5d 2-Chern-Simons theory and 3d integrable field theories
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929601477869568 |
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| author | Schenkel, Alexander Vicedo, Benoit |
| author_facet | Schenkel, Alexander Vicedo, Benoit |
| contents | The $4$-dimensional semi-holomorphic Chern-Simons theory of Costello and Yamazaki provides a gauge-theoretic origin for the Lax connection of $2$-dimensional integrable field theories. The purpose of this paper is to extend this framework to the setting of $3$-dimensional integrable field theories by considering a $5$-dimensional semi-holomorphic higher Chern-Simons theory for a higher connection $(A,B)$ on $\mathbb{R}^3 \times \mathbb{C}P^1$. The input data for this theory are the choice of a meromorphic $1$-form $ω$ on $\mathbb{C}P^1$ and a strict Lie $2$-group with cyclic structure on its underlying Lie $2$-algebra. Integrable field theories on $\mathbb{R}^3$ are constructed by imposing suitable boundary conditions on the connection $(A,B)$ at the $3$-dimensional defects located at the poles of $ω$ and choosing certain admissible meromorphic solutions of the bulk equations of motion. The latter provides a natural notion of higher Lax connection for $3$-dimensional integrable field theories, including a $2$-form component $B$ which can be integrated over Cauchy surfaces to produce conserved charges. As a first application of this approach, we show how to construct a generalization of Ward's $(2+1)$-dimensional integrable chiral model from a suitable choice of data in the $5$-dimensional theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08083 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 5d 2-Chern-Simons theory and 3d integrable field theories Schenkel, Alexander Vicedo, Benoit High Energy Physics - Theory Mathematical Physics The $4$-dimensional semi-holomorphic Chern-Simons theory of Costello and Yamazaki provides a gauge-theoretic origin for the Lax connection of $2$-dimensional integrable field theories. The purpose of this paper is to extend this framework to the setting of $3$-dimensional integrable field theories by considering a $5$-dimensional semi-holomorphic higher Chern-Simons theory for a higher connection $(A,B)$ on $\mathbb{R}^3 \times \mathbb{C}P^1$. The input data for this theory are the choice of a meromorphic $1$-form $ω$ on $\mathbb{C}P^1$ and a strict Lie $2$-group with cyclic structure on its underlying Lie $2$-algebra. Integrable field theories on $\mathbb{R}^3$ are constructed by imposing suitable boundary conditions on the connection $(A,B)$ at the $3$-dimensional defects located at the poles of $ω$ and choosing certain admissible meromorphic solutions of the bulk equations of motion. The latter provides a natural notion of higher Lax connection for $3$-dimensional integrable field theories, including a $2$-form component $B$ which can be integrated over Cauchy surfaces to produce conserved charges. As a first application of this approach, we show how to construct a generalization of Ward's $(2+1)$-dimensional integrable chiral model from a suitable choice of data in the $5$-dimensional theory. |
| title | 5d 2-Chern-Simons theory and 3d integrable field theories |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2405.08083 |