NUPHB13190S0550-3213(14)00302-210.1016/j.nuclphysb.2014.10.004The AuthorsHigh Energy Physics – TheoryThe classical Yang–Baxter equation and the associated Yangian symmetry of gauged WZW-type theoriesDedicated to the memory of Sotirios Bonanos whose MATHEMATICA software has helped numerous researchersGeorgiosItsiosagitsios@upatras.grKonstantinosSfetsosbksfetsos@phys.uoa.grKonstantinosSiamposc⁎konstantinos.siampos@umons.ac.beAlessandroTorriellida.torrielli@surrey.ac.ukaDepartment of Mathematics, University of Patras, 26110 Patras, GreeceDepartment of MathematicsUniversity of PatrasPatras26110GreecebDepartment of Nuclear and Particle Physics, Faculty of Physics, University of Athens, 15771 Athens, GreeceDepartment of Nuclear and Particle PhysicsFaculty of PhysicsUniversity of AthensAthens15771GreececMécanique et Gravitation, Université de Mons, 7000 Mons, BelgiumMécanique et GravitationUniversité de MonsMons7000BelgiumdDepartment of Mathematics, University of Surrey, Guildford GU2 7XH, UKDepartment of MathematicsUniversity of SurreyGuildfordGU2 7XHUK⁎Corresponding author.Editor: Stephan StiebergerAbstractWe construct the Lax-pair, the classical monodromy matrix and the corresponding solution of the Yang–Baxter equation, for a two-parameter deformation of the Principal chiral model for a simple group. This deformation includes as a one-parameter subset, a class of integrable gauged WZW-type theories interpolating between the WZW model and the non-Abelian T-dual of the principal chiral model. We derive in full detail the Yangian algebra using two independent methods: by computing the algebra of the non-local charges and alternatively through an expansion of the Maillet brackets for the monodromy matrix. As a byproduct, we also provide a detailed general proof of the Serre relations for the Yangian symmetry.1Introduction and motivationA class of σ-models was recently constructed via a gauging procedure involving the WZW action and the general Principal Chiral model (PCM) action for a group G [1]. The end result is the action(1.1)Sk,λ(g)=SWZW,k(g)+kπ∫J+a(λ−1−DT)ab−1J−b, where SWZW,k(g) is the WZW action at level k of a group element g∈G and λ is a general dim(g) square real matrix. In addition, we have employed the standard definitions(1.2)J+a=Tr(Ta∂+gg−1),J−a=Tr(Tag−1∂−g),Dab=Tr(TagTbg−1), with Ta, a=1,2,…,dim(g) being the generators of the Lie algebra g satisfying the commutation rules, normalization and Killing form[Ta,Tb]=fabcTc,Tr(TaTb)=δab,Kba=δba. The key property of this action arises when λ is proportional to the identity, i.e. λab=λδab, since then it becomes integrable. This was shown in [1] by explicitly demonstrating that the current components I±=I±aTa obey the standard integrability conditions(1.3)∂+I−+∂−I+=0,∂+I−−∂−I++[I+,I−]=0. The explicit realization in terms of the σ-model action variables is(1.4)H=14e2∫−∞+∞dσ(I+aI+a+I−aI−a),I+a=2λ1+λ(I−λD)ab−1J+b,I−a=−2λ1+λ(I−λDT)ab−1J−b, where we have also included the expression for the Hamiltonian corresponding to (1.1). The general proof was done in [1] by explicitly demonstrating that certain integrability algebraic constraints provided in [2,3] were satisfied. A simpler way to prove the integrability of (1.1) has been given more recently in [4] by utilizing the fact that the construction involves, as mentioned, a gauging procedure reminiscent of the gauged WZW models.As discussed in detail in [1] a motivation for studying this action relates to the global properties of the variables in σ-models arising via non-Abelian T-duality. The latter generalizes, in a certain sense, Abelian T-duality [5] and was initiated by [6–8]. It is easily seen that when the elements λab→0 then (1.1) becomes the WZW SWZW,k(g). Also recall [1] that when λ approaches the identity matrix, k→∞ and g∈G is appropriately expanded around the identity group element, then (1.1) becomes the non-Abelian T-dual for the general PCM.11For the isotropic case see the derivation in [9,10] and for the general anisotropic case in [11]. Recent developments in non-Abelian T-duality in the presence of RR flux fields initiated with the work in [12]. For relations to the AdS/CFT correspondence, a discussion of global issues and further developments and references see [13–15]. Hence (1.1) interpolates between these two extreme cases and a way of thinking to the non-Abelian T-duality of the PCM is as a limiting case of (1.1). In the latter action the group element g∈G is parametrized by compact variables. Hence the non-compactness displayed by the variables in the non-Abelian model is attributed to the zooming-limiting procedure we mentioned.The perturbation away from the WZW point is driven by the term λabJ+aJ−b which for generic λab preserves no isometries (enhanced to GL×GR when λab=λδab). Based on that, on the matching of global symmetries and on the result of the computation of the renormalization group flow equations for the matrix λ in [16,17], one concludes that (1.1) provides the effective action for the bosonized anisotropic non-Abelian Thirring model valid to all orders in λ and to leading order in the 1/k expansion. In the same papers the following remarkable symmetry was also noticed(1.5)S−k,λ−1(g−1)=Sk,λ(g), which in fact mathematically dictates the form of all the aforementioned properties.In this paper we will further investigate the integrable structure of a two-parameter deformation of the PCM, which includes as a one-parameter subset (1.1) for the prototypical isotropic case λab=λδab. In particular, based on the underlying algebraic structure, we will show the existence of a Yangian algebra [18] (for reviews see [19–21]) of classically conserved non-local charges in the spirit of a similar computation for the (generalized) Gross–Neveu and the isotropic PCM in [22]. In the isotropic case, the Yangian algebra corresponds to the adjoint action on g, i.e. g↦Λ0−1gΛ0,Λ0∈G. In addition, we will provide the Lax pair and we will compute the Poisson brackets of its spatial part which take the Maillet form [23,24]. This will provide an array of coefficients which, as required for consistency, solve a classical modified Yang–Baxter equation. This allows for the derivation of the Maillet brackets of the monodromy matrix [23,24]. An expansion of these brackets will provide an alternative derivation of the Yangian algebra.This work is organized as follows: In Section 2 we review the derivation of the Lax pair and the corresponding (classical) monodromy matrix for a general class of two-dimensional systems. In Section 3 we compute the Maillet brackets of the spatial part of the Lax pair. Using this, we derive a class of solutions of the modified classical Yang–Baxter equation and the Maillet brackets of the monodromy matrix. In Section 4 we explore the realization of the Yangian algebra through the charge algebra and through an expansion of the Maillet brackets of the monodromy matrix. Details of the derivation are given in Appendices B–D respectively. In Section 5 we conclude with a discussion on possible future directions. Besides Appendices B–D we also include Appendix A where we revisit the proof of Drinfeld's relations.2Lax pair and the classical monodromy matrixThe purview of this section is to construct the Lax pair and the monodromy matrix of a class of integrable σ-models which were constructed in [1] and reviewed in Section 1. We will provide a rather general discussion by assuming that the equations of motion and the flat connection identities are given by22The world-sheet coordinates (σ+,σ−) and (τ,σ) are related byσ±:=τ±σ,∂0:=∂τ=∂++∂−,∂1:=∂σ=∂+−∂−, so that ⋆dσ±=±dσ± and ⋆dτ=dσ, ⋆dσ=dτ in Lorentzian signature.(2.1)(1+ρ)∂+I−+(1−ρ)∂−I+=0,∂+I−−∂−I++[I+,I−]=0,where I is a Lie algebra valued one-form(2.2)I=IaTa,Ia=I+adσ++I−adσ−, which for ρ=0 describe the integrable (isotropic) σ-models reviewed in Section 1, whereas for ρ≠0 the form of the action is not known. We can rewrite (2.1) in a differential-form notation as(2.3)I:=I−ρ⋆I,d(⋆I)=0,dI+I∧I=0, which makes manifest the classical integrability even when ρ≠0. Note that, even though the redefinition (2.3) has made the parameter ρ disappear from the integrability conditions, it may very well be present in the Poisson brackets for I±a.Using the above and assuming fields vanish at spatial infinity, we can construct the first two conserved charges [25](2.4)Q0:=∫−∞+∞dσI0(σ),Qˆ:=∫−∞+∞dσI1(σ)+∫−∞+∞dσI0(σ)∫−∞σdσ′I0(σ′), and another (still conserved and, as we will see, particularly convenient) combination of them(2.5)Q1:=Qˆ−12Q02=∫−∞+∞dσI1(σ)+12∫−∞+∞dσ∫−∞σdσ′[I0(σ),I0(σ′)]. In the above formula, we have rewritten the second term – corresponding to Q02 – by splitting the double integral in the two domains σ>σ′ and σ′>σ, and changed variables σ↔σ′ in one of the pieces.It is well known that the (infinite number) of conserved charges can be methodically constructed from the Lax pair(2.6)∂0L1−∂1L0=[L0,L1]ordL=L∧L. Using the latter we can show that the monodromy matrix (see, for instance, [26])33The path ordered exponential readsPexp∫−∞+∞dσf(σ):=1+∫−∞+∞dσf(σ)+∫−∞+∞dσ∫−∞σdσ′f(σ)f(σ′)+⋯.(2.7)M(ν):=Pexp∫−∞∞dσL1(σ;ν) is conserved for all values of the complex spectral parameter ν, namely ∂0M(ν)=0.In our case the Lax pair reads(2.8)L0(σ;ν)=(I1+ν˜I0)ν˜1−ν˜2,L1(σ;ν)=(I0+ν˜I1)ν˜1−ν˜2,ν˜=ν+ρ1+νρ, or, equivalently, L±(σ;ν)=−ν˜ν˜∓1I±. By expanding M in powers of ν′:=ν+ρ, we find an infinite set of classically conserved charges:(2.9)M(ν′)=1+ν′1−ρ2Q0+ν′2(1−ρ2)2(Qˆ−ρQ0)+O(ν′3). One recognizes combinations of the charges (2.4) in the coefficients of the expansion.3The Maillet brackets and the Yang–Baxter equationIn this section we prove that when the above quantities are supplied by an appropriate algebraic structure, this allows to find explicit solutions to a modified classical Yang–Baxter equation. Consequently, the monodromy matrix obeys the associated Maillet brackets.Following Sklyanin [27], we compute the Poisson brackets by first writing L1=L1aTa and then44The superscript in parenthesis stands for the notation of tensor products of spacesM(1)=M⊗I,M(2)=I⊗M,m(12)=mabTa⊗Tb⊗I,m(13)=mabTa⊗I⊗Tb,m(23)=mabI⊗Ta⊗Tb, for an arbitrary matrix m=mabTa⊗Tb in the tensor product algebra.(3.1){L1(1)(σ1;μ),L1(2)(σ2;ν)}={L1a(σ1;μ),L1b(σ2;ν)}Ta⊗Tb. The Poisson brackets assume the Maillet-type form [24](3.2)([r−μν,L1(1)(σ1;μ)]+[r+μν,L1(2)(σ1;ν)])δ12+δ12′(r−μν−r+μν), where r±μν (as a shorthand notation for r±(μ,ν)) are matrices in the basis Ta⊗Tb. This is guaranteed to give a consistent Poisson structure, provided the Jacobi identities for these brackets are obeyed. This enforces r±μν to satisfy the modified classical Yang–Baxter relation(3.3)[r+ν1ν3(13),r−ν1ν2(12)]+[r+ν2ν3(23),r+ν1ν2(12)]+[r+ν2ν3(23),r+ν1ν3(13)]=0. The non-vanishing coefficient of the δ′ term in (3.2) is responsible for the above modification of the classical Yang–Baxter relation. Using (3.2), one can derive the Poisson brackets for the (classical) monodromy matrix [23](3.4){M(1)(μ),M(2)(ν)}=[rμν,M(μ)⊗M(ν)]−M(2)(ν)sμνM(1)(μ)+M(1)(μ)sμνM(2)(ν), which is consistent with the Jacobi identity, if we define the equal-point limits of the Poisson brackets through a generalized symmetric limit procedure [23].Returning to the case at hand, it was pointed out in [28] that the Poisson structure of the isotropic PCM admits a one-parameter family of deformations (with parameter denoted by x). Subsequently, in [2] this deformation was further extended by introducing a second parameter ρ. In our conventions such two-parameter algebra reads55Alternatively, in light-cone coordinates the algebra reads{I±a,I±b}=e2fabc[(1∓ρ)2I∓c−((1∓ρ)2+2x(1−ρ2))I±c]δσσ′±2e2(1∓ρ)2δabδσσ′′,{I±a,I∓b}=−e2fabc[(1−ρ)2I−c+(1+ρ)2I+c]δσσ′.(3.5){I0a,I0b}=−2e2fabc(1+ρ2+(1−ρ2)x)I0cδσσ′−8e2ρδabδσσ′′,{I1a,I1b}=2e2fabc(4ρI1c+(1+ρ2+x(ρ2−1))I0c)δσσ′−8e2ρδabδσσ′′,{I0a,I1b}={I1a,I0b}=−2e2fabc(1+ρ2+(1−ρ2)x)I1cδσσ′+4e2(1+ρ2)δabδσσ′′. When ρ=0, the action (1.1) provides a realization of this algebra for a general group G [1] with x=λ2+12λ (for the SU(2) case this realization was found by a brute force computation in [2]). There is no known action realizing the above algebra for ρ≠0.66The algebra for the PCM (pseudo-PCM) corresponds to choosing the parameters x=1 (x=−1) and ρ=0. The value x=1 corresponds to taking the parameter λ in this paper to unity. The fact that then the action (1.1) does not become that for a PCM but rather for its non-Abelian T-dual is consistent since non-Abelian T-duality can be cast as a canonical transformation in phase space [9–11]. We also note that for ρ=0 and e→0, x→∞, we obtain under an appropriate rescaling of the currents the algebra for the (generalized) Gross–Neveu model when e2x is finite, and for the conformal case when ex is finite. Plugging (2.8) into (3.2) and using the algebra (3.5), we find that the matrix r±μν readr±μν↦r±μνΠ,Π:=∑aTa⊗Ta,r+μν=2e2(1+μ2+x(1−μ2))(μ+ρ)(ν+ρ)(ν−μ)(1−μ2),r−μν=2e2(1+ν2+x(1−ν2))(μ+ρ)(ν+ρ)(ν−μ)(1−ν2)=−r+νμ, and henceforth r±μν denotes the scalar. As for (3.3), it reduces to the single algebraic condition(3.6)r+ν2ν3r+ν1ν2=r+ν1ν3r−ν1ν2+r+ν2ν3r+ν1ν3, extracted from the coefficient in front of the combination fabcTa⊗Tb⊗Tc. For completeness, we provide the values of rμν and sμν, obtained by rewriting r±μν=rμν±sμν:(3.7)rμν=−2e2(1−μ2ν2+x(1−μ2)(1−ν2))(μ+ρ)(ν+ρ)(μ−ν)(1−μ2)(1−ν2),sμν=−2e2(μ+ν)(μ+ρ)(ν+ρ)(1−μ2)(1−ν2), which are generically non-vanishing.4The realization of the Yangian algebraThe scope of this section is to explicitly realize the Yangian algebra by performing the mutual Poisson commutators between Q0a and Q1a and, alternatively, via an expansion of the Maillet brackets for the conserved monodromy matrix M.The Yangian algebra YC(g) is an associative Hopf algebra generated by the elements Ja and Qa obeying [18](4.1)[Ja,Jb]=FabcJc,[Ja,Qb]=[Qa,Jb]=FabcQc. In addition, the request that the co-product map (which we call f to avoid conflicts of notations) on Ja and Qa, namely(4.2)f(Ja)=Ja⊗I+I⊗Ja,f(Qa)=Qa⊗I+I⊗Qa+α2FabcJb⊗Jc,α∈C, acts as a homomorphism,77A homomorphism is a structure-preserving map between two algebraic structures (such as groups)f:A↦Bwithf(a1+a2)=f(a1)+f(a2),f(a1a2)=f(a1)f(a2),∀a1,a2∈A. implies the Serre relations – see Appendices A.1 and A.2 for details. The first Serre relation reads(4.3)[Qa,[Qb,Jc]]−[Ja,[Qb,Qc]]=α224aabcdefJ(dJeJf),aabcdef=FadkFbelFcfmFklm, where J(aJbJc) denotes the sum of all permutations of JaJbJc,88This sum explicitly expands asJ(aJbJc)=JaJbJc+JcJaJb+JbJcJa+JaJcJb+JbJaJc+JcJbJa. which for (classical) commuting quantities simplifies to 6JaJbJc. The first Serre relation is trivially satisfied for the su(2) case, as it turns out that aabcdef=εabeεcfd−εdbeεcfa. Using the Jacobi identity on the second term of the l.h.s. of (4.3), and using the second of the relations (4.1), we easily find that (4.3) can be written as(4.4)Fdab[Qc,Qd]+Fdca[Qb,Qd]+Fdbc[Qa,Qd]=α224aabcdefJ(dJeJf), a form which is particularly convenient for our purposes. In addition, the second Serre relation reads(4.5)Fkcd[[Qa,Qb],Qk]+Fkab[[Qc,Qd],Qk]=α224(aabkgefFkcd+acdkgefFkab)J(gJeQf).The first Serre implies the second one (for details see Appendix A.2), except for the su(2) case where it reads(4.6)[[Qa,Qb],[Jc,Qd]]+[[Qc,Qd],[Ja,Qb]]=α224(εeab(δfcδgd−δfdδcg)+εecd(δfaδgb−δfbδag))J(dJeQf). Hence, for the su(2) case only this relation is non-trivial.4.1Yangian algebra through the algebra of chargesNext we work out the algebra of the classical charges Q0 and Q1 defined in (2.4) and (2.5). Their components read99We use the definitionε12=ε(σ1−σ2)={1if σ1>σ2−1if σ1<σ2,andε′(x)=2δ(x).(4.7)Q0a=∫−∞+∞dσI0a(σ),Q1a=∫−∞+∞dσI1a(σ)+14fabc∫−∞+∞∫−∞+∞d2σ12ε12I0b(σ1)I0c(σ2). A comment is in order regarding the form of the charges. This should be in agreement with the co-product (4.2) and realized as half-positive and half-negative axis splitting (see [22] for details), upon the identifications(4.8)Ja↦Q0a,Qa↦Q1a,Fabc↦1αfabc. Using (3.5) we compute the Poisson brackets for the zeroth level charges1010To avoid ambiguities arising from the non-utralocal terms, like∫dσ1dσ2∂1δ12≠∫dσ2dσ1∂1δ12 we follow [25] and we define the Poisson bracket{Q0,1a,Q0,1b}=limL2→∞limL1→∞{Q0,1a,L1,Q0,1b,L2}, where Q0,1a,L are volume cutoff charges (the same as Q0,1a, with range from −L to L).(4.9){Q0a,Q0b}=−2e2(1+ρ2+x(1−ρ2))fabcQ0c. Using the Jacobi identity we find that(4.10){Q0a,Q1b}=2Δfabc∫−∞+∞dσI1c(σ)+Δ(fbeffaed+fbdefaef)∫−∞+∞dσ1I0d(σ1)∫−∞σ1dσ2I0f(σ2)⟹{Q0a,Q1b}=−2e2(1+ρ2+x(1−ρ2))fabcQ1c. For notational convenience we define(4.11)Δ:=−e2(1+ρ2+x(1−ρ2)). Finally, we can compute {Q1a,Q1b} as follows:(4.12)Q1a:=xa+ya,{Q1a,Q1b}={xa,xb}+{xa,yb}−{xb,ya}+{ya,yb},{xa,xb}+{xa,yb}−{xb,ya}=fabcQ2(1)c,Q2(1)a=2e2(1+ρ2+x(ρ2−1))Q0a+8e2ρ∫dσI1a+Δfabc∫d2σ12ε12I0b(σ1)I1c(σ2), where xa and ya correspond to the first and second term in the expression of Q1a, respectively. Furthermore, we can show that(4.13){ya,yb}=2Δfacdfbrefdrℓ×14Iceℓ,Iceℓ=∫d3σ123ε13ε32I0c(σ1)I0e(σ2)I0ℓ(σ3)=Iecℓ, which can be further simplified with the use of (B.8), proven in Appendix B and reported here for convenience:facdfbrefdrℓIceℓ=−13facdfbrefdrℓQ0cQ0eQ0ℓ−13fabrfrcdfdeℓIceℓ. Putting all together, the Poisson brackets of two Q1a's read(4.14){Q1a,Q1b}=fabc(Q2(1)c+Q2(2)c)−2Δ×facdfbrefdrℓ×112Q0cQ0eQ0ℓ,where Q2(2)a=−Δ6facdfdeℓIceℓ. By use of the Jacobi identity, we find the first Serre relation (4.4):(4.15)12fd[ab{Q1c],Q1d}=2e2(1+ρ2+(1−ρ2)x)×124faipfbjqfckrfijkQ0(pQ0qQ0r), where we have used the identity (C.1) proven in Appendix C(4.16)12fd[abfc]pmfdnqfmnrQ0pQ0qQ0r=3faipfbjqfckrfijkQ0pQ0qQ0r. In total, the charges (4.7) form a classical Yangian algebra in the sense of Poisson brackets, namely Eqs. (4.9), (4.10) and (4.15), under the correspondence (4.8) with(4.17)α=12Δ=−12e2(1+ρ2+(1−ρ2)x). The above is a generalization of a proof originally given in [22] for the isotropic PCM and for the (generalized) Gross–Neveu model, whose corresponding algebras for the I±a's are particular cases of (3.5) (see footnote 6).The appearance of the classical Yangian algebra was guaranteed in the first place by the existence of the Yang–Baxter equation (3.3) and the realization of the co-product (4.2) by (4.7) and (4.8). The only additional step which needed to be made was to compute the value of α through the Poisson brackets of the level-zero charges (4.9).4.1.1The su(2) caseFor the su(2) case, the first Serre relation is trivially satisfied, therefore we only have to study the second one (4.6). Using (4.10) we can rewrite the l.h.s. of (4.6) (once again understood in its classical version of Poisson brackets) as(4.18)2Δ(εcde{{Q1a,Q1b},Q1e}+εabe{{Q1c,Q1d},Q1e}). Next we note that (4.14) trivializes to(4.19){Q1a,Q1b}=εabcQc,Qa=Q2(1)a+Δ4(Q0aQ0dQ0d+Idda), where Q2(1)a is given in (4.12) with fabc replaced by εabc. Using these specialized expressions we find(4.20){{Q1a,Q1b},{Q0c,Q1d}}+{{Q1c,Q1d},{Q0a,Q1b}}=2e6(1+ρ2+(1−ρ2)x)3Q0e(εabeQ0[cQ1d]+εcdeQ0[aQ1b]), which is in agreement with (4.6), (4.8) and (4.17).4.2Yangian algebra through the Maillet brackets of the monodromy matrixAn alternative derivation of the Yangian algebra is obtained through the Maillet brackets (3.4) and the expansion of the monodromy matrix M (2.9).Rewriting (2.9) in terms of Q0,1 we find that(4.21)M(ν′)=1+ν′1−ρ2Q0+ν′2(1−ρ2)2(Q1−ρQ0+12Q02)+O(ν′3). Plugging (4.21) into (3.4), expanding first in ν′ and only afterwards in μ′, where μ′:=μ+ρ, and keeping all the terms up to the order O(ν′2μ′), produces, after a good deal of algebra (and writing Q=QaTa), (4.9) and (4.10), respectively(4.22){Q0a,Q0b}=−2e2(1+ρ2+(1−ρ2)x)fabcQ0c,{Q0a,Q1b}=−2e2(1+ρ2+(1−ρ2)x)fabcQ1c. Next we consider the expansion of the Maillet brackets (3.4) up to O(ν′2μ′2). We will see that, in order to study this term, it is necessary to expand the monodromy matrix up to the order O(ν′3). By manipulating the O(ν′2μ′2) term in the brackets, after a rather tedious computation, we obtain the first Serre relation in (4.15). The related technical details are presented in Appendix D.5Conclusions and outlookThe purview of the present paper is the construction of the Lax pair L0,1 for isotropic coupling matrices λ of the action (1.1) and the corresponding symmetry algebra (3.5) with ρ=0. Using its spatial part L1 we built the conserved classical monodromy matrix, derived the corresponding Poisson (Maillet-type [24,23]) brackets and the emerging modified Yang–Baxter equation as the Jacobi identity on these Poisson brackets. Employing the classical monodromy matrix we constructed the first two conserved charges and obtained their Yangian algebra, both through the charge algebra and also from an expansion of the Poisson brackets for the monodromy matrix. In addition, the renormalizability of this action at one-loop in the 1/k expansion [16] ensures that the above construction remains applicable at this order.It would be interesting to study generalizations of the construction we have provided for vanishing ρ and anisotropic coupling matrices λ, whose action was given in (1.1). These σ-models generically interpolate from the WZW to the non-Abelian T-dual of the anisotropic PCM, and so a good place to start this study are cases which possess an integrable anisotropic PCM endpoint, like the su(2) case [30–32]. Also note that the Yangian symmetries are preserved for the deformed WZW model on squashed spheres [33,34].It is possible to replace the WZW term in (1.1) by a coset CFT with action realization in terms of a gauged WZW model. In these case the end point of the deformation corresponds to the non-Abelian T-dual of PCM for coset instead of group spaces [1]. In that respect, and for symmetric coset spaces, the deformation has been convincingly argued to correspond to a quantum deformation of the bosonic sector of the string theory, when the deformation parameter is a root of unity [4]. When instead it is real, achieved by analytic continuation, the models are those of [35–37], based on the construction of [38,39] and realized as σ-models in [40]. We believe that our treatment is generalizable to these cases as well.AcknowledgementsWe would like to thank Nicolas Boulanger, Ben Hoare, Marc Magro and Vidas Regelskis for useful correspondence. The research of G. Itsios has been co-financed by the ESF (2007–2013) and Greek National Funds through the Operational Program “Education and Lifelong Learning” of the NSRF – Research Funding Program: “Heracleitus II. Investing in knowledge in society through the European Social Fund”. The research of K. Sfetsos is implemented under the ARISTEIA action (D.654 of GGET) of the operational programme education and lifelong learning and is co-funded by the European Social Fund (ESF) and National Resources (2007–2013). The work of K. Siampos has been supported by Actions de recherche concertées (ARC) de la Direction générale de l'Enseignement non obligatoire et de la Recherche scientifique – Direction de la Recherche scientifique – Communauté française de Belgique (AUWB-2010-10/15-UMONS-1), and by IISN-Belgium (convention 4.4511.06). A. Torrielli thanks EPSRC for funding under the First Grant project EP/K014412/1 “Exotic quantum groups, Lie superalgebras and integrable systems”. K. Sfetsos and K. Siampos would like to thank the University of Patras for hospitality, where part of this work was developed. A. Torrielli acknowledges useful conversations with the participants of the ESF and STFC supported workshop “Permutations and Gauge–String duality” (STFC-4070083442) (Queen Mary U. of London, July 2014).Appendix AThe Serre relationsThe scope of this appendix is to provide an explicit proof of the Serre relations for pedagogical reasons.A.1The first Serre relationThe proof goes along the lines suggested in [20]. Let us define the co-products of Ja and Qa as in (4.2), and the quantity(A.1)Zab:=f([Qa,Qb])−[Qa,Qb]⊗I−I⊗[Qa,Qb], on which f acts as a homomorphism (see footnote 7). Next we introduce(A.2)uab:=Fcdavcdb−Fcdbvcda,Fabcuab=0, where vabc is totally antisymmetric. Contracting (A.1) with uab, using (4.1) and the Jacobi identity on the term proportional to α, we find that(A.3)uabZab=α2uabFabeFcde(Qc⊗Jd−Jc⊗Qd)+α24uabFacdFbmnFcmr(Jr⊗JdJn+JdJn⊗Jr) with the first term vanishing due to (A.2). Substituting the value of uab we find(A.4)α24(Fijavijb−Fijbvija)FacdFbmnFcmr(Jr⊗JdJn+JdJn⊗Jr)=(A−B)×α24(Jr⊗JdJn+JdJn⊗Jr), where(A.5)A=vijbFijaFacdFbmnFcmr=2vijb(FajdFbmnFcmiFcar+FajdFbmnFcamFcir),B=vijaFijbFacdFbmnFcmr=2vijb(FajnFbcdFmciFamr+FajnFbcdFmacFimr), and we employed the Jacobi identity twice for each term. We then rewrite A−B as(A.6)A−B=C+D,C=2vijb(FajdFbmnFcmiFcar−FajnFbcdFmciFamr),D=2vijb(FajdFbmnFcamFcir−FajnFbcdFmacFimr). Applying the Jacobi identity on C and relabelling the indices of D, we find(A.7)D=4vijbFajdFbmnFcamFcir,C=−D2. Thus we proved that(A.8)uabFacdFbmnFcmr=2vijbFajdFbmnFcamFcir. Using (A.8), we can rewrite the r.h.s. of (A.3) as(A.9)−α22vijaaijardn(Jr⊗JdJn+JdJn⊗Jr),aabcdef=FadkFbelFcfmFklm. Thus (A.3) reads(A.10)2vijaFijbZab=α22vijaaijardn(Jr⊗JdJn+JdJn⊗Jr). Due to the contraction with a totally antisymmetric tensor we can rewrite (A.10) as(A.11)2vijaFb[ijZa]b=α22vijaa[ija]rdn(Jr⊗JdJn+JdJn⊗Jr). Next, we note that(A.12)a[ija]rdn=aija(rdn),aija(rdn)(Jr⊗JdJn+JdJn⊗Jr)=aijardn(J(r⊗JdJn)+J(dJn⊗Jr)), where (.) denotes the sum of all permutations (see footnote 8). Using (A.11) and (A.12) we find(A.13)vijaFb[ijZa]b=α24vijaaijardn(J(r⊗JdJn)+J(dJn⊗Jr)). In addition, using (4.2) we can easily prove that(A.14)J(r⊗JdJn)+J(dJn⊗Jr)=13(f(J(rJdJn))−J(rJdJn)⊗I−I⊗J(rJdJn)). Using (A.13) and (A.14) we find that(A.15)vijaFb[ijZa]b=α212vijaaijardn(f(J(rJdJn))−J(rJdJn)⊗I−I⊗J(rJdJn)). Finally, we make use of the properties (A.12) to manipulate the r.h.s. of (A.15) into(A.16)vijaaijardnW(rdn)=vijaaija(rdn)Wrdn=vijaa[ija]rdnWrdn,Wrdn=f(JrJdJn)−JrJdJn⊗I−I⊗JrJdJn. Using (A.16) we can write (A.15) as(A.17)vijaFb[ijZa]b=vija×α212a[ija]rdnWrdn. Since the latter holds for every antisymmetric tensor vija we conclude that(A.18)Fb[ijZa]b=α212a[ija]rdnWrdn⟹Fb[ij[Qa],Qb]=α212a[ija]rdnJrJdJn=α212aijardnJ(rJdJn). Expanding the antisymmetric part on the l.h.s. of (A.18) we find the first Serre relation (4.4), namely(A.19)Fmij[Qk,Qm]+Fmki[Qj,Qm]+Fmjk[Qi,Qm]=α224aijkrdnJ(rJdJn), where we used that Fa[bc]=2Fabc.A.2The second Serre relationApplying the Jacobi identity we can easily prove that(A.20)[[Qc,Qd],Qe]=[[Qc,Qe],Qd]−[[Qd,Qe],Qc],FcrkFrde−FdrkFrce=FcdrFrek. Using the first of (A.20) we can prove that(A.21)2(Lab|cd+Lbc|ad+Lca|bd)=Fe[ab[[Qc],Qe],Qd]−Fe[ab[[Qd],Qe],Qc]+Fe[cd[[Qa],Qe],Qb]−Fe[cd[[Qb],Qe],Qa], where Lab|cd denotes the l.h.s. of (4.5). Using (4.4), the second relation in (4.1) and the second equation in (A.20), we can rewrite the r.h.s. of (A.21) as(A.22)Rab|cd+Rbc|ad+Rca|bd, where Rab|cd denotes the r.h.s. of (4.5). Combining (A.21) and (A.22) we find(A.23)Lab|cd+Lbc|ad+Lca|bd=Rab|cd+Rbc|ad+Rca|bd. The solution to this equation is the second Serre relation (4.5). In fact, we may suppose it is not, by assuming that there exists another choice of Xab|cd such that(A.24)Lab|cd=Rab|cd+Xab|cd, where Xab|cd is such that(A.25)Xab|cd=−Xba|cd=−Xab|dc=Xcd|ab and(A.26)X[ab|c]d=0,Xa[b|cd]=0. Applying the Jacobi identity on (A.24) and using (A.25) we find(A.27)Fb[mnXq]b|[rd̲Fst]d=0⟹Xab|cd=FabeYe|cd+FcdeYe|ab,Ya|bc=−Ya|cb, where d̲ is excluded from the anti-symmetrization. Using these relations and (A.26), we find(A.28)Ya|bc∼Fabc⟹Xab|cd=εFabeFcde, where ε is an arbitrary constant. Thus (A.24) reads(A.29)Lab|cd=Rab|cd+εFabeFcde. Contracting the latter with FabℓFcdℓ and using the Jacobi identity on commutators, we find(A.30)0=0+εcG2dim(g)⟹ε=0, where FacdFbcd=cgδab, a=1,2,…,dim(g). This completes the proof of the redundancy of the second Serre relation (4.5).Appendix BThe triple integralThe scope of this appendix is to simplify (4.13). Let us first define the triple integral as(B.1)Iceℓ=Iecℓ=∫d3σ123ε13ε32I0c(σ1)I0e(σ2)I0ℓ(σ3), where d3σ123 stands for dσ1dσ2dσ3. This can be rewritten as follows1111For manipulations of similar integrals, see [29].:(B.2)Iceℓ=−12∫d3σ123dwε31I0c(σ1)ε32I0e(σ2)∂σ3(ε3wI0ℓ(w)). Integrating by parts we can easily prove that(B.3)Iceℓ+Iℓce+Ieℓc=−Q0cQ0eQ0ℓ⟹(facdfbrefdrℓ+faℓdfbrcfdre+faedfbrℓfdrc)Iceℓ=−facdfbrefdrℓQ0cQ0eQ0ℓ. This formula could equivalently be found through the identity(B.4)ε13ε32+ε21ε13+ε32ε21=−1. Using the Jacobi identity we can prove that(B.5)faℓdfdrefbrc=facdfbrefdrℓ+faedfbrcfdrℓ−facdfbrℓfdre−fabrfrcdfdℓe,faedfbrℓfdrc=facdfbrefdrℓ+faedfbrcfdrℓ−faℓdfbrefdrc−fabrfredfdℓc. Using the latter we can rewrite (B.3) as(B.6)(3facdfbrefdrℓ+2faedfbrcfdrℓ−facdfbrℓfdre−faℓdfbrefdrc+fabr(fdrcfdeℓ−fdrefdℓc))Iceℓ=−facdfbrefdrℓQ0cQ0eQ0ℓ. Using (B.4) we can prove that(B.7)(2faedfbrcfdrℓ−facdfbrℓfdre−faℓdfbrefdrc)Iceℓ=facdfbrefdrℓ(3Iceℓ+Q0cQ0eQ0ℓ). Combining (B.6) and (B.7) we find(B.8)facdfbrefdrℓIceℓ=−13facdfbrefdrℓQ0cQ0eQ0ℓ−13fabrfrcdfdeℓIceℓ.Appendix CSerre structure constantsIn this appendix we prove (4.16). In order to do so, we use the equivalent rewriting(C.1)12fd[abfc]d3=3faifbjfckfijk,fij:=fijkQ0k. We start from the r.h.s. and use the Jacobi identity to rewrite fckfijk as(C.2)fckfijk=−fcikfjk−fjckfik. Then we make use again of the Jacobi identity to deduce the following rewritings:(C.3)faifcik=fcaifik−fiakfci,fbjfjck=fbcjfjk+fjbkfcj. Using (C.2) and (C.3), we can rewrite the r.h.s. of (C.1) as(C.4)faifbjfckfijk=faj3fbcj+fbj3fcaj+fbj2fcifiaj−faj2fcifibj, or, equivalently, as(C.5)faifbjfckfijk=fbj3fcaj+fcj3fabj+fcj2faifibj−fbj2faificj,(C.6)faifbjfckfijk=fcj3fabj+faj3fbcj+faj2fbificj−fcj2fbifiaj. Adding (C.4), (C.5) and (C.6) together we find(C.7)3faifbjfckfijk=fd[abfc]d3−faj2(fcifibj−fbificj)−fbj2(faificj−fcifiaj)−fcj2(fbifiaj−faifibj). Using the Jacobi identity we can rewrite the terms in parentheses as(C.8)fcifibj−fbificj=fjdfdbc,faificj−fcifiaj=fjdfdca,fbifiaj−faifibj=fjdfdab. Using (C.8), we can rewrite (C.7) as(C.9)3faifbjfckfijk=fd[abfc]d3−12fd[abfc]d3=12fd[abfc]d3, which completes the proof of (C.1) or equivalently (4.16).Appendix DMaillet brackets and the first Serre relationLet us first consider the l.h.s. of (3.4) at the order O(ν′2μ′2) of the expansion of the monodromy matrix (4.21). Specifically, if we define(D.1)g1:=ν′1−ρ2,g2:=μ′1−ρ2,q=Q02,qˆ=Q12, we get(D.2)4g12g22{(qˆ−ρq+q2),(qˆ−ρq+q2)}. Expanding the Poisson brackets and using (4.22) written in terms of q,qˆ, i.e.(D.3){qa,qb}=Δfabcqc,{qa,qˆb}=Δfabcqˆc,Δ=−e2(ρ2+1+x(1−ρ2)), we obtain(D.4)4g12g22[{qˆa,qˆb}Ta⊗Tb+2ρΔfbacqˆcTa⊗Tb−ΔfbadqˆdqcTa⊗{Tb,Tc}+ρ2ΔfabcqcTa⊗Tb−ρΔfabdqdqcTa⊗{Tb,Tc}+Δfacdqˆdqb{Ta,Tb}⊗Tc−ρΔfbcdqaqd{Ta,Tb}⊗Tc+Δfbceqaqeqd{Ta,Tb}⊗{Tc,Td}], where {Ta,Tb}:=TaTb+TbTa.We now need to consider the r.h.s. of (3.4) at the same order. Let us define(D.5)q1:=q⊗I,q2:=I⊗q, as in footnote 4. There are several contributions, which we list here below:•both r(ν′,μ′) and s(ν′,μ′) are taken at the order O(ν′μ′), which means that both M(ν′) and M(μ′) are taken at the linear order. This term contributes(D.6)16e2ρg12g22(−[Π,q1q2]+q1Πq2−q2Πq1);•r(ν′,μ′) is taken at the order O(ν′μ′0), which means that M(ν′) is taken at the linear and M(μ′) at the quadratic order. No s(ν′,μ′) contribution is present at this order. We get(D.7)−8Δg12g22[Π,q1(qˆ2−ρq2+q22)];•r−(ν′,μ′) is taken at the order O(ν′μ′2), which means that M(ν′) is taken at the linear and M(μ′) at the zeroth order (making r+ immaterial). We get(D.8)8e2(1+3ρ2)g12g22[Π,q1];•r+(ν′,μ′) is taken at the order O(ν′2μ′0), which means that M(ν′) is taken at the zeroth and M(μ′) at the quadratic order (making r− irrelevant). We get(D.9)−16e2ρg12g22[Π,(qˆ2−ρq2+q22)];•r+(ν′,μ′) is taken at the order O(ν′2μ′), which means that M(ν′) is taken at the zeroth and M(μ′) at the linear order. We will show that this term does not contribute to the final result, upon applying the procedure (D.12) we will introduce shortly.•r+(ν′,μ′) is taken at the order O(ν′2μ′−1), which means that M(ν′) is taken at the zeroth and M(μ′) at the cubic order. The presence of this negative power in the expansion of r+ forces us to go to the third order in the expansion of the monodromy matrix, which we will perform later on – see the discussion around (D.12).Putting all the terms together and performing a few manipulations, we get for the r.h.s. of the Poisson relations(D.10)16e2ρg12g22qaqb(facdfbceTd⊗Te+Δ2e2fcbdTaTc⊗Td−fcbeTc⊗TaTe+[Δ2e2−1]facdTc⊗TdTb)−8Δg12g22qaqˆb(fcbdTaTc⊗Td+fcadTd⊗TcTb)−8Δg12g22qaqbqd(fcbeTcTa⊗TeTd+fcdeTcTa⊗TbTe+faceTe⊗TbTdTc)−16e2ρg12g22qˆbfcbaTc⊗Ta+8e2(1+3ρ2)g12g22qa[fabcTb⊗Tc−2ρ2(1+3ρ2)fabcTb⊗Tc].The strategy we will now follow is to bring everything on one side of the equation, namely to calculate(D.11)l.h.s.–r.h.s.g12g22, and to act upon it with the following operation:(D.12)Δ2fδ[αβtr(Tγ]⊗Tδ∘), where the three indices α,β and γ are totally antisymmetrized (without the 16 factor). Upon performing the operation (D.12), and by using the Jacobi identity, the very first term of the l.h.s. as contributing to (D.11), namely 4{qˆa,qˆb}Ta⊗Tb, can be seen to coincide with(D.13)4{qˆα,{qˆβ,qγ}}−4{qα,{qˆβ,qˆγ}}, which is the desired combination appearing in the Serre relations. It is therefore a matter of analyzing all the other terms after this operation is performed. One thing to notice is that anything looking like(D.14)fγδbΩb will vanish upon this operation, as can be seen by using the Jacobi identity. This is the reason why the contribution of r+(ν′,μ′) taken at the order O(ν′2μ′) is absent, as we commented earlier, since it is precisely of the form (D.14). Disregarding this type of terms as irrelevant to the final result, we combine the remaining terms in (D.11) and perform quite extensive manipulations and simplifications. Performing then the operation (D.12) on the result of this simplification produces some terms that we will call unwanted, since they do not look like the standard terms appearing in the Serre relations, and some that we call wanted, since they have the desired form.D.1Unwanted termsLet us begin with the unwanted terms. They come in two fashions:•We get a quadratic contribution with level zero charges, specifically(D.15)−8Δρqaqbfγebtr(Tδ{Te,Ta})+contrib.r+(ν′,μ′)atO(ν′2μ′−1);•We get a quadratic term with level zero and one charges, specifically(D.16)4Δqaqˆb[−feγbtr(Tδ{Te,Ta})−feγatr(Tδ{Te,Tb})]+contrib. fromr+(ν′,μ′)atO(ν′2μ′−1).The respective first terms in (D.15) and (D.16) vanish upon the operation (D.12). In order to see this, one needs to proceed in steps. Let us consider the first unwanted term. The first step consists of repeatedly using(D.17)fabc=tr(Ta[Tb,Tc]), to re-write the first term in (D.15), after the action of (D.12), as (indicating only the matrix part)(D.18)−12fδ[αβfγ]ebtr(Tδ{Te,Ta})=tr({Te,Ta}[Tα,Tβ])tr(Te[Tγ,Tb])+“2”, where “2” means that we have to add the other two cyclic permutations βγα and γαβ of the same structure on the l.h.s. of (D.18). At this point, it is convenient to open up the anti-commutator, involving the generator Te, and move Te close to the other trace by using cyclicity of the trace. When this is done, it produces two terms, in each of which one recognizes a structure of the type(D.19)tr(xTe)tr(Tey),y∈Lie algebra. The fact that y is Lie-algebra valued makes it possible to fuse the traces producing tr(xy). For the matrix part of the first unwanted term, we are therefore left with(D.20)tr(Ta[Tα,Tβ][Tγ,Tb])+tr([Tα,Tβ]Ta[Tγ,Tb])+“2”. Adding the “2” explicitly to this term, symmetrizing a↔b given the qaqb in front of the first term in (D.15) and eventually expanding all the (anti-)commutators explicitly, one sees that all terms cancel and the total contribution vanishes. We will calculate the contribution from r+(ν′,μ′) at O(ν′2μ′−1) later on.With regards to the first term in (D.16), let us re-write it as(D.21)4Δ(qaqˆb+qbqˆa)fγebtr(Tδ{Te,Ta}). From this we see that, due to the a↔b symmetry of the pre-factor, perfectly analogous considerations apply as for the term we have just shown to vanish. Once again, we will study the contribution from r+(ν′,μ′) at O(ν′2μ′−1) later on.D.2Third order of the monodromyWe have seen that, to be able to calculate the two unwanted terms left-over from (D.15) and (D.16), and also the related contribution to the wanted terms, we need the monodromy matrix up to the third order in the spectral parameter. We will derive this term in the expansion in this section.From Section 2 we have learned that the monodromy matrix M admits an expansion (adapted to the parameter g used in this section)(D.22)M(g)=Pexpg∫I0−sgI1(1+ρg)2−g2, where we note that both I0 and I1 appear and(D.23)s=ρ2−1. We need to isolate the third order term in g. If we recall the density (2.4) of the level-zero charge(D.24)2j(σ)=I0(σ), we see that the third-order term we are after reads, in compact notation,(D.25)2g3(4ρ2−s)∫j+2ρsg3∫I1−16ρg3∫∫σj(σ)j(σ′)−2sg3∫∫σ[j(σ)I1(σ′)+I1(σ)j(σ′)]+8g3∫∫σ∫σ′j(σ)j(σ′)j(σ″):=M3. We have to put this term into a form which is ready to be used for the Serre relations, therefore we parameterize(D.26)M3g3=AQ03+B2Q0Qˆ+C2QˆQ0+DQ02+E2Qˆ+FQ0+“Lie”, where we recall that (in the notation of this section) Qˆ reads(D.27)Qˆ=∫(−2ρj−sI1)+4∫∫σj(σ)j(σ′). “Lie” carries such a name because it is something which is not easily expressed in terms of Q or Qˆ, nevertheless it is Lie-algebra valued, hence it will drop after operation (D.12). In particular, we choose(D.28)“Lie”=H∫∫σ[j(σ),j(σ′)]+V2∫∫σ[j(σ),I1(σ′)]+U2∫∫σ[I1(σ),j(σ′)]+N∫∫σ∫σ′[j(σ),[j(σ′),j(σ″)]]+P∫∫σ∫σ[j(σ),[j(σ′),j(σ″)]]+R∫∫σ∫σσ′[j(σ),[j(σ′),j(σ″)]]. By appropriately splitting the integration domains and taking into account the ordering of the generators, one can show that the terms we use in our parametrization of M3 are enough to reconstruct the most general integral appearing at this order. In fact, we find in this way that they are more than sufficient, as we find a family of solutions when we try and match with (D.25):(D.29)A=−83,F=ρE+2(4ρ2−s),G=s(E+4ρ)V=s(B−4),U=−sB,C=4−B,R=N−83,P=163−2B−N,H=−E−8ρ,D=−E−4ρ, from which we see that we can set(D.30)B=E=N=0 as a convenient choice.The only contribution from the third order of the monodromy that can survive the operation (D.12) is then(D.31)g3[43q3+4qˆq−4ρq2].•The q2 term in (D.31) represents the contribution to the unwanted term (D.15) from the third order expansion in the monodromy. After acting with (D.12) and performing a few manipulations on the indexes, one can see that this contribution reproduces the same structure as the first addendum in (D.15), hence it vanishes for the same reasons.•The qˆq term in (D.31) combines with the order O(ν′2μ′−1) of r+(ν′,μ′) to give(D.32)−2Δg12g22qˆbqd[fcbeTc⊗TeTd+fcdeTc⊗TbTe]. After acting with (D.12), one can see that this term as well reduces the same structure as the first addendum in (D.16), hence vanishing by the same token.D.3Wanted termsWe now need to calculate the contribution we do expect to appear in the first Serre relation, namely we evaluate the cubic term in the level-zero charges in the expression (D.11). This amounts to the following – after acting with (D.12):(D.33)2Δ2qaqbqdfδ[αβ[4tr(Tγ]TaTe)tr(TδTcTd)fecb+2fγ]eafecbtr(TδTcTd)+2tr(Tγ]TaTe)fecbfdcδ+fγ]eafecbfdcδ+2tr(Tγ]TcTa)tr(TδTeTd)fcbe+2tr(Tγ]TcTa)tr(TδTbTe)fcde−2fγ]actr(TδTbTdTc)]+contribution fromr+(ν′,μ′)atO(ν′2μ′−1). Let us start with the part in (D.33) which is not coming from r+(ν′,μ′) at O(ν′2μ′−1). Exploiting the total symmetry of the pre-factor qaqbqc, after repeated use of the Jacobi identity, and by use of reconstructing commutators inside the traces to reduce the length of the traces as much as possible, we can recast that contribution into(D.34)−4Δ2qaqbqcfδ[αβfγ]betr(TδTaTeTc)+2Δ2qaqbqcfδ[αβfγ]brfrcefaeδ. Before proceeding with the calculation, let us compare with the contribution from r+(ν′,μ′) at O(ν′2μ′−1) and see whether the difficult-to-handle length-four trace cancels. It does indeed, since, by performing similar manipulations, the contribution from r+(ν′,μ′) at O(ν′2μ′−1) term results into(D.35)4Δ2qaqbqcfδ[αβfγ]betr(TδTaTeTc)−43Δ2qaqbqcfδ[αβfγ]brfrcefaeδ.We are then left with the two purely structure-constant contribution, which, by repeated use of the Jacobi identity, can be combined and manipulated into(D.36)−4Δ2qaqbqcfαaefβbdfγcrfedr. 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