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A deformation of quantum affine algebra in squashed WZNW models
High Energy Physics - Theory · Submitted 19 Nov 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We reveal here that an infinite-dimensional extension of U(1)_R is a deformation of quantum affine algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term.
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The abstract, sentence by sentence
We proceed to study infinite-dimensional symmetries in two-dimensional squashed Wess-Zumino-Novikov-Witten (WZNW) models at the classical level. The target space is given by squashed S^3 and the isometry is SU(2)_L x U(1)_R. It is known that SU(2)_L is enhanced to a couple of Yangians. We reveal here that an infinite-dimensional extension of U(1)_R is a deformation of quantum affine algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term. Then we consider the relation between the deformed quantum affine algebra and the pair of Yangians from the viewpoint of the left-right duality of monodromy matrices. The integrable structure is also discussed by computing the r/s-matrices that satisfy the extended classical Yang-Baxter equation. Finally two degenerate limits are discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We proceed to study infinite-dimensional symmetries in two-dimensional squashed Wess-Zumino-Novikov-Witten (WZNW) models at the classical level. | contribution 0.91 | 0.83 | 0.04 | asserted 0.71 |
| s1 | The target space is given by squashed S^3 and the isometry is SU(2)_L x U(1)_R. | background 0.87 | 0.16 | 0.02 | asserted 1.00 |
| s2 | It is known that SU(2)_L is enhanced to a couple of Yangians. | background 1.00 | 0.04 | 0.04 | asserted 1.00 |
| s3 | We reveal here that an infinite-dimensional extension of U(1)_R is a deformation of quantum affine algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term. | result 0.87 | 0.94 | 0.81 | asserted 1.00 |
| s4 | Then we consider the relation between the deformed quantum affine algebra and the pair of Yangians from the viewpoint of the left-right duality of monodromy matrices. | contribution 0.92 | 0.83 | 0.08 | asserted 0.83 |
| s5 | The integrable structure is also discussed by computing the r/s-matrices that satisfy the extended classical Yang-Baxter equation. | method 0.55 | 0.89 | 0.06 | asserted 0.98 |
| s6 | Finally two degenerate limits are discussed. | contribution 0.90 | 0.85 | 0.04 | asserted 0.69 |
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The paper
Original abstract
We proceed to study infinite-dimensional symmetries in two-dimensional squashed Wess-Zumino-Novikov-Witten (WZNW) models at the classical level. The target space is given by squashed S^3 and the isometry is SU(2)_L x U(1)_R. It is known that SU(2)_L is enhanced to a couple of Yangians. We reveal here that an infinite-dimensional extension of U(1)_R is a deformation of quantum affine algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term. Then we consider the relation between the deformed quantum affine algebra and the pair of Yangians from the viewpoint of the left-right duality of monodromy matrices. The integrable structure is also discussed by computing the r/s-matrices that satisfy the extended classical Yang-Baxter equation. Finally two degenerate limits are discussed.
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References
How the literature uses it
22 arXiv papers cite this in 43 sentences
Papers citing it · 32 sentences without a confident label
Builds on it 2 papers
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The second of these [ref] , extending the initial work [ref] done for $SU(2)$, is to add a Wess-Zumino term[The next section on $λ$-deformations gives more details of the WZ coupling and WZW model.] to the PCM with the action S_η,ζ = - 1/2 π ∫d^2 σ ⟨∂_+g g^-1 ( α1 +βR +γR^2 ) ∂_- g g^-1 ⟩+ k/24 π ∫_M_3 ⟨g̅^-1 d g̅ , [ g̅^-1 d g̅,g̅^-1 d g̅] ⟩ .
An Introduction to Generalised Dualities and their Applications to Holography and Integrability · $⋆$ Multi-parameter $η$-deformations $⋆$ · 0.90
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The deformed model furnishes a higher rank generalisation of the `squashed WZNW-model' introduced in [ref] .
Integrable double deformation of the principal chiral model · Conclusion · 0.94
Uses its data or values 1 paper
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The coefficients $Ã_Ł,$ take the values [ref] _Ł = √( 1 -^2/1 + ^2 ) , _ = √( 1 -^2/1 + ^2 ) .
Three-parameter integrable deformation of Z_4 permutation supercosets · Three-parameter deformation of Z2 permutation cosets · 0.93
Compares with it 2 papers
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The resulting dependence coincides with the analogous expressions in [ref] .
Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable sigma-model · Action · 0.99
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They correspond to the Poisson brackets found in [ref] and to the ones in [ref] for the $𝔰𝔲(2)$ case.
Integrable double deformation of the principal chiral model · Double deformation of the · 0.97
Cites it as support 1 paper
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In the examples of Yang-Baxter deformations [ref] and in the cases studied by Bernard and LeClair [ref] , we can find specific cases of nonlocality generated by deformations (although in different contexts) which justify this second hypothesis.
Nonlocal charges from marginal deformations of 2D CFTs: Holographic $T \bar T$, $T \bar J$ and Yang-Baxter deformations · Currents conservation · 1.00
Cites it as background 3 papers
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In the presence of $H$-flux, it is not straightforward to define the YB sigma model [There are several works [ref] where YB deformations of the WZ(N)W model based on the mCYBE have been studied.], and we shall concentrate only on $β$-deformations.
Integrable deformations of string sigma models and generalized supergravity · “YB deformations” of the $3×$S$^3×T^4$ superstring · 0.93
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Nevertheless, even in the bosonic setting integrable deformations of two-dimensional $σ$-models have received much attention [ref] and a vast web of connections and dualities between them has been uncovered over the years [ref] , see also [ref] for recent reviews.
Deformed WZW Models and Hodge Theory -- Part I · Introduction · 0.94
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In this case, due to the presence of $H$-flux, it is not straightforward to perform YB deformations.[There are several works [ref] where YB deformations of the WZ(N)W model based on the mCYBE have been studied.] Therefore, we do not show the equivalence in this paper.
Local $\beta$-deformations and Yang-Baxter sigma model · Introduction · 0.97
What it cites · 19 sentences without a confident label
Builds on 1 paper
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The analysis here is a generalization of the result obtained in [ref] .
On the classical equivalence of monodromy matrices in squashed sigma model · The left-right duality · 1.00
Uses the method or tool of 1 paper
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Let us find an appropriate gauge function[ The strategy is similar to the one to consider the Jordanian twists in the Schrödinger sigma models [ref] .].
Schrodinger Sigma Models and Jordanian Twists · $α=πi/2$ · 0.93
Compares with 4 papers
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Thus the $r$/$s$-matrices obtained in [ref] are reproduced.
The classical origin of quantum affine algebra in squashed sigma models · The $r$/$s$-matrices · 0.95
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Thus the classification of the $r$/$s$-matrice are subtle.], [equation] It should be noted that, when $K=0$ , the function $l^L_C,K(λ)$ is reduced to [equation] Thus the $r$/$s$-matrices in ([ref]) reproduce the results without the Wess-Zumino term [ref] .
Hybrid classical integrability in squashed sigma models · Lax pairs · 0.89
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The improved currents satisfy the following flatness condition, [equation] When $K=0$ , $A=√(C)$ and the improved currents constructed in [ref] are reproduced.
Hidden Yangian symmetry in sigma model on squashed sphere · Lax pairs · 0.84
Cites as background 18 papers
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This part contains a short review of the previous work [ref] .
Yangian symmetry in deformed WZNW models on squashed spheres · The left description · 0.99
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It is known that there exist non-local conserved currents which correspond to the broken components $T^±$ in the case without the Wess-Zumino term [ref] .
Hybrid classical integrability in squashed sigma models · $q$-deformation of $su(2)_R$ · 0.86
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The one is an algebraic approach based on $q$-deformations of the world-sheet S-matrix [ref] .
The Quantum Deformed Mirror TBA I · Introduction · 0.88
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