The $r$-matrix structure on the moduli space of framed Higgs pairs
M. Bertola
TL;DR
This work extends the classical Lie-Poisson $r$-matrix framework to Hitchin-type systems on higher-genus Riemann surfaces by modeling framed vector bundles and framing Higgs fields on the moduli space $\mathfrak{E}_{n,ng,\infty}$. Central to the construction is the non-abelian Cauchy kernel $\mathbf{K}(p,q)$, which provides a reproducing kernel for framed Higgs fields and enables an explicit, self-contained genus-$g$ $r$-matrix in terms of $\mathbf{K}$; the authors also derive the corresponding Poisson brackets for the Higgs fields and show the spectral-curve coordinates Poisson-commute. The elliptic (genus one) case is worked out in detail, with gauge-fixed formulas, comparisons to Rubtsov, and the emergence of an elliptic Calogero-Moser leaf, linking Hitchin systems to known integrable many-body dynamics via a Calogero-type Lax matrix. These results broaden the Poisson-geometric framework for Higgs bundles to arbitrary genus and lay groundwork for isomonodromic deformations and tau-function constructions on higher-genus curves. The work thus integrates moduli-theoretic Higgs fields, Tyurin data, and explicit $r$-matrix structures in a coherent higher-genus setting with tangible links to classical integrable systems.
Abstract
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.