Integrability of the Bi-Yang–Baxter σ-Model

Abstract

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].

A new class of integrable deformations of CFTs

“…Indeed, the computation of the corresponding RG flow equations using gravitational methods was performed in [2,3] and is in agreement with results from field theoretical methods [8][9][10]. Furthermore, it was shown that these models, named generically as λ-deformed, can be embedded in specific cases of low dimensionality to supergravity [11][12][13][14][15], while their relation to η-deformations for group and coset spaces introduced in [16][17][18] and [19][20][21] respectively, via Poisson-Lie T-duality [22] and appropriate analytic continuations, was uncovered in [23][24][25][26][27]. More recently, the computation of the all-loop correlators for the…”

Yang-Baxter deformations of Minkowski spacetime

“…It would be very interesting to clarify the correspondence between the list in [90][91][92] and our results. 13 It would be most important to generalize our argument to 10D Minkowski spacetime in order to extend our argument to a consistent string theory. For this purpose, we have to consider the 10D conformal group SO(2, 10) and realize 10D Minkowski spacetime as a slice of 11D AdS space.…”

“…Deformations of S 3 and AdS 3 have been well investigated [8][9][10][11][12][13][14][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. 1 A systematic way is the Yang-Baxter sigma model approach proposed by Klimcik [11][12][13].…”

Lax pairs for deformed Minkowski spacetimes

“…The Yang-Baxter sigma-model description is known as a systematic way to study integrable deformations of 2D integrable non-linear sigma models [1][2][3]. It was originally proposed for principal chiral models on the basis of the modified classical Yang-Baxter equation (mCYBE) .…”