On the Hamiltonian integrability of the bi-Yang-Baxter σ-model

Delduc, F.; Lacroix, Sylvain; Magro, Marc; Vicedo, Benoit

doi:10.1007/jhep03(2016)104

Cited by 37 publications

(57 citation statements)

References 84 publications

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“…This formula matches perfectly the last formula of Section 6 of Ref. [11] where the strong integrability of the bi-YB deformation of the principal chiral model was first established. To see it, we must relate the parameters appearing respectively in our action (2.8) and in that of Ref.…”

supporting

confidence: 88%

“…[20,27,29,44], that some of those integrable deformations are related by Poisson-Lie T-duality [30].The strong integrability of the principal model was proved in Ref. [38], that of the single and of the double Yang-Baxter deformations [25,26] was established in [11,12] and that of the single Yang-Baxter deformation with WZ term was proved in [13]. On the other hand, the strong integrability of the λdeformed σ-model [43] was demonstrated in Ref.…”

mentioning

confidence: 99%

“…To see it, we must relate the parameters appearing respectively in our action (2.8) and in that of Ref. [11]…”

mentioning

confidence: 99%

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Integrability of the Bi-Yang–Baxter σ-Model

2014

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We identify the r-matrix governing the Poisson brackets of the matrix elements of the Lax operator of the bi-YB-WZ model. IntroductionStrong integrability is the property of certain Lax integrable theories. It amounts to Poissoncommutativity of the integrals of motions which are obtained as the spectral invariants of the Lax operator. In this paper, we focus on the issue of the strong integrability of Lax integrable non-linear σ-models.The epitome of the Lax integrable non-linear σ model is the principal chiral model [45], which can be formulated on any quadratic Lie group. First examples of integrable deformations of the principal chiral model were obtained in Refs. [1,6,14] for the group SU (2) and it took one more decade to understand the algebraic structure behind those theories which permitted to generalize them to live on every simple compact group target. In particular, the models [6,14] were generalized in this way in Refs. [24][25][26] in the framework of so-called η-deformation (or Yang-Baxter deformation) procedure, while the model [1] was generalized in Ref. [43] in the framework of λ-deformations. Since then, many other developments have followed, proving the Lax integrability of complex structure-induced deformations [5], of homogeneous Yang-Baxter deformations [4,23,41], of deformations with a WZ term [8,13], of coset spaces [12], of combined η and λ deformations [44] or of coupled η and λ deformations [2,9,10,[16][17][18]35]. Moreover, it was established in Refs. [20,27,29,44], that some of those integrable deformations are related by Poisson-Lie T-duality [30].The strong integrability of the principal model was proved in Ref. [38], that of the single and of the double Yang-Baxter deformations [25,26] was established in [11,12] and that of the single Yang-Baxter deformation with WZ term was proved in [13]. On the other hand, the strong integrability of the λdeformed σ-model [43] was demonstrated in Ref. [21,22] and of the coupled λ-deformed models [16][17][18] in Ref. [19]. Finally, the construction of the Lax pairs and the proof of the strong integrability of the coupled η and λ deformations [9,10,35] was completed in Ref. [2].

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Integrability of the Bi-Yang–Baxter σ-Model

2014

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“…This is an arduous task whose end result takes the form ϕ λ (z) = 1 ζ 2 8ks 1 s 2 ξz 2. Using the results of [31,30] along with (3.55), the monodromy matrix when evaluated at the poles z i± generates a q-deformed Poisson algebra with q 1 = e −i/k 1 , q 2 = e −i/k 2 , (3.56)…”

Section: The Unequal Level Casementioning

confidence: 99%

“…This property will be the guideline in the unequal level case which follows. Secondly, it was argued in[30], using the results of[31], that the one-parameter deformation of the coset σ-model G × G/G diag admits such a description in terms of the twist function (see Eq. (3.3) of[30])…”

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Strong integrability of λ-deformed models

2020

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We study the notion of strong integrability for classically integrable λ-deformed CFTs and coset CFTs. To achieve this goal we employ the Poisson brackets of the spatial Lax matrix which we prove that it assumes the Maillet r/s-matrix algebra. As a consequence the system in question are integrable in the strong sense. Furthermore, we show that the derived Maillet r/s-matrix algebras can be realized in terms of twist functions, at the poles of which we recover the underlying symmetry algebras.

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Three-parameter integrable deformation of ℤ4 permutation supercosets

Delduc

Hoare

Kameyama

et al. 2019

J. High Energ. Phys.

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A three-parameter integrable deformation of Z 4 permutation supercosets is constructed.These supercosets are of the form F /F 0 where F 0 is the bosonic diagonal subgroup of the product supergroup F = G × G. They include the AdS 3 × S 3 and AdS 3 × S 3 × S 3 supercosets. This deformation encompasses both the bi-Yang-Baxter deformation of the semi-symmetric space σ-model on Z 4 permutation supercosets and the mixed flux model.Truncating the action at the bosonic level, we show that one recovers the bi-Yang-Baxter deformation of the principal chiral model plus Wess-Zumino term. in [6] rely on the deformation being governed by a solution of the modified classical Yang-Baxter equation. The two-parameter bi-Yang-Baxter deformation of the principal chiral model was introduced in [5] and shown to be integrable in [7,8]. As demonstrated in [9], the bi-Yang-Baxter deformation of the SU (2) principal chiral model is equivalent to the two-parameter deformation of S 3 found in [10]. In [6] the Yang-Baxter deformation of the symmetric space σ-model was constructed and shown to be integrable. For symmetric spaces that take the form of Z 2 permutation cosets (1.2), the deformed model is equivalent to a particular one-parameter model contained within the bi-Yang-Baxter deformation of the principal chiral model on F 0 .Adding the standard topological Wess-Zumino term [11][12][13] to the principal chiral model is well-known to preserve its classical integrability. Up to the expected quantization of the level, the resulting model interpolates between the principal chiral and conformal Wess-Zumino-Witten models. Generalising the SU (2) construction of [14,15], the Yang-Baxter deformation of the principal chiral model plus Wess-Zumino term for general Lie groups was derived in [16]. In order to construct this deformation it was assumed that the solution of the modified classical Yang-Baxter deformation governing the deformation cubes to its negative. This assumption is natural since it is satisfied by the standard Drinfel'd-Jimbo R-matrix for a non-split real form of a semi-simple Lie algebra [17][18][19]. In this article we will continue to focus on deformations governed by such R-matrices.By allowing for an antisymmetric term in the action, the two-parameter deformation of [10] was generalised to an integrable four-parameter deformation of S 3 in [2]. Observing that this model also contains the TsT transformation of the SU (2) Wess-Zumino-Witten model [20, 21], it was proposed in [9] that the four-parameter model should be understood as the combined bi-Yang-Baxter deformation and TsT transformation of the σ-model on S 3 plus Wess-Zumino term. The bi-Yang-Baxter deformation of the principal chiral model plus Wess-Zumino term was constructed in [1]. Applying a TsT transformation in the directions of the Cartan subalgebra, which is associated with adding a compatible abelian solution of the classical Yang-Baxter equation [22-29] to the Drinfel'd-Jimbo R-matrix, and considering the SU (2) case it was shown that this model indeed genera...

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On the Hamiltonian integrability of the bi-Yang-Baxter σ-model

Cited by 37 publications

References 84 publications

Integrability of the Bi-Yang–Baxter σ-Model

Integrability of the Bi-Yang–Baxter σ-Model

Strong integrability of λ-deformed models

Three-parameter integrable deformation of ℤ4 permutation supercosets

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