Hitchin systems, higher Gaudin operators and $r$-matrices

Abstract

We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of $\CC P^{1}$ and $S{L}_n$-bundles, they are equivalent to systems studied by Garnier. The corresponding quantum systems were identified by B. Feigin, E. Frenkel and N. Reshetikhin with Gaudin systems. We give a formula for the higher Gaudin operators, using results of R. Goodman and N. Wallach on the center of the enveloping algebras of affine algebras at the critical level. Finally we construct a dynamical $r$-matrix for Hitchin systems for a punctured elliptic curve, and $G{L}_n$-bundles, and (for $n=2$) the corresponding quantum system.

References in corpus (3)

• Gaudin Model, Bethe Ansatz and Critical Level
• $R$-matrices for Elliptic Calogero-Moser Models
• Dynamical $r$-matrices for the Elliptic Calogero-Moser Model

Cited by in corpus (13)

• Integrability and Seiberg-Witten Theory: Curves and Periods
• Integrable Structures in Supersymmetric Gauge and String Theory
• Multiscale N=2 SUSY field theories, integrable systems and their stringy/brane origin -- I
• Affine Algebras, Langlands Duality and Bethe Ansatz
• Elliptic quantum groups $E_{\tau ,\eta }(s{l}_2)$ and quasi-Hopf algebras
• Elliptic Calogero-Moser Systems and Isomonodromic Deformations
• 1+1 Gaudin Model
• Argument Shift Method and Gaudin Model
• Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the elliptic Hitchin systems
• Self-duality of the SL_2 Hitchin integrable system at genus two
• Limits of Gaudin Systems: Classical and Quantum Cases
• Painlevé - Calogero correpondence
• Quasi-Topological Gauged Sigma Models, The Geometric Langlands Program, And Knots