Mathematical Research Letters

Volume 3, Issue 3, May 1996 pp. 343-357.

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

Authors: B. Enriquez and V. Rubtsov
Author institution: Ecole Polytechnique, and ITEP

Summary: We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of ${\bold C} P^{1}$ 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.}

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