R-matrix structure of Hitchin system in Tyurin parameterization (Q1416895)

The author presents a classical \(r\)-matrix for the Hitchin system without marked points on a non-degenerate algebraic curve of genus \(g\geq2\) using Tyurin parametrization of the moduli space of rank \(n\) holomorphic vector bundles of degree \(ng\) [\textit{A.~N.~Tyurin}, Am. Math. Soc., Translat., II. Ser. 63, 245--279 (1967); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 29, 657--688 (1965; Zbl 0207.51603)]. More precisely, if \(\Sigma\) is a non-degenerate algebraic curve of genus \(g\geq2\) then, by Tyurin's result, an open dense subset of the moduli space of rank \(n\) holomorphic vector bundles of degree \(ng\) over \(\Sigma\) is parametrized by points in the quotient \([\Sigma\times {\mathbb P}({\mathbb C}^n)]^{(ng})/SL_n({\mathbb C})\). As a consequence, the phase space of the Hitchin system without marked points can be obtained via symplectic reduction in the space \({\mathcal P}=T^*[\Sigma\times {\mathbb P}({\mathbb C}^n)]^{ng}\). The main statement in the paper is that the Krichever-Lax matrix of the Hitchin system can be extended to the symplectic manifold \({\mathcal P}\) in such a way that the extended Krichever-Lax matrix admits a simple \(r\)-matrix structure, defined by a matrix-valued meromorphic section of the bundle \(\Sigma\times T^*\Sigma\to \Sigma\times\Sigma\), i.e., by a matrix-valued differential of the form \(r_{ij}(z,w)dw\). It is also shown how to derive the classical \(r\)-matrix for the original Krichever-Lax matrix of the Hitchin system from the \(r\)-matrix corresponding to the extended Krichever-Lax matrix. In particular, this gives an alternative proof of the commutativity of classical Hitchin Hamiltonians.