valued functions M : P → L ( V ) and r : P → L ( V ⊗ V ) such that { L, H } = [ L, M ] , (120) { L ⊗ Id , Id ⊗ L } = [ r, L ⊗ Id] − [ r p , Id ⊗ L ] . (121)In Eqs.(120) and (121), Id is the identity operator on V , the notation { ., . } stands forthe Poisson brackets on P and [ ., . ] means the commutator of linear operators actingeither on the space V or on the space V ⊗ V . If r ∈ L ( V ⊗ V ) can be written as r = (cid:88) α A α ⊗ B α (122)for some family of linear operators A α , B α ∈ L ( V ), then the notation r p means r p = (cid:88) α B α ⊗ A α . (123) Remark 4.1 : The fulfillment of the condition (120) guarantees that the spectral in-variants of the Lax operator L commute with the Hamiltonian [39] and it is sometimesreferred to as the weak Lax integrability. On the other hand, the fulfillment of thecondition (121) guarantees also that the spectral invariants commute among themselves[54, 42, 7]. In this case it is said that the dynamical system is strongly
Lax integrable.It is perhaps instructive to rewrite Eqs.(120) and (121) in some basis of the vectorspace V in which the operators L, M, r become matrices with appropriate subscripts andsuperscripts: { L ij , H } = L ik M kj − M ik L kj , (124) L ia δ mb , δ aj L bp } = r imab L aj δ bp − L ia δ mb r abjp − r miba δ aj L bp + δ ia L mb r bapj . (125)Of course, it is more convenient to rewrite Eq.(125) as { L ij , L mp } = r imap L aj − r amjp L ia − r mibj L bp + r bipj L mb . (126)It can be easily verified from Eq.(120) (or, equivalently, from Eq.(124)) that thespectral invariants of the Lax matrix L are the constants of motion since they Poisson-commute with the Hamiltonian: { tr ( L s ) , H } = { L i i . . . L i s − i s L i s i , H } == (cid:16) L i k M ki − M i k L ki (cid:17) . . . L i s − i s L i s i + · · · + L i i . . . L i s − i s (cid:16) L i s k M ki − M i s k L ki (cid:17) == tr [ L s , M ] = 0 . (127)In a similar way, it follows from Eq.(121) (or, equivalently, from Eq.(126)) that thespectral invariants of the Lax matrix L Poisson commute among themselves, that is { tr ( L s ) , tr ( L k ) } = 0 . (128) Remark 4.2
The existence of the matrices
L, M, r guarantees the Poisson-commutativityof the eigenvalues of the matrix L but it does not guarantee that those eigenvalues besufficiently numerous to generate some polarization B . Indeed, if the dimension of thevector space V is small, for example just equal to 2, and the dimension 2 n of the phasespace P is big, then the polarization, which must be locally generated by n functionallyindependent functions, cannot be generated by just two independent spectral invari-ants of the 2 × L . In practice, the matrix L must act on the vector spaceof sufficiently big dimension in order that its spectral invariants be able to generate apolarization. Let us illustrate this phenomenon on the example of the Calogero-Mosersystem which is Lax integrable and the Lax matrix L of which is given by the expression(108) L jk = p k δ jk − (1 − δ jk ) i gq j − q k . (129)We have already established in Section 3.5 that the spectral invariants (i.e. the eigenval-ues) of the matrix L generate the polarization B containing the Calogero-Moser Hamil-tonian and we now observe that the size n × n of the matrix L is the minimal possibleto accomplish this task.For completeness, we give also the explicit expressions for the matrices M and r inthe Calogero-Moser case (cf. [46, 2, 49]) M jk = − (1 − δ jk ) i g ( q j − q k ) + δ jk (cid:88) l (cid:54) = j ig ( q j − q l ) , (130) r imkp = 1 − δ im q i − q m δ ip δ mk + 12 δ ik (cid:32) − δ ip q p − q i δ im − − δ im q m − q i δ ip (cid:33) . (131)Technically, it is not always practical to work with one Lax matrix of big size and it issometimes more convenient to work with many Lax matrices of small size which together rovide a sufficient number of Poisson-commuting constants of motion to generate apolarization of P . The parameter λ distinguishing the members of the family of thosesmall Lax matrices is called the spectral parameter and it often takes complex values.Every single member of the family of the Lax matrices is then denoted as L ( λ ) andthe dependence of this object on the phase space variable is tacitly understood albeitsuppressed in the notations.The concept of the Lax matrix with spectral parameter makes possible to generalizethe definition of the Lax integrable system given before. The Hamiltonian dynamicalsystem ( P, H ) is thus referred to be the Lax integrable if there exists a one-parameterfamily of L ( V )-valued functions L ( ξ ) on P such that the spectral invariants of all Laxmatrices in the family together generate a polarization B ⊂
Fun( P ) containing H .The eigenvalues of all matrices in the family L ( ξ ) Poisson-commute with the Hamil-tonian H and Poisson-commute among themselves if there exist families of matrix-valuedfunctions M ( ξ ) : P → L ( V ) and r ( ξ, ζ ) : P → L ( V ⊗ V ) such that it holds { L ( ξ ) , H } = [ L ( ξ ) , M ( ξ )] , (132) { L ( ξ ) ⊗ Id , Id ⊗ L ( ζ ) } = [ r ( ξ, ζ ) , L ( ξ ) ⊗ Id] − [ r p ( ζ, ξ ) , Id ⊗ L ( ζ )] . (133)We shall not expand further this text by reviewing finite dimensional systems whichare Lax integrable by the Lax matrix with spectral parameter; the interested reader canfind examples in the book [6]. However, the Lax matrix with spectral parameter will bethe main object of our interest in the next section devoted to the integrability of systemswith infinite number of degrees of freedom. In that case, we do give concrete examplesof such Lax matrices, in particular those relevant for the dynamics of E -models. In what follows, we switch our attention to the systems with infinite number of degreesof freedom. The criterion that the spectral invariants of the Lax matrix for all valuesof the spectral parameter should generate the polarization containing the Hamiltonianis now problematic to impose, because the very notion of the polarization is difficult tohandle in the infinite dimensional case. For that reason one requires instead that theLax condition (132) must encode the complete set of equations of motion of the model.Having in mind this circumstance, we concentrate in what follows on the issue of the Laxintegrability of the so called non-linear σ -models in 1 + 1 spacetime dimensions whichare field theories naturally associated to the Riemannian-Kalb-Ramond manifolds.Working in some coordinate patch x i , we usually denote by g ij ( x ) the components ofthe Riemannian tensor on M and by h ijk ( x ) the components of the Kalb-Ramond fieldstrength. Locally, h ijk ( x ) can be written in terms of the anti-symmetric Kalb-Ramondpotential b jk ( x ) as h ljk ( x ) = ∂ l b jk ( x ) + ∂ j b kl ( x ) + ∂ k b lj ( x ) (134)and the σ -model action looks like (cf. Eq.(23)) S [ x ( τ, σ )] = (cid:90) dτ (cid:73) ( g ij ( x ) + b ij ( x )) ∂ + x i ∂ − x j ≡ (cid:90) dτ (cid:73) e ij ( x ) ∂ + x i ∂ − x j . (135) he first known example of the Lax integrable σ -model was introduced in [56] andit is know under the name of principal chiral model . This theory can live on everyquadratic Lie group K but in the present paper K will always stand for a simple com-pact connected and simply connected Lie group. The Riemannian structure on K isproportional to the bi-invariant metric equal to the (negative-definite) Killing-Cartanform ( ., . ) K at the group origin and there is no Kalb-Ramond field in this case. Denotingthe field configuration k ( τ, σ ) ∈ K , the action of the principal chiral model is given bythe expression S [ k ( τ, σ )] = − (cid:90) dτ (cid:73) ( k − ∂ + k, k − ∂ − k ) K . (136)Occasionally, the action (136) is written in some suitable coordinates x i on the groupmanifold as follows S [ x ( τ, σ )] = − (cid:90) dτ (cid:73) ( E i ( x ) , E j ( x )) K ∂ + x i ∂ − x j , (137)where E i ( x ) dx i is the K -valued left-invariant Maurer-Cartan form on the group manifold K expressed in the coordinates x i .The phase space P = LK × L K of the principal chiral model is the direct productof the loop group LK with the loop Lie algebra L K . We denote the points of P as pairs( k ( σ ) , J ( σ )) ≡ ( k, J ); the symplectic form is then given by the formula ω = − δ (cid:73) ( δkk − , J ) K (138)and the positive definite Hamiltonian by H := − (cid:73) ( ∂ σ kk − , ∂ σ kk − ) K − (cid:73) ( J , J ) K . (139)We associate to the data ( P, ω , H ) the first order action in the standard way: S [ k, J ] = (cid:90) dτ (cid:73) (cid:18) − ( ∂ τ kk − , J ) K + 12 ( J , J ) K + 12 ( ∂ σ kk − , ∂ σ kk − ) K (cid:19) . (140)The current J appears in this action quadratically and can be easily eliminated to yieldthe second order σ -model action (136).The Lax integrability of the principal chiral model was established by Zakharov andMikhailov in [56]. The Lax pair L ( ξ ) , M ( ξ ) depending on a spectral parameter ξ ∈ C isgiven by linear operators acting on the loop Lie algebra V ≡ L K C as follows L ( ξ ) = ∂ σ + 11 − ξ ad ( ξ J − ∂ σ kk − ) , M ( ξ ) = 11 − ξ ad ( ξ∂ σ kk − −J ) . (141)We must verify that it holds { L ( ξ ) , H } = [ L ( ξ ) , M ( ξ )] . (142)To do that, it is necessary to determine the Poisson brackets { ., . } generated by thesymplectic form (138). They read { Ad k ( σ ) ⊗ Id , Id ⊗ Ad k ( σ ) } = 0 , (143a) { Ad k ( σ ) , ( J ( σ ) , x ) K } = − ad x Ad k ( σ ) δ ( σ − σ ) , x ∈ K (143b) { ( J ( σ ) , x ) K , ( J ( σ ) , y ) K } = − ( J ( σ ) , [ x, y ]) K δ ( σ − σ ) , x, y ∈ K . (143c) et us now verify the relation (142). We first set R := ∂ σ kk − , (144)and then we infer from Eqs.(143) { ( J ( σ ) , x ) K , ( R ( σ ) , y ) K } = − ( R ( σ ) , [ x, y ]) K δ ( σ − σ ) − ( x, y ) K δ (cid:48) ( σ − σ ) . (145)Then we find { L ( ξ )( σ ) , ( J ( σ ) , y ) K } = [ ξ J ( σ ) − R ( σ ) , y ] δ ( σ − σ ) + yδ (cid:48) ( σ − σ )1 − ξ , (146) { L ( ξ )( σ ) , ( R ( σ ) , y ) K } = ξ [ R ( σ ) , y ] δ ( σ − σ ) − ξyδ (cid:48) ( σ − σ )1 − ξ , (147)which gives the desired result { L ( ξ ) , H } = − − ξ ad ( J − ξ R ) (cid:48) +[ J , R ] = [ L ( ξ ) , M ( ξ )] . (148)Finally, we find from Eqs.(141), (146) and (147) { tr ( L ( ξ )( σ )ad x ) , tr ( L ( ζ )( σ )ad y ) } == ( − ξζJ ( σ ) + ( ξ + ζ ) R ( σ ) , [ x, y ]) K δ ( σ − σ ) + ( ξ + ζ )( x, y ) K δ (cid:48) ( σ − σ )(1 − ξ )(1 − ζ ) , (149)The r -matrix satisfying the required relation { L ( ξ ) ⊗ Id , Id ⊗ L ( ζ ) } = [ r ( ξ, ζ ) , L ( ξ ) ⊗ Id] − [ r p ( ζ, ξ ) , Id ⊗ L ( ζ )] (150)is therefore given by the expression r ( ξ, ζ ) = ζ − ζ C δ ( σ − σ ) ξ − ζ . (151) Remark 4.3 : The symbol C in Eq. (151) stands for the Casimir element. The sim-plest way to define C consists in picking a basis T a ∈ K and define a Casimir matrixcharacterized by its matrix elements C ab := ( T a , T b ) K ≡ tr (ad T a ad T b ) . (152)Then in the defining relation of the Casimir element C appears the inverse Casimir matrix C := C ab ad T a ⊗ ad T b , (153)where the Einstein summation convention applies. The Casimir element C defined inthis way does not depend on the choice of the basis T a .We note finally, that the first order Hamiltonian equations of motions of the principalchiral model can be written as ∂ τ R = {R , H } = ∂ σ J + [ J , R ]; ∂ τ J = {J , H } = ∂ σ R . (154) Yang-Baxter deformations σ -model The Yang-Baxter σ -model was introduced in Ref.[25] as the model exhibiting a par-ticularly rich Poisson-Lie T-duality pattern and it was proven to be Lax integrable inRef.[26]. In the present subsection, we give the definition of the model, we describe itsHamiltonian formalism and identify the Lax matrix L ( λ ) and the r -matrix r ( λ, ζ ).The Lagrangian of the Yang-Baxter σ -model is given by the formula S β [ k ( τ, σ )] = −
12 cos ( β ) (cid:90) dτ (cid:73) ( k − ∂ + k, − tan ( β ) R k − ∂ − k ) K . (155)Here 0 ≤ β < π denotes a deformation parameter and R : K C → K C is the so calledYang-Baxter operator defined as RE α = − sign( α )i E α , RH j = 0 , (156)where E α , H j is the standard Chevalley basis of K C .The phase space P = LK × L K of the Yang-Baxter σ -model is the same as that ofthe principal chiral one but the symplectic form (138) is now renormalized: ω β = − ( β ) δ (cid:73) ( δkk − , J ) K . (157)The Hamiltonian (139) turns out to be also deformed and it reads H β := − (cid:73) ( R β , R β ) K − (cid:73) ( J , J ) K , (158)where R β := 1cos ( β ) ∂ σ kk − − tan ( β ) kR ( k − J k ) k − . (159)If we associate to the data ( P, ω β , H β ) the first order action in the standard way S β [ k, J ] = − ( β ) (cid:90) dτ (cid:73) ( ∂ τ kk − , J ) K + 12 (cid:90) dτ (cid:73) (( J , J ) K + ( R β , R β ) K ) (160)and integrate away the current J , we recover the second order σ -model action (155).The important fact is that the Hamiltonian equation of motions of the Yang-Baxter σ -model have the same form as those of the principal chiral model. Indeed, we find easily ∂ τ R β = {R β , H β } β = ∂ σ J + [ J , R β ] , ∂ τ J = {J , H β } β = ∂ σ R β , (161)where { ., . } β is the Poisson bracket { ., . } multiplied by the factor cos ( β ). The form ofthe equations (161) suggests the Lax pair of the Yang-Baxter σ -model: L β ( ξ ) = ∂ σ + 11 − ξ ad ( ξ J −R β ) , M β ( ξ ) = 11 − ξ ad ( ξ R β −J ) . (162)We must verify that it holds indeed { L β ( ξ ) , H β } β = [ L β ( ξ ) , M β ( ξ )] , (163) hich can be done straightforwardly by using the brackets of the currents J and R β : { ( J ( σ ) , x ) K , ( J ( σ ) , y ) K } β = − cos ( β ) ( J ( σ ) , [ x, y ]) K δ ( σ − σ ) , (164a) { ( R β ( σ ) , x ) K , ( R β ( σ ) , y ) K } β = sin ( β ) ( J ( σ ) , [ x, y ]) K δ ( σ − σ ) , (164b) { ( J ( σ ) , x ) K , ( R β ( σ ) , y ) K } β = − cos ( β ) ( R β ( σ ) , [ x, y ]) K δ ( σ − σ ) − ( x, y ) K δ (cid:48) ( σ − σ ) . (164c)Note that the brackets (164) follow from the brackets (143) as well as from thedefinition (159), due to the following important identities fulfilled by the Yang-Baxteroperator R : [ Rx, Ry ] = R [ Rx, y ] + R [ x, Ry ] + [ x, y ] , ∀ x, y ∈ K , (165)( Rx, y ) K = − ( x, Ry ) K , x, y ∈ K . (166)We also find from Eqs.(162) and (164) { tr ( L ( ξ )( σ )ad x ) , tr ( L ( ζ )( σ )ad y ) } == ((sin ( β ) − ξζ cos ( β )) J + ( ξ + ζ ) cos ( β ) R , [ x, y ]) K δ ( σ − σ ) + ( ξ + ζ )( x, y ) K δ (cid:48) ( σ − σ )(1 − ξ )(1 − ζ ) . (167)The r -matrix r β ( ξ, ζ ) satisfying the required relation { L β ( ξ ) ⊗ Id , Id ⊗ L β ( ζ ) } β = [ r β ( ξ, ζ ) , L β ( ξ ) ⊗ Id] − [ r pβ ( ζ, ξ ) , Id ⊗ L β ( ζ )] (168)is then given by the expression r β ( ξ, ζ ) = φ − β ( ζ ) C δ ( σ − σ ) ξ − ζ , (169)where the function φ β ( ζ ), called the twist one in [11], is given by φ β ( ζ ) = 1 − ζ cos ( β ) ζ + sin ( β ) . (170) E -model formulation of the Yang-Baxter σ -model It turns out that the action (155) of the Yang-Baxter σ -model can be derived from anappropriate E -model by the procedure detailed in Section 2.5. As the result, the action(155) is the special case of the action (51). Let us see how it works in more detail.For the underlying Drinfeld double D , we take the Lu-Weinstein one, that is D isthe special complex linear group SL ( n, C ) viewed as real group (i.e. it has the dimension2( n −
1) as the real manifold). The non-degenerate symmetric ad-invariant bilinear form( ., . ) D on the Lie algebra sl ( n, C ) is defined by taking a suitable normalized imaginarypart of the trace( X, Y ) D = − β ) cos ( β ) (cid:61) tr ( XY ) , X, Y ∈ sl ( n, C ) . (171) he two half-dimensional isotropic subgroups K and ˜ K are respectively the specialunitary group SU ( n ) and the upper-triangular group AN with real positive numbers onthe diagonal the product of which is equal to 1.The one-parametric family of the E -operators is given by E β X = − X + 1 + i tan ( β )2i tan ( β ) (cid:16) (1 + i tan ( β )) X + (1 − i tan ( β )) X † (cid:17) . (172)Recall that the second order σ -model action corresponding to the operator E β reads S β ( k ) = 12 (cid:90) dτ (cid:73) (cid:16) ( E β + Π( k )) − ∂ + kk − , ∂ − kk − (cid:17) D ., (173)where the linear operator E β : ˜ K → K is defined by the operator E β : D → D in theway that its graph { ˜ x + E β ˜ x, ˜ x ∈ ˜ K} coincides with the image of the operator Id+ E β .Moreover, the k -dependent operator Π( k ) : ˜ K → K is given byΠ( k ) = −J Ad k ˜ J Ad k − ˜ J , (174)where J , ˜ J are projectors; J projects to K with the kernel ˜ K and ˜ J projects to ˜ K withthe kernel K .To find out what is the linear operator E β : ˜ K → K , it is convenient to parametrizethe elements of the Lie algebra ˜ K by those of the Lie algebra K with the help of theYang-Baxter operator R . Every element of ˜ K can be thus described in a unique way as( R − i) y , y ∈ K and we then find from Eq.(172) E β ( R − i) y = − Ry − y tan ( β ) . (175)Furthermore, from Eq. (174) we obtainΠ( k )( R − i) y = ( R − Ad k R Ad k − ) y. (176)Combining Eqs.(175) and (176), we find( E β + Π( k )) − ∂ + kk − = − ( R − i) 1cot ( β ) + Ad k R Ad k − ∂ + kk − . (177)Inserting (177) into (173) and using (171), we recover indeed the action (155) of theYang-Baxter σ -model S β ( k ) = −
12 cos ( β ) (cid:90) dτ (cid:73) ( k − ∂ + k, − tan ( β ) R k − ∂ − k ) K . (178)The fact that the first-order Hamiltonian dynamics of the Yang-Baxter σ -model can beexpressed in terms of the E -model (172) explains the naturaleness of the current observ-ables J , R β employed in the previous section. Indeed, the crucial Poisson brackets (164)can be obtained from the following general Poisson bracket derived from the symplecticform (45) { ( j ( σ ) , X ) D , ( j ( σ (cid:48) ) , X (cid:48) ) D } = ( j ( σ ) , [ X, X (cid:48) ]) D +( X, X (cid:48) ) D ∂ σ δ ( σ − σ (cid:48) ) , X, X (cid:48) ∈ D , (179) here j ( σ ) = ∂ σ ll − . (180)Working with the Lu-Wenstein double K C and the bilinear form (171), we recover fromEq.(179) the Poisson brackets (164) upon the identification j ( σ ) = cos ( β ) (cid:0) cos ( β ) R β ( σ ) + i sin ( β ) J ( σ ) (cid:1) . (181)For example, setting X = x , X (cid:48) = i y , x, y ∈ K in (179) and using the ansatz (181), werecover the formula (164c).By the way, the formula (159) can be also easily derived from the E -model formalism.Indeed, decomposing l = k ˜ h like in Eq. (47), we find j = ∂ σ ll − = ∂ σ kk − + Ad k ( ∂ σ ˜ h ˜ h − ) = ∂ σ kk − + Ad k (( R − i) y ) , y ∈ K . (182)Comparing Eqs.(181) and (182), we recover the formula (159). Finally, we obtain theHamiltonian (158) from the formulas (46), (171) and (181). σ -model and its special limits The Yang-Baxter σ -model introduced in the previous two sections is the prototypicalrepresentative of the family of the so-called Yang-Baxter integrable deformations of theprincipal chiral model. Since 2002, many multi-parametric Yang-Baxter deformations were subsequently constructed e.g. in [27, 11, 53, 22, 47, 3, 8, 20, 19, 5, 30, 14]. Here wereview explicitely an example of the three-parametric deformation living on the simplecompact group target K which was introduced in Ref.[19, 30, 31] and bears the name ofthe bi-YB-WZ σ -model. The action of this model reads S bi − YB − WZ ( k ) = κ (cid:90) dτ (cid:73) tr (cid:18) k − ∂ + k α + e ρ r R k e ρ l R α − e ρ r R k e ρ l R k − ∂ − k (cid:19) ++ κ (cid:90) δ − (cid:73) tr ( k − δk, [ k − ∂ σ k, k − δk ]) . (183)Here the operator R k : K → K is defined as R k := Ad − k R Ad k , (184)furthermore α ∈ ] − ,
1[ and ρ l , ρ r ∈ ] − π, π [ are the deformation parameters and the realpositive level κ is quantized as usual so that the WZ term exhibits the 2 π ambiguity.Note that the case α = 0 corresponds to the standard WZNW model.Some other Yang-Baxter deformations previously constructed in the literature areappropriate special limits of the bi-YB-WZ σ -model (183). In particular, this is the casefor the so called YB-WZ σ -model introduced in Ref.[12]. Several equivalent expressionswere obtained for this YB-WZ deformation in Refs.[29], [13] and [30]. We reproducehere the parametrization given in [30]: S YB − WZ ( k ) = κ (cid:90) dτ (cid:73) tr (cid:18) k − ∂ + k α + e ρ l R α − e ρ l R k − ∂ − k (cid:19) + The integrable σ -models constructed previously on the target SU (2) in Refs.[9, 4, 15, 41]were later recognized to be of the Yang-Baxter type in [26, 21, 19]. κ (cid:90) δ − (cid:73) tr ( k − δk, [ k − ∂ σ k, k − δk ]) . (185)Note that this is the special case of the action (183) obtained by setting ρ r = 0.Furthermore, setting ρ r = 2 κb r , ρ l = 2 κb l , α = e − κa , (186)and taking limit κ →
0, we recover from the action (183) the bi-Yang-Baxter integrabledeformation of the principal chiral model introduced in [25, 27]: S bi − YB ( k ) = − (cid:90) dτ (cid:73) tr (cid:18) k − ∂ + k ( a + b r R k + b l R ) − k − ∂ − k (cid:19) . (187)Taking moreover b r = 0, we recover the Yang-Baxter σ -model (178) S YB ( k ) = − (cid:90) dτ (cid:73) tr (cid:18) k − ∂ + k ( a + b l R ) − k − ∂ − k (cid:19) , (188)with a = 2 cos ( β ), b L = − β ) cos ( β ). E -model formulation of the bi-YB-WZ σ -model Recall that the E -model is the first order dynamical system living on the loop group ofthe Drinfeld double where the symplectic form and the Hamiltonian read respectively ω = − (cid:73) (cid:0) l − δl ∧ , ( l − δl ) (cid:48) (cid:1) D , (189) H E = 12 (cid:73) (cid:0) l (cid:48) l − , E l (cid:48) l − (cid:1) D ≡ (cid:73) (cid:0) j, E j (cid:1) D . (190)The Poisson brackets of the currents j = l (cid:48) l − derived from the symplectic form (189)then read { ( j ( σ ) , X ) D , ( j ( σ (cid:48) ) , X (cid:48) ) D } = ( j ( σ ) , [ X, X (cid:48) ]) D +( X, X (cid:48) ) D ∂ σ δ ( σ − σ (cid:48) ) , X, X (cid:48) ∈ D . (191)Whenever there is given a maximally isotropic Lie subalgebra ˜ K ⊂ D , we can write downthe second order σ -model action S E ( l ) = 14 (cid:90) δ − (cid:73) (cid:18) δll − , [ ∂ σ ll − , δll − ] (cid:19) D ++ 14 (cid:90) dτ (cid:73) (cid:18) ∂ + ll − , Q − l ∂ − ll − (cid:19) D − (cid:90) dτ (cid:73) (cid:18) Q + l ∂ + ll − , ∂ − ll − (cid:19) D . (192)Here l ( τ, σ ) ∈ D is a field configuration and Q ± l : D → D are the projectors fullycharacterized by their respective images and kernelsIm( Q ± l ) = Ad l ( ˜ K ) , Ker( Q ± l ) = (1 ± E ) D . (193)The upshot is that the first order Hamiltonian dynamics of the σ -model (192) is givenby the data (189), (190) and (191). t was shown in Ref.[31] that the first order Hamiltonian dynamics of the bi-YB-WZ σ -model (183) can be formulated in terms of an appropriate E -model. Indeed,the underlying Drinfeld double is D = K C as in the one-parametric Yang-Baxter case,however, the bilinear form ( ., . ) D now reads (cid:0) X, X (cid:48) (cid:1) D := 4 κ sin ( ρ l ) (cid:61) tr (cid:0) e i ρ l XX (cid:48) (cid:1) , X, X (cid:48) ∈ K C . (194)Here the symbol (cid:61) means the imaginary part of a complex number and κ, ρ l are theparameters appearing in the action (183).The operator E is given by the formula E α,ρ l ,ρ r X = − X + 1 α − α − ( α − e − i ρ l e − ρ r R ) (cid:18) X − X † − cos ( ρ l ) − α − e ρ r R sin ( ρ l ) i( X + X † ) (cid:19) , (195)where again the parameters α, ρ l and ρ r are those appearing in the action (183).The maximally isotropic Lie subalgebra ˜ K ⊂ K C defined as˜ K = (cid:26) e − i ρ l − e − ρ l R sin ρ l y, y ∈ K (cid:27) . (196)The fact that the subspace of D defined by Eq.(196) is indeed the Lie subalgebra of D is the consequence of the properties of the Yang-Baxter operator, namely of the identity(165) rewritten as (cid:20) e − i ρ l − e − ρ l R sin ρ l x, e − i ρ l − e − ρ l R sin ρ l y (cid:21) = e − i ρ l − e − ρ l R sin ρ l [ x, y ] R,ρ l , x, y ∈ K . (197)Here [ ., . ] R,ρ L is a new Lie bracket on the vector space K which is defined with the helpof the standard Lie bracket [ ., . ] and of the Yang-Baxter operator as[ x, y ] R,ρ l := (cid:20) cos ρ l − e − ρ l R sin ρ l x, y (cid:21) + (cid:20) x, cos ρ l − e − ρ l R sin ρ l y (cid:21) . (198)As it was explained in Ref.[31], the subgroup ˜ K ⊂ D corresponding to the Liesubalgebra ˜ K turns out to be the semi-direct product of a suitable real form of thecomplex Cartan torus T C with the nilpotent subgroup N ⊂ K C generated by the positivestep operators E α . The space of cosets then D/ ˜ K turns out to be just the group K ,therefore the gauge fixing l = k transforms the model (192) into some σ -model livingon the group K . To see which one, we must evaluate the expressions Q ± k ∂ ± kk − . To dothat we first find from Eq.(195)(1 ± E ) D = (cid:8) ( α ± − e − i ρ l e − ρ r R ) x, x ∈ K (cid:9) (199)and then we write down the identity ∂ ± kk − = (cid:0) e − i ρ l − e − ρ l R k − (cid:1) (cid:0) α ± e ρ r R − e − ρ l R k − (cid:1) − ∂ ± kk − ++( α ± − e − i ρ l e − ρ r R )( α ± − e − ρ l R k − e − ρ r R ) − ∂ ± kk − . (200) aking into account the definition (193) of the projectors Q ± k as well as Eqs.(196),(199)and (200), we infer Q ± k ∂ ± kk − = (cid:0) e − i ρ l − e − ρ l R k − (cid:1) (cid:0) α ± e ρ r R − e − ρ l R k − (cid:1) − ∂ ± kk − . (201)Inserting the expressions (201) into the action (192) written for l = k we find preciselythe bi-YB-WZ action S bi − YB − WZ ( k ) = κ (cid:90) dτ (cid:73) tr (cid:18) k − ∂ + k α + e ρ r R k e ρ l R α − e ρ r R k e ρ l R k − ∂ − k (cid:19) ++ κ (cid:90) δ − (cid:73) tr ( k − δk, [ k − ∂ σ k, k − δk ]) . (202) σ -model The Lax pair operators L ( ξ ) and M ( ξ ) of the bi-YB-WZ σ -model were identified inRef.[31]. In this paper, we rewrite them in an equivalent but simpler way as follows L ( ξ ) = ∂ σ − ad O ( ξ ) j , (203) M ( ξ ) = − ad O ( ξ ) E j . (204)Here the involution E is given by Eq.(195), the R -linear operator O ( ξ ) : D → D is definedas O ( ξ ) j = (1 + h ( ξ ) e − ρ r R ) e i ρ l j + e − i ρ l j ∗
2i sin ( ρ l ) + g ( ξ ) j + j ∗
2i sin ( ρ l ) (205)and the meromorphic functions h ( ξ ), g ( ξ ) must verify the relation h ( ξ ) + g ( ξ ) + ( α + α − ) h ( ξ ) g ( ξ ) + 2 cos ( ρ l ) g ( ξ ) + 2 cos ( ρ r ) h ( ξ ) + 1 = 0 . (206)We can now straighforwardly verify the first condition (132) of the Lax integrability,that is the validity of the relation { L ( ξ ) , H E } = { L ( ξ ) , (cid:73) (cid:0) j, E j (cid:1) D } = [ L ( ξ ) , M ( ξ )] . (207)Using the current Poisson brackets (191) and the definition of the Hamiltonian (190),we first find { j, H E } = E ∂ σ j + [ E j, j ] , (208)which means that for verifying the condition (207) we must just prove the identity O ( ξ )[ E j, j ] = [ O ( ξ ) E j, O ( ξ ) j ] . (209)The validity of the identity (209) is then the straightforward consequence of the relation(206) as well as of the following identity proved in Ref.[31][ e ρ r R x, e ρ r R y ] = e ρ r R (cid:0) [ e ρ r R x, y ] + [ x, e ρ r R y ] − ρ r )[ x, y ] (cid:1) + [ x, y ] , x, y ∈ K . (210) The identity (209) appeared already in Ref.[37] as the reformulation of the identity introducedin Ref.[52]. y the way, for the meromorphic functions g ( ξ ) and h ( ξ ) verifying the condition (206),we can take e.g. g ( ξ ) = 4 cos ( ρ l ) − ρ r )( α + α − )( α − α − ) + 2 Aξ − ξ + A ( α + α − )( α − α − ) 1 + ξ − ξ , (211) h ( ξ ) = 4 cos ( ρ r ) − ρ l )( α + α − )( α − α − ) − A ( α − α − ) 1 + ξ − ξ , (212)where A = (cid:115) × cos ( ρ l ) + cos ( ρ r ) − ( α + α − ) cos ( ρ l ) cos ( ρ r )( α − α − ) . (213)Recall that the second condition (133) of the Lax integrability reads { L ( ξ ) ⊗ Id , Id ⊗ L ( ζ ) } = [ r ( ξ, ζ ) , L ( ξ ) ⊗ Id] − [ r p ( ζ, ξ ) , Id ⊗ L ( ζ )] . (214)Fixing a basis T a ∈ K , we make the following ansatz for the r -matrix r ( ξ, η ) = C ab ad ˆ r ( ξ,η ) T a ⊗ ad T b δ ( σ − σ ) , (215)so that the operator ˆ r ( ξ, η ) : K → K is the quantity that we wish to find.Contracting the condition (214) with an element ad x ⊗ ad y , x, y ∈ K , considering ξ, ζ ∈ R and using Eq.(203), we obtain { ( O ( ξ ) j ( σ ) , x ) K , ( O ( ζ ) j ( σ ) , y ) K } == − (cid:16) ( O ( ξ ) j ( σ ) , [ x, ˆ r ( ξ, ζ ) y ]) K + ( O ( ζ ) j ( σ ) , [ˆ r ( ζ, ξ ) x, y ]) K (cid:17) δ ( σ − σ ) − (cid:16) ( x, ˆ r ( ξ, ζ ) y ) K + (ˆ r ( ζ, ξ ) x, y ) K (cid:17) δ (cid:48) ( σ − σ ) . (216)Note that for ξ ∈ R , the image of the operator O ( ξ ) is just the Lie algebra K , moreover,it exists the operator O † ( ξ ) : K → D , which is ”adjoint” to O ( ξ ) : D → K in the senseof the equality ( O ( ξ ) j, x ) K = ( j, O † ( ξ ) x ) D . (217)Indeed, O † ( ξ ) is given by the formula O † ( ξ ) x = (Id + h ( ξ ) e ρ r R ) x + e − i ρ l g ( ξ ) x κ . (218)Now because of the relations (217) and (191), we have also { ( O ( ξ ) j ( σ ) , x ) K , ( O ( ζ ) j ( σ ) , y ) K } == ( j, [ O † ( ξ ) x, O † ( ζ ) y ]) D + ( O † ( ξ ) x, O † ( ζ ) y ) D ∂ σ δ ( σ − σ (cid:48) ) , x, y ∈ K . (219)Finally, comparing Eq.(216) with Eq.(219), we infer that the operator ˆ r must fulfil thefollowing conditions[ O † ( ξ ) x, O † ( ζ ) y ] = − O † ( ξ )[ x, ˆ r ( ξ, ζ ) y ] − O † ( ζ )[ˆ r ( ζ, ξ ) x, y ] , (220)( O † ( ξ ) x, O † ( ζ ) y ) D = − ( x, ˆ r ( ξ, ζ ) y ) K − (ˆ r ( ζ, ξ ) x, y ) K . (221) sing the identities (206) and (210), it is then straightforward to work out that theconditions (220) and (221) are fulfilled by the following operator ˆ r ( ξ, η )ˆ r ( ξ, ζ ) = h ( ζ )4 κ (cid:16) w ( ξ, ζ )Id − e ρ r R (cid:17) , (222)where w ( ξ, ζ ) = h ( ξ ) g ( ζ ) + h ( ζ ) g ( ξ ) + ( α + α − ) g ( ξ ) g ( ζ ) + 2 cos ( ρ r ) g ( ξ ) g ( ξ ) − g ( ζ ) . (223)The strong Lax integrability of the bi-YB-WZ model is thus established. Sufficient conditions for the weak Lax integrability of non-linear σ -models were formu-lated in Refs.[43],[44] and they turned out to be useful for finding new examples of theintegrable σ -models in Ref.[45]. As far as the weak Lax integrability of the E -modelsis concerned, the sufficient conditions were formulated in Ref.[52] and they turned outto be useful for constructing new examples of integrable E -models in Ref.[37]. Actually,the results of Section 5.5 show that the weak integrability condition of Refs.[52, 37] canbe supplemented by further conditions imposed on the family of operators O ( ξ ) whichguarantee also the strong Lax integrability of the theory. Indeed, if for a given E -modelwe find the families of linear operators O ( ξ ), O † ( ξ ) and ˆ r ( ξ, ζ ) verifying the conditions O ( ξ )[ E j, j ] = [ O ( ξ ) E j, O ( ξ ) j ] , (224)[ O † ( ξ ) x, O † ( ζ ) y ] = − O † ( ξ )[ x, ˆ r ( ξ, ζ ) y ] − O † ( ζ )[ˆ r ( ζ, ξ ) x, y ] , (225)( O † ( ξ ) x, O † ( ζ ) y ) D = − ( x, ˆ r ( ξ, ζ ) y ) K − (ˆ r ( ζ, ξ ) x, y ) K , (226)then the E -model is strongly Lax integrable. The results of Section 5.5 can be then alsointerpreted in the way, that the general sufficient conditions (224),(225) and (226) of thestrong Lax integrability have the particular nontrivial solution given by the operators(195), (205), (218) and (222). Acknowledgement : I am indebted to Simon Ruijsenaars for reading the manuscriptand suggesting several important improvements in particular in Section 3.
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