By calculating inequivalent classical r-matrices for the  $$gl(2,{\mathbb {R}})$$  <math> <mrow> <mi>g</mi> <mi>l</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mi>R</mi> <mo>)</mo> </mrow> </math>   Lie algebra as solutions of (modified) classical Yang-Baxter equation ((m)CYBE), we classify the YB deformations of Wess-Zumino-Witten (WZW) model on the  $$GL(2,{\mathbb {R}})$$  <math> <mrow> <mi>G</mi> <mi>L</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mi>R</mi> <mo>)</mo> </mrow> </math>   Lie group in twelve inequivalent families. Most importantly, it is shown that each of these models can be obtained from a Poisson-Lie T-dual  $$\sigma $$  <math> <mi>σ</mi> </math>  -model in the presence of the spectator fields when the dual Lie group is considered to be Abelian, i.e. all deformed models have Poisson-Lie symmetry just as undeformed WZW model on the  $$GL(2,{\mathbb {R}})$$  <math> <mrow> <mi>G</mi> <mi>L</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mi>R</mi> <mo>)</mo> </mrow> </math>  . In this way, all deformed models are specified via spectator-dependent background matrices. For one case, the dual background is clearly found.

Iran

Ali Eghbali

Tayebe Parvizi

Adel Rezaei-Aghdam

Yang-Baxter deformations of the  $$GL(2,{\mathbb {R}})$$  <math> <mrow> <mi>G</mi> <mi>L</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mi>R</mi> <mo>)</mo> </mrow> </math>   WZW model and non-Abelian T-duality

We proceed to generalize the Yang-Baxter (YB) deformation of Wess-Zumino-Witten (WZW) model to the Lie supergroups case. This generalization enables us to utilize various kinds of solutions of the (modified) graded classical Yang-Baxter equation ((m)GCYBE) to classify the YB deformations of WZW models based on the Lie supergroups. We obtain the inequivalent solutions (classical r-matrices) of the (m)GCYBE for the <math><mi>g</mi><mi>l</mi><mo>(</mo><mn>1</mn><mo>|</mo><mn>1</mn><mo>)</mo></math> and <math><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>A</mi><mo>)</mo></math> Lie superalgebras in the non-standard basis, in such a way that the corresponding automorphism transformations are employed. Then, the YB deformations of the WZW models based on the <math><mi>G</mi><mi>L</mi><mo>(</mo><mn>1</mn><mo>|</mo><mn>1</mn><mo>)</mo></math> and <math><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>A</mi><mo>)</mo></math> Lie supergroups are specified by skew-supersymmetric classical r-matrices satisfying (m)GCYBE. In some cases for both families of deformed models, the metrics remain invariant under the deformation, while the components of B-fields are changed. After checking the conformal invariance of the models up to one-loop order, it is concluded that the <math><mi>G</mi><mi>L</mi><mo>(</mo><mn>1</mn><mo>|</mo><mn>1</mn><mo>)</mo></math> and <math><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>A</mi><mo>)</mo></math> WZW models are conformal theories within the classes of the YB deformations preserving the conformal invariance. However, our results are interesting in themselves, but at a constructive level, may prompt many new insights into (generalized) supergravity solutions.

Yang-Baxter deformation of WZW model based on Lie supergroups: The cases of (1|1) and ( + )

The Yang-Baxter (YB) deformations of Wess-Zumino-Witten (WZW) model on the Heisenberg Lie group (<math><msub><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msub></math>) are examined. We proceed to obtain the nonequivalent solutions of (modified) classical Yang-Baxter equation ((m)CYBE) for the <math><msub><mrow><mi>h</mi></mrow><mrow><mn>4</mn></mrow></msub></math> Lie algebra by using its corresponding automorphism transformation. Then we show that YB deformations of <math><msub><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msub></math> WZW model are split into ten nonequivalent backgrounds including metric and B-field such that some of the metrics of these backgrounds can be transformed to the metric of <math><msub><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msub></math> WZW model while the antisymmetric B-fields are changed. The rest of the deformed metrics have a different isometric group structure than the <math><msub><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msub></math> WZW model metric. As an interesting result, it is shown that all new integrable backgrounds of the YB deformed <math><msub><mrow><mi>H</mi></mrow><mrow><mn>4</mn></mrow></msub></math> WZW model are conformally invariant up to two-loop order. In this way, we obtain the general form of the dilaton fields satisfying the vanishing beta-function equations of the corresponding σ-models.