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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.02184 |
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| _version_ | 1866914701256949760 |
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| author | Parra, Juan Carlos Morales Schroers, Bernd |
| author_facet | Parra, Juan Carlos Morales Schroers, Bernd |
| contents | Classical dynamical $r$-matrices arise naturally in the combinatorial description of the phase space of Chern-Simons theories, either through the inclusion of dynamical sources or through a gauge-fixing procedure involving two punctures. Here we consider classical dynamical $r$-matrices for the family of Lie algebras which arise in the Chern-Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical $r$-matrices in this case, and show that they can be viewed as generalised complexifications, in a sense which we define, of the equations governing dynamical $r$-matrices for $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$. We obtain explicit families of solutions and relate them, via Weierstrass factorisation, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02184 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classical dynamical $r$-matrices for the Chern-Simons formulation of generalised 3d gravity Parra, Juan Carlos Morales Schroers, Bernd High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics Classical dynamical $r$-matrices arise naturally in the combinatorial description of the phase space of Chern-Simons theories, either through the inclusion of dynamical sources or through a gauge-fixing procedure involving two punctures. Here we consider classical dynamical $r$-matrices for the family of Lie algebras which arise in the Chern-Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical $r$-matrices in this case, and show that they can be viewed as generalised complexifications, in a sense which we define, of the equations governing dynamical $r$-matrices for $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$. We obtain explicit families of solutions and relate them, via Weierstrass factorisation, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models. |
| title | Classical dynamical $r$-matrices for the Chern-Simons formulation of generalised 3d gravity |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2403.02184 |