The $r$-matrix structure on the moduli space of framed Higgs pairs
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909839207170048 |
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| author | Bertola, M. |
| author_facet | Bertola, M. |
| contents | On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson $r$-matrix structure. The known $r$-matrices are defined on the Riemann sphere (rational), the cylinder (trigonometric), or the torus (elliptic). We extend the formalism to the case of a Riemann surface $\mathcal C$ of higher genus $g$: we consider the moduli space of framed vector bundles of rank $n$ and degree $ng$, where the framing consists in a choice of basis of $n$ independent holomorphic sections chosen to trivialize the fiber at a given point $\infty\in \mathcal C$. The co-tangent space is known to be identified with the set of Higgs fields, i.e., one-forms on $\mathcal C$ with values in the endomorphisms of the vector bundle, with an additional simple pole at $\infty$. The natural symplectic structure on the co-tangent bundle of the moduli space induces a Poisson structure on the Higgs fields. The result is then an explicit $r$--matrix that generalizes the known ones. A detailed discussion of the elliptic case with comparison to the literature is also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11408 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $r$-matrix structure on the moduli space of framed Higgs pairs Bertola, M. Exactly Solvable and Integrable Systems Symplectic Geometry On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson $r$-matrix structure. The known $r$-matrices are defined on the Riemann sphere (rational), the cylinder (trigonometric), or the torus (elliptic). We extend the formalism to the case of a Riemann surface $\mathcal C$ of higher genus $g$: we consider the moduli space of framed vector bundles of rank $n$ and degree $ng$, where the framing consists in a choice of basis of $n$ independent holomorphic sections chosen to trivialize the fiber at a given point $\infty\in \mathcal C$. The co-tangent space is known to be identified with the set of Higgs fields, i.e., one-forms on $\mathcal C$ with values in the endomorphisms of the vector bundle, with an additional simple pole at $\infty$. The natural symplectic structure on the co-tangent bundle of the moduli space induces a Poisson structure on the Higgs fields. The result is then an explicit $r$--matrix that generalizes the known ones. A detailed discussion of the elliptic case with comparison to the literature is also provided. |
| title | The $r$-matrix structure on the moduli space of framed Higgs pairs |
| topic | Exactly Solvable and Integrable Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2509.11408 |