The 3d mixed BF Lagrangian 1-form: a variational formulation of Hitchin's integrable system
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915733500329984 |
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| author | Caudrelier, Vincent Harland, Derek Singh, Anup Anand Vicedo, Benoit |
| author_facet | Caudrelier, Vincent Harland, Derek Singh, Anup Anand Vicedo, Benoit |
| contents | We introduce the concept of gauged Lagrangian $1$-forms, extending the notion of Lagrangian $1$-forms to the setting of gauge theories. This general formalism is applied to a natural geometric Lagrangian $1$-form on the cotangent bundle of the space of holomorphic structures on a smooth principal $G$-bundle $\mathcal{P}$ over a compact Riemann surface $C$ of arbitrary genus $g$, with or without marked points, in order to gauge the symmetry group of smooth bundle automorphisms of $\mathcal{P}$. The resulting construction yields a multiform version of the $3$d mixed BF action with so-called type A and B defects, providing a variational formulation of Hitchin's completely integrable system over $C$. By passing to holomorphic local trivialisations and going partially on-shell, we obtain a unifying action for a hierarchy of Lax equations describing the Hitchin system in terms of meromorphic Lax matrices. The cases of genus $0$ and $1$ with marked points are treated in greater detail, producing explicit Lagrangian $1$-forms for the rational Gaudin hierarchy and the elliptic Gaudin hierarchy, respectively, with the elliptic spin Calogero-Moser hierarchy arising as a special subcase. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_05127 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The 3d mixed BF Lagrangian 1-form: a variational formulation of Hitchin's integrable system Caudrelier, Vincent Harland, Derek Singh, Anup Anand Vicedo, Benoit Mathematical Physics High Energy Physics - Theory Exactly Solvable and Integrable Systems We introduce the concept of gauged Lagrangian $1$-forms, extending the notion of Lagrangian $1$-forms to the setting of gauge theories. This general formalism is applied to a natural geometric Lagrangian $1$-form on the cotangent bundle of the space of holomorphic structures on a smooth principal $G$-bundle $\mathcal{P}$ over a compact Riemann surface $C$ of arbitrary genus $g$, with or without marked points, in order to gauge the symmetry group of smooth bundle automorphisms of $\mathcal{P}$. The resulting construction yields a multiform version of the $3$d mixed BF action with so-called type A and B defects, providing a variational formulation of Hitchin's completely integrable system over $C$. By passing to holomorphic local trivialisations and going partially on-shell, we obtain a unifying action for a hierarchy of Lax equations describing the Hitchin system in terms of meromorphic Lax matrices. The cases of genus $0$ and $1$ with marked points are treated in greater detail, producing explicit Lagrangian $1$-forms for the rational Gaudin hierarchy and the elliptic Gaudin hierarchy, respectively, with the elliptic spin Calogero-Moser hierarchy arising as a special subcase. |
| title | The 3d mixed BF Lagrangian 1-form: a variational formulation of Hitchin's integrable system |
| topic | Mathematical Physics High Energy Physics - Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2509.05127 |