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

TL;DR

This paper extends Hitchin's integrable systems to punctured curves, linking classical and quantum Gaudin models, and constructs new dynamical r-matrices for these systems, including explicit formulas for higher Gaudin operators.

Contribution

It adapts Hitchin systems to punctured curves, derives formulas for higher Gaudin operators, and constructs dynamical r-matrices for Hitchin systems on punctured elliptic curves.

Findings

Derived formulas for higher Gaudin operators.

Constructed dynamical r-matrices for punctured elliptic curves.

Connected classical Hitchin systems with quantum Gaudin models.

Abstract

We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of $\CC P^{1}$ and $SL_{n}$-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 $GL_{n}$-bundles, and (for $n=2$) the corresponding quantum system.