Deformations of $T^{1,1}$ as Yang-Baxter sigma models

Crichigno, P. Marcos; Matsumoto, Takuya; Yoshida, Kentaroh

doi:10.48550/arxiv.1406.2249

Cited by 22 publications

(35 citation statements)

References 29 publications

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“…In fact, a four-parameter deformation of the SU(2) principal chiral model has already been constructed in [43]. Yet from the point of view of the twist function we only expect to be able to construct a three-parameter deformation in the case of an arbitrary Lie group G. However, recall that it has also been suggested in [44] that the fourth parameter of the deformation in [43] is related to a TsT-transformation, and therefore shall correspond to a deformation where the twist function is not modified [23,25,30,20].…”

Section: Discussionmentioning

confidence: 99%

“…Nevertheless, in the Hamiltonian framework, the procedure for obtaining these alternative deformations may also be interpreted as deforming the corresponding twist functions [11,20]. For completeness, let us also mention that within the Yang-Baxter class of integrable deformations there is also a way to deform a given σ-model by using a solution of the classical Yang-Baxter equation [21][22][23][24][25][26][27][28][29][30][31], but without changing its twist function [20].…”

Section: Introductionmentioning

confidence: 99%

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

Delduc

Lacroix

Magro

et al. 2016

J. High Energ. Phys.

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Abstract:The bi-Yang-Baxter σ-model is a certain two-parameter deformation of the principal chiral model on a real Lie group G for which the left and right G-symmetries of the latter are both replaced by Poisson-Lie symmetries. It was introduced by C. Klimčík who also recently showed it admits a Lax pair, thereby proving it is integrable at the Lagrangian level. By working in the Hamiltonian formalism and starting from an equivalent description of the model as a two-parameter deformation of the coset σ-model on G × G/G diag , we show that it also admits a Lax matrix whose Poisson bracket is of the standard r/s-form characterised by a twist function which we determine. A number of results immediately follow from this, including the identification of certain complex Poisson commuting KacMoody currents as well as an explicit description of the q-deformed symmetries of the model. Moreover, the model is also shown to fit naturally in the general scheme recently developed for constructing integrable deformations of σ-models. Finally, we show that although the Poisson bracket of the Lax matrix still takes the r/s-form after fixing the G diag gauge symmetry, it is no longer characterised by a twist function.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Delduc

Lacroix

Magro

et al. 2016

J. High Energ. Phys.

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show abstract

“…has been discussed in [11]. Yang-Baxter deformations [12][13][14][15][16][17] of T 1,1 are discussed in [11,18,19] (For a short summary, see [20]).…”

Section: Introductionmentioning

confidence: 99%

Chaotic string dynamics in deformed $T^{1,1}$

Ishii,

Kushiro,

Yoshida

2021

Preprint

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Recently, Arutyunov, Bassi and Lacroix have shown that 2D non-linear sigma model with a deformed T 1,1 background is classically integrable [arXiv:2010.05573 [hep-th]]. This background includes a Kalb-Ramond two-form with a critical value. Then the sigma model has been conjectured to be non-integrable when the twoform is off critical. We confirm this conjecure by explicitly presenting classical chaos. With a winding string ansatz, the system is reduced to a dynamical system described by a set of ordinary differential equations. Then we find classical chaos, which indicates non-integrability, by numerically computing Poincaré sections and Lyapunov spectra for some initial conditions.

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“…Further remarkably, these deformations may work for non-integrable backgrounds, such as a Sasaki-Einstein manifold T 1,1 [25]. TsT transformations of T 1,1 [13,26] are reproduced as Yang-Baxter deformations [27]. Thus the connection between gravity solutions and classical r-matrices may deserve to be called the gravity/CYBE correspondence (For a short summary see [28]).…”

Section: Introductionmentioning

confidence: 99%

Lax pairs for deformed Minkowski spacetimes

Kyono

Sakamoto

Yoshida

2016

J. High Energ. Phys.

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Abstract:We proceed to study Yang-Baxter deformations of 4D Minkowski spacetime based on a conformal embedding. We first revisit a Melvin background and argue a Lax pair by adopting a simple replacement law invented in 1509.00173. This argument enables us to deduce a general expression of Lax pair. Then the anticipated Lax pair is shown to work for arbitrary classical r-matrices with Poincaré generators. As other examples, we present Lax pairs for pp-wave backgrounds, the Hashimoto-Sethi background, the SpradlinTakayanagi-Volovich background.

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Deformations of $T^{1,1}$ as Yang-Baxter sigma models

Cited by 22 publications

References 29 publications

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

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

Chaotic string dynamics in deformed $T^{1,1}$

Lax pairs for deformed Minkowski spacetimes

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