A deformation of quantum affine algebra in squashed WZNW models
aa r X i v : . [ h e p - t h ] N ov KUNS-2467
A deformation of quantum affine algebrain squashed WZNW models
Io Kawaguchi ∗ and Kentaroh Yoshida † Department of Physics, Kyoto UniversityKyoto 606-8502, Japan
Abstract
We proceed to study infinite-dimensional symmetries in two-dimensional squashedWess-Zumino-Novikov-Witten (WZNW) models at the classical level. The target spaceis given by squashed S and the isometry is SU (2) L × U (1) R . It is known that SU (2) L is enhanced to a couple of Yangians. We reveal here that an infinite-dimensionalextension of U (1) R is a deformation of quantum affine algebra, where a new deformationparameter is provided with the coefficient of the Wess-Zumino term. Then we considerthe relation between the deformed quantum affine algebra and the pair of Yangiansfrom the viewpoint of the left-right duality of monodromy matrices. The integrablestructure is also discussed by computing the r / s -matrices that satisfy the extendedclassical Yang-Baxter equation. Finally two degenerate limits are discussed. ∗ E-mail: io at gauge.scphys.kyoto-u.ac.jp † E-mail: kyoshida at gauge.scphys.kyoto-u.ac.jp ontents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The classical action of the squashed WZNW models . . . . . . . . . . . . . . 4 q -deformation of su (2) R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Expansions of the monodromy matrix . . . . . . . . . . . . . . . . . . . . . . 154.4 An infinite-dimensional extension of q -deformed su (2) R . . . . . . . . . . . . 194.5 The r / s -matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 α = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.2 α = πi/ J aµ and j L ± µ A.1 The Poisson brackets of J aµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.2 The current algebra of j L ± µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 B A prescription to treat non-ultra local terms 41
B.1 Yangian Y ( su (2) L ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42B.2 q -deformed su (2) R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49B.3 A deformation of quantum affine algebra . . . . . . . . . . . . . . . . . . . . 52 Introduction
The AdS/CFT correspondence [1–3] has been a fascinating topic in the study of stringtheory over the decade after Maldacena’s proposal. A tremendous amount of works havebeen devoted to test and generalize it. Nowadays, the research area covers diverse subjects.A great achievement in the recent progress is the discovery of the integrable structure behindthe AdS/CFT correspondence (For a comprehensive review on this subject, see [4]).In the string-theory side, the integrable structure of two-dimensional string sigma-modelswith target spacetime AdS × S plays an important role [5]. It is closely related to the factthat AdS × S is described as a symmetric coset. It leads to an infinite number of theconserved charges constructed, for example, by following the pioneering works [6–10] (Fora comprehensive textbook see [11]). The symmetric cosets that potentially may lead to aholographic interpretation are classified, including spacetime fermions [12].The next interesting issue is to consider integrable deformations of the AdS/CFT cor-respondence. There are two approaches. The one is an algebraic approach based on q -deformations of the world-sheet S-matrix [13–17]. The deformed S-matrices are explicitlyconstructed, while the target-space geometry is unclear. The other is a geometric approachbased on deformations of target spaces of the sigma models. It seems likely that the de-formed geometries are represented by non-symmetric cosets [18] in comparison to AdS × S and hence the prescription for symmetric cosets is not available any more. It is necessary todevelop a new method to argue the integrability.In the latter approach, there is a long history (For classic papers see [19–21]). Motivatedby integrable deformations of AdS/CFT, a series of works have been done on squashedS and warped AdS [22–30]. Some specific higher-dimensional cases are discussed in [31].Remarkably, the classical integrable structure of deformed sigma models was recently shownfor arbitrary compact Lie groups and the coset cousins by Delduc, Magro and Vicedo [32].Then, they successively presented a q -deformation of the AdS × S superstring [33].Based on the latter approach, we are here concerned with the classical integrable structureof two-dimensional Wess-Zumino-Novikov-Witten (WZNW) models with the target spacesquashed S . The isometry is given by SU (2) L × U (1) R . It is partly explained in [24]that there exist a couple of Yangian algebras based on SU (2) L by explicit constructions ofnon-local conserved charges and direct computations of the Poisson brackets of the charges.1n this paper we consider an infinite-dimensional extension of U (1) R . In the case withoutthe Wess-Zumino term, it is just a classical analogue of a quantum affine algebra [26].When the Wess-Zumino term is added, an additional constant parameter is introduced asits coefficient. A natural question is what happens to the quantum affine algebra. As onemay easily guess, a new kind of deformation is induced by the presence of the Wess-Zuminoterm. The resulting algebra is a classical analogue of the deformed quantum affine algebra.So far, it is not clear what is the mathematical formulation of the deformed quantum affinealgebra, though it seems likely to be a two-parameter quantum toroidal algebra [34]. Inorder to answer the question, it is necessary to see the first realization of the two-parameterquantum toroidal algebra.This paper is organized as follows. In section 2 the classical action of the squashedWZNW models is introduced. In section 3 we consider the classical integrable structurebased on SU (2) L . This is called the left description . We present a couple of Lax pairs andthe associated monodromy matrices. The classical r / s -matrices are shown to satisfy theextended classical Yang-Baxter equation. The infinite-dimensional extensions of SU (2) L areYangians. In section 4 we argue the classical integrable structure based on U (1) R . This iscalled the right description . A Lax pair and the associated monodromy matrix are presented.The classical r / s -matrices satisfy the extended classical Yang-Baxter equation. Remarkably,an infinite-dimensional extension of U (1) R is shown to be a deformation of quantum affinealgebra, where a new deformation parameter is provided by the coefficient of the Wess-Zumino term. In section 5 the gauge equivalence between the left and right descriptions isproven. Under an identification between the spectral parameters, the left Lax pair is relatedto the right
Lax pair via a gauge transformation. In section 6 we argue two degeneratelimits in the right description. At some special points in the parameter space, the deformedquantum affine algebra degenerates to a Yangian, according to the enhancement of U (1) R to SU (2) R . Section 7 is devoted to conclusion and discussion.In Appendix A, we explain the computation of the current algebra in detail. Appendix Bprovides a prescription to treat non-ultra local terms in computing of the Poisson bracketsof the conserved charges. 2 Preliminary
Let us begin with the setup to fix our notation and convention. The metric of squashed S isfirst provided in terms of an SU (2) group element. Then the classical action of the squashedWZNW models is introduced. For later convenience, the equations of motion are explicitlywritten down. The metric of round S can be described as a U (1) fibration over S . The squashing isone-parameter deformations of the U (1) direction. The metric of squashed S is given by ds = L (cid:2) dθ + sin θdφ + (1 + C ) ( dψ + cosh θdφ ) (cid:3) . (2.1)When C = 0 , the metric is reduced to that of S with radius L and the isometry is SU (2) L × SU (2) R . When C = 0 , the isometry is reduced to SU (2) L × U (1) R .In order to rewrite the metric (2.1) , let us introduce an SU (2) group element g like g = e T φ e T θ e T ψ ∈ SU (2) . (2.2)Here the su (2) generators T a ( a = 1 , ,
3) satisfy the following relations (cid:2) T a , T b (cid:3) = ε abc T c , (2.3)and normalized as Tr (cid:0) T a T b (cid:1) = − δ ab . (2.4)Note that ε abc is the totally anti-symmetric tensor normalized as ε = +1 . The su (2)indices are raised and lowered by δ ab and its inverse, respectively.It is useful to define T ± as T ± ≡ √ (cid:0) T ± iT (cid:1) . (2.5)Then the commutation relations in (2.3) are rewritten as (cid:2) T ± , T ∓ (cid:3) = ∓ iT , (cid:2) T ± , T (cid:3) = ± iT ± , (2.6)3nd the normalization of the generators in (2.4) is given byTr (cid:0) T ± T ∓ (cid:1) = Tr (cid:0) T T (cid:1) = − . (2.7)Then the metric (2.1) is rewritten in terms of the SU (2) group element as ds = − L h Tr (cid:0) J (cid:1) − C (cid:0) Tr (cid:2) T J (cid:3)(cid:1) i , (2.8)where we have introduced the left-invariant one-form J ≡ g − dg . (2.9)Note that J can be represented by the angle variables ( θ, φ, ψ ) as follows: J = T (sin ψdθ − cos ψ sin θdφ ) + T (cos ψdθ + sin ψ sin θdφ ) + T ( dψ + cos θdφ )= T + √ iψ ( − idθ − sin θdφ ) + T − √ − iψ ( idθ − sin θdφ ) + T ( dψ + cos θdφ ) . With the metric (2.8) , it is easy to see the invariance under SU (2) L × U (1) R . The SU (2) L and U (1) R transformations are the left- and right- multiplications, g → e β a T a · g · e − αT . (2.10)Here β a and α are real parameters. First of all, let us introduce two-dimensional non-linear sigma models whose target space isgiven by squashed S . The classical action is S σM = 1 λ Z ∞−∞ dt Z ∞−∞ dx η µν (cid:2) tr ( J µ J ν ) − C tr (cid:0) T J µ (cid:1) tr (cid:0) T J ν (cid:1)(cid:3) , (2.11)where the parameter λ is the coupling constant and the base space is a two-dimensionalMinkowski spacetime with the coordinates t (time) and x (space) and the metric − η tt = η xx = +1 . (2.12)Note that the region of the parameter C is restricted C > − , S W Z = n π Z ds Z ∞−∞ dt Z ∞−∞ dx ǫ ˆ µ ˆ ν ˆ ρ tr (cid:16) e J ˆ µ e J ˆ ν e J ˆ ρ (cid:17) , e J ≡ e g − d e g , (2.13)where n is an integer. Note that the above integral is performed on a three-dimensional basemanifold spanned by ( t, x, s ) . The totally anti-symmetric tensor ǫ ˆ µ ˆ ν ˆ ρ is normalized as ǫ txs = +1 . (2.14)The SU (2) element e g is defined on this three-dimensional manifold. It interporates betweena constant element at s = 0 and g ( t, x ) at s = 1 : e g ( t, x, s = 0) = g : const. , e g ( t, x, s = 1) = g ( t, x ) . (2.15)Note that the Wess-Zumino term (2.13) is the same as in the case of round S and hence itis invariant under the SU (2) L × U (1) R symmetry of the sigma model action (2.11) .Let us consider the Wess-Zumino-Novikov-Witten models defined on squashed S , whichhenceforth are called “ squashed WZNW models ”. The action is given by the sum of S σM in(2.11) and S W Z in (2.13) : S = S σM + S W Z . (2.16)The action (2.16) is also SU (2) L × U (1) R -invariant.From the action (2.16) , the equations of motion are obtained, ∂ µ J µ − C tr (cid:0) T ∂ µ J µ (cid:1) T − C tr (cid:0) T J µ (cid:1) (cid:2) J µ , T (cid:3) + Kǫ µν ∂ µ J ν = 0 , (2.17)where the new constant K is defined as K ≡ nλ π , (2.18)and the totally anti-symmetric tensor ǫ µν is normalized as ǫ tx = +1 . (2.19)The su (2) components of the left-invariant one-form J a are defined as J a ≡ − T a J ) , (2.20)5r equivalently J = T + J − + T − J + + T J . (2.21)In terms of J a , the equations of motion are rewritten as(1 + C ) ∂ µ J µ + Kǫ µν ∂ µ J ν = 0 , (2.22) ∂ µ J ± µ ∓ iCJ µ J ± ,µ + Kǫ µν ∂ µ J ± ν = 0 . By definition, the left-invariant one-form J satisfies the flatness condition: ǫ µν ( ∂ µ J ν + J µ J ν ) = 0 . (2.23)This condition can also be rewritten in terms of the components J a as ǫ µν (cid:0) ∂ µ J ν + iJ + µ J − ν (cid:1) = 0 , (2.24) ǫ µν (cid:0) ∂ µ J ± ν ± iJ µ J ± ν (cid:1) = 0 . The flatness condition (2.25) enables us to rewrite the equations of motion (2.23) as(1 + C ) ∂ µ J µ − iKǫ µν J + µ J − ν = 0 , (2.25) ∂ µ J ± µ ∓ iCJ µ J ± ,µ ∓ iKǫ µν J µ J ± ν = 0 . The expressions in (2.25) play an important role in our later discussion.
In this section, we discuss the classical integrable structure of squashed WZNW models basedon the SU (2) L symmetry. We call it left description . This part contains a short review ofthe previous work [24].First of all, we construct an SU (2) L conserved current which satisfies the flatness con-dition. With the flat and conserved current, we obtain a Lax pair and the correspondingmonodromy matrix. Then we compute the classical r / s -matrices for the Lax pair. Finally,we show that the SU (2) L symmetry is enhanced to the Yangian algebra Y ( su (2) L ) .6 .1 Lax pairs The classical action (2.16) has the SU (2) L symmetry and the associated conserved currentis given by j Lµ = gJ µ g − − C tr (cid:0) T J µ (cid:1) gT g − − Kǫ µν gJ ν g − . (3.1)The conservation law of this current is equivalent to the equations of motion in (2.17) like ∂ µ j Lµ = g (cid:2) ∂ µ J µ − C tr (cid:0) T ∂ µ J µ (cid:1) T − C tr (cid:0) T J µ (cid:1) (cid:2) J µ , T (cid:3) + Kǫ µν ∂ µ J ν (cid:3) g − (3.2)= 0 . Note that j Lµ does not satisfy the flatness condition due to the deformation.One may consider to improve j Lµ so as to satisfy the flatness condition. This requirementleaves two improved currents [24], j L ± µ = gJ µ g − − C tr (cid:0) T J µ (cid:1) gT g − − Kǫ µν gJ ν g − ∓ Aǫ µν ∂ ν (cid:0) gT g − (cid:1) , with the coefficient A represented by A = s C (cid:18) − K C (cid:19) . (3.3)The subscripts ± denote the degeneracy of the improved currents. The improved currentssatisfy the following flatness condition, ǫ µν (cid:0) ∂ µ j L ± ν − j L ± µ j L ± ν (cid:1) = 0 . (3.4)When K = 0 , A = √ C and the improved currents constructed in [22] are reproduced.With the flat SU (2) L currents, two Lax pairs are constructed as L L ± t ( x ; λ L ± ) = 12 h L L ± + ( x ; λ L ± ) + L L ± − ( x ; λ L ± ) i , (3.5) L L ± x ( x ; λ L ± ) = 12 h L L ± + ( x ; λ L ± ) − L L ± − ( x ; λ L ± ) i ,L L ± + ( x ; λ L ± ) = 11 + λ L ± j L ± + , L L ± − ( x ; λ L ± ) = 11 − λ L ± j L ± − . Here λ L ± are spectral parameters and j L ± + and j L ± − are defined as j L ± + ≡ j L ± t + j L ± x , j L ± − ≡ j L ± t − j L ± x . (3.6)7he following commutation relation h ∂ t − L L ± t ( x ; λ L ± ) , ∂ x − L L ± x ( x ; λ L ± ) i = 0 . (3.7)gives rise to the conservation law of the flat SU (2) L current (equivalently, the equations ofmotion) and the flatness condition.Then the monodromy matrices are defined as U L ± ( λ L ± ) ≡ P exp (cid:20)Z ∞−∞ dx L L ± x ( x ; λ L ± ) (cid:21) . (3.8)The symbol P denotes the path ordering. Due to the flatness of the Lax pairs in (3.7) , themonodromy matrices are conserved, ddt U L ± ( λ L ± ) = 0 . (3.9)Thus an infinite set of conserved charges can be obtained by expanding the monodromymatrices with respect to λ L ± around appropriate points. For example, the monodromymatrices can be expanded around λ L ± = ∞ as U L ± ( λ L ± ) = exp " ∞ X n =0 λ − n − L ± Q L ± ( n ) . (3.10)In the next subsection, we will discuss the algebra generated by Q L ± ( n ) .Before closing this subsection, let us discuss the r / s -matrices computed from the Laxpairs in (3.5) by following the Maillet formalism [35]. One can read off them from thePoisson brackets between the spatial components of the Lax pairs, n L L ± x ( x ; λ L ± ) ⊗ , L L ± x ( y ; µ L ± ) o P (3.11)= h r L ± ( λ L ± , µ L ± ) , L L ± x ( x ; λ L ± ) ⊗ ⊗ L L ± x ( y ; µ L ± ) i δ ( x − y ) − h s L ± ( λ L ± , µ L ± ) , L L ± x ( x ; λ L ± ) ⊗ − ⊗ L L ± x ( y ; µ L ± ) i δ ( x − y ) − s L ± ( λ L ± , µ L ± ) ∂ x δ ( x − y ) . To compute the above Poisson bracket, we have to use the current algebra for j L ± µ , n j L ± ,at ( x ) , j L ± ,bt ( y ) o P = ε abc j L ± ,ct ( x ) δ ( x − y ) − Kδ ab ∂ x δ ( x − y ) , (3.12) n j L ± ,at ( x ) , j L ± ,bx ( y ) o P = ε abc j L ± ,cx ( x ) δ ( x − y ) + (cid:0) K + A (cid:1) δ ab ∂ x δ ( x − y ) , j L ± ,ax ( x ) , j L ± ,bx ( y ) o P = − (cid:0) K + A (cid:1) ε abc j L ± ,ct ( x ) δ ( x − y ) − Kε abc j L ± ,cx ( x ) δ ( x − y ) − Kδ ab ∂ x δ ( x − y ) . The explicit expressions of r / s -matrices are given by r L ± ( λ ± , µ ± ) = h LC,K ( λ ± ) + h LC,K ( µ ± )2( λ ± − µ ± ) (cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T (cid:3) , (3.13) s L ± ( λ ± , µ ± ) = h LC,K ( λ ± ) − h LC,K ( µ ± )2( λ ± − µ ± ) (cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T (cid:3) , where a scalar function h LC,K ( λ ) is defined as h LC,K ( λ ) ≡ A + ( λ + K ) − λ . The r / s -matrices satisfy the extended classical Yang-Baxter equation ∗ , h ( r − s ) L ± ( λ L ± , µ L ± ) , ( r + s ) L ± ( λ L ± , ν L ± ) i (3.14)+ h ( r + s ) L ± ( λ L ± , µ L ± ) , ( r + s ) L ± ( µ L ± , ν L ± ) i + h ( r + s ) L ± ( λ L ± , ν L ± ) , ( r + s ) L ± ( µ L ± , ν L ± ) i = 0 . It should be noted that, when K = 0 , the function l LC,K ( λ ) is reduced to h LC, ( λ ) ≡ C + λ − λ . Thus the r / s -matrices in (3.13) reproduce the results without the Wess-Zumino term [25] . So far, the monodromy matrices U L ± ( λ L ± ) have been introduced and an infinite number ofthe conserved charges Q L ± ( n ) are obtained by expanding them with respect to λ L ± .For the first three levels, the explicit expressions of Q L ± ( n ) are given by Q L ± ,a (0) = Z ∞−∞ dx j L ± ,at ( x ) , (3.15) Q L ± ,a (1) = 14 Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) ε abc j L ± ,bt ( x ) j L ± ,ct ( y ) − Z ∞−∞ j L ± ,ax ( x ) , ∗ The r / s -matrices depend on λ L ± and µ L ± individually (not only λ L ± − µ L ± ) and they satisfy theextended classical Yang-Baxter equation. Thus the classification of the r / s -matrice are subtle. L ± ,a (2) = 112 Z ∞−∞ dx Z ∞−∞ dy Z ∞−∞ dz ǫ ( x − y ) ǫ ( x − z ) δ bc × h j L ± ,bt ( x ) j L ± ,at ( y ) j L ± ,ct ( z ) − j L ± ,at ( x ) j L ± ,bt ( y ) j L ± ,ct ( z ) i − Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) ε abc j L ± ,bt ( x ) j L ± ,cx ( y ) + Z ∞−∞ dx j L ± ,at ( x ) . Note that these charges can be directly constructed from the flat SU (2) L currents j L ± µ ( x )recursively by following the BIZZ construction [8] .The next task is to show that the conserved charges satisfy the defining relations ofYangian Y ( su (2) L ) . The Poisson brackets of the first two levels are given by [24] n Q L ± ,a (0) , Q L ± ,b (0) o P = ε abc Q L ± ,c (0) , (3.16) n Q L ± ,a (1) , Q L ± ,b (0) o P = ε abc Q L ± ,c (1) , n Q L ± ,a (1) , Q L ± ,b (1) o P = ε abc (cid:20) Q L ± ,c (2) + 112 (cid:16) Q L ± (0) (cid:17) Q L ± ,c (0) + 2 KQ L ± ,c (1) (cid:21) . The Serre relations are shown as n Q L ± , , n Q L ± , +(1) , Q L ± , − (1) o P o P = 14 Q L ± , (cid:16) Q L ± , +(0) Q L ± , − (1) − Q L ± , − (0) Q L ± , +(1) (cid:17) , (3.17) n Q L ± , ± (1) , n Q L ± , ± (1) , Q L ± , ∓ (1) o P o P − n Q L ± , , n Q L ± , ± (1) , Q L ± , o P o P = 14 Q L ± , ± (0) (cid:16) Q L ± , ± (0) Q L ± , ∓ (1) − Q L ± , ∓ (0) Q L ± , ± (1) (cid:17) − Q L ± , (cid:16) Q L ± , ± (0) Q L ± , − Q L ± , Q L ± , ± (1) (cid:17) , n Q L ± , ± (1) , n Q L ± , ± (1) , Q L ± , o P o P = 14 Q L ± , ± (0) (cid:16) Q L ± , ± (0) Q L ± , − Q L ± , Q L ± , ± (1) (cid:17) . Thus the defining relations of Yangian Y ( su (2) L ) at the classical level are satisfied in thesense of Drinfeld’s first realization [36, 37].Here we should comment on the treatment of non-ultra local terms contained in thecurrent algebra of j L ± µ ( x ) . They develop ambiguities in computing the Poisson bracketsof the conserved charges and there might be the possibility that the defining relations of Y ( su (2) L ) are spoiled. In the present case, the presence of the Wess-Zumino term make thesituation worse. It develops non-ultra local terms even in the Poisson brackets of j L ± t ( x ) andhence cause ambiguities in computing the usual SU (2) L Lie algebra part. The treatment ofthe non-ultra local terms is argued in Appendix B in detail.10
The right description
The classical integrable structure of the squashed WZNW models can also be described basedon U (1) R as another description. This description is called right description . A Lax pair andthe associated monodromy matrix are presented. Then an infinite-dimensional extensionof U (1) R is argued. The resulting algebra is a deformation of the standard quantum affinealgebra. In the right description, the r / s -matrices are deformed by an additional term, incomparison to the case without the Wess-Zumino term. A Lax pair which respects U (1) R is given by † L Rt ( x ; λ R ) = 12 (cid:2) L R + ( x ; λ R ) + L R − ( x ; λ R ) (cid:3) , (4.1) L Rx ( x ; λ R ) = 12 (cid:2) L R + ( x ; λ R ) − L R − ( x ; λ R ) (cid:3) ,L R ± ( x ; λ R ) = − sinh ( α ± β )sinh [ α ± ( β + λ R )] (cid:20) T + J −± + T − J + ± + cosh ( α ± λ R )cosh α T J ± (cid:21) . New constants α and β are related to C and K liketanh α = iCA , tanh β = iCKA (1 + C ) . (4.2)Note that α and β have the periodicities, α ∼ α + πi , β ∼ β + πi . (4.3)In the Lax pair (4.1), a spectral parameter λ R has been introduced. This is seeminglyindependent of λ L ± at this stage, but eventually there is a relation between them as we willsee later. The Lax pair (4.1) is referred to as the right Lax pair hereafter.The relations given in (4.2) imply the inequalities for ( C, K ) : − tanh α = C A = C (1 + C )1 + C − K > − , (4.4) − tanh β = C K A (1 + C ) = CK (1 + C )(1 + C − K ) > − . With the kinematic restriction
C > − C − K + 1)( C + K + 1)( C − K + 1) > . † The anisotropic Lax pair with K = 0 is constructed originally by Cherednik [19]. See also [20]. The allowed region of C and K . In the red region, α and β are real. On the other hand, theseare purely imaginary in the blue region. Due to the relations in (4.2) and the reality of C and K , α and β must be real or purelyimaginary. When C ( C − K + 1) < α and β are real (up to the shift of iπn with n ∈ Z ) .On the other hand, when C ( C − K + 1) > α and β are purely imaginary. Thus theallowed region of C and K can be expressed on the ( C, K )-plane as depicted in Fig. 1.The relations given in (4.2) can be solved for α and β , and hence C , K (and A ) arewritten in terms of α and β , K = sinh 2 β sinh 2 α , C = − sinh( α + β ) sinh( α − β )cosh α , (4.5) A = − i sinh( α + β ) sinh( α − β )sinh 2 α . Note that (
C, K, A ) are invariant under the shift of α and β by iπ .The equations of motion in (2.17) and the flatness condition of J are reproduced fromthe commutation relation, (cid:2) ∂ t − L Rt ( x ; λ R ) , ∂ x − L Rx ( x ; λ R ) (cid:3) = 0 . (4.6) Monodromy matrix
With the right Lax pair given in (4.1) , the associated monodromy matrix is defined as U R ( λ R ) ≡ P exp (cid:20)Z ∞−∞ dx L Rx ( x ; λ R ) (cid:21) . (4.7)12he flatness of the Lax pair (4.6) ensures that the monodromy matrix is a conserved quantity, ddt U R ( λ R ) = 0 . (4.8)Thus, by expanding U R ( λ R ), an infinite number of conserved charges are constructed. q -deformation of su (2) R In the squashed WZNW models, the SU (2) R symmetry, which is preserved by round S , isbroken to U (1) R , due to the deformation term. The Noether current for U (1) R is given by j R, µ = − (cid:2) (1 + C ) J µ + Kǫ µν J ,ν (cid:3) (4.9)= − cosh β cosh α (cid:18) cosh β cosh α J µ + sinh β sinh α ǫ µν J ,ν (cid:19) . For later convenience, the expressions have been given in terms of α and β , as well as interms of C and K .It is known that there exist non-local conserved currents which correspond to the brokencomponents T ± in the case without the Wess-Zumino term [25]. Let us show that this is thecase even in the squashed WZNW models.For this purpose, it is helpful to introduce a non-local function χ ( x ) , χ ( x ) = − Z ∞−∞ dy ǫ ( x − y ) j R, t ( y ) . (4.10)This function satisfies the differential equation, ∂ µ χ = − ǫ µν j R, ,ν = (1 + C ) ǫ µν J ,ν + KJ µ . (4.11)This relation ensures the conservation law of the non-local currents, as we will see later.With χ ( x ) , the conserved, non-local currents are constructed as j R, ± µ = e γ ± χ i R, ± µ , (4.12) i R, ± µ = − (cid:0) J ± µ + ( K ± iA ) ǫ µν J ± ,ν (cid:1) = − α (cid:2) sinh 2 αJ ± µ ± (cid:0) cosh 2 α − e ∓ β (cid:1) ǫ µν J ± ,ν (cid:3) . Here γ ± are defined as γ ± ≡ (cid:18) CC A ∓ iK (cid:19) − = − i e ± β sinh 2 α β . (4.13)13he non-local currents j R, ± µ give rise to the conserved charges Q R, ± = Z ∞−∞ dx j R, ± t ( x ) , (4.14)as well as the standard Noether charge of U (1) R , Q R, = Z ∞−∞ dx j R, t ( x ) . (4.15)As a next step, let us compute the algebra of Q R, ± and Q R, . For this purpose, it isnecessary to evaluate the Poisson brackets of i R, ± t ( x ) and j R, t ( x ) , n j R, t ( x ) , j R, t ( y ) o P = 2 K∂ x δ ( x − y ) , (4.16) n i R, ± t ( x ) , j R, t ( y ) o P = ± ii R, ± t ( x ) δ ( x − y ) , n i R, ± t ( x ) , i R, ∓ t ( y ) o P = ∓ ij R, t ( x ) δ ( x − y ) + 2 K∂ x δ ( x − y ) . Because the Poisson brackets contain non-ultra local terms even for the first bracket due tonon-vanishing K , an appropriate prescription to treat the non-ultra local terms is requiredto evaluate the Poisson brackets of Q R, ± and Q R, , as in the case of the su (2) L algebra inthe left description. The prescription is argued in Appendix B in detail.The resulting algebra can be regarded as a classical analogue of q -deformed su (2) R [37,38] , n Q R, ± , Q R, o P = ± iQ R, ± , (4.17) n Q R, + , Q R, − o P = − i ( γ + ) + ( γ − ) γ + γ − γ + + γ − ) sinh (cid:20) ( γ + + γ − )2 Q R, (cid:21) . The q -parameter is given by q = e ( γ + + γ − ) / = e γ , (4.18)where γ is a new parameter defined as γ ≡ γ + + γ − − i α . (4.19)Note that there is an unfamiliar ovarall factor (( γ + ) + ( γ − ) ) / γ + γ − in the right-hand sideof the Poisson bracket between Q R, + and Q R, − . However, this factor can be absorbed byrescaling Q R, ± without changing the Poisson structure.14imilarly, there is another set of conserved, non-local currents, e j R, ± µ = e − γ ∓ χ e i R, ± µ , (4.20) e i R, ± µ = − (cid:0) J ± µ + ( K ∓ iA ) ǫ µν J ± ,ν (cid:1) = − α (cid:2) sinh 2 αJ ± µ ∓ (cid:0) cosh 2 α − e ± β (cid:1) ǫ µν J ± ,ν (cid:3) , and the associated charges are e Q R, ± = Z ∞−∞ dx e j R, ± t ( x ) . (4.21)Note that e j R, ± are related to j R, ± µ ( x ) by flipping the sign of A . The sign flip of A is translatedto the following transformation laws,( α, β ) → ( − α, − β ) , γ ± → − γ ∓ . (4.22)Thus, with the sign flip of A , the algebra of Q R, and e Q R, ± is obtained as n e Q R, ± , Q R, o P = ± i e Q R, ± , (4.23) n e Q R, + , e Q R, − o P = − i ( γ + ) + ( γ − ) γ + γ − γ + + γ − ) sinh (cid:20) ( γ + + γ − )2 Q R, (cid:21) . The algebra is also the q -deformed su (2) R with the same q -parameter. The next task is to argue an infinite-dimensional extension of q -deformed su (2) R . For thispurpose, let us consider two expansions of the monodromy matrix U R ( λ R ) with respect tothe spectral parameter λ R . The expansion points are the same as in the case of squashedsigma models [26] and the monodromy matrix is expanded around Re λ R = ±∞ . Note thatthe spectral parameter λ R is periodically identified as λ R ∼ λ R + 2 πi , (4.24)and hence the spectral parameter λ R is regarded as living on a cylinder. As a result, the pointRe λ R = + ∞ is certainly different from the point Re λ R = −∞ . Thus the two expansionpoints give rise to two different sets of conserved charges. Here the expressions of theconserved charges are explicitly obtained. 15irst of all, recall the concrete expression of L Rx ( x ; λ R ) : L Rx ( x ; λ R ) (4.25)= − α + β + λ R ) sinh( α − β − λ R ) × n T h − sinh α cosh α (cid:0) cosh β cosh λ R sinh λ R + sinh β cosh β sinh λ R (cid:1) J t + (cid:0) sinh( α + β ) sinh( α − β ) cosh λ R − sinh β cosh β cosh λ R sinh λ R − sinh α sinh λ R (cid:1) J x i + T + h − sinh α cosh α sinh λ R J − t + (sinh( α + β ) sinh( α − β ) cosh λ R − sinh β cosh β sinh λ R ) J − x i + T − h − sinh α cosh α sinh λ R J + t + (sinh( α + β ) sinh( α − β ) cosh λ R − sinh β cosh β sinh λ R ) J + x io . This expression is useful to expand U R ( λ R ) . Expansion around Re λ R = −∞ Let us first expand U R ( λ R ) around Re λ R = −∞ . It is convenient to perform a transforma-tion for the spectral parameter as z R = e λ R . (4.26)The expansion around Re λ R = −∞ corresponds to the one around z R = 0 . Then theexpanded monodromy matrix is expected to be the following form, U R ( λ R ) = e q (0) / exp " ∞ X n =1 z nR q ( n ) e q (0) / . (4.27)Here q ( n ) ( n ≥
0) are conserved charges to be determined by a concrete expansion of U R ( λ R ) .Note that L Rx ( λ R ) is expanded around z R = 0 as L Rx ( x ; λ R ) = iγ + T j R, t − z R e β sinh 2 α (cid:16) T − i R, + t + T + e i R, − t (cid:17) − z R e β sinh α T (cid:18) j R, x + cosh 2 α sinh 2 α j R, t (cid:19) − z R e β sinh α (cid:20) T − (cid:18) i R, + x + cosh 2 α sinh 2 α i R, + t (cid:19) + T + (cid:18)e i R, − x + cosh 2 α sinh 2 α e i R, − t (cid:19)(cid:21) O ( z R ) . (4.28)By comparing the expansion of the expected form (4.27) U R ( λ R ) = e q (0) / (cid:20) z R q (1) + z R (cid:18) q (2) + 12 q (cid:19) + z R (cid:18) q (3) + 12 (cid:8) q (1) , q (2) (cid:9) + 16 q (cid:19) + O ( z R ) (cid:21) e q (0) / , with the direct expansion of U R ( λ R ) using (4.28) , the expressions of q ( n ) are fixed as follows: q (0) = iγ + T Q R, , (4.29) q (1) = − e β sinh 2 α (cid:16) T − Q R, +(1) + T + e Q R, − (1) (cid:17) ,q (2) = − i β sinh αT Q R, ,q (3) = −
14 e β sinh α (cid:16) T − Q R, +(3) + T + e Q R, − (3) (cid:17) . where the following quantities have been introduced, Q R, = Z ∞−∞ dx j R, t ( x ) ( = Q R, ) , (4.30) Q R, +(1) = Z ∞−∞ dx j R, + t ( x ) ( = Q R, + ) , e Q R, − (1) = Z ∞−∞ dx e j R, − t ( x ) ( = e Q R, − ) ,Q R, = − Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) j R, + t ( x ) e j R, − t ( y ) − i e − β Z ∞−∞ dx j R, x ( x ) − i e − β cosh 2 α sinh 2 α Q R, ,Q R, +(3) = 12 Z ∞−∞ dx Z ∞−∞ dy Z ∞−∞ dz ǫ ( x − y ) ǫ ( x − z ) e j R, − t ( x ) j R, + t ( y ) j R, + t ( z )+2 i e − β Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) j R, + t ( x ) (cid:18) j R, x ( y ) + cosh 2 α sinh 2 α j R, t ( y ) (cid:19) + 4e − β sinh 2 α Z ∞−∞ dxj R, + x ( x ) − e Q R, +(1) (cid:16) Q R, +(1) (cid:17) + 4e − β cosh 2 α sinh α Q R, +(1) , e Q R, − (3) = 12 Z ∞−∞ dx Z ∞−∞ dy Z ∞−∞ dz ǫ ( x − y ) ǫ ( x − z ) j R, + t ( x ) e j R, − t ( y ) e j R, − t ( z ) − i e − β Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) e j R, − t ( x ) (cid:18) j R, x ( y ) + cosh 2 α sinh 2 α j R, t ( y ) (cid:19) + 4e − β sinh 2 α Z ∞−∞ dx e j R, − x ( x ) − Q R, +(1) (cid:16) e Q R, − (1) (cid:17) + 4e − β cosh 2 α sinh α e Q R, − (1) . ...17hese are also conserved charges and will play an important role later in studying thetower structure of the conserved charges that indicates an infinite-dimensional extension of q -deformed su (2) R . Expansion around Re λ R = + ∞ Let us next consider another expansion around Re λ R = + ∞ . It is helpful to introduce anew parametrization z ′ R defined as z ′ R = z − R = e − λ R . (4.31)The expansion around Re λ R = + ∞ corresponds to the one around z ′ R = 0 . Then theexpected form of the expanded monodromy matrix U R ( λ R ) is given by U R ( λ R ) = e ¯ q (0) / exp " ∞ X n =1 z ′ Rn ¯ q ( n ) e ¯ q (0) / . (4.32)Here ¯ q ( n ) ( n ≥
0) are conserved charges.The Lax pair L Rx ( λ R ) is expanded around z ′ R = 0 as L Rx ( x ; λ R ) = iγ − T j R, t + z ′ R e − β sinh 2 α (cid:16) T + i R, − t + T − e i R, + t (cid:17) − z ′ R e − β sinh α T (cid:18) j R, x − cosh 2 α sinh 2 α j R, t (cid:19) − z ′ R e − β sinh α (cid:20) T + (cid:18) i R, − x − cosh 2 α sinh 2 α i R, − t (cid:19) + T − (cid:18)e i R, + x − cosh 2 α sinh 2 α e i R, + t (cid:19)(cid:21) + O ( z ′ R ) . (4.33)Thus, by comparing the expansion of the expected monodromy matrix U R ( λ R ) = e ¯ q (0) / (cid:20) z ′ R ¯ q (1) + z ′ R (cid:18) ¯ q (2) + 12 ¯ q (cid:19) + z ′ R (cid:18) ¯ q (3) + 12 (cid:8) ¯ q (1) , ¯ q (2) (cid:9) + 16 ¯ q (cid:19) + O ( z ′ R ) (cid:21) e ¯ q (0) / , with the direct expansion using (4.33) , ¯ q ( n ) are obtained as¯ q (0) = iγ − T Q R, , (4.34)¯ q (1) = e − β sinh 2 α (cid:16) T + Q R, − (1) + T − e Q R, +(1) (cid:17) , ¯ q (2) = − i − β sinh αT ¯ Q R, , q (3) = 14 e − β sinh α (cid:16) T + Q R, − (3) + T − e Q R, +(3) (cid:17) . Here the conserved quantities have been introduced as Q R, = Z ∞−∞ dx j R, t ( x ) ( = Q R, ) , (4.35) Q R, − (1) = Z ∞−∞ dx j R, − t ( x ) ( = Q R, − ) , e Q R, +(1) = Z ∞−∞ dx e j R, + t ( x ) ( = e Q R, + ) , ¯ Q R, = Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) j R, − t ( x ) e j R, + t ( y ) − i e β Z ∞−∞ dx j R, x ( x ) + 2 i e β cosh 2 α sinh 2 α Q R, ,Q R, − (3) = 12 Z ∞−∞ dx Z ∞−∞ dy Z ∞−∞ dz ǫ ( x − y ) ǫ ( x − z ) e j R, + t ( x ) j R, − t ( y ) j R, − t ( z ) − i e β Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) j R, − t ( x ) (cid:18) j R, x ( y ) − cosh 2 α sinh 2 α j R, t ( y ) (cid:19) − β sinh 2 α Z ∞−∞ dxj R, − x ( x ) − e Q R, +(1) (cid:16) Q R, − (1) (cid:17) + 4e β cosh 2 α sinh α Q R, − (1) , e Q R, +(3) = 12 Z ∞−∞ dx Z ∞−∞ dy Z ∞−∞ dz ǫ ( x − y ) ǫ ( x − z ) j R, − t ( x ) e j R, + t ( y ) e j R, + t ( z )+2 i e β Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) e j R, + t ( x ) (cid:18) j R, x ( y ) − cosh 2 α sinh 2 α j R, t ( y ) (cid:19) − β sinh 2 α Z ∞−∞ dx e j R, + x ( x ) − Q R, +(1) (cid:16) e Q R, +(1) (cid:17) + 4e β cosh 2 α sinh α e Q R, +(1) . ...These are also important to see the tower structure of the conserved charges in addition tothe previous expansion. q -deformed su (2) R The next is to compute the Poisson brackets of Q R ( n ) and e Q R ( n ) ( n ≥
0) .For this purpose, the following Poisson brackets are useful, n j R, t ( x ) , j R, t ( y ) o P = 2 K∂ x δ ( x − y ) , (4.36) n i R, ± t ( x ) , j R, t ( y ) o P = ± ii R, ± t ( x ) δ ( x − y ) , n i R, ± t ( x ) , i R, ∓ t ( y ) o P = ∓ ij R, t ( x ) δ ( x − y ) + 2 K∂ x δ ( x − y ) . ne i R, ± t ( x ) , j R, t ( y ) o P = ± i e i R, ± t ( x ) δ ( x − y ) , e i R, ± t ( x ) , e i R, ∓ t ( y ) o P = ∓ ij R, t ( x ) δ ( x − y ) + 2 K∂ x δ ( x − y ) , n i R, ± t ( x ) , e i R, ∓ t ( y ) o P = ∓ i h(cid:16) ∓ α sinh β cosh β (cid:17) j R, t ( x ) ± α cosh αj R, x ( x ) i δ ( x − y )+2 ( K ± iA ) ∂ x δ ( x − y ) , n j R, x ( x ) , i R, ± t ( y ) o P = ∓ i h e ± β i R, ± t ( x ) ± e ± β (1 − cosh 2 α cosh 2 β )sinh 2 α cosh β i R, ± t ( x ) i δ ( x − y ) , n j R, x ( x ) , e i R, ± t ( y ) o P = ∓ i h e ∓ β e i R, ± t ( x ) ∓ e ∓ β (1 − cosh 2 α cosh 2 β )sinh 2 α cosh β e i R, ± t ( x ) i δ ( x − y ) . Then, with the above relations, the Poisson brackets are computed as n Q R, ± (1) , Q R, o P = ± iQ R, ± (1) , n Q R, +(1) , Q R, − (1) o P = − i ( γ + ) + ( γ − ) γ + γ − γ + + γ − ) sinh (cid:20) ( γ + + γ − )2 Q R, (cid:21) , n e Q R, ± (1) , Q R, o P = ± iQ R, ± (1) , n e Q R, +(1) , e Q R, − (1) o P = − i ( γ + ) + ( γ − ) γ + γ − γ + + γ − ) sinh (cid:20) ( γ + + γ − )2 Q R, (cid:21) , n Q R, − (1) , e Q R, − (1) o P = 0 , n Q R, +(1) , e Q R, +(1) o P = 0 , n Q R, − (1) , e Q R, +(1) o P = i γ − γ + γ ¯ Q R, , n Q R, +(1) , e Q R, − (1) o P = i γ + γ − γQ R, , n ¯ Q R, , Q R, − (1) o P = − i γ − γ + γ (cid:20) Q R, − (3) + 23 e Q R, +(1) (cid:16) Q R, − (1) (cid:17) (cid:21) , n ¯ Q R, , e Q R, +(1) o P = i γ − γ + γ (cid:20) e Q R, +(3) + 23 Q R, − (1) (cid:16) e Q R, +(1) (cid:17) (cid:21) , n Q R, , e Q R, − (1) o P = i γ + γ − γ (cid:20) e Q R, − (3) + 23 Q R, +(1) (cid:16) e Q R, − (1) (cid:17) (cid:21) , n Q R, , Q R, +(1) o P = − i γ + γ − γ (cid:20) Q R, +(3) + 23 e Q R, − (1) (cid:16) Q R, +(1) (cid:17) (cid:21) , n Q R, − (3) , Q R, − (1) o P = − i γ − γ + γ (cid:16) Q R, − (1) (cid:17) ¯ Q R, , n e Q R, +(3) , e Q R, +(1) o P = i γ − γ + γ (cid:16) e Q R, +(1) (cid:17) ¯ Q R, , n e Q R, − (3) , e Q R, − (1) o P = i γ + γ − γ (cid:16) e Q R, − (1) (cid:17) Q R, , Q R, +(3) , Q R, +(1) o P = − i γ + γ − γ (cid:16) Q R, +(1) (cid:17) Q R, . Again, we should be careful for non-ultra local terms. For the detail, see Appendix B.In addition, the Serre-like relations are evaluated as follows: n Q R, ± (1) , n Q R, ± (1) , n Q R, ± (1) , e Q R, ∓ (1) o P o P o P = − (cid:18) γ ± γ ∓ γ (cid:19) (cid:16) Q R, ± (1) (cid:17) n Q R, ± (1) , e Q R, ∓ (1) o P , (4.37) n e Q R, ± (1) , n e Q R, ± (1) , n e Q R, ± (1) , Q R, ∓ (1) o P o P o P = − (cid:18) γ ∓ γ ± γ (cid:19) (cid:16) e Q R, ± (1) (cid:17) n e Q R, ± (1) , Q R, ∓ (1) o P . The relations in (4.37) may be interpreted as deformations of the classical analogue of q -Serrerelations in the standard quantum affine algebra U q ( [ su (2) R ) and ensure that the resultingalgebra exhibits the similar tower structure. The higher-level charges are obtained fromthe level 1 charges by taking the Poisson bracket repeatedly as with U q ( [ su (2) R ) . The de-formed algebra obtained here may be lifted up to the quantum level. Although we have notyet succeeded to find out its mathematical formulation, a two-parameter quantum toroidalalgebra [34] would be a possible candidate of it.The tower structure of the conserved charges is depicted in Fig. 2. The overall coefficientof the Poisson bracket is affected by the deformation. When going up the tower of charges,the factor γ − /γ + is multiplied to the Poisson bracket. On the other hand, when going downthe tower, the factor γ + /γ − is multiplied to the Poisson bracket.When γ + /γ − = e β = ± q -Serre relations. The first condition e β = 1 means that K = 0 and corresponds to squashedsigma models without the Wess-Zumino term. The second condition e β = − K = 0and C = − C → − O (3) non-linear sigma models are reproduced as itis obvious from the classical action. However, because this limit is singular, the secondcondition would also be singular. r / s -matrices The r / s -matrices associated with the right Lax pair in (4.1) are computed from the Poissonbracket between the spatial components of the Lax pair, n L Rx ( x ; λ R ) ⊗ , L Rx ( y ; µ R ) o P = h r R ( λ R , µ R ) , L Rx ( x ; λ R ) ⊗ ⊗ L Rx ( y ; µ R ) i δ ( x − y ) − h s R ( λ R , µ R ) , L Rx ( x ; λ R ) ⊗ − ⊗ L Rx ( y ; µ R ) i δ ( x − y )21 evel k Rules:
Figure 2:
The tower structure of the conserved charges. − s R ( λ R , µ R ) ∂ x δ ( x − y ) . The resulting r / s -matrices are r R ( λ R , µ R ) = h Rα,β ( λ R ) + h Rα,β ( µ R )2 sinh( λ R − µ R ) (cid:2) T − ⊗ T + + T + ⊗ T − + cosh( λ R − µ R ) T ⊗ T (cid:3) − h Rα,β ( λ R ) − h Rα,β ( µ R )2 tanh βT ⊗ T ,s R ( λ R , µ R ) = h Rα,β ( λ R ) − h Rα,β ( µ R )2 sinh( λ R − µ R ) (cid:2) T − ⊗ T + + T + ⊗ T − + cosh( λ R − µ R ) T ⊗ T (cid:3) − h Rα,β ( λ R ) + h Rα,β ( µ R )2 tanh βT ⊗ T , (4.38)where a scalar function h Rα,β ( λ ) is defined as h Rα,β ( λ ) ≡ sinh 2 α sinh λ α + β + λ ) sinh( α − β − λ ) . (4.39)It is straightforward to show that the classical r / s -matrices given in (4.38) satisfy the ex-tended classical Yang-Baxter equation, h ( r − s ) R ( λ R , µ R ) , ( r + s ) R ( λ R , ν R ) i + h ( r + s ) R ( λ R , µ R ) , ( r + s ) R ( µ R , ν R ) i h ( r + s ) R ( λ R , ν R ) , ( r + s ) R ( µ R , ν R ) i = 0 . Thus the classical integrability has been ensured ‡ .In the previous subsection, a deformation of the quantum affine algebra has been shownexplicitly. According to the deformation, the r / s -matrices given in (4.38) are also deformedby additional terms proportional to tanh β T ⊗ T .It seems likely that the additional terms cannot be eliminated by any gauge transforma-tions. This would indicate that the deformed quantum affine algebra cannot be mapped tothe standard quantum affine algebra.When β = 0 and α = 0 (i.e., K = 0) , h Rα,β ( λ ) is reduced to h Rα, ( λ ) ≡ sinh 2 α sinh λ α + λ ) sinh( α − λ ) , (4.40)and the deformation terms proportional to tanh β T ⊗ T vanish. Thus the r / s -matricesobtained in [26] are reproduced. Let us show the gauge-equivalence, which is referred to as left-right duality , between theright Lax pair L Rµ ( λ R ) and a pair of the left Lax pairs L L ± µ ( λ L ± ) under a certain relationbetween the spectral parameters and the rescaling of sl (2) generators. The equivalence isshown in the case without the Wess-Zumino term [27]. The analysis here is a generalizationof the result obtained in [27]. The fundamental domains of the spectral parameters
It is useful to realize the fundamental domains of the spectral parameters.Recall that a pair of the Lax pairs in the left description [See (3.5)] are given by L L ± t ( x ; λ L ± ) = 12 h L L ± + ( x ; λ L ± ) + L L ± − ( x ; λ L ± ) i , ‡ The r / s -matrices depend on λ R and µ R individually (not only λ R − µ R ), though they satisfy theextended classical Yang-Baxter equation. Hence it is unclear to classify the r / s -matrices in a well-knownmanner. Two Riemann spheres with two punctures. L L ± x ( x ; λ L ± ) = 12 h L L ± + ( x ; λ L ± ) − L L ± − ( x ; λ L ± ) i ,L L ± + ( x ; λ L ± ) = 11 + λ L ± j L ± + , L L ± − ( x ; λ L ± ) = 11 − λ L ± j L ± − , where each of the spectral parameters take the values on a Riemann sphere with two punc-tures: λ L ± ∈ C ∪ {∞} . The punctures come from the fact that each Lax pair has two polesat λ L ± = ± L Rt ( x ; λ R ) = 12 (cid:2) L R + ( x ; λ R ) + L R − ( x ; λ R ) (cid:3) ,L Rx ( x ; λ R ) = 12 (cid:2) L R + ( x ; λ R ) − L R − ( x ; λ R ) (cid:3) ,L R ± ( x ; λ R ) = − sinh ( α ± β )sinh [ α ± ( β + λ R )] (cid:20) T + J −± + T − J + ± + cosh ( α ± λ R )cosh α T J ± (cid:21) . The spectral parameter λ R is periodically identified as λ R ∼ λ R + 2 πi , (5.1)and hence it takes the values on a cylinder. Because the right Lax pair is regular in theRe λ R → ±∞ limit, the fundamental domain of spectral parameter can be regarded as aRiemann sphere under the map λ R → z R = e λ R . (5.2)It is obvious that the right Lax pair has four poles at z R = e ± α − β , − e ± α − β . (5.3)24igure 4: The Riemann sphere with four punctures for α and β are purely imaginary. Thus the domain of the spectral parameter z R is regarded as a Riemann sphere with fourpunctures as depicted in Fig. 4 § .In the following, we show that the Riemann sphere of z R is related to a pair of theRiemann spheres of λ L ± . The reduced right descriptions
As a next step, we introduce the reduced right descriptions . Lax pairs in the reduced rightdescriptions are obtained from the original right Lax pair through the following isomorphismsof the sl (2) algebra, T ± → e ± λ R T ± = e − iλ R T T ± e iλ R T (5.4)or T ± → e ∓ λ R T ± = e iλ R T T ± e − iλ R T . (5.5)Because T is invariant under the isomorphisms, the generators can be rewritten into T a → e ∓ iλ R T T a e ± iλ R T . (5.6)One of the resulting Lax pairs is L R + ± ( x ; λ R + ) = − sinh ( α ± β )sinh (cid:2) α ± (cid:0) β + λ R + (cid:1)(cid:3) (5.7) § When α and β are real, the four poles are on the real axis of the z R -plane. When α and β are purelyimaginary, the four poles are on the unit circle with the center at the origin. " e + λ R + T + J −± + e − λ R + T − J + ± + cosh (cid:0) α ± λ R + (cid:1) cosh α T J ± . For later convenience, we change the subscript of the spectral parameter from R to R + .This reduced right Lax pair corresponds to the former isomorphism (5.4) . Similarly, theother reduced right Lax pair is defined as L R − ± ( x ; λ R − ) = − sinh ( α ± β )sinh (cid:2) α ± (cid:0) β + λ R − (cid:1)(cid:3) (5.8) × " e − λ R − T + J −± + e + λ R − T − J + ± + cosh (cid:0) α ± λ R − (cid:1) cosh α T J ± . For this reduced right Lax pair, its spectral parameter is denoted as λ R − . This reduced rightLax pair corresponds to the later isomorphism (5.5) .Note that the domains of the spectral parameters λ R ± have the periodicities, λ R ± ∼ λ R ± + πi , (5.9)and hence the squares of z R ± z R ± = e λ R ± (5.10)live on the Riemann spheres ¶ . Each of the reduced right Lax pairs L R ± µ ( λ R ± ) has two polesin its fundamental domain, z R ± = e α − β and e − α − β . (5.11)As a result, the fundamental domains of z R ± are a pair of Riemann spheres with two punc-tures.So far, we have shown that the right description is decomposed to a couple of the reducedright descriptions. The statement we want to show is that each of the Lax pair in the reducedright description is equivalent to each of the Lax pairs in the left description through a gaugetransformation with a certain identification of the spectral parameters. A relation of the spectral parameters
Then the next task is to find out a relation of the spectral parameters in the left and rightdescriptions. Because both λ L ± and z R ± live on the two-punctured Riemann spheres, they ¶ Note that z R ± and − z R ± cannot be distinguished and hence z R ± takes the value on a Riemann sphere. z R ± = aλ L ± + bcλ L ± + d . (5.12)The constant parameter should be fixed by the pole structure and the correspondence of theexpansion points for Yangians.The first requirement is that the position of the poles should be mapped each other. Thiscondition leads to e ± α − β = a ± bc ± d . (5.13)Now we have two possibilities: the pole at z R ± = e α − β corresponds to 1) the pole at λ L ± = 1 , or 2) the pole at λ L ± = − K = 0 .Then, recall that the Yangian charges Q L ± ( n ) are obtained by expanding the (reduced) rightLax pairs around z R ± = 1 . From this information, the following relation is obtained,1 = ac . (5.14)Thus the final result is given by z R ± = sinh 2 αλ L ± + cosh 2 α − e − β sinh 2 αλ L ± − cosh 2 α + e β = λ L ± + K + iAλ L ± + K − iA , (5.15)or equivalently, λ L ± + KiA = 1tanh λ R ± . (5.16)Here we comment on the connection between the two-punctured λ L -Riemann spheresand the four-punctured z R -Riemann sphere. The relation between z R and λ L is also givenby (5.15) : z R = λ L ± + K + iAλ L ± + K − iA . (5.17)However the interpretation is more involved because − z R is certainly a different from thepoint z R . As a relation between z R and λ L ± (not z R and λ L ± ) , the relation in (5.17) indicatesthat there exists a cut between λ L ± = − K + iA and λ L ± = − K − iA . Two-punctured λ L -Riemann spheres are connected along the cut and combined into a single Riemann spherewith four punctures. The resulting four-punctured Riemann sphere is depicted in Fig.5.27igure 5: Two-punctured λ L -Riemann spheres are connected along the cut. The gauge equivalence
We concentrate on showing the gauge equivalence between L R + µ ( x ; λ R + ) and L L + µ ( x ; λ L + ) .The analysis for the case with (-)-subscript is also similar. The gauge transformation isgenerated by g and the relation between the spectral parameters is given by λ L + + KiA = 1tanh λ R + . (5.18)First of all, the left Lax pair L L + µ ( x ; λ L + ) can be rewritten as L L + ± ( x ; λ L + ) (5.19)= 11 ± λ L + g (cid:0) J ± − C tr (cid:0) T J ± (cid:1) T ∓ KJ ± ∓ A (cid:2) J ± , T (cid:3)(cid:1) g − = 11 ± λ L + g (cid:2) T + (1 ∓ K ∓ iA ) J −± + T − (1 ∓ K ± iA ) J + ± + T (1 + C ∓ K ) J ± (cid:3) g − . Then a gauge-transformation of the left Lax pair is given by h L L + ± ( x ; λ L + ) i g (5.20) ≡ g − L L + ± ( x ; λ L + ) g − g − ∂ ± g = − J ± + 11 ± λ L + (cid:2) T + (1 ∓ K ∓ iA ) J −± + T − (1 ∓ K ± iA ) J + ± + T (1 + C ∓ K ) J ± (cid:3) = − (cid:20) λ L + + K + iAλ L + ± T + J −± + λ L + + K − iAλ L + ± T − J + ± + λ L + + K ∓ Cλ L + ± T J ± (cid:21) . h L L + ± ( x ; λ L + ) i g (5.21)= − sinh ( α ± β )sinh (cid:2) α ± (cid:0) β + λ R + (cid:1)(cid:3) " e + λ R + T + J −± + e − λ R − T − J + ± + cosh (cid:0) α ± λ R + (cid:1) cosh α T J ± = L R + ± ( x ; λ R + ) . Similarly, with the spectral-parameter relation λ L − + KiA = 1tanh λ R − , (5.22)we can show the gauge equivalence between the other left Lax pair L L − µ ( x ; λ L − ) and theother reduced right Lax pair L Rµ ( x ; λ R − ) : h L L − ± ( x ; λ L − ) i g = L R − ± ( x ; λ R − ) . (5.23) Summary
Let us summarize the results obtained so far. The two reduced right Lax pairs L R ± µ ( x ; λ R ± )have been introduced. Then they are obtained from the right Lax pair through the sl (2)isomorphisms, T a → e ∓ iλ R T T a e ± iλ R T . (5.24)The reduced right Lax pairs L R ± µ ( x ; λ R ± ) are related to the left Lax pairs L L ± µ ( x ; λ L ± ) throughthe gauge transformation generated by g , h L L ± µ ( x ; λ L ± ) i g = L R ± µ ( x ; λ R ± ) , (5.25)under the relations between the spectral parameters λ L ± + KiA = 1tanh λ R ± . (5.26)So far we have shown that U L ± ( λ L ± ) are gauge-equivalent to U R ± ( λ R ± ) and hence thecharges for the deformed U q ( [ su (2) R ) can also be obtained from U L ± ( λ L ± ) . One can readoff the expansion points from the relation (5.26) and the fact that the expansion pointsfor U R ± ( λ R ± ) are z R ± = 0 and z R ± = ∞ . Thus the charges for the deformed U q ( [ su (2) R )are obtained by expanding U L ± ( λ L ± ) with respect to λ L ± around λ L ± = − K − iA and λ L ± = − K + iA . The locations of poles and the expansion points for the deformed U q ( [ su (2) R )and Y ( su (2) L ) are summarized in Tab. 1 . 29harges \ Monodromies U R ± ( λ R ± ) U L ± ( λ L ± ) Q R, , Q R, +(1) , e Q R, − (1) − K − iAQ R, , Q R, − (1) , e Q R, +(1) ∞ − K + iAQ L,a (0) , Q
L,a (1) ∞ local charges e ± α − β ± The conserved charges and the expansion points of monodromy matrices are listed. The expansionpoints of U R ± ( λ R ± ) are described in terms of z R ± . The gauge transformation of the r/s -matrices
The r / s -matrices for the reduced right Lax pairs can be obtained from the r / s -matrices forthe left Lax pairs. Recall the gauge-transformation laws of the r / s -matrices,( r + s ) R ± ( λ R ± , µ R ± ) δ ( x − y ) (5.27)= g − ( x ) ⊗ g − ( y ) × " ( r + s ) L ± ( λ L ± , µ L ± ) δ ( x − y ) − n L L ± x ( x ; λ L ± ) ⊗ , g ( y ) o P ⊗ g − ( y ) × g ( x ) ⊗ g ( y ) , ( r − s ) R ± ( λ R ± , µ R ± ) δ ( x − y ) (5.28)= g − ( x ) ⊗ g − ( y ) × " ( r − s ) L ± ( λ L ± , µ L ± ) δ ( x − y ) − n g ( x ) ⊗ , L L ± x ( y ; µ L ± ) o P ⊗ g − ( y ) × g ( x ) ⊗ g ( y ) . Thus the Poisson brackets n L L ± x ( x ; λ L ) ⊗ , g ( y ) o P ⊗ g − ( y ) (5.29)= − − λ L (cid:26)(cid:18) − λ L − K C (cid:19) (cid:0) T − ⊗ T + + T + ⊗ T − + T ⊗ T (cid:1) ± A (cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T , ⊗ gT g − ( y ) (cid:3) + CK C (cid:2)(cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T , ⊗ gT g − ( y ) (cid:3) , ⊗ gT g − ( y ) (cid:3)(cid:27) × δ ( x − y ) , n g ( x ) ⊗ , L L ± x ( y ; µ L ) o P g − ( x ) ⊗ − µ L (cid:26)(cid:18) − µ L − K C (cid:19) (cid:0) T − ⊗ T + + T + ⊗ T − + T ⊗ T (cid:1) ± A (cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T , gT g − ( x ) ⊗ (cid:3) + CK C (cid:2)(cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T , gT g − ( x ) ⊗ (cid:3) , gT g − ( x ) ⊗ (cid:3)(cid:27) × δ ( x − y ) , are needed to compute the r / s -matrices for the reduced right Lax pair.With the explicit expression of the r / s -matrices for the left Lax pairs (3.13) , the relation(5.29) and the maps between spectral parameters (5.26) , we can evaluate the right handside of (5.27) as ( r + s ) R ± ( λ R ± , µ R ± ) δ ( x − y ) (5.31)= 1sinh( λ R ± − µ R ± ) sinh 2 α sinh λ R ± α + β + λ R ± ) sinh( α − β − λ R ± ) × ((cid:2) cosh( λ R ± − µ R ± ) − tanh β sinh( λ R ± − µ R ± ) (cid:3) T ⊗ T +e ± ( λ R ± − µ R ± ) T − ⊗ T + + e ∓ ( λ R ± − µ R ± ) T + ⊗ T − ) δ ( x − y ) . Similarly, the following can be obtained( r − s ) R ± ( λ R ± , µ R ± ) δ ( x − y ) (5.32)= 1sinh( λ R ± − µ R ± ) sinh 2 α sinh µ R ± α + β + µ R ± ) sinh( α − β − µ R ± ) × ((cid:2) cosh( λ R ± − µ R ± ) + tanh β sinh( λ R ± − µ R ± ) (cid:3) T ⊗ T +e ± ( λ R ± − µ R ± ) T − ⊗ T + + e ∓ ( λ R ± − µ R ± ) T + ⊗ T − ) δ ( x − y ) . Thus the resulting r / s -matrices for the reduced right Lax pairs are r R ± ( λ R ± , µ R ± ) = h Rα,β ( λ R ± ) + h Rα,β ( µ R ± )2 sinh( λ R ± − µ R ± ) × h e ± ( λ R ± − µ R ± ) T − ⊗ T + + e ∓ ( λ R ± − µ R ± ) T + ⊗ T − + cosh( λ R ± − µ R ± ) T ⊗ T (cid:3) − (cid:0) h Rα,β ( λ R ± ) − h Rα,β ( µ R ± ) (cid:1) βT ⊗ T , R ± ( λ R ± , µ R ± ) = h Rα,β ( λ R ± ) − h Rα,β ( µ R ± )2 sinh( λ R ± − µ R ± ) × h e ± ( λ R ± − µ R ± ) T − ⊗ T + + e ∓ ( λ R ± − µ R ± ) T + ⊗ T − + cosh( λ R ± − µ R ± ) T ⊗ T (cid:3) − (cid:0) h Rα,β ( λ R ± ) + h Rα,β ( µ R ± ) (cid:1) βT ⊗ T . The r / s -matrices for the right Lax pair (4.38) are obtained from these r / s -matrices for thereduced right Lax pairs by taking the isomorphism of sl (2) (5.24) into account. So far, it has been shown that U (1) R is enhanced to q -deformed su (2) R . Then let us arguethe q → q -deformed su (2) R degenerates to the original su (2) R .In fact, there are two kinds of the q → the degeneratelimits . Since the q -parameter is written as q = e γ , the limits are specified by the condition, γ = 0 . From (4.19) , this condition is equivalent to α = 0 or α = πi . (6.1)In the following, we will argue each of the limits. α = 0 We begin with the α → C, K )and the parameters ( α, β ) , (4.5) . Due to the relation and the finiteness of the originalparameters, the condition α = 0 requires the condition β = 0 .Let us rescale α and β as α → ǫ α , β → ǫ β , (6.2)and take the ǫ → K = βα , C = 0 , A = 0 . (6.3)Thus this limit reproduces the undeformed SU (2) WZNW models.32ext we consider the limit for the right Lax pair. Let us perform the redefinition ofspectral parameter, λ R → ǫαλ R − ǫβ , (6.4)and take the ǫ → L R ± ( x ; λ R ) = − ± K ± λ R (cid:2) T + J −± + T − J + ± + T J ± (cid:3) . (6.5)The same procedure gives rise to the r / s -matrices, r R ( λ R , µ R ) = h R ( λ R ) + h R ( µ R )2( λ R − µ R ) (cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T (cid:3) , (6.6) s R ( λ R , µ R ) = h R ( λ R ) − h R ( µ R )2( λ R − µ R ) (cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T (cid:3) . Here the function h R ( λ ) is given by h R ( λ ) = ( λ − K ) − λ . (6.7)Since γ ± = 0 and A = 0 , two sets of non-local currents j R, ± µ and e j R, ± µ degenerate into j R, ± µ = e j R, ± µ = − (cid:0) J ± µ + Kǫ µν J ± ,ν (cid:1) ≡ J R, ± µ . (6.8)For later convenience, let us introduce the following matrix valued conserved current: J Rµ = T + J R, − µ + T − J R, + µ + T J R, µ , (6.9)where the T component J R, µ is defined as J R, µ ≡ j R, µ = − (cid:0) J µ + Kǫ µν J ,ν (cid:1) . (6.10)It is an easy task to show that this conserved current satisfies the flatness condition: ǫ µν (cid:0) ∂ µ J Rν − J Rµ J Rν (cid:1) = 0 , (6.11)and the right Lax pair can be written in terms of the conserved current J Rµ as L R ± ( x ; λ R ) = J R ± ± λ R . (6.12)33hen it is possible to construct an infinite number of conserved charges from J Rµ , byfollowing the BIZZ construction [8]: Y R,a (0) = Z ∞−∞ dx J R,at ( x ) , (6.13) Y R,a (1) = 14 Z ∞−∞ dx Z ∞−∞ dy ε abc ǫ ( x − y ) J R.bt ( x ) J R.ct ( y ) − Z ∞−∞ dx J R.ax ( x ) , ...The current algebra for J R,aµ is given by n J R,at ( x ) , J R,bt ( y ) o P = ε abc J R,ct ( x ) + 2 Kδ ab ∂ x δ ( x − y ) , (6.14) n J R,at ( x ) , J R,bx ( y ) o P = ε abc J R,cx ( x ) + (cid:0) K (cid:1) δ ab ∂ x δ ( x − y ) , n J R,ax ( x ) , J R,bx ( y ) o P = − K ε abc J R,ct ( x ) + 2 Kε abc J R,cx ( x ) δ ( x − y )+2 Kδ ab ∂ x δ ( x − y ) . With this current algebra, one can show that the algebra formed by a set of Y R ( n ) is Yangian Y ( su (2) L ) with an appropriate regularization of non-ultra local terms. α = πi/ The other degenerate limit is next considered. Again, due to the relation (4.5) and thefiniteness of (
C, K ) , the condition α = πi/ β = πi/ α and β as α → πi ǫ α , β → πi ǫ β . (6.15)The ǫ → K = βα , C = β α − , A = 0 . (6.16)This limit describes the points specified by C = K − α = 0 ,the coefficient of the improvement term vanishes, A = 0 . At the points, the effect ofthe squashing parameter C is canceled by the Wess-Zumino term. As a result, the rightdescription becomes isotropic even though the metric of target space is deformed.To see the right Lax pair at the points, the spectral parameter should be redefined as λ R → ǫ α λ R − ǫ β . (6.17)34hen the ǫ → L R ± ( x ; λ R ) = − ± K ± λ R (cid:2) T + J −± + T − J + ± + (1 ∓ K ± λ R ) T J ± (cid:3) . (6.18)Similarly, the r / s -matrices are obtained as r R ( λ R , µ R ) = h Rπi/ ( λ R ) + h Rπi/ ( µ R )2( λ R − µ R ) (cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T (cid:3) (6.19) − h Rπi/ ( λ R ) − h Rπi/ ( µ R )2 K T ⊗ T ,s R ( λ R , µ R ) = h Rπi/ ( λ R ) − h Rπi/ ( µ R )2( λ R − µ R ) (cid:2) T − ⊗ T + + T + ⊗ T − + T ⊗ T (cid:3) − h Rπi/ ( λ R ) + h Rπi/ ( µ R )2 K T ⊗ T , where the function h Rπi/ ( λ ) is defined as h Rπi/ ( λ ) ≡ ( λ − K ) − λ . (6.20)In this limit, the following relations are satisfied, γ ± = − γ ∓ = ± i/K , A = 0 . Then e j R, ± µ coincide with j R, ± µ as j R, ± µ = e j R, ± µ = − e ± iχ/K (cid:0) J ± + Kǫ µν J ± ,ν (cid:1) ≡ J R, ± µ . (6.21)Note that J R, ± µ remain non-local currents, in contrast to the α = 0 case. It is useful todefine a matrix valued conserved current as J Rµ = T + J R, − µ + T − J R, + µ + T J R, µ , (6.22)where the T component J R, µ is defined as J R, µ ≡ − j R, µ = (cid:0) K J µ + Kǫ µν J ,ν (cid:1) . (6.23)Here the sign of j R, µ is flipped for later convenience. Although J Rµ is non-local, it satisfiesthe flatness condition, ǫ µν (cid:0) ∂ µ J Rν − J Rµ J Rν (cid:1) = 0 . (6.24)35hus an infinite number of conserved charges can be constructed from J Rµ like Y R,a (0) = Z ∞−∞ dx J R,at ( x ) , (6.25) Y R,a (1) = 14 Z ∞−∞ dx Z ∞−∞ dy ε abc ǫ ( x − y ) J R.bt ( x ) J R.ct ( y ) − Z ∞−∞ dx J R.ax ( x ) , ...Again, the higher-level charges Y R ( n ≥ are obtained recursively by the BIZZ procedure [8].In addition, another Lax pair can be obtained from the flat current J Rµ as L R ± ( λ R ) = J R ± ± λ R , (6.26)where we use the same spectral parameter λ R , for later convenience. This Lax pair isnon-local and hence it is different from the Lax pair (6.18) . However there exists a gaugetransformation which connect the non-local Lax pair (6.26) to the right Lax pair (6.18) .Let us find an appropriate gauge function k . An important observation is that, at λ R = ∞ ,the right Lax pair (6.18) does not vanish but approaches to a finite quantity as follows: L R ± ( λ R = ∞ ) = − T (1 ± K ) J ± = − T ∂ ± h χK i . (6.27)The gauge transformation must be chosen to eliminate this finite part because the non-localLax pair (6.26) vanishes at λ R = ∞ . Thus the gauge transformation is generated by F ( x ) = e − T χ ( x ) K . (6.28)Note that a constant SU (2) element may be multiplied from the right and thus the function F ( x ) is not determined uniquely. However, the choice in (6.28) is appropriate to relate theright Lax pair (6.18) to the non-local Lax pair (6.26) , as we will see just below.By performing a non-local gauge transformation generated by F ( x ) in (6.28) , the rightLax pair (6.18) can be rewritten as L R ± ( λ R ) → (cid:2) L R ± ( λ R ) (cid:3) F = F − L R ± ( λ R ) F − F − ∂ ± F (6.29)= J R ± ± λ R = L R ± ( λ R ) . k The strategy is similar to the one to consider the Jordanian twists in the Schr¨odinger sigma models [29]. U R ( λ R ) = (cid:2) U R ( λ R ) (cid:3) F = F − ( x = ∞ ) U R ( λ R ) F ( x = −∞ ) (6.30)= e − T QR, K U R ( λ R )e − T QR, K . This relation may be interpreted as a classical analogue of the Reshetikhin twist [39]. It is in-teresting to reveal the relation to the quantum twist from the viewpoint of the mathematicalformulation.In addition, it is worth to note that the flat and conserved current J Rµ can be written as J Rµ = − (cid:0) G − ∂ µ G + Kǫ µν G − ∂ ν G (cid:1) , G = gF . (6.31)Then the current algebra of J R,aµ is given by n J R,at ( x ) , J R,bt ( y ) o P = ε abc J R,ct ( x ) + 2 Kδ ab ∂ x δ ( x − y ) , (6.32) n J R,at ( x ) , J R,bx ( y ) o P = ε abc J R,cx ( x ) + (cid:0) K (cid:1) δ ab ∂ x δ ( x − y ) , n J R,ax ( x ) , J R,bx ( y ) o P = − K ε abc J R,ct ( x ) + 2 Kε abc J R,cx ( x ) δ ( x − y )+2 Kδ ab ∂ x δ ( x − y ) . Again, the infinite-dimensional algebra of Y R ( n ) is Y ( su (2) R ) . We have considered the classical integrable structure of the squashed WZNW models basedon an infinite-dimensional extension of U (1) R . The system contains two deformation pa-rameters. The one is provided by the coefficient of the Wess-Zumino term. The other is thesquashing parameter of target space. We have constructed an anisotropic Lax pair and com-puted the associated r / s -matrices, which may be regarded as one-parameter deformationsof the results without the Wess-Zumino term [25, 26].Due to the presence of the Wess-Zumino term, an infinite-dimensional extension of su (2) R is given by a deformation of the standard quantum affine algebra, which contain two param-eters. The deformed algebra contains two sets of q -deformed su (2) R and the tower structureof the conserved charges is quite similar to that of the standard quantum affine algebra.The only modification appears in the coefficient of the Poisson bracket. It also changes the37lassical q -Serre relations. The resulting algebra seems likely to be the classical analogue ofa two-parameter quantum toroidal algebra [34]. However, it is not sure so far for this pointbecause it seems that the first realization of the algebra has not been constructed yet.The left-right duality has also been revealed by showing that the left Lax pairs are gauge-equivalent to the right Lax pair with the relation between the spectral parameters and theisomorphism of the sl (2) algebra. This is a generalization of the duality in the case withoutthe Wess-Zumino term. The right Lax pair can also be decomposed to a pair of the reducedright Lax pairs, each of which is equivalent to the corresponding left Lax pair.In addition, two degenerate limits have been considered. They are realized at the points C = 0 (corresponding to α = β = 0 ) and the points C = K − α = β = πi/ SU (2) R conserved current sat-isfying the flatness condition. With this current, another Lax pair can be constructed. Forthe later case, the construction is more involved and a non-local gauge-transformation isneeded. In addition, Yangian generators have also been constructed.There are some open issues. It would be interesting to study the fast-moving limit[40] of the squashed WZNW models (For squashed S and warped AdS see [41] and [42],respectively). Some applications of the results may be considered in the context of stringtheory. In this direction, for a deformation of AdS /CFT , see an earlier paper [43]. Forwarped AdS and squashed S geometries in string theory, for example, see [44–48].We hope that some applications of the deformed quantum affine algebra would be foundand be a key ingredient in studying the integrability of the AdS/CFT correspondence. Acknowledgment
We would like to thank Takuya Matsumoto for useful discussions. The work of IK wassupported by the Japan Society for the Promotion of Science (JSPS) .38 ppendixA The Poisson brackets of J aµ and j L ± µ The Poisson brackets of J aµ play an important role in computing the current algebra of j L ± µ .The computation is straightforward but very messy. In order to use the canonical Poissonbrackets of the dynamical variables, the computation is described in terms of angle variables.Then, by using the Poisson brackets, the current algebra of j L ± µ is computed. A.1 The Poisson brackets of J aµ The classical action (2.16) is composed of the two parts S = S σM + S W Z . Each part isexpressed in terms of the angle variables ( θ, φ, ψ ) as follows: S σM = 12 λ Z ∞−∞ dt Z ∞−∞ dx (cid:20) ˙ θ + sin θ ˙ φ + (1 + C ) (cid:16) ˙ ψ + cos θ ˙ φ (cid:17) − θ ′ − sin θφ ′ − (1 + C ) ( ψ ′ + cos θφ ′ ) i ,S W Z = − n π Z ∞−∞ dt Z ∞−∞ dx (cid:16) cos θ ˙ φψ ′ − cos θ ˙ ψφ ′ (cid:17) . (A.1)The symbols “dot” and “prime” denote the derivatives with respect to t and x , respectively.The conjugate momenta (Π θ , Π φ , Π ψ ) for ( θ, φ, ψ ) are given byΠ θ = ˙ θ , Π φ = sin θ ˙ φ + (1 + C ) cos θ (cid:16) ˙ ψ + cos θ ˙ φ (cid:17) − K cos θψ ′ , Π ψ = (1 + C ) (cid:16) ˙ ψ + cos θ ˙ φ (cid:17) + K cos θφ ′ . (A.2)The “velocity” variables are˙ θ = b Π θ , ˙ φ = 1sin θ b Π φ − cos θ sin θ b Π ψ , ˙ ψ = (cid:18)
11 + C − cos θ sin θ (cid:19) b Π ψ − cos θ sin θ b Π φ , (A.3)where ( b Π θ , b Π φ , b Π ψ ) are defined as b Π θ ≡ Π θ , b Π φ ≡ Π φ + K cos θψ ′ , b Π ψ ≡ Π ψ − K cos θφ ′ . (A.4)39 φ ψ b Π θ b Π φ b Π ψ θ φ ψ b Π θ − K sin θψ ′ − K sin θφ ′ b Π φ − − K sin θψ ′ K sin θθ ′ b Π ψ − K sin θφ ′ − K sin θθ ′ J are expressed as J ± t = 1 √ ∓ iψ (cid:18) ± i b Π θ − θ b Π φ + cos θ sin θ b Π ψ (cid:19) ,J t = 11 + C b Π ψ ,J ± x = 1 √ ∓ iψ ( ± iθ ′ − sin θφ ′ ) ,J x = ψ ′ + cos θφ ′ . (A.5)By using the Poisson brackets in Tab. 2 and the expressions given in (A.5), the Poissonbrackets of J aµ are computed like (cid:8) J ± t ( x ) , J ∓ t ( y ) (cid:9) P = ∓ i (cid:2) − (1 + C ) J t + KJ x (cid:3) ( x ) δ ( x − y ) , (A.6) (cid:8) J ± t ( x ) , J t ( y ) (cid:9) P = ± i (cid:20) −
11 +
C J ± t + K C J ± x (cid:21) ( x ) δ ( x − y ) , (cid:8) J ± t ( x ) , J ∓ x ( y ) (cid:9) P = ∓ i (cid:2) − J x (cid:3) ( x ) δ ( x − y ) + ∂ x δ ( x − y ) , (cid:8) J ± t ( x ) , J x ( y ) (cid:9) P = ± i (cid:2) − J ± x (cid:3) ( x ) δ ( x − y ) , (cid:8) J t ( x ) , J ± x ( y ) (cid:9) P = ∓ i (cid:20) −
11 +
C J ± x (cid:21) ( x ) δ ( x − y ) , (cid:8) J t ( x ) , J x ( y ) (cid:9) P = 11 + C ∂ x δ ( x − y ) , (cid:8) J ± x ( x ) , J ∓ t ( y ) (cid:9) P = ∓ i (cid:2) − J x (cid:3) ( x ) δ ( x − y ) + ∂ x δ ( x − y ) , (cid:8) J ± x ( x ) , J t ( y ) (cid:9) P = ± i (cid:20) −
11 +
C J ± x (cid:21) ( x ) δ ( x − y ) , (cid:8) J x ( x ) , J ± t ( y ) (cid:9) P = ∓ i (cid:2) − J ± x (cid:3) ( x ) δ ( x − y ) , J x ( x ) , J t ( y ) (cid:9) P = 11 + C ∂ x δ ( x − y ) . (A.7)Similarly, the Poisson brackets between g and J aµ are evaluated as (cid:8) g ( x ) , J ± t ( y ) (cid:9) P = g ( x ) T ± δ ( x − y ) , (cid:8) g ( x ) , J t ( y ) (cid:9) P = 11 + C g ( x ) T δ ( x − y ) . (A.8) A.2 The current algebra of j L ± µ To compute the current algebra of j L ± µ , it is convenient to rewrite j L ± µ in terms of J as j L ± µ = g (cid:8) J µ − C tr (cid:0) T J µ (cid:1) T − Kǫ µν J ν ∓ Aǫ µν (cid:2) J ν , T (cid:3)(cid:9) g − = g (cid:8) T + (cid:2) J − µ − ( K ± iA ) ǫ µν J − ,ν (cid:3) + T − (cid:2) J + µ − ( K ∓ iA ) ǫ µν J + ,ν (cid:3) + T (cid:2) (1 + C ) J µ − Kǫ µν J ,ν (cid:3)(cid:9) g − . (A.9)With the relations in (A.7) and (A.8) , the current algebra is computed as n j L ± ,at ( x ) , j L ± ,bt ( y ) o P = ε abc j L ± ,ct ( x ) δ ( x − y ) − Kδ ab ∂ x δ ( x − y ) , n j L ± ,at ( x ) , j L ± ,bx ( y ) o P = ε abc j L ± ,cx ( x ) δ ( x − y ) + (cid:0) K + A (cid:1) δ ab ∂ x δ ( x − y ) , n j L ± ,ax ( x ) , j L ± ,bx ( y ) o P = − (cid:0) K + A (cid:1) ε abc j L ± ,ct ( x ) δ ( x − y ) − Kε abc j L ± ,cx ( x ) δ ( x − y ) − Kδ ab ∂ x δ ( x − y ) . In addition, the Poisson brackets between g and j L + µ are also obtained as n g ( x ) , j L + ,at ( y ) o P = T a g ( x ) δ ( x − y ) , (A.10) n g ( x ) , j L + ,ax ( y ) o P = (cid:26) − K C T a + A (cid:2) T a , gT g − (cid:3) + CK C (cid:2)(cid:2) T a , gT g − (cid:3) , gT g − (cid:3)(cid:27) g ( x ) δ ( x − y ) . B A prescription to treat non-ultra local terms
We present the computations of 1) Yangian, 2) q -deformed su (2) R , and 3) a deformedquantum affine algebra. In particular, a prescription to treat non-ultra local terms is carefullydescribed in each case. 41 .1 Yangian Y ( su (2) L ) Let us first compute Yangian Y ( su (2) L ) . Non-ultra local terms appear in the computationsof the Poisson brackets of Q L ± ( n ) and hence we should be careful of the order of limits. Thesubscripts of L , ( ± ) are omitted for simplicity henceforth. The Poisson brackets at level 0
The first is the Poisson brackets of the level 0 charges. The charges are regularized as Q L,a (0) ( X , X ) = Z X − X dx j L,at ( x ) . (B.1)Then the Poisson brackets of the regularized charges are given by n Q L,a (0) ( X , X ) , Q L,b (0) ( Y , Y ) o P = Z X − X dx Z Y − Y dy h − Kδ ab ∂ x δ ( x − y ) + ε abc j L,ct ( x ) δ ( x − y ) i . The first term contains a derivative of the delta function, called a non-ultra local term. Thisterm develops an ambiguity depending on the order of limits as follows: n Q L,a (0) ( X , X ) , Q L,b (0) ( Y , Y ) o P (B.2)= − Kδ ab [ θ ( X − Y ) − θ ( X − Y )] + ε abc Q L,c (0) (min( X , Y ) , min( X , Y )) . In the following, we will not write down unambiguous terms explicitly as follows: n Q L,a (0) ( X , X ) , Q L,b (0) ( Y , Y ) o P (B.3)= − Kδ ab [ θ ( X − Y ) − θ ( X − Y )] + (no ambiguity) . Note that this ambiguity arises due to the presence of the Wess-Zumino term. When K = 0 ,there is no ambiguity at the level 0 as usual.Now, one may follow a prescription proposed in [10] , X = X ≡ X , Y = Y ≡ Y , (B.4)so as to make the ambiguous term vanish. Then, by taking the limits X → ∞ and Y → ∞ ,the following is obtained, n Q L,a (0) , Q
L,b (0) o P = ε abc Q L,c (0) . (B.5)42 he Poisson brackets at level 1 The next is to regularize the level 1 charges. The level 1 charges are regularized as Q L,a (1) ( X, X ′ , X ′′ ) = 14 Z X − X dx Z X ′ − X ′ dx ′ ǫ ( x − x ′ ) ε abc j L,bt ( x ) j L,ct ( x ′ ) − Z X ′′ − X ′′ dx ′′ j L,ax ( x ′′ ) . The Poisson brackets of the regularized level 1 and level 0 charges are n Q L,a (1) ( X, X ′ , X ′′ ) , Q L,b (0) ( Y ) o P = 2 K ε abc h θ ( X − Y ) Q L,c (0) (min( X ′ , Y )) + θ ( X ′ − Y ) Q L,c (0) (min(
X, Y )) i +(no ambiguity) . (B.6)Now the unambiguous terms are given by ε abc Q L,c (1) , (B.7)and hence the ambiguous terms have to vanish. Thus the Y → ∞ limit should be takenbefore the X, X ′ , X ′′ → ∞ limits. As a result, the following is obtained, n Q L,a (1) , Q
L,b (0) o P = ε abc Q L,a (1) . (B.8) The Poisson brackets at level 2
The Poisson brackets of the regularized level 1 charges are computed here. Those give riseto the brackets at level 2, n Q L,a (1) ( X, X ′ , X ′′ ) , Q L,b (1) ( Y, Y ′ , Y ′′ ) o P (B.9)= − K θ ( X − Y ) Z X ′ − X ′ dx ′ Z Y ′ − Y ′ dy ′ × [ ǫ ( x ′ + Y ) ǫ ( y ′ + Y ) − ǫ ( x ′ − Y ) ǫ ( y ′ − Y )] × h δ ab δ cd j L,ct ( x ′ ) j L,dt ( y ′ ) − j L,bt ( x ′ ) j L,at ( y ′ ) i − K θ ( X ′ − Y ) Z X − X dx Z Y ′ − Y ′ dy ′ × [ ǫ ( x + Y ) ǫ ( y ′ + Y ) − ǫ ( x − Y ) ǫ ( y ′ − Y )] × h δ ab δ cd j L,ct ( x ) j L,dt ( y ′ ) − j L,bt ( x ) j L,at ( y ′ ) i − K θ ( X − Y ′ ) Z X ′ − X ′ dx ′ Z Y − Y dy [ ǫ ( x ′ + Y ′ ) ǫ ( y + Y ′ ) − ǫ ( x ′ − Y ′ ) ǫ ( y − Y ′ )] × h δ ab δ cd j L,ct ( x ′ ) j L,dt ( y ) − j L,bt ( x ′ ) j L,at ( y ) i − K θ ( X ′ − Y ′ ) Z X − X dx Z Y − Y dy × [ ǫ ( x + Y ′ ) ǫ ( y + Y ′ ) − ǫ ( x − Y ′ ) ǫ ( y − Y ′ )] × h δ ab δ cd j L,ct ( x ) j L,dt ( y ) − j L,bt ( x ) j L,at ( y ) i + (1 + C ) + K C ) θ ( X − Y ′′ ) ε abc Z X ′ − X ′ dx ′ [ ǫ ( x ′ + Y ′′ ) − ǫ ( x ′ − Y ′′ )] j L,ct ( x ′ )+ (1 + C ) + K C ) θ ( X ′ − Y ′′ ) ε abc Z X − X dx [ ǫ ( x + Y ′′ ) − ǫ ( x − Y ′′ )] j L,ct ( x )+ (1 + C ) + K C ) θ ( Y − X ′′ ) ε abc Z Y ′ − Y ′ dy ′ [ ǫ ( y ′ + X ′′ ) − ǫ ( y ′ − X ′′ )] j L,ct ( y ′ )+ (1 + C ) + K C ) θ ( Y ′ − X ′′ ) ε abc Z Y − Y dy [ ǫ ( y + X ′′ ) − ǫ ( y − X ′′ )] j L,ct ( y )+(no ambiguity) . By taking the
Y, Y ′ , Y ′′′ → ∞ limits before sending X, X ′ , X ′′ to infinity, the followingexpression is obtained, n Q L,a (1) ( X, X ′ , X ′′ ) , Q L,b (1) o P (B.10)= min Y,Y ′ ,Y ′′ →∞ n Q L,a (1) ( X, X ′ , X ′′ ) , Q L,b (1) ( Y, Y ′ , Y ′′ ) o P = (1 + C ) + K C ) ε abc Z ∞−∞ dy [ ǫ ( y + X ′′ ) − ǫ ( y − X ′′ )] j L,ct ( y ) + (no ambiguity) . Then the ambiguous terms are given by(1 + C ) + K C ε abc Q L,c (0) , (B.11)by taking the X, X ′ , X ′′ → ∞ limits.On the other hand, since the unambiguous terms are represented by ε abc (cid:20) Q L,c (2) + 112 (cid:0) Q L (0) (cid:1) Q L,c (0) + 2 KQ L,c (1) − (1 + C ) + K C Q
L,c (0) (cid:21) , (B.12)the net result is n Q L,a (1) , Q
L,b (1) o P = ε abc (cid:20) Q L,c (2) + 112 (cid:0) Q L (0) (cid:1) Q L,c (0) + 2 KQ L,c (1) (cid:21) . (B.13)44 erre relations The Serre-relations are finally considered. One of them is represented by n Q L, , n Q L, +(1) , Q L, − (1) o P o P = 14 Q L, h Q L, +(0) Q L, − (1) − Q L, − (0) Q L, +(1) i . (B.14)Let us first introduce the following quantity: n Q L, +(1) , Q L, − (1) o P ( X, X ′ , X ′′ , X ′′′ , X ′′′′ , X ′′′′′ ) (B.15)= − i Z X − X dx Z X ′ − X ′ dx ′ Z X ′′ − X ′′ dx ′′ ǫ ( x − x ′ ) ǫ ( x − x ′′ ) × h j L, + t ( x ) j L, t ( x ′ ) j L, − t ( x ′′ ) + j L, − t ( x ) j L, t ( x ′ ) j L, + t ( x ′′ )+ j L, t ( x ) j L, t ( x ′ ) j L, t ( x ′′ ) i − Z X ′′′ − X ′′′ dx ′′′ Z X ′′′′ − X ′′′′ dx ′′′′ ǫ ( x ′′′ − x ′′′′ ) × h j L, + t ( x ′′′ ) j L, − x ( x ′′′′ ) − j L, − t ( x ′′′ ) j L, + x ( x ′′′′ ) i − i Z X ′′′′′ − X ′′′′′ dx ′′′′′ j L, t ( x ′′′′′ )+ K Z X ′′′ − X ′′′ dx ′′′ Z X ′′′′ − X ′′′′ dx ′′′′ ǫ ( x ′′′ − x ′′′′ ) j L, + t ( x ′′′ ) j L, − t ( x ′′′′ )+2 iK Z X ′′′′′ − X ′′′′′ dx ′′′′′ j L, x ( x ′′′′′ ) . Note that the regularized Q L, is given by Q L, ( Y, Y ′ , Y ′′ ) = i Z Y − Y dy Z Y ′ − Y ′ dy ′ ǫ ( y − y ′ ) j L, + t ( y ) j L, − t ( y ′ ) − Z Y ′′ − Y ′′ dy ′′ j L, x ( y ′′ ) . (B.16)By using the quantities in (B.15) and (B.16) , the following relation is obtained, n Q L, ( Y, Y ′ , Y ′′ ) , n Q L, +(1) , Q L, − (1) o P ( X, X ′ , X ′′ , X ′′′ , X ′′′′ , X ′′′′′ ) o P (B.17)= K Z min( X,X ′′ ,Y ) − min( X,X ′′ ,Y ) dx Z X ′ − X ′ dx ′ Z Y ′ − Y ′ dy ′ ǫ ( x − x ′ ) ǫ ( x − y ′ ) j L, + t ( x ) j L, t ( x ′ ) j L, − t ( y ′ ) − K Z min( X,X ′′ ,Y ′ ) − min( X,X ′′ ,Y ′ ) dx Z X ′ − X ′ dx ′ Z Y − Y dy ǫ ( x − x ′ ) ǫ ( x − y ) j L, − t ( x ) j L, t ( x ′ ) j L, + t ( y )+ K Z min( X,X ′ ,Y ′ ) − min( X,X ′ ,Y ′ ) dx ′ Z X ′′ − X ′′ dx ′′ Z Y − Y dy ǫ ( x ′ − y ) ǫ ( x ′ − x ′′ ) j L, t ( x ′ ) j L, − t ( x ′′ ) j L, + t ( y )+ K Z X ′ − X ′ dx ′ Z min( X,X ′′ ,Y ′ ) − min( X,X ′′ ,Y ′ ) dx ′′ Z Y − Y dy ǫ ( x ′′ − y ) ǫ ( x ′′ − x ′ ) j L, − t ( x ′′ ) j L, t ( x ′ ) j L, + t ( y )45 K Z min( X,X ′ ,Y ) − min( X,X ′ ,Y ) dx ′ Z X ′′ − X ′′ dx ′′ Z Y ′ − Y ′ dy ′ ǫ ( x ′ − y ′ ) ǫ ( x ′ − x ′′ ) j L, t ( x ′ ) j L, + t ( x ′′ ) j L, − t ( y ′ ) − K Z X ′ − X ′ dx ′ Z min( X,X ′′ ,Y ) − min( X,X ′′ ,Y ) dx ′′ Z Y ′ − Y ′ dy ′ ǫ ( x ′′ − y ′ ) ǫ ( x ′′ − x ′ ) j L, + t ( x ′′ ) j L, t ( x ′ ) j L, − t ( y ′ )+ K θ ( Y − X ′′ ) Z X − X dx Z X ′ − X ′ dx ′ Z Y ′ − Y ′ dy ′ ǫ ( x − x ′ ) j L, + t ( x ) j L, t ( x ′ ) j L, − t ( y ′ ) × [ ǫ ( y ′ + X ′′ ) ǫ ( x + X ′′ ) − ǫ ( y ′ − X ′′ ) ǫ ( x − X ′′ )] − K θ ( Y ′ − X ′′ ) Z X − X dx Z X ′ − X ′ dx ′ Z Y − Y dy ǫ ( x − x ′ ) j L, − t ( x ) j L, t ( x ′ ) j L, + t ( y ) × [ ǫ ( y + X ′′ ) ǫ ( x + X ′′ ) − ǫ ( y − X ′′ ) ǫ ( x − X ′′ )] − K θ ( Y ′ − X ) Z X ′ − X ′ dx ′ Z X ′′ − X ′′ dx ′′ Z Y − Y dy j L, t ( x ′ ) j L, − t ( x ′′ ) j L, + t ( y ) × [ ǫ ( y + X ) ǫ ( x ′ + X ) ǫ ( x ′′ + X ) − ǫ ( y − X ) ǫ ( x ′ − X ) ǫ ( x ′′ − X )] − K θ ( Y − X ) Z X ′ − X ′ dx ′ Z X ′′ − X ′′ dx ′′ Z Y ′ − Y ′ dy ′ j L, t ( x ′ ) j L, + t ( x ′′ ) j L, − t ( y ′ ) × [ ǫ ( y ′ + X ) ǫ ( x ′ + X ) ǫ ( x ′′ + X ) − ǫ ( y ′ − X ) ǫ ( x ′ − X ) ǫ ( x ′′ − X )]+ i C ) + K C Z min( X ′′′ ,X ′′′′ ,Y ) − min( X ′′′ ,X ′′′′ ,Y ) dx ′′′ Z Y ′ − Y ′ dy ′ ǫ ( x ′′′ − y ′ ) j L, + t ( x ′′′ ) j L, − t ( y ′ ) − i C ) + K C Z min( X ′′′ ,X ′′′′ ,Y ′ ) − min( X ′′′ ,X ′′′′ ,Y ′ ) dx ′′′ Z Y − Y dy ǫ ( x ′′′ − y ) j L, + t ( y ) j L, − t ( x ′′′ )+ iK Z min( X ′′′ ,X ′′′′ ,Y ′ ) − min( X ′′′ ,X ′′′′ ,Y ′ ) dx ′′′′ Z Y − Y dy ǫ ( y − x ′′′′ ) j L, + t ( y ) j L, − x ( x ′′′′ ) − iK Z min( X ′′′ ,X ′′′′ ,Y ) − min( X ′′′ ,X ′′′′ ,Y ) dx ′′′′ Z Y ′ − Y ′ dy ′ ǫ ( x ′′′′ − y ′ ) j L, + x ( x ′′′′ ) j L, − t ( y ′ ) − i C ) + K C Z min( X,X ′ ,Y ′′ ) − min( X,X ′ ,Y ′′ ) dx Z X ′′ − X ′′ dx ′′ ǫ ( x − x ′′ ) j L, + t ( x ) j L, − t ( x ′′ ) − i C ) + K C Z min( X,X ′ ,Y ′′ ) − min( X,X ′ ,Y ′′ ) dx Z X ′′ − X ′′ dx ′′ ǫ ( x − x ′′ ) j L, − t ( x ) j L, + t ( x ′′ )+ i C ) + K C Z min( X,X ′ ,Y ′′ ) − min( X,X ′ ,Y ′′ ) dx ′ Z X ′′ − X ′′ dx ′′ ǫ ( x ′ − x ′′ ) j L, t ( x ′ ) j L, t ( x ′′ )+ i C ) + K C Z X ′ − X ′ dx ′ Z min( X,X ′′ ,Y ′′ ) − min( X,X ′′ ,Y ′′ ) dx ′′ ǫ ( x ′′ − x ′ ) j L, t ( x ′′ ) j L, t ( x ′ ) − i C ) + K C Z min( X,X ′ ,Y ′′ ) − min( X,X ′ ,Y ′′ ) dx Z X ′′ − X ′′ dx ′′ ǫ ( x − x ′′ ) j L, t ( x ) j L, t ( x ′′ ) − i C ) + K C Z min( X,X ′′ ,Y ′′ ) − min( X,X ′′ ,Y ′′ ) dx Z X ′ − X ′ dx ′ ǫ ( x − x ′ ) j L, t ( x ) j L, t ( x ′ )46 i C ) + K C θ ( Y − X ′′′′ ) Z X ′′′ − X ′′′ dx ′′′ Z Y ′ − Y ′ dy ′ j L, + t ( x ′′′ ) j L, − t ( y ′ ) × [ ǫ ( x ′′′ + X ′′′′ ) ǫ ( y ′ + X ′′′′ ) − ǫ ( x ′′′ − X ′′′′ ) ǫ ( y ′ − X ′′′′ )]+ i C ) + K C θ ( Y ′ − X ′′′′ ) Z X ′′′ − X ′′′ dx ′′′ Z Y − Y dy j L, − t ( x ′′′ ) j L, + t ( y ) × [ ǫ ( x ′′′ + X ′′′′ ) ǫ ( y + X ′′′′ ) − ǫ ( x ′′′ − X ′′′′ ) ǫ ( y − X ′′′′ )] − i Kθ ( Y ′ − X ′′′ ) Z X ′′′′ − X ′′′′ dx ′′′′ Z Y − Y dy j L, − x ( x ′′′′ ) j L, + t ( y ) × [ ǫ ( x ′′′′ + X ′′′ ) ǫ ( y + X ′′′ ) − ǫ ( x ′′′′ − X ′′′ ) ǫ ( y − X ′′′ )] − i Kθ ( Y − X ′′′ ) Z X ′′′′ − X ′′′′ dx ′′′′ Z Y ′ − Y ′ dy ′ j L, + x ( x ′′′′ ) j L, − t ( y ′ ) × [ ǫ ( x ′′′′ + X ′′′ ) ǫ ( y ′ + X ′′′ ) − ǫ ( x ′′′′ − X ′′′ ) ǫ ( y ′ − X ′′′ )]+ i C ) + K C θ ( Y ′′ − X ′ ) Z X − X dx Z X ′′ − X ′′ dx ′′ j L, + t ( x ) j L, − t ( x ′′ ) × ǫ ( x − x ′′ ) [ ǫ ( x + X ′ ) − ǫ ( x − X ′ )]+ i C ) + K C θ ( Y ′′ − X ′ ) Z X − X dx Z X ′′ − X ′′ dx ′′ j L, − t ( x ) j L, + t ( x ′′ ) × ǫ ( x − x ′′ ) [ ǫ ( x + X ′ ) − ǫ ( x − X ′ )]+ i C ) + K C θ ( Y ′′ − X ) Z X ′ − X ′ dx ′ Z X ′′ − X ′′ dx ′′ j L, t ( x ′ ) j L, t ( x ′′ ) × [ ǫ ( x ′ + X ) ǫ ( x ′′ + X ) − ǫ ( x ′ − X ) ǫ ( x ′′ − X )]+ i C ) + K C θ ( Y ′′ − X ′ ) Z X − X dx Z X ′′ − X ′′ dx ′′ j L, t ( x ) j L, t ( x ′′ ) × ǫ ( x − x ′′ ) [ ǫ ( x + X ′ ) − ǫ ( x − X ′ )]+ i C ) + K C θ ( Y ′′ − X ′′ ) Z X − X dx Z X ′ − X ′ dx ′ j L, t ( x ) j L, t ( x ′ ) × ǫ ( x − x ′ ) [ ǫ ( x + X ′′ ) − ǫ ( x − X ′′ )]+(no ambiguity) . Then, by taking the limits in which X , X ′ , X ′′ , X ′′′ , X ′′′′ and X ′′′′′ are sent to infinity, theexpression is simplified as n Q L, ( Y, Y ′ , Y ′′ ) , n Q L, +(1) , Q L, − (1) o P o P (B.18)= K Z Y − Y dx Z ∞−∞ dx ′ Z Y ′ − Y ′ dy ′ ǫ ( x − x ′ ) ǫ ( x − y ′ ) j L, + t ( x ) j L, t ( x ′ ) j L, − t ( y ′ ) − K Z Y ′ − Y ′ dx Z ∞−∞ dx ′ Z Y − Y dy ǫ ( x − x ′ ) ǫ ( x − y ) j L, − t ( x ) j L, t ( x ′ ) j L, + t ( y )47 K Z Y ′ − Y ′ dx ′ Z ∞−∞ dx ′′ Z Y − Y dy ǫ ( x ′ − y ) ǫ ( x ′ − x ′′ ) j L, t ( x ′ ) j L, − t ( x ′′ ) j L, + t ( y )+ K Z ∞−∞ dx ′ Z Y ′ − Y ′ dx ′′ Z Y − Y dy ǫ ( x ′′ − y ) ǫ ( x ′′ − x ′ ) j L, − t ( x ′′ ) j L, t ( x ′ ) j L, + t ( y ) − K Z Y − Y dx ′ Z ∞−∞ dx ′′ Z Y ′ − Y ′ dy ′ ǫ ( x ′ − y ′ ) ǫ ( x ′ − x ′′ ) j L, t ( x ′ ) j L, + t ( x ′′ ) j L, − t ( y ′ ) − K Z ∞−∞ dx ′ Z Y − Y dx ′′ Z Y ′ − Y ′ dy ′ ǫ ( x ′′ − y ′ ) ǫ ( x ′′ − x ′ ) j L, + t ( x ′′ ) j L, t ( x ′ ) j L, − t ( y ′ )+ i C ) + K C Z Y − Y dx ′′′ Z Y ′ − Y ′ dy ′ ǫ ( x ′′′ − y ′ ) j L, + t ( x ′′′ ) j L, − t ( y ′ ) − i C ) + K C Z Y ′ − Y ′ dx ′′′ Z Y − Y dy ǫ ( x ′′′ − y ) j L, + t ( y ) j L, − t ( x ′′′ )+ iK Z Y ′ − Y ′ dx ′′′′ Z Y − Y dy ǫ ( y − x ′′′′ ) j L, + t ( y ) j L, − x ( x ′′′′ ) − iK Z Y − Y dx ′′′′ Z Y ′ − Y ′ dy ′ ǫ ( x ′′′′ − y ′ ) j L, + x ( x ′′′′ ) j L, − t ( y ′ ) − i C ) + K C Z Y ′′ − Y ′′ dx Z ∞−∞ dx ′′ ǫ ( x − x ′′ ) j L, + t ( x ) j L, − t ( x ′′ ) − i C ) + K C Z Y ′′ − Y ′′ dx Z ∞−∞ dx ′′ ǫ ( x − x ′′ ) j L, − t ( x ) j L, + t ( x ′′ )+ i C ) + K C Z Y ′′ − Y ′′ dx ′ Z ∞−∞ dx ′′ ǫ ( x ′ − x ′′ ) j L, t ( x ′ ) j L, t ( x ′′ )+ i C ) + K C Z ∞−∞ dx ′ Z Y ′′ − Y ′′ dx ′′ ǫ ( x ′′ − x ′ ) j L, t ( x ′′ ) j L, t ( x ′ ) − i C ) + K C Z Y ′′ − Y ′′ dx Z ∞−∞ dx ′′ ǫ ( x − x ′′ ) j L, t ( x ) j L, t ( x ′′ ) − i C ) + K C Z Y ′′ − Y ′′ dx Z ∞−∞ dx ′ ǫ ( x − x ′ ) j L, t ( x ) j L, t ( x ′ )+(no ambiguity) . Finally, by sending Y , Y ′ and Y ′′ to infinity, the ambiguous terms are written as n Q L, , n Q L, +(1) , Q L, − (1) o P o P (B.19)= iK Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) h j L, + t ( x ) j L, − x ( y ) − j L, + x ( x ) j L, − t ( y ) i + (no ambiguity) . The unambiguous terms are evaluated as14 Q L, h Q L, +(0) Q L, − (1) − Q L, − (0) Q L, +(1) i (B.20)48 iK Z ∞−∞ dx Z ∞−∞ dy ǫ ( x − y ) h j L, + t ( x ) j L, − x ( y ) − j L, + x ( x ) j L, − t ( y ) i . Thus the relation (B.14) has been shown, though the expression was quite messy in themiddle of the computation. The other Serre relations can also be shown in a similar way.
B.2 q -deformed su (2) R The Poisson brackets of Q R, and Q R, ± are next computed.The level 0 charge Q R, and the non-local field χ ( x ) are regularized as, respectively, Q R, ( X , X ) = Z X − X dx j R, t ( x ) , (B.21) χ ( x ; X , X ) = − Z X − X dx ′ ǫ ( x − x ′ ) j R, t ( x ′ ) , x ∈ [ − X , X ] . Note that for χ ( x ; X , X ) the domain of x is restricted. The restriction prevents us fromtaking the x → ±∞ limit before − X and X are sent to infinity. Hence the relationlim x →±∞ χ ( x ; X , X ) = ∓ Q R, ( X , X )is not valid any more. The Poisson brackets of the regularized charges Q R, and χ ( x ) are n Q R, ( X , X ) , Q R, ( Y , Y ) o P = 2 K h θ ( X − Y ) − θ ( X − Y ) i , (B.22) n χ ( x ; X , X ) , Q R, ( Y , Y ) o P = − K h θ ( x − Y ) − θ ( x + Y )+ θ ( Y − X ) + θ ( Y − X ) i , n χ ( x ; X , X ) , χ ( y ; Y , Y ) o P = K h − ǫ ( x − y ) + θ ( x − Y ) + θ ( x + Y ) − θ ( y − X ) − θ ( y + X ) + 12 θ ( X − Y ) − θ ( X − Y ) i . Next we define the regularized level 1 charges as Q R, ± ( X , X ; X ′ , X ′ ) = Z X − X dx e γ ± χ ( x ; X ′ ,X ′ ) i R, ± t ( x ) , [ − X , X ] ⊂ [ − X ′ , X ′ ] . Note that the interval [ − X , X ] is included in the interval [ − X ′ , X ′ ] . Then the Poissonbrackets of the regularized charges Q R, and Q R, ± are given by n Q R, ± ( X , X ; X ′ , X ′ ) , Q R, ( Y , Y ) o P (B.23)49 ± iQ R, ± (cid:0) min( X , Y ) , min( X , Y ); X ′ , X ′ (cid:1) ± i (cid:18) − γ ± γ ∓ (cid:19) (cid:2) Q R, ± (cid:0) min( X , Y ) , min( X , Y ); X ′ , X ′ (cid:1) − [ θ ( Y − X ′ ) + θ ( Y − X ′ )] Q R, ± ( X , X ; X ′ , X ′ ) (cid:3) , n Q R, + ( X , X ; X ′ , X ′ ) , Q R, − ( Y , Y ; Y ′ , Y ′ ) o P = i γ − γ + γ + + γ − h e γ + χ (min( X ,Y ); X ′ ,X ′ )+ γ − χ (min( X ,Y ); Y ′ ,Y ′ ) − e γ + χ ( − min( X ,Y ); X ′ ,X ′ )+ γ − χ ( − min( X ,Y ); Y ′ ,Y ′ ) i − i γ + − γ − γ + γ − h θ ( X − Y )e γ + χ ( − Y ; X ′ ,X ′ )+ γ − χ ( − Y ; Y ′ ,Y ′ ) − θ ( X − Y )e γ + χ ( Y ; X ′ ,X ′ )+ γ − χ ( Y ; Y ′ ,Y ′ ) i − (cid:0) γ + − γ − (cid:1) Q R, − ( Y , Y ; Y ′ , Y ′ ) × Z X − X dx [ θ ( x − Y ′ ) + θ ( x + Y ′ )] e γ + χ ( x ; X ′ ,X ′ ) i R, + t ( x )+ 12 (cid:0) γ + − γ − (cid:1) Q R, + ( X , X ; X ′ , X ′ ) × Z Y − Y dy [ θ ( y − X ′ ) + θ ( y + X ′ )] e γ − χ ( y ; Y ′ ,Y ′ ) i R, − t ( y ) − (cid:0) γ + − γ − (cid:1) [ θ ( X ′ − Y ′ ) − θ ( X ′ − Y ′ )] × Q R, + ( X , X ; X ′ , X ′ ) Q R, − ( Y , Y ; Y ′ , Y ′ ) . The order of limits
Let us consider the order of limits in the Poisson brackets computed so far.The first is the Poisson brackets in (B.22) , which depend on the order of limits. Note thatthe bracket { Q R, , Q R, } P should vanish. A prescription is to take a specific regularizationas follows: Q R, ( X ) = Q R, ( X, X ) . (B.24)Then the Poisson bracket vanishes properly, n Q R, ( X ) , Q R, ( Y ) o P = 0 . (B.25)Thus, after taking the limits, the desired result is obtained as n Q R, , Q R, o P = 0 . (B.26)50imilarly, the level 1 charges can be regularized as Q R, ± ( X ; X ′ ) = Q R, ± ( X, X ; X ′ , X ′ ) . (B.27)The Poisson brackets of Q R, ± ( X, X ′ ) and Q R, ( Y ) are n Q R, ± ( X ; X ′ ) , Q R, ( Y ) o P = ± iQ R, ± (cid:0) min( X, Y ); X ′ (cid:1) (B.28) ± i (cid:18) − γ ± γ ∓ (cid:19) (cid:2) Q R, ± (cid:0) min( X, Y ); X ′ (cid:1) − θ ( Y − X ′ ) Q R, ± ( X ; X ′ ) (cid:3) , and the Poisson bracket between Q R, ± ( X, X ′ ) is n Q R, + ( X ; X ′ ) , Q R, − ( Y ; Y ′ ) o P (B.29)= i γ − γ + γ + + γ − h e γ + χ (min( X,Y ); X ′ )+ γ − χ (min( X,Y ); Y ′ ) − e γ + χ ( − min( X,Y ); X ′ )+ γ − χ ( − min( X,Y ); Y ′ ) i − i γ + − γ − γ + γ − θ ( X − Y ) h e γ + χ ( − Y ; X ′ )+ γ − χ ( − Y ; Y ′ ) − e γ + χ ( Y ; X ′ )+ γ − χ ( Y ; Y ′ ) i − (cid:0) γ + − γ − (cid:1) Q R, − ( Y ; Y ′ ) Z X − X dx [ θ ( x − Y ′ ) + θ ( x + Y ′ )] e γ + χ ( x ; X ′ ) i R, + t ( x )+ 12 (cid:0) γ + − γ − (cid:1) Q R, + ( X ; X ′ ) Z Y − Y dy [ θ ( y − X ′ ) + θ ( y + X ′ )] e γ − χ ( y ; Y ′ ) i R, − t ( y ) , where χ ( x ; X ) is defined as χ ( x ; X ) ≡ χ ( x ; X, X ) . (B.30)Thus the Poisson brackets in (B.28) and (B.29) depend on the order of limits. The appear-ance of the order of limits is due to the presence of the Wess-Zumino term. When K = 0 , γ + = γ − and no ambiguity exists, as discussed in [25].Let us consider the order of limits for the Poisson bracket (B.28) . A prescription is totake the limit Y → ∞ before the limit X ′ → ∞ : n Q R, ± ( X ; X ′ ) , Q R, o P = ± iQ R, ± (cid:0) X ; X ′ (cid:1) . (B.31)Then one can obtain the desired expression, n Q R, ± , Q R, o P = ± iQ R, ± , (B.32)by sending X and X ′ to infinity. 51he next is the order of limits in (B.29) . A prescription is to take the X ′ → ∞ limitbefore the X → ∞ limit. Thus the regularized charge is introduced as Q R, ± ( X ) = lim X ′ →∞ Q R, ± ( X ; X ′ ) . (B.33)The Poisson bracket between Q R, ± is n Q R, + ( X ) , Q R, − ( Y ) o P = i γ − γ + γ + + γ − h e ( γ + + γ − ) χ (min( X,Y )) − e ( γ + + γ − ) χ ( − min( X,Y )) i − i γ + − γ − γ + γ − θ ( X − Y ) h e ( γ + + γ − ) χ ( − Y ) − e ( γ + + γ − ) χ ( Y ) i . The second term depends on the order of limits. This ambiguity appears as only the overallfactor and hence it can be absorbed by rescaling the charges. Here the order of limits isfixed in the following way: n Q R, + , Q R, − o P ≡ lim X →∞ n Q R, + ( X ) , Q R, − ( X ) o P = i γ + ) + ( γ − ) γ + γ − ( γ + + γ − ) lim X →∞ h e ( γ + + γ − ) χ ( X ) − e ( γ + + γ − ) χ ( − X ) i = − i ( γ + ) + ( γ − ) γ + γ − γ + + γ − ) sinh (cid:20) ( γ + + γ − )2 Q R, (cid:21) . (B.34)Thus the q -deformed su (2) algebra has been shown with some prescriptions for the order oflimits, where the q -parameter is defined as q ≡ exp (cid:20) γ + + γ − (cid:21) . B.3 A deformation of quantum affine algebra
Let us here present how to compute the deformed quantum affine algebra presented inSec. 4.4. In the middle of the computation, again, it is necessary to introduce some prescrip-tions to treat the ambiguities coming from the order of limits.The Poisson bracket between Q R, − (1) and e Q R, +(1) is first computed. For the regularizedcharges, it is given by n Q R, − (1) ( X ) , e Q R, +(1) ( Y ) o P (B.35)= 12 e − β sinh 2 α (cid:20)Z X − X dx Z Y − Y dy ǫ ( x − y ) j R, − t ( x ) e j R, + t ( y )52 i e β Z min( X,Y ) − min( X,Y ) dx j R, x ( x ) + 2 i e β cosh 2 α sinh 2 α Q R, (min( X, Y )) = 12 e − β sinh 2 α ¯ Q R, ( X, Y, min(
X, Y )) . Here the level 2 charge ¯ Q R, is regularized as¯ Q R, ( X, X ′ , X ′′ ) = Z X − X dx Z X ′ − X ′ dy ǫ ( x − y ) j R, − t ( x ) e j R, + t ( y ) (B.36) − i e β Z X ′′ − X ′′ dx j R, x ( x ) + 2 i e β cosh 2 α sinh 2 α Q R, ( X ′′ ) . Note that no ambiguous term is contained in this Poisson bracket (B.35) . Thus, by takingthe
X, Y → ∞ limits, we obtain the following: n Q R, − (1) , e Q R, +(1) o P = 12 e − β sinh 2 α ¯ Q R, . (B.37)As a next step, the Poisson bracket between ¯ Q R, and Q R, − (1) is computed. For the regu-larized charges, it is evaluated as n ¯ Q R, ( X, X ′ , X ′′ ) , Q R, − (1) ( Y ) o P (B.38)= −
12 e − β sinh 2 α × "Z X − X dx Z X ′ − X ′ dx ′ Z Y − Y dy ǫ ( x − x ′ )[ ǫ ( x − y ) − ǫ ( x ′ − y )] j R, − t ( x ) e j R, + t ( x ′ ) j R, − t ( y ) − i e β Z X − X dx Z min( X ′ ,Y ) − min( X ′ ,Y ) dx ′ ǫ ( x − x ′ ) j R, − t ( x ) (cid:18) j R, x ( x ′ ) − cosh 2 α sinh 2 α j R, t ( x ′ ) (cid:19) − β sinh 2 α Z min( X ′′ ,Y ) − min( X ′′ ,Y ) dx ′′ j R, − x ( x ′′ ) + 4e β cosh 2 α sinh α Q R, − (1) (min( X ′′ , Y )) +4( K − iA ) n Q R, − (1) (min( X, Y )) − θ ( Y − X ′ ) Q R, − (1) ( X ) o = −
12 e − β sinh 2 α × " Q R, − (3) ( X, X ′ , Y, X, min( X ′ , Y ) , min( X ′′ , Y ))+ 23 Q R, − (1) ( X ) e Q R, +(1) ( X ′ ) Q R, − (1) ( Y )+ 12 Z X − X dx Z X ′ − X ′ dx ′ Z Y − Y dy ǫ ( x − y )[ ǫ ( x − x ′ )+ ǫ ( y − x ′ )] j R, − t ( x ) e j R, + t ( x ′ ) j R, − t ( x ′′ ) +4( K − iA ) n Q R, − (1) (min( X, Y )) − θ ( Y − X ′ ) Q R, − (1) ( X ) o . K − iA ) n Q R, − (1) (min( X, Y )) − θ ( Y − X ′ ) Q R, − (1) ( X ) o (B.39)depends on the order of limits. Note that the level 3 charge Q R, − (3) is regularized as Q R, − (3) ( X, X ′ , X ′′ , X ′′′ , X ′′′′ , X ′′′′′ ) (B.40)= 12 Z X − X dx Z X ′ − X ′ dx ′ Z X ′′ − X ′′ dx ′′ ǫ ( x ′ − x ) ǫ ( x ′ − x ′′ ) j R, − t ( x ) e j R, + t ( x ′ ) j R, − t ( x ′′ ) − i e β Z X ′′′ − X ′′′ dx ′′′ Z X ′′′′ − X ′′′′ dx ′′′′ ǫ ( x ′′′ − x ′′′′ ) j R, − t ( x ′′′ ) (cid:18) j R, x ( x ′′′′ ) − cosh 2 α sinh 2 α j R, t ( x ′′′′ ) (cid:19) − β sinh 2 α Z X ′′′′′ − X ′′′′′ dx ′′′′′ j R, − x ( x ′′′′′ ) + 4e β cosh 2 α sinh α Q R, − (1) ( X ′′′′′ ) − Q R, − (1) ( X ) e Q R, +(1) ( X ′ ) Q R, − (1) ( X ′′ ) . A prescription is to take the Y → ∞ limit before X is sent to infinity so that the ambiguousterm vanishes: n ¯ Q R, ( X, X ′ , X ′′ ) , Q R, − (1) o P = lim Y →∞ n ¯ Q R, ( X, X ′ , X ′′ ) , Q R, − ( Y ) o P = −
12 e − β sinh 2 α h Q R, − (3) ( X, X ′ , ∞ , X, X ′ , X ′′ ) + 23 Q R, − (1) ( X ) e Q R, +(1) ( X ′ ) Q R, − (1) i −
14 e − β sinh 2 α Z X − X dx Z X ′ − X ′ dx ′ Z ∞−∞ dy ǫ ( x − y ) [ ǫ ( x − x ′ ) + ǫ ( y − x ′ )] × j R, − t ( x ) e j R, + t ( x ′ ) j R, − t ( y ) . By taking the remaining limits, we obtain the following result: n ¯ Q R, , Q R, − (1) o P = −
12 e − β sinh 2 α (cid:20) Q R, − (3) + 23 e Q R, +(1) (cid:16) Q R, − (1) (cid:17) (cid:21) . (B.41)It is interesting to see a higher-level Poisson bracket, for example, the Poisson bracketbetween Q R, − (3) and Q R, − (1) . For the regularized charges, it is given by n Q R, − (3) ( X, X ′ , X ′′ , X ′′′ , X ′′′′ , X ′′′′′ ) , Q R, − (1) ( Y ) o P (B.42)= −
14 e − β sinh 2 α × " − Q R, − (1) ( X ) Q R, − (1) ( Y ) Z X ′ − X ′ dx ′ Z X ′′ − X ′′ dx ′′ ǫ ( x ′ − x ′′ ) e j R, + t ( x ′ ) j R, − t ( x ′′ )54 12 Q R, − (1) ( X ′′ ) Q R, − (1) ( Y ) Z X − X dx Z X ′ − X ′ dx ′ ǫ ( x − x ′ ) j R, − t ( x ) e j R, + t ( x ′ )+ 12 Z X − X dx Z X ′ − X ′ dx ′ Z X ′′ − X ′′ dx ′′ Z Y − Y dy × n ǫ ( x ′ − x ′′ ) ǫ ( y − x ) [ ǫ ( y − x ′ ) + ǫ ( x − x ′ )]+ ǫ ( x ′ − x ) ǫ ( y − x ′′ ) [ ǫ ( y − x ′ ) + ǫ ( x ′′ − x ′ )] o × j R, − t ( x ) e j R, + t ( x ′ ) j R, − t ( x ′′ ) j R, − t ( y ) − i e β Q R, − (1) ( X ) Q R, − (1) ( X ′′ ) Z min( X ′ ,Y ) − min( X ′ ,Y ) dx ′ (cid:20) j R, x ( x ′ ) − cosh 2 α sinh 2 α j R, t ( x ′ ) (cid:21) +4 i e β Z X − X dx Z min( X ′ ,Y ) − min( X ′ ,Y ) dx ′ Z X ′′ − X ′′ dx ′′ ǫ ( x − x ′ ) ǫ ( x − x ′′ ) × j R, − t ( x ) (cid:20) j R, x ( x ′ ) − cosh 2 α sinh 2 α j R, t ( x ′ ) (cid:21) j R, − t ( x ′′ ) − i e β Z X ′′′ − X ′′′ dx ′′′ Z X ′′′′ − X ′′′′ dx ′′′′ Z Y − Y dyǫ ( x ′′′ − x ′′′′ ) ǫ ( x ′′′ − y ) × j R, − t ( x ′′′ ) (cid:20) j R, x ( x ′′′′ ) − cosh 2 α sinh 2 α j R, t ( x ′′′′ ) (cid:21) j R, − t ( y ) − β sinh 2 α Z X ′′′ − X ′′′ dx ′′′ Z min( X ′′′′ ,Y ) − min( X ′′′′ ,Y ) dx ′′′′ ǫ ( x ′′′ − x ′′′′ ) j R, − t ( x ′′′ ) j R, − x ( x ′′′′ ) − β sinh 2 α Z X ′′′′′ − X ′′′′′ dx ′′′′′ Z Y − Y dy ǫ ( x ′′′′′ − y ) j R, − x ( x ′′′′′ ) j R, − t ( y )+ 8e β cosh 2 α sinh α Z X ′′′ − X ′′′ dx ′′′ Z min( X ′′′′ ,Y ) − min( X ′′′′ ,Y ) dx ′′′′ ǫ ( x ′′′ − x ′′′′ ) × j R, − t ( x ′′′ ) j R, − t ( x ′′′′ ) − Q R, − (1) ( X ) Q R, − (1) ( X ′′ ) ¯ Q R, ( Y, X ′ , min( X ′ , Y )) +( K − iA ) θ ( X ′ − Y ) × Z X − X dx Z X ′′ − X ′′ dx ′′ [ ǫ ( x + Y ) ǫ ( x ′′ + Y ) − ǫ ( x − Y ) ǫ ( x ′′ − Y )] j R, − t ( x ) j R, − t ( x ′′ ) . Note that the last term( K − iA ) θ ( X ′ − Y ) Z X − X dx Z X ′′ − X ′′ dx ′′ (B.43) × [ ǫ ( x + Y ) ǫ ( x ′′ + Y ) − ǫ ( x − Y ) ǫ ( x ′′ − Y )] j R, − t ( x ) j R, − t ( x ′′ ) . is proportional to the step function and hence seems likely to be an ambiguous term. How-ever, this is not the case. This term does not depend on the order of limits and it is reduced55o zero after taking all of the limits. Thus the resulting Poisson bracket is given by n Q R, − (3) , Q R, − (1) o P = −
16 e − β sinh 2 α (cid:16) Q R, − (1) (cid:17) ¯ Q R, . (B.44)Similarly, the other Poisson brackets are computed, though those are not touched here. References [1] J. M. Maldacena, “The large N limit of superconformal field theories and supergrav-ity,” Adv. Theor. Math. Phys. (1998) 231 [Int. J. Theor. Phys. (1999) 1113].[arXiv:hep-th/9711200].[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators fromnon-critical string theory,” Phys. Lett. B (1998) 105 [arXiv:hep-th/9802109].[3] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. (1998)253 [arXiv:hep-th/9802150][4] N. Beisert et al. , “Review of AdS/CFT Integrability: An Overview,” arXiv:1012.3982[hep-th].[5] I. Bena, J. Polchinski and R. Roiban, “Hidden symmetries of the AdS × S superstring,”Phys. Rev. D (2004) 046002 [hep-th/0305116].[6] M. L¨uscher, “Quantum nonlocal charges and absence of particle production in the two-dimensional nonlinear sigma model,” Nucl. Phys. B (1978) 1.[7] M. L¨uscher and K. Pohlmeyer, “Scattering of massless lumps and nonlocal charges inthe two-dimensional classical nonlinear sigma model,” Nucl. Phys. B (1978) 46.[8] E. Brezin, C. Itzykson, J. Zinn-Justin and J. B. Zuber, “Remarks about the existenceof nonlocal charges in two-dimensional models,” Phys. Lett. B (1979) 442.[9] D. Bernard, “Hidden Yangians in 2-D massive current algebras,” Commun. Math. Phys. (1991) 191.[10] N. J. MacKay, “On the classical origins of Yangian symmetry in integrable field theory,”Phys. Lett. B (1992) 90 [Erratum-ibid. B (1993) 444].5611] E. Abdalla, M. C. Abdalla and K. Rothe, “Non-perturbative methods in two-dimensional quantum field theory,” World Scientific, 1991.[12] K. Zarembo, “Strings on Semisymmetric Superspaces,” JHEP (2010) 002[arXiv:1003.0465 [hep-th]].[13] N. Beisert and P. Koroteev, “Quantum Deformations of the One-Dimensional HubbardModel,” J. Phys. A (2008) 255204 [arXiv:0802.0777 [hep-th]].[14] N. Beisert, W. Galleas and T. Matsumoto, “A Quantum Affine Algebra for the DeformedHubbard Chain,” J. Phys. A (2012) 365206 [arXiv:1102.5700 [math-ph]].[15] B. Hoare, T. J. Hollowood and J. L. Miramontes, “ q -Deformation of the AdS × S Su-perstring S-matrix and its Relativistic Limit,” JHEP (2012) 015 [arXiv:1112.4485[hep-th]]; “Bound States of the q -Deformed AdS × S Superstring S-matrix,” JHEP (2012) 076 [arXiv:1206.0010 [hep-th]]; “Restoring Unitarity in the q -DeformedWorld-Sheet S-Matrix,” arXiv:1303.1447 [hep-th].[16] M. de Leeuw, V. Regelskis and A. Torrielli, “The Quantum Affine Origin of theAdS/CFT Secret Symmetry,” J. Phys. A (2012) 175202 [arXiv:1112.4989 [hep-th]].[17] G. Arutyunov, M. de Leeuw and S. J. van Tongeren, “The Quantum Deformed MirrorTBA I,” JHEP (2012) 090 [arXiv:1208.3478 [hep-th]]; “The Quantum DeformedMirror TBA II,” JHEP [JHEP (2013) 012] [arXiv:1210.8185 [hep-th]].[18] S. Schafer-Nameki, M. Yamazaki and K. Yoshida, “Coset Construction for Duals ofNon-relativistic CFTs,” JHEP (2009) 038 [arXiv:0903.4245 [hep-th]].[19] I. V. Cherednik, “Relativistically Invariant Quasiclassical Limits Of Integrable Two-Dimensional Quantum Models,” Theor. Math. Phys. (1981) 422 [Teor. Mat. Fiz. (1981) 225].[20] L. D. Faddeev and N. Y. Reshetikhin, “Integrability of the principal chiral field modelin (1+1)-dimension,” Annals Phys. (1986) 227.[21] J. Balog, P. Forgacs and L. Palla, “A Two-dimensional integrable axionic sigma modeland T duality,” Phys. Lett. B (2000) 367 [hep-th/0004180].5722] I. Kawaguchi and K. Yoshida, “Hidden Yangian symmetry in sigma model on squashedsphere,” JHEP (2010) 032. [arXiv:1008.0776 [hep-th]].[23] D. Orlando, S. Reffert and L. I. Uruchurtu, “Classical integrability of the squashedthree-sphere, warped AdS and Schr¨odinger spacetime via T-Duality,” J. Phys. A (2011) 115401. [arXiv:1011.1771 [hep-th]].[24] I. Kawaguchi, D. Orlando and K. Yoshida, “Yangian symmetry in deformed WZNWmodels on squashed spheres,” Phys. Lett. B (2011) 475. [arXiv:1104.0738 [hep-th]].[25] I. Kawaguchi and K. Yoshida, “Hybrid classical integrability in squashed sigma models,”Phys. Lett. B (2011) 251 [arXiv:1107.3662 [hep-th]], “Hybrid classical integrablestructure of squashed sigma models: A short summary,” J. Phys. Conf. Ser. (2012)012055 [arXiv:1110.6748 [hep-th]].[26] I. Kawaguchi, T. Matsumoto and K. Yoshida, “The classical origin of quantum affinealgebra in squashed sigma models,” JHEP (2012) 115 [arXiv:1201.3058 [hep-th]].[27] I. Kawaguchi, T. Matsumoto and K. Yoshida, “On the classical equivalence of mon-odromy matrices in squashed sigma model,” JHEP (2012) 082 [arXiv:1203.3400[hep-th]].[28] I. Kawaguchi and K. Yoshida, “Classical integrability of Schrodinger sigma models and q -deformed Poincare symmetry,” JHEP (2011) 094 [arXiv:1109.0872 [hep-th]],“Exotic symmetry and monodromy equivalence in Schrodinger sigma models,” JHEP (2013) 024 [arXiv:1209.4147 [hep-th]];[29] I. Kawaguchi, T. Matsumoto and K. Yoshida, “Schroedinger sigma models and Jorda-nian twists,” JHEP (2013) 013 [arXiv:1305.6556 [hep-th]].[30] D. Orlando and L. I. Uruchurtu, “Integrable Superstrings on the Squashed Three-sphere,” JHEP (2012) 007 [arXiv:1208.3680 [hep-th]].[31] B. Basso and A. Rej, “On the integrability of two-dimensional models with U (1) × SU ( N ) symmetry,” Nucl. Phys. B (2013) 337 [arXiv:1207.0413 [hep-th]].[32] F. Delduc, M. Magro and B. Vicedo, “On classical q-deformations of integrable sigma-models,” arXiv:1308.3581 [hep-th]. 5833] F. Delduc, M. Magro and B. Vicedo, “An integrable deformation of the AdS × S su-perstring action,” arXiv:1309.5850 [hep-th].[34] V. Ginzburg, M. Kapranov and E. Vasserot, “Langlands reciprocity for algebraic sur-faces,” Math. Res. Lett. (1995) 147.[35] J. M. Maillet, “New integrable canonical structures in two-dimensional models,” Nucl.Phys. B (1986) 54.[36] V. G. Drinfel’d, “Hopf algebras and the quantum Yang-Baxter equation,” Sov. Math.Dokl. (1985) 254.[37] V. G. Drinfel’d, “Quantum groups,” J. Sov. Math. (1988) 898 [Zap. Nauchn. Semin. , 18 (1986)].[38] M. Jimbo, “A q difference analog of U ( g ) and the Yang-Baxter equation,” Lett. Math.Phys. (1985) 63.[39] N. Reshetikhin, “Multiparameter quantum groups and twisted quasitriangular Hopfalgebras,” Lett. Math. Phys. (1990) 331.[40] M. Kruczenski, “Spin chains and string theory,” Phys. Rev. Lett. (2004) 161602[hep-th/0311203].[41] W. -Y. Wen, “Spin chain from marginally deformed AdS × S ,” Phys. Rev. D (2007)067901 [hep-th/0610147].[42] T. Kameyama and K. Yoshida, “String theories on warped AdS backgrounds and inte-grable deformations of spin chains,” JHEP (2013) 146 [arXiv:1304.1286 [hep-th]].[43] D. Israel, C. Kounnas, D. Orlando and P. M. Petropoulos, “Electric/magnetic de-formations of S and AdS , and geometric cosets,” Fortsch. Phys. (2005) 73[hep-th/0405213].[44] D. Orlando and L. I. Uruchurtu, “Warped anti-de Sitter spaces from brane intersectionsin type II string theory,” JHEP (2010) 049 [arXiv:1003.0712 [hep-th]].[45] W. Song and A. Strominger, “Warped AdS3/Dipole-CFT Duality,” JHEP (2012)120 [arXiv:1109.0544 [hep-th]]. 5946] S. Detournay, J. M. Lapan and M. Romo, “SUSY Enhancements in (0,4) Deformationsof AdS /CFT ,” JHEP (2012) 006 [arXiv:1109.4186 [hep-th]].[47] S. Detournay and M. Guica, “Stringy Schroedinger truncations,” JHEP (2013)121 [arXiv:1212.6792].[48] P. Karndumri and E. O Colgain, “3D Supergravity from wrapped D3-branes,” JHEP1310