Integrable $\mathbb{Z}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915142485147648 |
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| author | Aizawa, Naruhiko Ito, Ren Kuznetsova, Zhanna Tanaka, Toshiya Toppan, Francesco |
| author_facet | Aizawa, Naruhiko Ito, Ren Kuznetsova, Zhanna Tanaka, Toshiya Toppan, Francesco |
| contents | In this paper we present a general framework to construct integrable $\mathbb{Z}_2^2$-graded extensions of classical, two-dimensional Toda and conformal affine Toda theories. The scheme is applied to define the extended Liouville and Sinh-Gordon models; they are based on $\mathbb{Z}_2^2$-graded color Lie algebras and their fields satisfy a parabosonic statististics. The mathematical tools here introduced are the $\mathbb{Z}_2^2$-graded covariant extensions of the Lax pair formalism and of the Polyakov's soldering procedure. The $\mathbb{Z}_2^2$-graded Sinh-Gordon model is derived from an affine $\mathbb{Z}_2^2$-graded color Lie algebra, mimicking a procedure originally introduced by Babelon-Bonora to derive the ordinary Sinh-Gordon model. The color Lie algebras under considerations are: the $6$-generator $\mathbb{Z}_2^2$-graded $sl_2$, the $\mathbb{Z}_2^2$-graded affine ${\widehat{sl_2}}$ algebra with two central extensions, the $\mathbb{Z}_2^2$-graded Virasoro algebra obtained from a Hamiltonian reduction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_13503 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integrable $\mathbb{Z}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories Aizawa, Naruhiko Ito, Ren Kuznetsova, Zhanna Tanaka, Toshiya Toppan, Francesco Mathematical Physics High Energy Physics - Theory Exactly Solvable and Integrable Systems In this paper we present a general framework to construct integrable $\mathbb{Z}_2^2$-graded extensions of classical, two-dimensional Toda and conformal affine Toda theories. The scheme is applied to define the extended Liouville and Sinh-Gordon models; they are based on $\mathbb{Z}_2^2$-graded color Lie algebras and their fields satisfy a parabosonic statististics. The mathematical tools here introduced are the $\mathbb{Z}_2^2$-graded covariant extensions of the Lax pair formalism and of the Polyakov's soldering procedure. The $\mathbb{Z}_2^2$-graded Sinh-Gordon model is derived from an affine $\mathbb{Z}_2^2$-graded color Lie algebra, mimicking a procedure originally introduced by Babelon-Bonora to derive the ordinary Sinh-Gordon model. The color Lie algebras under considerations are: the $6$-generator $\mathbb{Z}_2^2$-graded $sl_2$, the $\mathbb{Z}_2^2$-graded affine ${\widehat{sl_2}}$ algebra with two central extensions, the $\mathbb{Z}_2^2$-graded Virasoro algebra obtained from a Hamiltonian reduction. |
| title | Integrable $\mathbb{Z}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories |
| topic | Mathematical Physics High Energy Physics - Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2406.13503 |